P H D TH E S I S - theses.fr

P H D T H E S I SUNIVERSITE DE PAU ET DES PAYS DE L’ADOUR

ECOLE DOCTORALE (ED211)

Defended by

Khaled Elkelanyon February 5, 2016

to obtain the title ofPhD Degree of Science

of Universite de Pau et des Pays de l’Adour

Specialty : Physical Chemistry

Design of Enhanced Piezoelectric Materialsfrom Quantum Chemical Calculations

Jury members:

Reviewers:Philippe D’ARCO Professeur / Universite Pierre et Marie Curie

Marie-Liesse DOUBLET Directeure de Recherche / Universite Montpellier

Examinators:Olivier CAMBON Professeur / Universite Montpellier

Gilberte CHAMBAUD Professeure / Universite Paris-Est

Roberto DOVESI Professeur / Universite de Turin

Supervisors:Philippe CARBONNIERE Maıtre de Conference HDR / Universite de Pau et des Pays de l’Adour

Michel RERAT Professeur / Universite de Pau et des Pays de l’Adour

Resume de la these

Une analyse exhaustive de la piezoelectricite a ete realisee par la modelisation

moleculaire basee sur l’application des principes de la mecanique quantique.

La piezoelectricite est la deuxieme derivee de l’energie par rapport aux deux

champs externes, electrique et mecanique. Elle peut etre theoriquement donnee

comme la somme des contributions electronique et nucleaire (vibrationnelle) dans

l’approximation de Born-Oppenheimer. La contribution electronique est induite

par la deformation du nuage electronique en raison de la perturbation mecanique.

Cette contribution peut etre individuellement calculee utilisant la condition dite

“clamped-ion” pour laquelle les coordonnees fractionnaires des atomes sont fixees.

La contribution nucleaire provient des deplacement relatifs des noyaux dans le

systeme perturbe; elle peut etre incluse dans le calcul si les coordonnees atomiques

sont laissees libres de se deplacer.

La calibration de la methode et des parametres du calcul est d’abord examinee

en comparant les resultats calcules concernant les oxydes de silicium et de ger-

manium a leurs homologues experimentaux. Ensuite, les parametres micro-

scopiques qui influencent chaque contribution de cette propriete macroscopique

de reponse sont distinctement rationalises. L’effet piezoelectrique est limite aux

systemes non-centrosymetriques, c’est-a-dire aux materiaux qui n’ont pas de centre

d’inversion. Pour un materiau non-centrosymetrique, la contribution electronique

a la piezoelectricite dependra de la difference d’energie entre les etats spectro-

scopiques de transition: fondamental et excites, et donc du gap electronique di-

rect. Le graphene est donc apparu comme un bon candidat pour l’examen de cet

effet en raison de son gap nul. Cependant, le graphene n’est pas intrinsequement

piezoelectrique en raison de l’existence d’un centre d’inversion. Le dopage du

graphene est ici suggere afin de briser le centre de symetrie pour induire une pro-

priete piezoelectrique. Cet aspect a ete illustre par l’exemple du nitrure de bore.

Nous avons montre qu’une grande piezoelectricite, dominee largement par la con-

tribution electronique, peut apparaıtre dans le graphene aux faibles concentrations

de dopant.

D’un autre cote, la contribution nucleaire de la propriete piezoelectrique dans un

materiau non-centrosymetrique sera affectee par les phonons. Elle sera d’autant

plus importante que les modes mous du materiau sont de bas nombre d’ondes

et actifs dans l’infrarouge (IR). La structure de SrTiO3 perovskite est consideree

comme un bon candidat pour examiner cet effet en raison de sa transition de phase

qui conduit a une grande variation des proprietes dielectriques. Un enorme effet

piezoelectrique est estime pour la phase ferroelectrique de SrTiO3 en raison du

mouvement des atomes de Titane, Ti, mais a tres basse temperature (au dessous

de 24 K). Une autre structure de perovskite de BaTiO3 a ete consideree pour

laquelle la phase rhomboedrique ferroelectrique qui conduit a une enorme reponse

piezoelectrique, pourrait etre ici trouvee a une temperature plus haute (au dessous

de 183 K).

Enfin, apres la rationalisation de la propriete piezoelectrique, la conception de

materiaux montrant un effet piezoelectrique eleve a ete tentee. Nous avons montre

que la grande piezoelectricite induite par le dopage de BN dans le plan du graphene

tendra vers une valeur unique, ni nulle ni infinie, et de facon independante

de la nature physique ou chimique particuliere du defaut. L’induction d’une

piezoelectricite hors du plan du graphene en brisant sa planeite selon la direc-

tion z est etudiee. La reponse piezoelectrique obtenue est largement amelioree

par rapport a la limite finie de la piezoelectricite dans le plan, mais aux grandes

concentrations du defaut. En effet, contrairement a la composante dans le plan de

la piezoelectricite induite dans le graphene, la composante hors du plan, depend

de la nature du defaut et diminue jusqu’a tendre vers zero a dilution infinie.

La brisure de symetrie par un defaut dans le plan ou hors du plan du graphene

permet de concevoir des materiaux piezoelectriques a base de graphene qui n’est

pas piezoelectrique par nature. Les directions des piezoelectricites induites ainsi

que leurs amplitudes permettent diverses applications technologiques significatives.

Abstract

An exhaustive analysis of the technologically important piezoelectric phenomena

is here done by applying quantum chemical simulations. Piezoelectricity as the

second energy derivative with respect to both external electric and mechanical

fields, can be theoretically given as the sum of electronic and nuclear (vibrational)

contributions. In general, the electronic contribution arises from the electronic

cloud deformation due to mechanical or/and field perturbation. This contribution

can be individually calculated using clamped-ion conditions, where the fractional

coordinates are fixed “not optimized” after the deformation. The nuclear (ionic)

contribution is initiated from the relative displacements of nuclei in perturbed

system; and it can be included in the calculations if the coordinates are left to

relax.

At first, the calibration of the assumed computational scheme is examined by

comparing our calculated piezoelectric properties of the well-known piezoelectric

quartz to their experimental counterparts. Secondly, the microscopic parameters

that influence each contribution of piezoelectric macroscopic property are distinctly

rationalized. Piezoelectric effect is indeed restricted to materials that don’t exhibit

an inversion center of symmetry. The electronic contribution to the piezoelectricity

in a non-centrosymmetric material, will basically depend on the energy differences

between initial and final spectroscopic transition states, and so generally the elec-

tronic direct gap if the corresponding transition is allowed. Exotic semiconducting

graphene materials have been studied in order to investigate such behaviour. How-

ever, graphene is not intrinsically piezoelectric due to the existence of inversion

symmetry center. Hence, doping with different patterns and concentrations of BN

is assumed in order to break the symmetry center and engineer piezoelectricity into

graphene sheet. In this case, a large piezoelectric effect is induced into graphene

at low concentrations of BN-dopant.

On the other side, the vibrational contribution to the piezoelectricity will essen-

tially affect by the vibrational phonons. It could be giant if the material has an

infrared-active (IR) soft phonon mode (i.e. mode has a small wavenumber and

large IR-intensity), and does not absolutely exhibit symmetry inversion center.

SrTiO3 perovskite structure is considered as a good candidate to investigate this

influence due to its phase transition that leads to a large variation of the dielectric

properties. A huge piezoelectric effect is estimated for the ferroelectric phase of

SrTiO3 due to the anomalously large zero-point vibrational motion of Ti atoms,

at however, very low temperature (below 24 K). Another perovskite structure of

barium tintante, BaTiO3, has been considered, where the ferroelectric phase that

showing a huge piezoelectric response, could be here found at higher temperature

conditions (below 183 K).

After the rationalization of the piezoelectric property, the design of materials that

presenting a high piezoelectric effect has been attempted. It has been shown that

the large in-plane piezoelectricity induced in BN-doped graphene can be acquired

by including any in-plane defect(s). Moreover, in the limit of vanishing defect

concentration, the piezoelectric response will tend toward a unique value, neither

null nor infinite, regardless of the particular chemical or physical nature of the

defect. The induction of an out-of-plane piezoelectricity in graphene by breaking

its planarity through the non-periodic z-direction is stated, where the obtained

piezoelectric response is largely improved compared to the finite in-plane piezo-

electric limit, at however higher concentration of the defect. Contrarily to what

has been discussed for the in-plane piezoelectric effect, the out-of-plane one even-

tually vanishes as far as the limit of infinite defect dilution is reached, and so it

relies ultimately on the nature of the defect.

Nevertheless, in-plane and out-of-plane symmetry-breaking defect will engineer a

piezoelectricity into this non-piezoelectric graphene, and have both of them their

significant technological applications.

To memory of the great man

who taught me the value of the science...

REMERCIEMENTS

Ce travail a ete realise au sein de l’Equipe de Chimie Physique - Institut des

Sciences Analytiques et de Physicochimie pour l’Environnement et les Materiaux

(ECP-IPREM), de l’Universite de Pau et des Pays de l’Adour.

Tout d’abord, je tiens a remercier mon Dieu pour m’avoir donne la force, la ca-

pacite, et le pouvoir d’accomplir cette these.

Mes remerciements s’adressent tout d’abord au professeur Michel Rerat, professeur

de chimie physique a l’Universite de Pau et des Pays de l’Adour et directeur de

cette these. Je lui adresse tout mon respect et ma profonde sympathie. Je lui

dois mes premiers pas dans le monde academique et je le remercie d’avoir toujours

cru en mes capacites. Je lui suis reconnaissant du temps et de la patience qu’il a

toujours manifeste a mon egard. J’espere de tout cœur que notre collaboration ne

s’arretera jamais.

Je tiens a remercier mon co-directeur de these Mr. Philippe Carbonniere, maıtre

de conference HDR a l’Universite de Pau et des Pays de l’Adour. Je lui exprime

toute ma gratitude pour les conseils et les encouragements qu’il m’a apportes tout

au long de cette these. Ses idees et ses conseils m’ont permis de mener a bien ces

recherches.

J’exprime mes sinceres remerciements a Mme. Marie-Liesse Doublet, directrice

de recherche a l’Universite Montpellier 2, et Mr. Philippe D’Arco, professeur a

l’Universite Pierre et Marie Curie a Paris, pour avoir accepte d’etre rapporteurs

de ma these, et egalement pour l’interet qu’ils ont porte a ce travail.

Je voudrais exprimer ma profonde reconnaissance a Mme. Gilberte Cham-

baud, professeure a l’Universite Paris-Est, et Mr. Olivier Cambon, professeur a

l’Universite Montpellier 2, pour l’honneur et le plaisir qu’ils m’ont fait d’accepter

de faire partie du jury de ma these.

De meme, je remercie plus particulierement Mr. Roberto Dovesi, professeur au

Laboratoire de Chimie Theorique de l’Universite de Turin (Italie) pour avoir ac-

cepte de participer a ce jury et pour m’avoir accueilli dans son laboratoire pendant

un mois. Je tiens egalement a lui exprimer ma profonde gratitude pour ses conseils

et pour toute l’aide qu’il m’a apportee. Ce fut un grand plaisir de collaborer avec

lui.

Je tiens aussi a remercier mes collaborateurs du groupe de Chimie Theorique de

Turin, Italie: Alessandro Erba, Marco De La Pierre, Matteo Ferrabone, Valentina

Lacivita, et Agnes Mahmoud, ainsi que ma collaboration interne avec Mr. Jean-

Marc Sotiropoulos, Directeur de Recherche, et Panos Karamanis, Charge de

Recherche, de l’IPREM.

Je tiens aussi a remercier Germain Vallverdu, Maıtre de Conferences, Jacques

Hertzberg, Ingenieur de Recherche, Mr. Alain Dargelos, professeur Emerite, et

Marc Odunlami, Ingenieur d’Etudes, pour leur disponibilite et pour toute l’aide

qu’ils m’ont apportee.

Je tiens a remercier tous les autres membres du Laboratoire de Chimie Physique de

l’IPREM qui ont contribues par leur gentillesse et leur sympathie a ma formation

de chercheur et au travail quotidien agreable durant cette periode.

Je voudrais exprimer plus particulierement mes sinceres remerciements a Mme.

Patricia Corno, pour sa gentillesse, son soutien logistique et amical tout au long

de mes quatre annees de these. Je lui souhaite plein de bonheur pour la suite.

Je tiens a remercier tout particulierement mes parents qui, par leurs encourage-

ments constants et leur ecoute, m’ont toujours donne les moyens d’arriver jusqu’au

bout de ce que je voulais faire. Merci egalement a mon frere, a mes sœurs ainsi

qu’a ma femme et mes cheres filles.

Je voudrais remercier enfin tous les amis qui m’ont soutenu, et egalement

l’ambassade d’Egypte et plus particulierement tous les membres du Bureau Cul-

turel a Paris.

. . .

Khaled E. El-Kelany

Contents

List of Figures i

List of Tables vi

Glossary xi

INTRODUCTION GENERALE 4

INTRODUCTION 7

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

A Theoretical Tools for Calculating Piezoelectric Proper-ties 12

I THEORETICAL APPROACH 14

1 Theoretical Approach for Electronic Structure Computations . . . . . . 14

1.1 Hartree Fock Self-Consistent Field Method (HF-SCF) . . . . . . 16

1.2 Density Functional Theory (DFT) . . . . . . . . . . . . . . . . . 19

1.2.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.2 Mathematical Description of Exchange and Correlation

Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.3 Periodic Treatment . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3.1 Bloch Function . . . . . . . . . . . . . . . . . . . . . . 28

1.3.2 Born-Von Karman Boundary Condition . . . . . . . . 30

1.3.3 Crystalline Orbitals . . . . . . . . . . . . . . . . . . . 30

1.4 Basis Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1.4.1 Atom-Centered Basis Sets . . . . . . . . . . . . . . . . 31

1.4.2 Pseudopotential (Ps) Basis Set . . . . . . . . . . . . . 35

2 Theory for Piezoelectricity and Associated Response Properties . . . . 36

2.1 Direct and Converse Piezoelectricity . . . . . . . . . . . . . . . 37

2.1.1 General Mathematical Definition . . . . . . . . . . . . 37

2.1.2 Piezoelectric Tensor and Symmetry . . . . . . . . . . . 42

2.1.3 Piezoelectric Tensor and Dimensionality . . . . . . . . 43

2.1.4 Quantum Chemical Point of View . . . . . . . . . . . . 44

2.2 Associated Response Properties . . . . . . . . . . . . . . . . . . 46

CONTENTS

2.2.1 Elasticity and Phonons . . . . . . . . . . . . . . . . . . 46

2.2.2 Infrared Intensity . . . . . . . . . . . . . . . . . . . . . 49

2.2.3 Polarizability and Dielectric Constant . . . . . . . . . 50

II COMPUTATIONAL APPROACH 52

1 The Software Used: CRYSTAL Code . . . . . . . . . . . . . . . . . . . 52

1.1 Structure of Input File . . . . . . . . . . . . . . . . . . . . . . . 53

1.2 The Keywords Applied in This Study . . . . . . . . . . . . . . . 54

1.3 Piezoelectricity Computations in CRYSTAL . . . . . . . . . . . 55

1.4 Capability in Treating Systems of Different Dimensionality: The

Example of MgO Nanotubes . . . . . . . . . . . . . . . . . . . . 57

2 The Optimization of The Basis Set . . . . . . . . . . . . . . . . . . . . 58

3 The Choice of Computational Parameters . . . . . . . . . . . . . . . . 64

3.1 The DFT-Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.2 The Number of k-Points . . . . . . . . . . . . . . . . . . . . . . 66

3.3 The Bielectronic Integrals Tolerances . . . . . . . . . . . . . . . 67

4 The Choice of Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 69

5 Calibration of Piezoelectric Property Computations: The Case of α-

Quartz Doped by Ge (Si1−xGexO2) . . . . . . . . . . . . . . . . . . . . 74

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

B Rationalization of Piezoelectric Property 94

I VIBRATIONAL CONTRIBUTION TO THE PIEZOELECTRIC-

ITY 97

1 Symmetry Conditions for Non-Zero Piezoelectricity . . . . . . . . . . . 97

1.1 Symmetry of Operators Associated to Electric and Mechanical

Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

1.2 Physical Aspect for Transition Moments . . . . . . . . . . . . . 98

2 IR-Active Soft Mode Contribution . . . . . . . . . . . . . . . . . . . . . 98

3 Strontium Titanate Perovskite Example . . . . . . . . . . . . . . . . . 100

3.1 Phase Transition . . . . . . . . . . . . . . . . . . . . . . . . . . 100

3.2 Phonon and Dielectric Properties . . . . . . . . . . . . . . . . . 103

3.3 Elastic and Piezoelectric Behaviour . . . . . . . . . . . . . . . . 105

CONTENTS

II ELECTRONIC CONTRIBUTION TO THE PIEZOELECTRICITY109

1 Symmetry Conditions for Non-Zero Piezoelectricity . . . . . . . . . . . 109

1.1 Symmetry of Associated Operators . . . . . . . . . . . . . . . . 109

1.2 Physical interpretation for Transition Moments . . . . . . . . . 110

2 Transition Energy Condition for Large Piezoelectricity . . . . . . . . . 110

3 BN-Doped Graphene Example . . . . . . . . . . . . . . . . . . . . . . . 112

3.1 Band Gap of BN-doped Graphene . . . . . . . . . . . . . . . . . 113

3.2 Elastic and Piezoelectric Properties . . . . . . . . . . . . . . . . 118

3.3 Independence of Piezoelectricity from DFT Functional . . . . . 124

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

C Design of Materials for Large Piezoelectricity: Applica-tion to Graphene 132

I INDEPENDANCE OF IN-PLANE PIEZOELECTRICITY FROM

DEFECT NATURE IN GRAPHENE 135

1 Holes in Graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

2 Si, Ge, and Sn doped Graphene . . . . . . . . . . . . . . . . . . . . . . 136

3 Finite Limit of In-Plane Piezoelectricity in Graphene . . . . . . . . . . 138

II OUT-OF-PLANE PIEZOELECTRICITY IN GRAPHENE 141

1 Carbon Substitution in Graphene . . . . . . . . . . . . . . . . . . . . . 143

2 H-Substitution in Pyrrolic-N-Doped Graphene . . . . . . . . . . . . . . 147

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

CONCLUSIONS 158

CONCLUSION GENERALE 161

Appendix 166

List of Figures

A.I.1 SCF cyclization involved in the Hartree Fock method. . . . . . . . . . 18

A.I.2 Possible distribution of the electrons for Single, Double and Triple

virtual excitation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

A.I.3 Lattice points of a space lattice in a square two dimensions. Panel (a)

shows primitive (only one lattice point, four lattice points where each

point is a four-shared), and Non-primitive (more than one lattice point,

six four-shared lattice points) unit cells. The lattice vectors and angles

are presented in (b). Panel (c) represents the Wigner-Sietz primitive

cell for a hexagonal 2D structure. . . . . . . . . . . . . . . . . . . . . 26

A.I.4 The matrix representation of Fock Hamiltonian operator in both di-

rect and reciprocal spaces. A block-diagonal matrix free from repeti-

tion could solely be obtained for the reciprocal lattice space. Fourier

transformations permit passing from direct to reciprocal spaces and

vice versa. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A.I.5 Definition of both electric and mechanical behaviours within a mate-

rial. Panel (a) represents the electric field vectors: D is the electric

displacement vector, P is the electric polarization, and E is the electric

field intensity (0 = 8.854× 10−12 C2.N−1.m−2, is the permittivity of

vacuum). The definition of mechanical strain and stress are given in

Panels (b and c) respectively. . . . . . . . . . . . . . . . . . . . . . . 39

A.I.6 Graphical scheme that introduces Voigt notation for mechanical strain

in 3D (bulk) and 2D (surface) systems. . . . . . . . . . . . . . . . . . 41

A.I.7 Diagram represents the electrical-mechanical inter-relations. . . . . . 42

A.I.8 Form of piezoelectric tensor as related to the group symmetry, the

image is taken from Nye 1985.(47) Key of notation is also presented. . 43

A.I.9 Introducing of tensor order: second, third, and fourth-rank-tensors

are considered. Examples and total number of components are stated

for each case. Note that, zero-rank-tensor is corresponding to scalars,

while first-rank-tensor is referred to vectors as the dipole moment, for

instance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

A.II.1 The optimized geometry of MgO systems of different dimensionality:

bulk ((a), 3D); primitive planar (unbuckled) and conventional non pla-

nar (buckled) cell of the monolayer ((b) and (c), 2D); (12,0) and (12,12)

nanotubes, ((d) and (e), 1D). . . . . . . . . . . . . . . . . . . . . . . 58

i

LIST OF FIGURES

A.II.2 Graphical representation of the structure of SrTiO3. In panel a), the

cubic Pm3m phase is represented in the xy plane; the conventional cu-

bic cell (thick continuous line) and the quadruple pseudo-cubic tetrag-

onal cell (dashed line) are shown which contain 5 and 20 atoms, respec-

tively. The same structure and cells are also represented in a different

view in panel b). Panel c) reports the structure of the I4/mcm tetrag-

onal phase in the xy plane; rotation of adjacent TiO6 octahedra along

the z direction can be inferred from comparison with panel a). . . . . 63

A.II.3 Graphical definition of (a) the intertetrahedral bridging angle θ and

(b) the tetrahedral tilting angle δ which is an order parameter for the

α - β phase transition in quartz. . . . . . . . . . . . . . . . . . . . . . 75

A.II.4 Electromechanical coupling coefficients as a function of the intertetra-

hedral bridging angle θ. The experimental values for pure end members

(full black diamonds) are also reported.(130) In the experimental work,

Ranieri et al. also predicted the coupling values for some intermediate

compositions with small substitutional fraction x assuming a linear

behavior; these data are reported as empty black diamonds. . . . . . . 83

B.I.1 Polarized IR reflectance at different temperatures of compressively

strained (001) SrTiO3 film on the (110) NdGaO3 substrate, image is

taken from Ref.(1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.I.2 The structure of an ABO3 perovskite with the origin centered at (a)

the B-site ion and (b) the A-site ion, the image is taken from Ref.(2). 100

B.II.1 Uv-visible spectra of pure graphene and h-BN monolayers, as the vari-

ation of the imaginary part of the electronic uncoupled contribution

(Sum Over States, SOS) of polarizability, through periodic αXX and

non-periodic αZZ directions, with respect to the wavenumber hν. . . . 111

B.II.2 BN-doped piezoelectric graphene in different configurations (R,W ).

The radius R of BN rings and the wall width W separating them are

graphically defined. The unit cell of each configuration is shown as

thick black lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

B.II.3 Energetic and electronic properties of (1,W ) BNG structures as a func-

tion of p = 2R+W . Upper panel: cohesive energy ∆E (the graphene,

p → ∞, limit is given by the horizontal line) and energy of mixing

∆Em (zero by definition at p=0 and p=∞). Lower panel: evolution of

the electronic band gap Eg; red and blue lines are just meant as eye-

guides. The inset shows the structure of (1,W ) BNG configurations

as p increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

ii

LIST OF FIGURES

B.II.4 Convergence of the band gap Eg (in eV) of BNG patterns to the

graphene limit (x = 0) as a function of the BN concentration x (or

3/p2). Values for the two cases of p, multiple or not of 3, are repre-

sented by empty and full circles, respectively. The parameters given in

the insets are obtained with the fitting functions Eg = a0 + a1x, and

Eg = b0 + b1x, for the two cases of p, where the highest concentration

points (x=0.33, and 0.19) are excluded. . . . . . . . . . . . . . . . . . 116

B.II.5 Energy band structure of (a) pristine graphene, (b) pristine h-BN,

(c-d) BNG (1,W ) structures with p multiple of 3, (e-f) BNG (1,W )

structures with p not multiple of 3. HOCO and LUCO energy levels

are drawn as dashed green and red lines, respectively. . . . . . . . . . 117

B.II.6 Upper panel: direct piezoelectric constant e11 of BNG structures as a

function of -ln(x) where x is the BN substitution fraction; total (full

symbols) and purely electronic (empty symbols) values are reported

for (1,W ) BNGs (circles), (2,W ) BNGs (triangles) and pure h-BN

(squares). Purely electronic and total data are fitted with a function

a+b/(−ln(x)). Lower panel: nuclear relaxation effect on the e11 piezo-

electric constant, ∆e11 = etot11 − eele11 as a function of -ln(x). . . . . . . . 120

B.II.7 Infrared (IR) spectrum, as the plot of IR-intensity in km/mol per atom

as a function of frequency for the IR-active modes. The modes with

intensity less than 0.1 km/mol are excluded as a reason of clarity. Data

for pristine h-BN and graphene (unit cell), and (1,4), (1,5), (1,6), and

(1,7) BNG are included. The frequency and intensity (x,y) for the

more and less intense modes for each structure are also displayed. . . 122

B.II.8 The variation of in-plane piezoelectricity as a function of electronic

band gap − ln(Eg) for the series (1,W ) of BN-doped graphene . . . . 123

C.I.1 Graphical representation of the structure of two defects, one of D3h

and one of C2v point-symmetry, of graphene (holes in this case) that

break its inversion symmetry. The unit cell of the p=4 case is sketched

in both cases with black continuous lines. . . . . . . . . . . . . . . . . 135

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LIST OF FIGURES

C.I.2 Dependence of the in-plane direct piezoelectric constant of graphene,

eele11 (purely electronic contribution), on defect concentration x. Four

different defects are considered: (1,W ) BNGs (full red circles), (2,W )

BNGs (empty black circles), D3h holes (full blue triangles) and C2v

holes (empty magenta triangles). For (1,W ) BNGs and D3h holes the

fitting a+ b/(−ln(x)) is also reported. All results are obtained at the

B3LYP level. For (1,W ) BNGs, a LDA result is also shown (red empty

circle) at lower defect concentration. . . . . . . . . . . . . . . . . . . . 136

C.I.3 Graphical representation for Si, Ge, and Sn doped graphene is given

in panel (a). In panel (b), the variation of substitutional fraction x of

dopant in graphene sheet is shown for Si-doped graphene, for instance.

The unit cell used in the calculations is highlighted. . . . . . . . . . . 137

C.II.1 The optimized structure of Si, Ge, Sn, and pyrrolic-N, piezoelectric

doped-graphene is shown in panels a, b, c, and d respectively. For

each configuration front and lateral views are represented, where the

deformation angle δ (in degree) and bond lengths (in A) are reported.

The unit cell used in the calculation is displayed for each configuration. 142

C.II.2 Atomic displacement corresponding to the vibrational mode, Si-atom

moves up where C-atoms move down and vice versa, see red arrows in

panel (a). Phonon wavenumbers, ν, and IR-intensity, Ip(z) that can

be expressed as ( ∂µz

∂Qp)2, corresponding to this displacement are given

for each SiG structure, where the concentration x is indicated at left

of the panel. panel (b) represents the variation of deformation angle

(δ) with respect to x, where a graphical sketch is introduced in panel

(c) to show how δ can be calculated. . . . . . . . . . . . . . . . . . . 145

C.II.3 (Top) Column representation of direct, e31, and converse, d31, piezo-

electric constants (relaxed-ion) for our simulated structure, x = 0.125

for all structures. (Bottom) The variation of induced piezoelectricity

related parameters; deformation angle, δ, where the angle initiated

by vibrational mode amplitude at absolute zero temperature (±∆) is

included, wavenumber ν, and IR-intensity through z-direction. The

inset in the figure represents the vibrational mode amplitude (at 0 K)

that seems almost symmetric for SiG with respect to graphene plane. 146

C.II.4 Graphical representation of optimized geometry of pyrrolic-N-doped

graphene, H-NG, where H is here substituted by other functional

groups (F-,Cl-, H3C-, and H2N-). The unit cell of each configuration

is shown as thick lines. The inset demonstrates the angle θ. . . . . . . 149

iv

LIST OF FIGURES

C.II.5 (a) Atomic displacement corresponding to the vibrational mode, where

arrows refer to the movement direction for each atom. The variation

of θ angle, that has been introduced in the previous figure, and phonon

wavenumber, ν, as a function of the structure are given in panels (b)

and (c), respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

v

List of Tables

A.I.1 Representation of piezoelectric tensor for different dimensionality sys-

tems, where Voigt’s notation is applied for mechanical deformation.

So, for instance, the component 11 in the piezoelectric tensor refers

to xxx, while 12 points out to xyy, and 36 to zxy. Dash refers to

polarized non-deformable 0D system. . . . . . . . . . . . . . . . . . . 44

A.II.1 Calculated properties of the (n,0) series of MgO nanotubes and of the

monolayer (l-MgO). ∆E, δE and ∆E are the energy difference be-

tween the (n,0) tube and the corresponding (n/2,n/2) one (they have

the same number of atoms), the relaxation energy for the rolled config-

uration and the energy difference between the relaxed tube and l-MgO,

respectively. Values reported are in µHa per MgO unit. Ru is the un-

relaxed radius (in A) of the Mg cations (the anions O are at Ru±0.38);

RMg and RO are the same distances after relaxation. The radius of

the oxygen atom “outside” the tube (at Ru + 0.38 before relaxation)

remains essentially unaltered. BG is the band gap in eV. α and α⊥ are

the longitudinal and transverse components of the electronic αe, and

static α0 polarizabilities per MgO unit (in A3). The unrelaxed (sum

over states) αSOS values are also reported. The Layer∗ row gives the

average of the monolayer perpendicular polarizabilities per MgO unit

using the relation α⊥n = 1

2(α⊥

l + αl ) (l stands for layer, n is the label

of the tube) which should be equal to the large radius limit (∞ row)

of the transverse nanotube polarizability. . . . . . . . . . . . . . . . . 59

A.II.2 The influence of the basis set on the computed structural and elec-

tronic properties for α-SiO2 and GeO2 hexagonal structures. Five op-

timized basis sets with different configuration of polarization orbitals

are studied: the basis set B1 with only one d polarization orbital for

Si/Ge atoms; B2 basis with two d polarization orbitals, whereas the

basis sets B3 and B4 have one d, one f and two d, one f polarization

orbitals, respectively; the data obtained by the pseudopotential B5 is

also reported. Si/Ge atoms are in 3a position (u, 0, 13) and O atoms are

in general 6c position (x, y, z) of space group P3221. θ is the interte-

trahedral angle (angle between SiO2 tetrahedrons). The percentage

errors are given in parenthesis with respect to the experimental data.

Calculations are performed at the PBE0 level. . . . . . . . . . . . . . 61

vi

LIST OF TABLES

A.II.3 The influence of the basis set on the computed structural parameters

and band gap (Eg) of the four structures of SrTiO3 is here considered.

The basis A corresponds to the basis used by Evarestov et al with

another polarization d (αd = 0.9 for Sr and Ti atoms and 1.2 for O-

atom). A more diffused polarization d orbital with α = 0.3 has been

added (all atoms) to basis A to give the basis set B. In the C basis,

a polarization f function has been added to all atoms with αf equals

0.9. Calculations are performed at the PBE0 level. . . . . . . . . . . . 65

A.II.4 The effect of the DFT integration grid and electronic integral tolerances

on computed structural and electronic properties. Three DFT grids

(nr,na) with nr radial points and a maximum of na angular points are

used: G1 = (55,434), G2 = (75,974) and G3 = (99,1454), with fixed

integral tolerance T2. Three sets of integral tolerances are considered:

T1 = (6 6 6 6 12), T2 = (8 8 8 8 16) and T3 = (10 10 10 10 20), with

fixed grid G3. Calculations are performed with the pseudopotential,

B5, basis sets and PBE0 hybrid functional. . . . . . . . . . . . . . . . 68

A.II.5 Influence of the DFT integration grid and electronic integral tolerances

on computed structural and electronic properties (as defined in the

text) of the four structures of SrTiO3 here considered. Three DFT

grids (nr,na) with nr radial points and a maximum of na angular points

are used: G1 = (55,434), G2 = (75,974) and G3 = (99,1454). Three

sets of integral tolerances are considered: T1 = (8 8 8 8 16), T2 = (10

10 10 12 24) and T3 = (12 12 12 15 30). Calculations are performed

with the PBE0 hybrid functional. . . . . . . . . . . . . . . . . . . . . 70

A.II.6 Effect of the adopted one-electron Hamiltonian on computed struc-

tural and electronic properties for α-SiO2 and α-GeO2. See text for a

definition of the quantities reported. . . . . . . . . . . . . . . . . . . . 71

A.II.7 Influence of the adopted one-electron Hamiltonian on computed struc-

tural and electronic properties (as defined in the text) of the four struc-

tures of SrTiO3 here considered. The pseudo-cubic structure is con-

sidered for comparison with the others. . . . . . . . . . . . . . . . . . 73

A.II.8 For any composition x, number Ntot of atomic configurations, number

of Nirr symmetry-irreducible configurations among them, multiplicity

M and number of symmetry operators Nops proper of each irreducible

configuration. The elongated supercell has only six Si-positions. . . . 76

vii

LIST OF TABLES

A.II.9 Structural and energetic properties of the Si1−xGexO2 solid solution

series, as a function of the substitutional content x. Calculations are

performed using the PBE0 hybrid functional. ∆E is the energy dif-

ference with respect to the two end-members; and is obtained by the

equation: ∆E = ESi1−xGexO2- (1-x)ESiO2

+ xEGeO2. All data re-

ported are per unit cell. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

A.II.10 Elastic (in GPa) and compliance (in TPa−1) constants of the

Si1−xGexO2 solid solution as a function of the composition x. Interte-

trahedral bridging angle θ (in deg) and bulk modulus Ks (in GPa) are

also reported. Experimental data for x = 0 are from Ref.(135), for

x =0.07 from Ref.(132) and for x = 1 from Ref.(136). Calculations

performed at PBE0 level. . . . . . . . . . . . . . . . . . . . . . . . . . 79

A.II.11 Direct and converse independent piezoelectric constants, electronic

and static dielectric constants (relative permittivity) and electrome-

chanical coupling coefficients of the Si1−xGexO2 solid solution, as a

function of the composition x. Experimental data, when available, are

reported for end-members. Calculations performed at PBE0 level. . . 81

B.I.1 Phonon wave-numbers ν (cm−1), infrared intensity Ip (km/mol) and

vibrational contribution to the dielectric tensor vib (for the first three

structures, parallel and perpendicular components refer to the z direc-

tion) for each mode p of the four SrTiO3 structures considered. Dashes

indicate null values. IR intensities and vibrational contributions to the

dielectric tensors are not reported for imaginary phonon frequencies

(crosses). The last two rows report the total vibrational and electronic

contributions to . Values obtained with the PBE0 hybrid functional.

Experimental values are from Refs.(36; 37; 39; 29). The symmetry la-

beling of the modes according to the irreps of the various structures

can be found in Refs.(17; 29; 35). . . . . . . . . . . . . . . . . . . . . 104

B.I.2 Elastic and compliance constants of the four structures considered of

SrTiO3. Electronic “clamped-ion” and total “relaxed”, with nuclear

contribution, constants are reported. The computed bulk modulus Ks

is also reported. Calculations performed at PBE0 level. . . . . . . . . 108

B.I.3 Direct and converse piezoelectric constants of the two ferroelectric

structures considered of SrTiO3. Electronic “clamped-ion” and total

“relaxed”, with nuclear contribution, constants are reported. Calcula-

tions performed at PBE0 level. . . . . . . . . . . . . . . . . . . . . . . 108

viii

LIST OF TABLES

B.II.1 Elastic properties of (BN)xG1−x structures in the whole composition

range. Elastic stiffness constants, Cvu, Young’s modulus, YS, and bulk

modulus, KS, are given in N/m. Compliance constants, svu, are given

in 10−3 m/N while Poisson’s ratio, ν, is dimensionless. Total values are

reported along with purely nuclear relaxation effects (in parentheses)

using the B3LYP level. . . . . . . . . . . . . . . . . . . . . . . . . . . 119

B.II.2 Direct and converse piezoelectric coefficients, e11 and d11, as calculated

by LDA/DFT and B3LYP levels of theory, and as a function of band

gap Eg (in eV). Values reported are in 10−10C/m for piezoelectric stress

coefficient e11 and pm/V for piezoelectric strain coefficient d11. Total

values are reported along with purely nuclear relaxation effects (in

parentheses). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

C.I.1 In-plane direct and converse piezoelectric constants of SiG as a func-

tion of substitutional fraction x. Electronic “clamped-ion” and total

“relaxed”, with nuclear contribution, constants are reported. e11 is

given in 2-dimensional unit of 10−10C/m, and d11 in pm/V. Calcula-

tions performed at B3LYP level. . . . . . . . . . . . . . . . . . . . . . 138

C.I.2 Elastic behaviour as a function of concentration fraction x for Si-doped

graphene sheets. Elastic and compliance constants are given in N/m

and 10−3 m/N, respectively. . . . . . . . . . . . . . . . . . . . . . . . 139

C.I.3 In-plane direct piezoelectric constants as a function of substitutional

fraction x for Cs-symmetry structures, GeG and SnG. Total values are

reported along with nuclear relaxation contributions (in parentheses).

Constant values are given in 10−10C/m. . . . . . . . . . . . . . . . . . 140

C.II.1 Out-of-plane direct (e31) and converse (d31) piezoelectric constants of

SiG as a function of x. Total values are reported along with nuclear

relaxation contributions (in parentheses). . . . . . . . . . . . . . . . . 144

C.II.2 Elastic properties of simulated structures of Si, Ge, Sn, and H-N doped

graphene, x is here equal to 0.125. Electronic and vibrational contri-

butions are separately reported, where total values are the sum of

both. Dashes refer to null value of Cs-symmetry structures. Elastic

constants, Cuv, are given in N/m, and compliance constants, suv, are

given in 10−3 m/N. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

C.II.3 Out-of-plane direct and converse piezoelectric constants, relaxed total

values, for H-substituted pyrrolic-N-doped graphene. e and d con-

stants are given in 10−10 C/m, and pm/V, respectively. . . . . . . . . 151

ix

LIST OF TABLES

C.II.4 In-plane and out-of-plane piezoelectric (direct and converse) constants

of perturbed graphene as computed in the present study and as com-

pared to experimental and theoretical values of other 2D and 3D piezo-

electric materials. Direct e constants are reported in 10−10C/m for 2D

system and in C/m2 for 3D system. Converse d constants are expressed

in pm/V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

x

Glossary

BF Bloch Function; a wavefunction for a

particle in a periodically-repeating envi-

roment

BZ Brillouin Zone; the primittive cell zone

in reciprocal space

CPHF/CPKS Coupled Perturbed Hartree

Fock/Coupled Perturbed Khon-Sham; a

successive coupled perturbed procedure

of wavefunction to compute dielectric

properties

DFT Density Functional Theory; theorem of

using the electron density to find the sys-

tem energy

ECPs Effective Core Pseudopotential; an in-

considerable core electrons basis set

HF Hartree Fock method; an applied theory

to solve Schrodinger equation

HOMO/HOCO Highest Occupied Molecular

Orbital/Highest Occupied Crystalline

Orbital

KS Khon-Sham scheme

LUMO/LUCO Lowest Unoccupied Molecular

Orbital/Lowest Unoccupied Crystalline

Orbital

SCF Self Consistent Field method; successive

iteration process till the convergence of

the property

SOS Sum Over States; it refers to the elec-

tronic uncoupled contribution, in other

words the unperturbed charge distribu-

tion interacts with the field

xi

INTRODUCTION

2

INTRODUCTION GENERALE

La piezoelectricite consiste en la conversion mutuelle des forces mecaniques et

electriques dans le materiau. Elle se decline en effet direct qui conduit a produire

un champ electrique par deformation mecanique ou effet inverse lorsqu’un mouvement

est produit par application d’une tension electrique. Ainsi une reponse electrique ap-

paraıt dans le materiau quand une deformation mecanique est appliquee, et inverse-

ment. Le mot piezoelectricite a une origine grecque, il est compose de deux par-

ties: piezo ou piezein qui signifie presser, et electrique ce qui signifie une source de

champ electrique. L’effet direct de la piezoelectricite est decouvert par les physiciens

francais Jacques et Pierre Curie en 1880, et il peut etre explique de la facon que le

deplacement des ions dans le materiau grace a la deformation a donne lieu a une po-

larisation electrique. L’effet inverse a toutefois ete decouvert plus tard par Gabriel

Lippmann en 1881 grace a l’aspect mathematique des principes fondamentaux de la

thermodynamique. Elle represente la distorsion quand un champ electrique est applique

a travers le materiau. Lorsqu’un champ electrique est applique, les ions sont deplaces

par des forces electrostatiques, et font donc la deformation mecanique de l’ensemble

du cristal.

Depuis sa decouverte, la piezoelectricite est la base d’une grande variete

d’applications technologiques (commutateurs de taille nanometrique, capteurs,

moteurs, collecteurs d’energie, actionneurs, etc.). Des applications modernes de

la piezoelectricite directe ont ete recemment developpees comme la production

d’electricite dans une ville faite par la pression des voitures sur une route, l’utilisation

de la force mecanique produite par les trains comme source d’energie dans les gares, ou

meme l’utilisation de la force musculaire dans les salles de sport ou les discotheques,

afin de generer un champ electrique via les capteurs piezoelectriques. Aussi, l’effet

piezoelectrique inverse a de nombreuses applications telles que les muscles artificiels,

les machines de nettoyage a ultrasons, et les sondeurs piezoelectriques.

Du point de vue experimental, les techniques de la mesure de la piezoelectricite

sont classees en deux categories: les mesures directe et indirecte. La mesure directe

repose sur une mesure du deplacement des atomes induit par un champ electrique

applique (permet de deduire les coefficients, d, de la piezoelectricite inverse), ou

sur une mesure du champ electrique produit par une pression (permet de deduire

les coefficients de la piezoelectricite directe, e). La mesure indirecte s’obtient par

la mesure d’une propriete induite par une contrainte electrique ou mecanique. Par

exemple, la mesure de l’amplitude de l’onde acoustique induite par un arc electrique.

Pour la mesure de la piezoelectricite d’un film fin, il existe par exemple en tout une

4


dizaine de techniques qui possedent chacune leur resolution propre, leurs avantages et

leurs inconvenients.

Du point de vue theorique, la piezoelectricite peut etre exprimee comme la somme

de deux contributions: une contribution electronique dite “clamped-ion” et une

autre nucleaire dite “internal-strain”. La contribution electronique provient de la

deformation du nuage electronique en raison du champ mecanique et/ou la pertur-

bation. Elle peut etre calculee separement en utilisant des conditions “clamped-ion”,

ou les coordonnees fractionnaires sont laissees fixes (non optimisees) apres l’application

du champ mecanique. La contribution nucleaire est obtenue a partir des deplacements

relatifs des noyaux dans le systeme perturbe et elle peut etre incluse dans le calcul si les

coordonnees atomiques sont relaxees. Chaque contribution peut etre exprimee comme

une somme de termes perturbatifs d’ordre deux de l’energie du systeme.

L’approche la plus utilisee pour calculer la reponse piezoelectrique de cristaux est

l’approche numerique de la phase de Berry. Cette methode consiste a calculer la

derivee premiere du moment dipolaire (ou vecteur de polarisation) par rapport a une

deformation mecanique. Ces derivees sont obtenues numeriquement par des differences

finies des phases de Berry.

Au cours des dernieres versions de crystal, le calcul de la piezoelectricite en

utilisant cette derniere approche numerique, etait decompose en quelques etapes. La

premiere etape consistait a effectuer un calcul preliminaire de l’energie du systeme

non deforme. La suivante reposait, pour plusieurs valeurs de la deformation (η) et

pour chacune de ses composantes, sur le calcul du vecteur de polarisation electrique

(P). Puis, des “fits” des valeurs obtenues etaient finalement faits afin d’obtenir les

composantes de la piezoelectricite (tangentes des courbes P en fonction de η). Cette

procedure a ete appliquee pour les exemples: ZnO et BeO.

Pour une manipulation plus pratique, une procedure entierement automatisee de

ces etapes a ete mise en œuvre dans la derniere version en developpement de crystal

par le groupe de chimie theorique de Turin. Celle-ci est utilisee dans le travail present

et a permis de predire les proprietes piezoelectriques de systemes complexes de facon

efficace. De plus, le calcul de la piezoelectricite adapte aux systemes 2D (et 1D) par

nos collegues turinois, est exploite pour concevoir des materiaux montrant un effet

piezoelectrique augmente. L’automatisation du calcul de la piezoelectricite ainsi que

la disponibilite des hybrides-DFT (donnant communement la meilleure estimation des

proprietes optiques et vibratoires), sont les raisons principales d’utiliser le software

crystal dans cette etude.

5


L’objectif principal de cette etude est donc de donner une analyse exhaustive de

ce phenomene piezoelectrique, de rationaliser les parametres principaux qui pourraient

affecter cette propriete, et enfin de concevoir des materiaux possedant une propriete

piezoelectrique induite et augmentee.

Ainsi, la these est classee en trois parties contenant chacune deux chapitres. La

premiere partie (Part A) introduit le contexte theorique, les outils de calcul de la

piezoelectricite et des proprietes de reponse associees. Elle est divisee en deux chapitres:

les approches theoriques et computationnelles. Nous y presentons les principes de calcul

de la structure electronique et de la piezoelectricite. Le code utilise et la calibration

des parametres de calcul sont egalement detailles.

La deuxieme partie, Part B, rend compte de la rationalisation de l’effet

piezoelectrique en mettant en evidence les parametres microscopiques qui affectent

la propriete macroscopique. Les contributions electronique et vibrationnelle de la

piezoelectricite y sont discutees dans les chapitres I et II, respectivement. Un ex-

emple illustratif est egalement donne pour chaque contribution: l’exemple du graphene

dope par BN pour la contribution electronique, ainsi que l’exemple de deux structures

de perovskite (SrTiO3 et BaTiO3) pour la partie vibrationnelle.

La derniere partie (Part C) est consacree a la conception des materiaux pouvant

presenter un fort effet piezoelectrique. L’etude est ici focalisee sur la fonctionnalisation

de la surface du graphene pour induire une telle propriete sur un materiau qui ne

la possede pas par nature. Conclusions et perspectives sont enfin donnees avant les

annexes.

6

INTRODUCTION

Piezoelectricity consists in the mutual conversion of mechanical and electrical forces

in the material, and so the electric response of the material when a mechanical defor-

mation is applied, and vice versa. The word piezoelectricity has a Greek origin, and

it is composed from two parts: piezo or piezein which means to squeeze or press, and

electric that means a source of electric field. So, the direct piezoelectric effect is defined

as the appearance of an electrical potential when the crystal is pressed. This effect is

discovered by French physicists Jacques and Pierre Curie in 1880,(1) and it can be

explained as that, the displacement of ions due to the mechanical pressure resulted in

an electric polarization of the crystal structural units. The converse effect was however

discovered later by Gabriel Lippmann in 1881 through the mathematical aspect from

the fundamental thermodynamic principles of the theory.(2) It represents the distor-

tion when an electrical field is applied through the material. When an electrical field

is applied, the ions are displaced by electrostatic forces resulting in the mechanical

deformation of the whole crystal.

Since its discovery, piezoelectricity is central to a wide variety of technological

applications (nanosized switches, sensors, motors, energy harvesters, actuators,

etc.).(3; 4) Modern applications of piezoelectricity as a source of electricity, are

recently considered such as: the generation of electricity in a city made from the

pressure of cars on a road, and the use of mechanical forces produced in train stations,

or even sport halls, to generate an electric field via the piezoelectric sensors. As

well, converse piezoelectric effect has many applications such as: artificial muscles,

ultrasonic cleaning machines, ultrasonic humidifier and piezoelectric sounders, among

others.(5; 6)

From an experimental point of view, the techniques for piezoelectric measurement

are classified into two categories: direct and indirect measurements. The former

one relies on a direct investigation of either the mechanical displacement induced

by applied electric field (converse piezoelectric coefficients, d, can be extracted), or

the amount of electric charge produced by imposing a load, from which the direct

piezoelectric coefficients, e, can be obtained. The indirect measurement depends on

the intrinsic linkage between the mechanical behaviour (stress and strain) and the

electrical ones (voltage or charge). Generally, applying of an electric or mechanical

fields through the material is assumed to generate an induced response property.

For example, variuos acoustic waves can be generated when an arc electric field is

applied through the material. The amplitudes of these generated acoustic waves can

be detected as signals. From the obtained signals an indirect determination of the

7

INTRODUCTION

piezoelectric properties of this material can be then deduced. Several techniques

(direct and indirect) for the measurement of piezoelectricity of a thin film, for

instance, can be found within the following citation(7) where the resolution, reliabil-

ity, advantages and disadvantages of each technique are explicitly stated and discussed.

Theoretically, piezoelectricity can be expressed as the sum of two contributions;

purely electronic “clamped-ion” and nuclear “internal-strain” contributions. Com-

monly, the electronic contribution arises from the electronic cloud deformation due

to mechanical or/and field perturbation. This contribution can be individually cal-

culated using clamped-ion conditions, where the fractional coordinates are fixed “not

optimized” after applying of the mechanical field. The nuclear (ionic) contribution is

initiated from the relative displacements of nuclei in the perturbed system; and it can

be included in the calculations if the coordinates are left to relax. So, the two contri-

butions can be obtained separately from the theoretical calculations, however, for the

experimental measurements only the static (total) value can be determined. Since the

piezoelectricity is considered as a second-order-perturbation of the system energy, so

each contribution can be expressed mathematically as a sum of fractional terms.

The most widely used approach to compute the piezoelectric response of crystals

is the Berry phase numerical scheme.(8; 9) This method consists in the calculation of

the first derivative of the dipole moment (or polarization vector) with respect to cell

deformation. The derivatives are obtained numerically by finite differences over Berry

phases computed at various strained configurations.

In the previous versions of crystal, the calculation of piezoelectric components of a

system using the latter numerical scheme, was decomposed to few steps. A preliminary

run for the energy of the undistorted system must be firstly performed. Then, for the

distorted system, a second preliminary run have to be carried out, followed by a third

one that calculates the polarization vector (P) with respect to the deformation (η).

For more accuracy, other runs are done for other distortions. Then a fit of obtained

values is made in order to get the value of different components of piezoelectric tensor

(slopes of the relation P as a function of η). This procedure has been applied for the

example of ZnO and BeO systems.(10)

For a more practical handling, a fully-automated procedure has been implemented

in the last versions of crystal (developed version of crystal09 and crystal14)

by the theoretical chemistry group of Torino. This procedure is applied in the

present work, and it allows to perform effectively the calculation of piezoelectric

tensor for more complicated systems. Moreover, the calculations of piezoelectricity

of low dimensional 2D systems (and 1D), that developed and implemented recently

in crystal by our collaborators, are here exploited to design 2D materials showing

8

INTRODUCTION

enhanced piezoelectric effect. The automation of piezoelectricity calculations as well

as the availability of hybrids DFT functionals that commonly well-estimate optic and

vibrational properties, are the main stimuli for employing crystal program in the

current study.

So, the main objective of the present study is to give an exhaustive analysis of the

piezoelectric phenomenon, to rationalize the principle parameters that could affect this

technologically important property, and finally to design a material that could exert a

huge piezoelectric and/or converse effect.

In this respect, the thesis is classified into three parts, where each part contains

two chapters:

⇒ The first part (Part A) introduces theoretical background and tools for calculating

the piezoelectricity and its associated response properties. This part is divided

into two chapters: theoretical and computational approaches. We have discussed

in this part the theory for both electronic structure computations and piezo-

electricity, as well as, the crystal code, the optimization of basis set, and the

calibration of piezoelectric computations. The calibration process is performed

on the well-known piezoelectric α-quartz, where silicon atoms are progressively

substituted with germanium in order to study the whole range of substitution for

Si1−xGexO2 solid solution.(11)

⇒ The second part, Part B, reports the rationalization of the piezoelectric effect,

where the microscopic parameters that could affect this macroscopic property

are given and discussed into details for both electronic and vibrational contri-

butions to the piezoelectricity; Chapters I and II, respectively. An illustrative

example is additionally nominated for each piezoelectric contribution: BN-doped

graphene for electronic contribution,(12) and two perovskite structures (SrTiO3

and BaTiO3) for vibrational one.(13; 14)

⇒ Final part (Part C) hands the design of materials for large piezoelectric and/or

converse effect, where functionalized graphene: Si, Ge, Sn, and pyrrolic-N-doped

graphene structures are considered for this purpose.(15)

Conclusions and perspectives are finally drawn before the appendix.

9

REFERENCES

REFERENCES

[1] J. Cuire and P. Curie. C. R. Acad. Sci., 91:294–295, 1880.

[2] G. Lippmann. Annales de chimie et de physique, 24:145, 1881.

[3] G. Gautschi. Piezoelectric Sensorics: Force, Strain, Pressure, Acceleration and

Acoustic Emission Sensors, Materials and Amplifiers. Engineering online library.

Springer, 2002.

[4] A. Nechibvute, A. Chawanda, and P. Luhanga. Smart Materials Research,

2012:Article ID 853481, 2012.

[5] J. D. Madden, N. A. Vandesteeg, P. A. Anquetil, P. G. Madden, A. Takshi, R. Z.

Pytel, S. R. Lafontaine, P. A. Wieringa, and I. W. Hunter. IEEE J. Oceanic Eng.,

29(3):706–728, 2004.

[6] A. Manbachi and R. S. C. Cobbold. Ultrasound, 19(4):187–196, 2011,

http://ult.sagepub.com/content/19/4/187.full.pdf+html.

[7] J.-M. Liu, B. Pan, H. Chan, S. Zhu, Y. Zhu, and Z. Liu. Materials Chemistry and

Physics, 75(13):12 – 18, 2002.

[8] R. D. King-Smith and D. Vanderbilt. Phys. Rev. B, 47:1651–1654, Jan 1993.

[9] R. Resta. Rev. Mod. Phys., 66:899–915, Jul 1994.

[10] Y. Noel, C. M. Zicovich-Wilson, B. Civalleri, P. D’Arco, and R. Dovesi. Phys.

Rev. B, 65:014111, Dec 2001.

[11] K. E. El-Kelany, A. Erba, P. Carbonniere, and M. Rerat. J. Phys.: Cond. Matter,

26(20):205401, 2014.

[12] K. E. El-Kelany, P. Carbonniere, A. Erba, and M. Rerat. J. Phys. Chem. C,

119(16):8966–8973, 2015.

[13] A. Erba, K. E. El-Kelany, M. Ferrero, I. Baraille, and M. Rerat. Phys. Rev. B,

88(3):035102 1–10, 2013.

[14] A. Mahmoud, A. Erba, K. E. El-Kelany, M. Rerat, and R. Orlando. Phys. Rev.

B, 89:045103, Jan 2014.

[15] K. E. El-Kelany, P. Carbonniere, A. Erba, J.-M. Sotiropoulos, and M. Rerat.

submitted, 2016.

10

Part A

Theoretical Tools for Calculating

Piezoelectric Properties

12

I. THEORETICAL APPROACH

1 Theoretical Approach for Electronic Structure

Computations

This chapter will be devoted to introducing the computational chemistry and some

quantum mechanics aspects which can help in understanding the basis required for

electronic structure calculations. In the late of the seventeenth century after the dis-

cover of Newton’s laws for classical mechanics, physicists found that these laws don’t

correctly describe the motion of the particles in the microscopic scale, such as the elec-

trons and nuclei in atoms and molecules. In order to describe the behaviour of these

particles, another mechanics (quantum mechanics) were required. We begin with the

fundamental postulate of quantum mechanics, the so-called wave function, Ψ, exists

for any (chemical) system, and appropriate operators which act upon Ψ return the

observable properties of the system.(1) The most important foundation of theoretical

chemistry is the Schrodinger equation, in which an operator acts upon Ψ to return the

system energy, E. This can be written as:

HΨ = EΨ (I.1)

where H is the Hamiltonian operator, Ψ is the wave function (eigenfunction for a given

Hamiltonian) and E is the energy of the system. The typical form of the Hamiltonian

operator takes into account five contributions to the total energy of a system: the ki-

netic energies of the electrons and nuclei, the attraction of the electrons to the nuclei,

and the inter-electronic and inter-nuclear repulsions. In more complicated situations

such as: the presence of an external electric or magnetic fields, considering the rela-

tivistic effects, etc., other terms are required in the Hamiltonian. So, the Hamiltonian

can be expressed mathematically as;

H = −

i

2

2me

∇2i −

k

2

2mk

∇2k −

i

k

e2Zk

rik+

i≺j

e2

rij+

k≺l

e2ZkZl

rkl(I.2)

where i and j run over electrons, k and l run over nuclei, is the Planck’s constant

divided by 2π, me is the mass of the electron, mk is the mass of the nucleus k. ∇2 is the

Laplacian operator, e is the charge of electron, Z is an atomic number and generally,

rab is the distance between two particles a and b. The wave function Ψ describes

the system and takes as variables the positions of electrons and nuclei in the system,

leading to the following equation:

HΨi( x1, ...., xNR1, ...., RM) = EΨi( x1, ...., xN

R1, ...., RM) (I.3)

14

1. Theoretical Approach for Electronic Structure Computations

xN describing the position of the electrons N, and RM describing the position of the

nuclei M. A knowledge of Ψ allows the properties of the system to be deduced. The

wave function in equation (I.1) chosen to be orthonormal (orthogonal and normal) over

all space, i.e.:

Ψi|Ψj = δij (I.4)

where δij is the Kronecker delta symbol and takes the values δij = 0 if i = j for

orthogonality and δij = 1 if i = j for normality of the function.

The Born-Oppenheimer Approximation

As noted previously, the wave functions Ψ are functions of the position of both the

nuclei and the electrons of the system. Since, nucleus is much heavier than electron

(approximately 1800 times more massive than electrons), and the mass is found in the

denominator of the kinetic energy terms of the Hamiltonian in equation (I.2).So, it is

convenient to decouple these two motions, and compute electronic energies for fixed

nuclear positions.(2) In this case, the kinetic energy of the nucleus can set to be zero

and the electronic Schrodinger equation is taken to be:

(Hel + VN)Ψel(qi; qk) = EelΨel(qi; qk) (I.5)

where, Hel = −

i2

2me∇2

i −

i

ke2Zk

rik+

i≺je2

rij, and the wave function depends

now explicitly on the electronic coordinates:

HelΨi(el)( x1, x2, ...., xi, xj, ...., xN) = EelΨi(el)( x1, x2, ...., xi, xj, ...., xN) (I.6)

All electrons are characterized by a spin quantum number, with two possible eigen-

values -1/2 or +1/2, that are defined by the alignment of the spin with respect to an

arbitrary axis. These two types of spin are called α and β (by convention α and β are

the spin functions for ms = +1/2 and -1/2, respectively) and are orthonormalised:

α|α = β|β = 1

α|β = β|α = 0

The wave function is described by both a spatial component and a spin component:

Ψ(x) = ψ(x).σ σ = α or β

The Slater Determinant

From the anti-symmetry principle, we can deduce that: the wave function Ψ is not

observable itself, but the expression |Ψ( x1, x2, ..., xN)|2d x1d x2....d xN , where d x1d x2...

is a small volume, represents the probability of finding an electron at a given point

15


in space. The electrons being indistinguishable, the exchange of two electrons doesn’t

change the probability:

|Ψ( x1, x2, ..., xi, xj, ..., xN)|2 = |Ψ( x1, x2, ..., xj, xi, ..., xN)|

2 (I.7)

However, the exchange of two electrons leads to a change of sign of the wave func-

tion, i.e. Ψ is anti-symmetric with respect to electron change. This represents the

quantum-mechanical generalization of the Pauli’s exclusion principle (two electrons

can not be characterized by the same set of quantum numbers).

The exact wave function is unknown, so it is necessary to generate a trial wave

function which obeys this anti-symmetry principle. To do that, the N-electron wave

function is expressed as an anti-symmetric product of N one-electron wave functions

χi(xi). The product is donated by ΦSD and is referred to as the Slater determinant,

which was first exploited by Slater in 1929:(3)

ΦSD =1√N !

χ1( x1) χ2( x1) . . . χN( x1)χ1( x2) χ2( x2) . . . χN( x2)

......

. . ....

χ1( xN) χ2( xN) . . . χN( xN)

(I.8)

Where N is the total number of electrons and χ is a spin-orbital. The columns are

single electron wave functions (orbitals) χ(x), while the rows are the electron indices.

1.1 Hartree Fock Self-Consistent Field Method (HF-SCF)

Much of the difficulty of solving the Schrodinger equation stems from the need to simul-

taneously determine the energy of each electron in the presence of all other electrons.

In the Hartree-Fock (HF) method this is avoided by calculating the energy of each

electron in the averaged static field of the others. Fock proposed first the extension of

Hartree’s SCF procedure to Slater determinantal wave functions just as with Hartree

product orbitals, the HF MOs can be individually determined as eigen functions of a

set of one-electron operators, but now the interaction of each electron with the static

field of all of the other electrons includes exchange effects on the Coulomb repulsion.(4)

So the Fock Hamiltonian can be expressed as:

fi = −1

2∇2

i −nuclei

k

Zk

rik+ V HF

i (I.9)

where V HFi is the HF potential, and represents the average repulsive potential ex-

perienced by each electron due to the other N-1 electrons. Thus, the complicated

16


two-electron repulsion operator 1/rij in the Hamiltonian is replaced by the simple one-

electron operator V HFi where the electron-electron repulsion is taken into account only

in an average way. And, V HFi has two components:

V HF ( x1) =N

j

Jj( x1)− Kj( x1)

. (I.10)

The Coulomb operator J is defined as;

Jj( x1) =

|χj( x2)|2 1

r12d x2 (I.11)

and represents the potential that an electron at position x1 experiences due to the

average charge distribution of another electron in spin orbital χj.

The second term in equation I.10 is the exchange contribution to the HF potential.

It has no classical analog and it describes the modification of the energy that can be

ascribed to the effects of spin correlation, as will be seen in the next section. It is

defined through its effect when operating on a spin orbital.

Kj( x1)χi( x1) =

χ∗j( x2)

1

r12χi( x2)d x2χj( x1) (I.12)

In order to calculate V HFi , which is non-local and depends on the spin orbitals, it

is necessary to know χ. So, initially a guess is made of the electron energies V HFi , then

χ is calculated and used to generate a new V HFi . The energy of each electron is then

calculated in the field of the initial electron configuration. This process continues until

the cycle converges, i.e. until successive potentials are identical, see Figure A.I.1.

To finally solve the Hartree-Fock equation two methods can be used depending on

the situation faced. If the system contains an even number of electrons, all of them

paired, the system is solved using the RHF formalism (Restricted Hartree Fock) or

if there is an odd number of electrons or an even number but with some electrons

unpaired, then the UHF formalism (Unrestricted Hartree-Fock) can be used. In the

first case, RHF, the equations of Roothan-Hall(5) are used to resolve the equation,

whereas in the second case, UHF, the equation is resolved using the Berthier-Pople-

Nesbert equations.

Electron Correlation from post HF Methods

The main problem emerging from the solutions obtained through the HF method

is that the total energy obtained is always higher than the real energy calculated from

Schrodinger equation. This is principally due to the fact that in the Hartree-Fock

method the electrons are considered to move in an average electronic field, so the

17


Figure A.I.1: SCF cyclization involved in the Hartree Fock method.

correlated motion of each electron with the others is omitted. The difference between

the real energy and the HF energy is designated as the correlation energy.

Ecorrelation = Etotal − EHF (I.13)

This gap represents the electronic correlation energy. The correlation constitutes,

in most cases, approximately 1% of the total energy but this 1% can have a large

influence on the properties calculated for the system. The usual way to introduce the

correlation is to take into account the excitation of one or more electrons from one

or more occupied orbitals to one or more virtual orbitals higher in energy, see Figure

A.I.2.

Each state is described by a Slater determinant, and the combination of them gives

the new trial function which should be closer to the real system than the original

determinant,

Ψ = c0ΦHF +

i=1

ciΦi (I.14)

where the sum is over all the possible electronic configurations and ci are the co-

efficients defining the contribution of each configuration “excited state” to the wave

function.

18


Figure A.I.2: Possible distribution of the electrons for Single, Double and Triple virtual

excitation.

1.2 Density Functional Theory (DFT)

1.2.1 Principles

In contrast to Hartree-Fock theory, DFT theory is based on electron density, rather

than on wave functions. Electron density can be easily found experimentally via X-

ray diffraction as well as from theory. By using electron density associated with the

correct Hamiltonian operator, the energy of the system can be completely described.

In 1964, Hohenberg and Kohn(6) established two fundamental theorems that marked

the beginning of modern DFT. Their first theorem states that “the external potential

Vext(r) applied on the system (Vext is an external potential to the system which is

due to the presence of the nuclei) is defined as a unique functional of the electronic

density, ρ(r)”. One particular external potential can be defined by one and only one

particular electron density and vice versa. In turn Vext fixes H which is therefore a

unique functional of ρ(r). We can simply add ρ0 as the property which contains the

information about N, ZA, RA and summarize this as:

ρ =⇒ N,ZA, RA =⇒ H =⇒ Ψ0 =⇒ E0 (and all other properties)

Since the complete ground state energy is a functional of the ground state electron

19


density, so must be its individual components and we can write:

E0[ρ0] = T [ρ0] + Eee[ρ0] + ENe[ρ0] (I.15)

where we revert to the subscript ‘Ne’ to specify the kind of external potential presented

in our case, which is fully defined by the attraction due to the nuclei. The T [ρ0]

and Eee[ρ0] parts of the equation are independent of the variables N, RA and ZA

(respectively: number of electrons, electron-nucleus distance and the nuclear charge)

whereas ENe[ρ0] is dependent upon those variables. So the previous equation can be

re-written as following:

E0[ρ0] = T[ρ0] + Eee[ρ0]

universally valid

+

ρ0(r)VNed(r)

systemdependent

(I.16)

The independent parts are gathered into a new quantity: the Hohenberg-Kohn func-

tional:

FHK [ρ] = T [ρ] + Eee[ρ] (I.17)

If the functional FHK was known exactly, it would allow the calculation of E0. How-

ever the explicit forms of the two terms which compose this functional are unknown.

The Eee term can be separated in two terms: a Coulomb part and a term containing

all the non-classical contributions to the electron-electron interaction.

Eee[ρ] = J [ρ] + Enucl[ρ] (I.18)

Enucl is the non-classical contribution to the electron-electron interaction containing

all the effects of self-interaction correction, exchange and Coulomb correlation. At this

point, the ground state density determines the Hamiltonian operator which character-

izes all states of the system, ground and excited states. The second Hohenberg-Kohn

theorem states that FHK [ρ], the functional that gives the lowest ground state energy

of the system, if and only if the input density is the true ground state density, ρ0. So,

this theorem simply applies the variational theory to the electronic density. And when

an approximate electronic density ρ(r), associated with an external potential Vext, is

used, the resulting energy, as in HF, will be always greater than or equal to the exact

ground state energy:

E[ρ] = T [ρ] + ENe[ρ] + Eee[ρ] ≥ Eexact (I.19)

The poor presentation of the kinetic energy appeared as a problem in the

Hohenberg-Kohn orbital-free model. In 1965, Kohn and Sham(7) realized that most

of the problems are connected with the way of the distribution of the kinetic energy.

20


So, they proposed to resolve the problem by introducing the idea of a fictitious system

(KS scheme) builds from a set of orbitals (one electron functions) where the electrons

are non interacting, i.e. each electron is submitted to an average repulsion field coming

from the other electrons. They divided the total energy in the following parts:

E[ρ] = TS[ρ] +

[Vext(r) + J(r)]ρ(r)dr + EXC [ρ] (I.20)

TS[ρ] corresponds to the electron kinetic energy of the hypothetical system, with ρ

equivalent to the real system but with the non-interacting electrons. J(r) represents

the classical Coulomb interaction between the electrons, and Vext is the potential arising

from the nuclei:

J(r) =

ρ(r)|r − r|

dr (I.21)

Vext =

A

ZA

|RA − r|(I.22)

r and r represent the coordinates of 2 electrons. The computation of the kinetic energy

can be expressed in terms of one electron function:

TS[ρ] = −1

2

N

i=1

< φi|2 |φi > (I.23)

Here the suffix emphasizes that this is not the true kinetic energy but is that of a

system of non-interacting electrons, which reproduce the true ground state density.

Finally, EXC is a term which encompasses all the other contributions to the energy

which are not accounted for in the previous terms such as electron exchange, correlation

energy and correction for the self-interaction included in the Coulomb term and the

portion of the kinetic energy which corresponds to the differences between the non-

interacting and the real system.

A new Hamiltonian can be created by taking into account only the non-interacting

system:

HS = −1

2

N

i

2i +

N

i

VS(ri) (I.24)

One-electron functions are reintroduced in density functional theory in the form of

Kohn-Sham orbitals, φi. As for HF, these orbitals are determined by:

fKSφi = εiφi (I.25)

with fKS = −1

22 +VS(r) (I.26)

21


where fKS is a one-electron operator, called the Kohn-Sham operator and the corre-

sponding orbitals are called Kohn-Sham orbitals. VS(r) describes the effective potential

of the non-interacting reference system. The non-interacting system is related to the

real system by choosing an effective potential, VS such that:

ρS(r) =N

i

|φi(r)|2 = ρ0(r) (I.27)

Then we can come back to the original system:

EDFT [ρ] = TS[ρ] + ENe[ρ] + J [ρ] + EXC [ρ] (I.28)

where:

EXC [ρ] = (T [ρ]− TS[ρ]) + (Eee[ρ]− J [ρ]) = TC [ρ] + Enucl[ρ] (I.29)

In this way, it becomes possible to compute the major part of the kinetic energy (the

rest being merged with the non-classical electron-electron repulsion). The Hohenberg-

Kohn functional then becomes:

F [ρ(r)] = TS[ρ(r)] + J [ρ(r)] + EXC [ρ(r)] (I.30)

where EXC contains the residual kinetic energy as well as the repulsion terms.

As stated previously, EXC is the only unknown term of the equation. To model

it, it is necessary to approximate it. In the next section the different approximation

methods used to constitute the functional will be described.

As will be shown later, all the calculations in the present work are done by using

the crystal code, the only program that permits the use of DFT-hybrids for the

treatment of periodic systems. Since, piezoelectricity is found to be very sensitive for

both optical and vibrational properties, as will be discussed in the next part, so we are

here interesting in this kind of Hamiltonian (DFT-hybrids) that give commonly a well

estimation for these properties.

1.2.2 Mathematical Description of Exchange and Correlation Parts

Exchange-Correlation Functional (Exc)

As noted earlier, the EXC [ρ] functional contains the non-classical contributions

to the potential energy due to the electron-electron interaction and the difference

between the kinetic energy of the real system and the kinetic energy related to the

non-interacting system. The “functionals” described here represent different approxi-

mations to this exchange-correlation functional. Development of new functionals is an

ongoing and active area of research.

22


LDA: Local Density Approximation

This is the base for most of the exchange-correlation functionals, and it is defined

the using of electronic density of a uniform electron gas. The constant value of the elec-

tronic density does not reflect the rapid variation of densities in a molecule. Although

LDA is a rough approximation, it is the only system for which the density is defined by

ρ = NV

(N represents the number of electrons and V represents the volume of the gas),

and the form of the exchange and correlation energy functionals are known exactly

or to a very high accuracy. In the case of open-shell systems the electronic density,

ρ is replaced by the spin electronic densities, ρα and ρβ such as, ρ = ρα + ρβ. This

approximation is called local spin-density approximation: LSDA. A famous example

of a LDA functional is the one developed by Vosko, Wilk and Nusair (VWN) based on

high-level quantum Monte Carlo calculations for uniform electron gases.(8) The use of

LDA gives more accurate results for the determination of molecular properties (struc-

tures, vibrational frequencies, charge moments, elastic moduli) than the HF method

but shows some flaws in the case of energetics details (bond energies, energy barriers

in chemical reaction) which are poorly characterized by using this type of functional.

GGA: Generalized Gradient Approximation

The LDA can be considered as a zeroth order approximation, but LDA describes

the energies rather badly so a new type of functional was introduced: the generalized

gradient approximation. These functionals include the gradient of the electron density,

ρ. This use of the electron density gradient describes the non-homogeneity of the

true electron density rather better.

The GGA is usually divided into exchange and correlation terms that can, then, be

solved individually.

EGGAXC = EGGA

X + EGGAC (I.31)

Here is a non exhaustive list of some of the most efficient GGA functionals com-

monly used in computational chemistry:

• B is an exchange functional developed by Becke.(9) It is a gradient correction

to the LSDA exchange energy. It includes a single parameter fitted on known

atomic data from the rare gas atoms.

• P86 is a correlation functional developed by Perdew.(10) It is a popular gradient

correction to LSDA which includes one empirical parameter fitted for Neon atom.

• PW91 is an exchange-correlation functional developed by Perdew then Perdew,

Wang and Burke.(11; 12) It is a modification of the P86 functional.

23


• B95 is a correlation functional (for meta-GGA) developed by Becke,(13) it does

not contain any empirical parameters and treats better the self-interaction error.

• PBE is an exchange-correlation functional developed by Perdew, Burke and

Ernzerhof.(14)

• LYP is a correlation functional developed by Lee, Yang and Parr.(15) It is the

most extensively used GGA correlation functional in molecular systems. It con-

tains four empirical parameters fitted to Helium atom.

From the previous GGA functionals a combination between exchange and correla-

tion functionals is made so as to try to describe completely the systems. Some of the

most common combinations are: BLYP, BP86 and BPW91.

Meta-GGA

The meta-GGA functionals are an expansion of the normal GGA. Contrary to

the GGA, the meta-GGA includes Laplacian of the electron density or the local

kinetic energy density, 2ρ. Common meta-GGA functionals include B1B95(13) and

PBEKCIS.(14)

Hybrid Functionals

The previous functional types all present a problem because the exchange part is

very poorly described due to a problem of electronic self-interaction. On the other hand,

the exchange part in HF is defined exactly. So an alternative approach would be to use

a mix of DFT and HF to describe the exchange energy. However, taking the correlation

part from DFT and the exchange part from HF gives poor results (worse than GGA).

A first approach to this problem would be to regroup the exchange and correlation

parts, so a functional that describes the system better than the GGA functionals can

be obtained.(16) The final solution to this problem is the use of a combination of HF,

GGA and LDA functionals to describe the exact exchange and correlation part of the

hybrid functional. Usually hybrid functionals are composed of a mixture of exact and

DFT exchanges. The main element of these functionals come from GGA functionals,

so they are often called GGA hybrid functionals.

• B3 contains exact exchange, and is an exchange functional developed by

Becke.(17) It is a combination of LDA and GGA functionals.

• PBE0 also called PBE1PBE, has been developed by Adamo and Barone.(18) It

is a combination of 75% PBE GGA exchange functional and 25% of HF exchange.

24


• B97 and B98 were developed first by Becke (B97),(19) then modified by Becke

and Schmider (B98).(20) Unlike PBE0 and B3 functionals, B98 and B97 are meta-

GGA hybrid functionals instead of GGA hybrids. They contain an exchange part

taken from HF method.

To describe correctly the exchange-correlation term, it is necessary to combine ex-

change and correlation functionals to obtain an hybrid functional such as: B1B95,(13)

B1LYP(21) or B3P86.(10)

Global hybrid functionals, in which a fraction of exact exchange is mixed with GGA

XC, were introduced by Becke(22) from considerations of the adiabatic connection

formula. In this context, he proposed a hybrid with three parameters,(17)

EhybXC = ELSD

XC + a0(EexactX − ELSD

X ) + aX(EGGAX − ELSD

X ) + aC(EGGAC − ELSD

C ), (I.32)

where EGGAXC is a generalized gradient approximation and ELSD

XC is its local spin density

part. In subsequent work,(13) the functional given by equation (I.32) was simplified by

setting aX = 1−a0 and aC = 1, leading to a hybrid with just one adjustable parameter,

EhybXC = EGGA

XC + a0(EexactX − EGGA

X ). (I.33)

Perdew, Ernzerhof, and Burke(16) later presented an argument based on Gorling-

Levy perturbation theory(23; 24) that suggests that a0 ≈ 1/4. With this value, a

hybrid based on PBE GGA XC, EGGAXC , and free from adjustable parameters, was

proposed.(25; 26) As noted earlier, this hybrid is known as PBE0.(26)

1.3 Periodic Treatment

Before discussing the description of electronic structure of crystalline solids, we will

briefly explain some of expressions related to the crystal lattices and lattice geometry:

The Unit Cell : Crystalline solids can be described as ordered repetitions of atoms

or groups of atoms in three dimensions. One can identify structural fragments that form

the smallest repeating units. Such a small repeating unit that reflects the symmetry

of the structure, is called a unit cell. The unit cell is considered as the building block

of the crystal structure defining the lattice (set of identical points). There are two

distinct types of unit cell: primitive and non-primitive. Primitive unit cells contain

only one lattice point, which is made up from the lattice points at each of the corners.

Non-primitive unit cells contain additional lattice points, either on a face or within

the unit cell, and so have more than one lattice point per unit cell, see panel (a) in

Figure A.I.3. In an ideal crystal, all repeating units are identical and they are related

25


by translational symmetry operations, corresponding to a set of three vectors a, b, and

c. These three co-planar vectors are not necessarily orthogonal, and they confine three

angles α, β, and γ, see Figure A.I.3, panel (b). The length of the unit cell can be

given by the value of a, b, and c vectors along the crystallographic axes x, y, and z,

respectively. a, b, c,α, β, γ are collectively know as the lattice parameters or even unit

cell parameters.

Figure A.I.3: Lattice points of a space lattice in a square two dimensions. Panel (a) shows

primitive (only one lattice point, four lattice points where each point is a four-shared),

and Non-primitive (more than one lattice point, six four-shared lattice points) unit cells.

The lattice vectors and angles are presented in (b). Panel (c) represents the Wigner-Sietz

primitive cell for a hexagonal 2D structure.

Crystal Structure, Lattice System, and Crystal System : The structure of a

crystal can be constructed by combining these elements: (i) the lattice type, which

defines the location of the lattice points within the unit cell; (ii) the lattice parameters,

that define the size and shape of the unit cell; (iii) the motif or the basis, which gives a

list of atoms associated with each lattice point, along with their fractional coordinates

relative to the lattice point. A lattice system is generally identified as a set of lattices

with the same shape according to the relative lengths of the unit cell vectors (a, b, c)

and the angles between them (α, β, γ). Each lattice is assigned to one of the following

classifications (lattice types) based on the positions of the lattice points within the

cell: primitive (P), body-centered (I), face-centered (F), base-centered (A, B, or C),

and rhombohedral (R). In total there are seven lattice systems: triclinic, monoclinic,

26


orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. A crystal system is a

set of point groups in which the point groups themselves and their corresponding space

groups are assigned to the same lattice system. In total there are seven crystal systems:

triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

Bravais Lattice : Since the lattice points within the unit cell may be arranged in

different way, different lattice types are possible within each of the crystal systems.

Altogether, there are fourteen different ways of distributing lattice points to make

space lattices in a three-dimensional lattice. The unit cells of each of these lattices are

distinguished by their pattern of lattice points. These fourteen lattices are called the

Bravais lattices.

Wigner-Seitz Cell and Brillouin Zone : A Wigner-Seitz cell is an example of

another kind of primitive cell, since it contains only one lattice point. The Wigner-

Seitz cell around a lattice point is defined as the locus of points in space that are closer

to that lattice point than to any of the other lattice points. The cell may be chosen

by first picking a lattice point. Then, lines are drawn to all nearby (closest) lattice

points. At the midpoint of each line, another line is drawn normal to each of the first

set of lines. The smallest area (or volume) is enclosed in this way and is called the

Wigner-Seitz primitive cell, see Figure A.I.3, panel (c). The Wigner-Seitz cell in the

reciprocal space is known as the first Brillouin zone (BZ).

Direct and Reciprocal Space : Any periodic distribution of an object (or motif)

can be described by the translations that repeat the object periodically. This set of

translations generate what we call a direct lattice. A lattice translation operation is

defined as the displacement of a crystal parallel to itself by a crystal translation vector

R,

R = ua+ vb+ wc (I.34)

where u, v, and w are three integers ranging from minus infinity to plus infinity,

including zero, and a, b, and c are the basis vector in three-dimensional space. The

parallelepiped formed by these three basis vectors is the unit cell and their directions

define the crystallographic axes: X, Y , and Z. For all atoms inside the unit cell, the

coordinate values are in the interval 0 to 1, and they are therefore called fractional

coordinates and are given by: x = X/a, y = Y/b, and z = Z/c.

Every periodic structure has two lattices associated with it. The first is the real

space lattice (direct space), and this describes the periodic structure. The second is

the reciprocal lattice,(27; 28) and this determines how the periodic structure interacts

with waves. A diffraction pattern of a crystal is a map of the reciprocal lattice of the

27


crystal, in contrary to a microscopic image, which is a map of the real crystal structure.

The reciprocal lattice is described by a new set of vectors a*, b*, and c*, satisfying

the condition:

eiK·R = 1, and K ·R = 2π · n (I.35)

Where, n is an integer, and R is a real space lattice vector that may be expressed in

terms of the lattice basis vectors: a, b, c as in equation I.34. The reciprocal lattice

basis vectors are defined in terms of the direct unit cell basis as follow:

a∗ =2π · b× c

|a · b× c|; b∗ =

2π · c× a

|a · b× c|; c∗ =

2π · a× b

|a · b× c|. (I.36)

Each vector in I.36 is orthogonal to two of the axes of the crystal lattice, and thus

has the property:

a∗ · a = 2π, b∗ · a = 0, c∗ · a = 0;

a∗ · b = 0, b∗ · b = 2π, c∗ · b = 0;

a∗ · c = 0, b∗ · c = 0, c∗ · c = 2π. (I.37)

Equations I.37, suggest that the roles of direct and reciprocal space may be inter-

changed; i.e. the reciprocal of the reciprocal lattice is the direct one. So, any reciprocal

lattice vector can be expressed as a linear sum of these reciprocal basis vectors:

K = ha∗ + kb∗ + lc∗, (h,k,l are integers) (I.38)

The vector K in equation I.38, gives the position of the k-points in the reciprocal space.

Note that vectors in the crystal lattice have the dimensions of [length], while vectors

in the reciprocal space have the dimensions of [length]−1.

1.3.1 Bloch Function

Bloch function (BF) or Bloch state is a type of wave function for a particle in a

periodically-repeating environment, most commonly an electron in a crystal. So, it

is a mathematical form of an electron wave function in the presence of a periodic

potential energy. The time independent Schrodinger equation for an electron in a

periodic potential will be:

− 2

2m∇2 + U(r)

ψ = εψ (I.39)

where the potential energy is invariant under a lattice translation vectorR: U(r+R) =

U(r), and R = ua + vb + wc. A periodic potential appears because the ions are

28


arranged with a periodicity of their Bravais lattice. According to the Bloch theorem,

the solution of Schrodinger equation, equation I.39, will be the product of a plane wave

and a function with the periodicity of the Bravais lattice.

ψkn(r) = uk

n(r)eik·r, where uk

n(r+R) = ukn(r) (I.40)

where r is position, ψ is the Bloch wave function, u is a periodic function with the

same periodicity as the crystal, k is a vector of real numbers called the crystal wave

vector, and i is the imaginary unit. In other words, if you multiply a plane wave by

a periodic function, you get a Bloch wave. The quantum number n appeared in the

last equation, is a discrete index called the band index and takes numbers n = 1, 2, 3,

. . . This quantum number corresponds to the appearance of independent eigenstates

of different energies but with the same k (each has a different periodic component u).

Within a band (i.e., for fixed n), ψkn varies continuously with k, as does its energy.

Also, for any reciprocal lattice vector K, ψkn = ψ

(k+K)n . Therefore, all distinct Bloch

waves occur for k-values within the first Brillouin zone of the reciprocal lattice.

The first Brillouin zone is a restricted set of k-vectors with the property that no

two of them are equivalent, yet every possible k is equivalent to one (and only one)

vector in the first Brillouin zone. Therefore, if we restrict k to the first Brillouin zone,

then every Bloch state has a unique k. Therefore the first Brillouin zone is often used

to depict all of the Bloch states without redundancy, see Figure A.I.4.

Figure A.I.4: The matrix representation of Fock Hamiltonian operator in both direct and

reciprocal spaces. A block-diagonal matrix free from repetition could solely be obtained for

the reciprocal lattice space. Fourier transformations permit passing from direct to reciprocal

spaces and vice versa.

29


1.3.2 Born-Von Karman Boundary Condition

Solid periodic model is finite system, but macroscopic; for simplicity we can suppose

that it is a parallelepiped containing N = N1 · N2 · N3 unit cells,(29) whose sides

are Nj · aj. The electrostatic potential, the atomic positions and the electron charge

distribution close to the surface are different from those at the center of the finite

crystal. However due to the macroscopic nature, the number of atoms perturbed by

surface effect is a small fraction. The Born-Von Karman boundary condition is a

periodic boundary condition which imposes the restriction that a wave function must

be periodic on a certain Bravais lattice. The condition can be stated as,

ψk(r+Njaj) = ψk(r), for j = 1, 2, 3 (I.41)

where aj are the primitive vectors andNj are integers of orderN1/3 whereN = N1N2N3

is the total number of primitive cells in the crystal. If this condition is applied to the

Bloch function, one gets:

ψk(r+Njaj) = e(iNjk·aj)ψk(r) = ψ(r) (I.42)

The quantum number k can be composed from the reciprocal lattice vectors with

(non-integer) coefficients xi,

k = x1b1 + x2b2 + x3b3 (I.43)

Since aj·bi = 2πδij, the Bloch theorem(30) then gives e(2πixjNj) = 1. Thus, xi = mj/Nj

and the allowed Bloch wave vectors are given with mj integers by:

k =3

j=1

mj

Nj

bj with − Nj

2≤ mj ≤

Nj

2(I.44)

mj is an integer with Nj assumed to be even. As for the free electron case, the volume

of the reciprocal lattice cell ∆k per allowed k is given by,

∆k =(2π)3

V(I.45)

where V is the volume of the direct lattice cell.

1.3.3 Crystalline Orbitals

In solid state computational chemistry and physics, the unknown single-particle crys-

talline wave function, ψkn(r), is expanded in a finite set of BFs, φk

n(r):

ψkn(r) =

µ

ckµnφkn(r) (I.46)

30


The coefficients, ckµn, are determined variationally by solving the set of couples matrix

equations:

HkCk = SkCkEk (I.47)

(Ck) + SkCk = I (I.48)

where Hk is the Hamiltonian matrix in the basis set of the φkn(r) functions; S

k is the

overlap matrix among these functions (Sk = I if the basis functions are orthogonal);

Ck is the matrix of the variational coefficients, ckµn; and Ek is the diagonal matrix

of the single particle eigenvalues, εkn. Two basic types of BFs can be used as will be

explained later: a) localized functions, or atomic orbital (AO) based BFs; and b) plane

wave.

1.4 Basis Sets

1.4.1 Atom-Centered Basis Sets

Basis functions are used to create the atomic orbitals (AO) or molecular orbitals and

are usually expanded as a linear combination of such functions with the coefficients to

be determined. These basis functions can be classified into two main types:

• Slater-type orbitals, also called STOs, have the exponential dependence: e−ζr

and are very close in their mathematical expression to the real AO:

ηSTO = Nrn−1e−ζrYlm(Θ,Φ) (I.49)

where N is a factor of normalization, ζ is the exponent. r,Θ and Φ are spheri-

cal coordinates and Ylm is the angular momentum part (function describing the

shape). Finally n, l and m are the classical quantum numbers: principal, angular

momentum and magnetic, respectively.

• Gaussian-type orbitals, also known as GTOs, which have the exponential depen-

dence: e−αr2 :

ηGTO = Nxlymzne−αr2 (I.50)

where N is, as previously, a normalization factor, x, y and z are Cartesian coor-

dinates.

Linear representations of one-electron functions are usually adopted, and these combi-

nations are often called contracted Gaussians:

f(r) =

p

µ=1

cµφµ(r) (I.51)

31


Despite the fact that, STOs reproduce much better the wave function in the proximity

of the nuclei, their use has become less and less frequent in favour of GTOs, for which

the calculation of multi-center two electron integrals is essentially simpler.

With few exceptions, the basic ingredients in solid state applications for construc-

tion of the basis-set functions, φµ(r), are plane-waves (PW) and/or Gaussian type

orbitals (GTO), and they are the only ones that are considered in the following. The

basic advantage obtained from the use of PWs and GTOs is related to the fact that

they make the computation of integrals in direct and/or reciprocal space very easy. If

numerical techniques are used to perform integrals throughout the calculation, much

greater freedom is possible in the choice of representative functions: these may include

Slater type orbital. A KS computational scheme based on a numerical approach has

been prepared recently by Baerends and collaborators.(31) All types of symmetry and

periodicity in one, two, and three dimensions can be handled.

When using linear representations, the problem of the errors related to the use

of an incomplete expansion set arises. There is the need of making such errors as

small as possible, while using manageable basis sets. Each application requires a

careful analysis: the art of devising good basis sets is a very important one, and

based on experience and competence. In this respect, even ab initio methods can

profit from empirical knowledge. One of the advantages that are obtained from the

use of Baerends’ numerical approach is that the problem of basis set incomplete-

ness, which often plagues schemes based on PWs and GTOs, becomes almost irrelevant.

Some of the terms used to describe localized atom-centered basis sets:

i. Minimal Basis Set

The minimal (or minimum) basis(32) set is one in which, a single basis function

is used for each orbital on each free atom. However, each atom in the second

period of the periodic table would have basis functions of p type which are added

to the basis functions corresponding to the 1s and 2s orbitals of the free atom.

So, a minimal set consists of a 1s function for hydrogen and 1s, 2s, and 2p (five

functions: two s functions and three p functions) for Li. . .Ne atoms. Minimal

basis sets are known to give surprisingly good results for geometry searches but

are large contaminated in energy calculations, however they are much cheaper

than their larger counterparts.

ii. Double-Zeta (DZ) and Triple-Zeta (TZ) Basis Set

The double-zeta basis set is obtained by considering two basis functions for each

atomic orbital of the occupied shells.(32) So, replacing each STO of a minimal

basis set by two basis functions differ in their orbital exponents ζ (zeta). For

32


instance, 2 functions for H or He, 10 functions for Li. . .Ne, and 18 functions for

Na. . .Ar. The triple-zeta is the same as double-zeta but three basis functions

differ in their orbital exponents are here applied.

iii. Split-Valence (SV) Basis Set

Since the valence electrons take principally part in the bonding, it is also common

to represent valence orbitals by more than one basis function (each of which can in

turn be composed of a fixed linear combination of primitive Gaussian functions).

This basis sets are called split-valence basis and it uses generally two STOs for each

valence atomic orbital and only one STO for each inner-shell atomic orbital.(33)

Since the different orbitals of the split have different spatial extents, the combi-

nation allows the electron density to adjust its spatial extent appropriate to the

particular molecular environment.

iv. Polarization (P) Basis Functions

To give additional flexability to the description of molecular orbitals (MOs), po-

larization functions(34; 35) can be added where for example a p-function is added

to light atoms (hydrogen and helium). Similarly, d-type functions can be added

to a basis set with valence p orbitals, and f-type functions to a basis set with d-

type orbitals, and so on. These are auxiliary functions with one additional node,

and are denoted by an asterisk, ∗. Two asterisks, ∗∗, indicate that polarization

functions are also added to light atoms (hydrogen and helium).

v. Diffuse Basis Functions

Another common addition to basis sets is the addition of diffuse functions,(32; 33)

denoted by a plus sign, +. They are formed by the addition of four highly diffuse

functions (s,px,py,pz) on each non-hydrogen atom. Two plus signs, ++, indicate

that a highly diffuse s functions are also added to light atoms (hydrogen and

helium). A highly diffuse function is one with a very small orbital exponent. This

type of addition is so applicable for anions and compounds with lone pairs of

electrons in order to have a significant electron density at large distances from the

nuclei and improve the accuracy of the basis.

Plane Waves Plane waves (PWs) are, in sense, the ideal basis functions for a periodic

system. If one denotes the general wave-vector as k and the translationally equivalent

vector in the first Brillouin Zone (BZ) as κ, we can write:

φk(r) = Ω−1/2 exp(ık.r) = Ω

−1/2 exp[ı(κ+ k).r] ≡ φk(r;κ) (I.52)

33


The last symbol indicates that the general PW is a Bloch function (BF) associated

with the point κ within the BZ and labeled with a discrete index k, corresponding to

a vector of the reciprocal lattice. PWs are an orthonormal, complete set: any function

belonging to the class of continuous, normalized functions (which are those of interest

in QM) can be expanded with arbitrary precision in the PW set.

In solid state applications, a finite number (p) of GTOs are attributed to the various

atoms in the reference zero cell (Aµ will denote the coordinate of the nucleus on which

ϕµ is centered); the same GTOs are then associated with all translationally equivalent

atoms in the crystal. In total, we have Np GTOs, from which we can construct Np

Gaussian-type Bloch orbitals:

φµ(r; κ) =

T

ϕµ(r−Aµ −T) exp(ıκ.T) (µ = 1, · · · , p;κ = 1, · · · , N) (I.53)

In particular, there are at least three reasons for avoiding the use of very diffuse

molecular basis sets primitives (low exponents) in solid state studies: first, the number

of integrals to be explicitly calculated increases explosively; secondly, the accuracy of

the calculations must be particularly high in order to avoid pseudo-linear dependence

catastrophes; thirdly, diffuse functions are not of much use in densely packed crys-

tals, because their tails are found in regions where there is large vibrational freedom

associated with functions on other atoms.

With respect to PWs, the use of suitably contracted GTOs permits us to describe

accurately electronic distributions both in the valence and in the core region with a

limited number of basis functions. The price is the loss of orthogonality, of universality

and the need for more sophisticated algorithms for the calculation of the required

integrals. The latter are expressed in terms of the primitives; their evaluation is made

simply by the fundamental property that the product of two Gaussians is a Gaussian

(Boys’ theorem). In particular for s-type primitives:

g(r−A;α, 0, 0)g(r−A;α, 0, 0) = Kg(r−A;α, 0, 0) (I.54)

with: α = α + α; A = (αA + αA)/α; and K = exp(−αα|A−A|2/α). This

result is easily generalized to higher quantum numbers.

The most complicated integrals that enter the calculations are the so-called two-

electron, four-center integrals which result from calculating the Coulomb or exchange

term in the Fock Hamiltonian or the corresponding energy term, after having expressed

the spin orbitals ψi(r) in the GTO basis set:

(µν|κλ) =

dr dr [ϕµ(r−Aµ −Tµ)ϕν(r−Aν −Tν)]×

× [ϕκ(r −Aκ −Tκ)ϕλ(r

−Aλ −Tλ)]/|r− r| (I.55)

34


This kind of integral, as well as the simpler nuclear attraction ones, are obtained by

starting from an auxiliary function of one variable:

F0(w) =

1

0

ds exp(−ws2) (I.56)

An additional advantage of GTOs is due to the fact that their Fourier transform is

another Gaussian [Fpexp(−αr2) ∝ exp(−p2/4α)], and their use in combination with

PW techniques is therefore easy.(36)

1.4.2 Pseudopotential (Ps) Basis Set

In many respects, core electrons are unimportant for determining the stability, structure

and low-energy response properties of molecules and crystals. It is a well-established

practice to modify the one-electron part of the Hamiltonian by replacing the bare

nuclear attraction with a pseudopotential operator, Vps, which permits us to restrict

the calculation to valence electrons, for both types of system. Consider a system

comprising a set of nuclei A and n electrons, of which n are valence electrons. We can

write equations of the form:

[−2 /2 +

ρ(r)/|r− r|dr + µxc(r; [ρ]) +

A

VpsA]ψi(r) = εiψ

i(r) (I.57)

Where the operator: Vps =

A VpsA. Primed symbols have been introduced to indicate

that the SCF solution of these equations is limited to valence electrons; that is, we will

consider only the n/2 eigenfunctions corresponding to the lowest eigenvalues, and

use them to calculate the valence density, ρ(r), which in turn is used to define the

electrostatic, exchange and correlation interactions between valence electrons. The

pseudopotential operator must reproduce screened nuclear attractions, but must also

account somehow for the Pauli exclusion principle, which requires that valence orbitals

are orthogonal to core ones. Consider an atom A at A, with ncA core electrons, whose

highest angular quantum number is L. Suppose L = 1, that is, the core contains only

s and p electrons. At long range, we must have: VpsA = −(ZA − ncA)/rA, with rA =

|r−A|. At short range, VpsA must act differently on functions of s and p symmetry,

while it operates equivalently on functions of higher angular quantum number, that is:

Vps = −(ZA − ncA)/rA +L

=0

U s−r (rA)

m=−

|mm|A +W s−r(rA) (I.58)

W s−r(rA) and U s−r (rA) are short-range functions; the term in braces is a projector,

which makes U s−r act only on functions which have -symmetry with respect to rA.

35


In establishing the explicit form of VpsA, a number of characteristics are sought (not

all of them can be optimally satisfied):

i) Pseudo-valence eigenvalues, εi, should coincide with the true ones, εi;

ii) Pseudo-orbitals, ψi(r), should resemble as closely as possible the true ones, ψi(r),

in an external region as well as being smooth and node-less in the core region;

iii) Pseudo-orbitals, ψi(r), should be properly normalized;

iv) The functional form of the Ps, though preserving non-local character, should be

designed so as to simplify as far as possible their use in computations;

v) Explicit expressions for the functions W s−r(rA) and U s−r (rA) should be provided

for the different atomic species, to be used independently of the environment;

The performance of Ps techniques in solid state physics is usually very good, except

for some critical cases (for instance, if core relaxation effects are important, which

may occur when simulating very high pressures, or when the electron configuration in

the crystalline environment is very different from that of the isolated atom).(37; 38)

Among Ps designed for KS-LDA calculations, the norm-conserving ones tabulated for

all atoms by Bachelet et al.(39; 40) are perhaps the most popular; ultrasoft Ps, which

ensure the very smooth behaviour of the pseudo-valence orbitals in the core region are

useful in applications where plane waves are used as a basis set.(41)

2 Theory for Piezoelectricity and Associated Re-

sponse Properties

The word piezoelectricity means electricity resulting from pressure. Piezoceramics are

materials that demonstrate what is known as the piezoelectric effect. This piezoelectric

effect is described by a linear electromechanical relationship between the mechanical

and the electrical state in crystalline materials with no inversion center of symmetry.

Piezoelectric Effect; appearance of an electrical potential across some faces of a crystal

when it is under pressure, and of distortion when an electrical field is applied. Pierre

Curie and Jacques Curie discovered the effect in 1880. It is explained by the displace-

ment of ions, causing the electric polarization of the crystal structural units. When

an electrical field is applied, the ions are displaced by electrostatic forces, resulting in

the mechanical deformation of the whole crystal. The converse effect however was dis-

covered later by Gabriel Lippmann in 1881 through the mathematical aspect from the

36

2. Theory for Piezoelectricity and Associated Response Properties

fundamental thermodynamic principles of the theory. The first applications were made

during World War I with piezoelectric ultrasonic transducers. Nowadays, piezoelectric-

ity is used in everyday life. For example, in the car’s airbag sensor where the material

detects the change in acceleration of the car by sending an electrical signal which trig-

gers the airbag.(42) Piezoelectric crystals are used in such devices as the transducer,

record-playing pickup elements, and the microphone.

The reason why piezoelectric material creates a voltage is because when a mechan-

ical stress is applied, the crystalline structure is disturbed and it changes the direction

of the polarization P of the electric dipoles. Depending on the nature of the dipole (if

it is induced by ion or molecular groups), this change in the polarization might either

be caused by a re-configuration of the ions within the crystalline structure or by a

re-orientation of molecular groups.(43) As a consequence, the bigger the mechanical

stress, the bigger the change in polarization and the more electricity is produced. The

change in P appears as a variation of surface charge density upon the crystal faces, i.e.

as a variation of the electrical field extending between the faces. For example, a 1 cm3

cube of quartz with 2 kN (500 Ibf) of correctly applied force can produce a voltage of

12500 V.(43; 44)

2.1 Direct and Converse Piezoelectricity

2.1.1 General Mathematical Definition

When a piezoelectric material strains, it develops an internal electric field; this effect

is known as the direct piezoelectric effect. On contrary, a piezoelectric material ex-

periences strain when an electrical field is applied to it, and this behaviour is defined

as the conversed effect. These reactions, electrical field and mechanical behaviour, are

direction-related properties, and as a result, the equations governing piezoelectricity

are usually expressed with vectors.

• The electric behaviour:

At microscopic scale, a material that has a volume (V) can be decomposed to

atomic or molecular groups, these groups may have individual dipole moments.

The total dipole moment can be defined as,

µn =

n

qnrn, (I.59)

where rn is the position vector of the charge qn. The overall polarization P

induced by these dipoles is defined as the total dipole moment averaged over the

37


volume of the cell;

P =1

V

n

µn. (I.60)

The electrical behaviour of a material appeared by applying an external electric

field, E, can be explained by;

P = χE, and (I.61)

D = E + 4πP = (1 + 4πχ) E = E =⇒ Di = ijEj (I.62)

where D is the electric charge density displacement (electric displacement), is

the permittivity, and χ is the dielectric susceptibility.

• The mechanical behaviour:

The mechanical behaviour expresses from Hook’s law as:

S = sT =⇒ Sij = sijklTkl (I.63)

where S is the strain, s is the compliance, and T is the stress.

FigureA.I.5 introduces the elementary definition for both the electric and mechan-

ical behaviours in a material. The electric behaviour in a dielectric (material which

contains no free electrons) is given in Panel (a). Electric polarization P is connected

with the polarization charge only when dielectric is placed in an external field, while the

electric displacement D is connected with free charge only, so that it is not altered by

the introduction of the dielectric into the applied electric field. Electric intensity, E, is

connected with all charges that are actually present, whether free or bound. The lines

of E indicate the presence of both kinds of charge. The definition of mechanical strain

(S) and stress (T) are presented in Panels (b and c), respectively. The transformation

between S and T via the elasticity are additionally given.

These two behaviours may be combined into so-called coupled equations,

Di = TijEj + dijkTjk (I.64)

Sij = sEijklTkl + dkijEk (I.65)

d is the matrix from the converse piezoelectric effect. The superscript E indicates a

zero, or constant, electric field; the superscript T indicates a zero, or constant, stress.

So, T is the permittivity under constant stress, and sE is the elastic compliance under

constant electric field. All indices range from 1 to 3. From equation I.64, the electric

charge density displacement (Di) is related to the electric field through the dielectric

constant (ij). And, a stress can be a source of electric displacement under the converse

38


Figure A.I.5: Definition of both electric and mechanical behaviours within a material.

Panel (a) represents the electric field vectors: D is the electric displacement vector, P is the

electric polarization, and E is the electric field intensity (0 = 8.854× 10−12 C2.N−1.m−2,

is the permittivity of vacuum). The definition of mechanical strain and stress are given in

Panels (b and c) respectively.

piezoelectric effect. One can note that in the absence of mechanical stress T , the

equation I.64 will only describe the electrical behaviour of the material. As shown in

the equation I.65, both a stress and an electric field can result in a strain. The elastic

compliance is represented by sEijkl. Likewise, in the absence of electric field E, the

equation I.65 will give S = sET which is Hooke’s law. These equations have Di and

Sij as independent variables.

Note that the equation I.63 can be as well written as:

T = C S =⇒ Tij = CijklSkl (I.66)

where C is the elastic constant, and is related to the compliance by inverting the

matrix; s = C−1. This will lead to another two of coupled equations,

Di = SijEj + eijkSjk (I.67)

Tij = CEijklSkl − ekijEk (I.68)

Here, the superscript S indicates a zero, or constant, strain.

39


From equations I.64 and I.65, one can calculate the piezoelectric strain constant:

dijk from the mechanical response results from applying electric field, or dkij from the

electric response ensues from applying mechanical field, respectively:

dijk =

∂Di

∂Tjk

E

, or dkij =

∂Sij

∂Ek

T

, (coulomb.newton−1) (I.69)

where from equations I.67 and I.68, direct piezoelectric stress constant eijk or ekij can

be similarly obtained:

eijk =

∂Di

∂Sjk

E

, or ekij = −∂Tij

∂Ek

S

, (coulomb.meter−2) (I.70)

A complete set of coupled equations can be found elsewhere,(45; 46; 47; 48) and addi-

tional two piezoelectric constants can be obtained.

gijk = −∂Ei

∂Tjk

D

, or gkij =

∂Sij

∂Dk

T

, (meter2.coulomb−1) (I.71)

hijk = −

∂Ei

∂Sjk

D

, or hkij = −∂Tij

∂Dk

S

, (newton.coulomb−1) (I.72)

gijk is also known as piezoelectric strain constant, where hijk is defined as piezo-

electric stress constant. The first set of 4 terms: ∂Di/∂Tjk, ∂Di/∂Sjk, ∂Ei/∂Tjk and

∂Ei/∂Sjk, correspond to the direct piezoelectric effect (electric response induced in the

material by applying mechanical field) and the second set of 4 terms correspond to

the converse piezoelectric effect (mechanical response initiated from applying electric

field).(49) For instance, the strain piezoelectric tensor, d, can be measured through the

direct piezoelectric effect by looking to the change in the charge displacement (∂Di)

when the material is stressed at constant electric field. Otherwise, it can be measured

through the converse piezoelectric effect by regarding the deformation (∂Sjk) occurring

with the material when an electric field is passed at constant stress. The unit for each

coefficient has been indicated to show the relations between the 4 coefficients.

To simplify the indices notation, Voigt(47) notation can be used alternatively to the

above tensorial notation. In this notation the subscripts ij and kl for both strain (S)

and stress (T ) in coupled equations, are replaced by u and v, where u or v = 1,. . . ,6

(1 = xx, 2 = yy, 3 = zz, 4 = yz, 5 = xz, 6 = xy), see Figure A.I.6.

Another fundamental parameter used in electromechanical applications is the elec-

tromechanical coupling factor k. The electromechanical coupling factor, which mea-

sures the ability of a material to interconvert electrical and mechanical energy, is ex-

pressed as:

k2 =ConvertedMechanicalEnergy

InputElectricalEnergy=

ConvertedElectricalEnergy

InputMechanicalEnergy(I.73)

40


Figure A.I.6: Graphical scheme that introduces Voigt notation for mechanical strain in

3D (bulk) and 2D (surface) systems.

By neglecting the indices and substituting equation I.64 in equation I.65, we can get:

S = sE(1− k2)T +d

TD, (I.74)

So,

k2 =d2

sET(I.75)

By applying Voigt notation, the electromechanical coupling coefficients kiv, which gives

the efficiency of transformation of mechanical into electrical energy, are defined as:(50;

51)

kiv =div

TiisEvv

or kiv =eiv

SiiCEvv

. (I.76)

In the case where the electric displacement is equal to zero, the formula I.74 becomes:

S = sE(1− k2)T (I.77)

The strain is still proportional to the stress, but the compliance is multiplied by the

term (1− k2). When k is equal to zero, the equation is simply Hooke’s Law, which is

logical as it means that all the energy in the material is strain energy, see Figure A.I.7.

41


Figure A.I.7: Diagram represents the electrical-mechanical inter-relations.

2.1.2 Piezoelectric Tensor and Symmetry

As mentioned in Section 2.1, when a stress is applied to certain crystals they develop

an electric moment whose magnitude is proportional to the applied stress. In general,

a state of stress (or strain) is specified by a second-rank tensor with nine components,

while the polarization of a crystal, being a vector, is specified by three components. So,

piezoelectricity is mostly specified by a third-rank tensor that has only 18 components

(for 3-dimensional material) instead of 27 due to the application of matrix notation.

More information about these notations: second and third-rank tensors, will be in-

troduced later. The point group symmetry of the material will again determine the

transformations under which the tensor must be invariant, defining specific piezoelec-

tric coefficients that can be nonzero. In Figure A.I.8, we report piezoelectric tensors

for some different symmetry point groups. For instance, a material which crystal-

lizes in cubic system, class 432, has all piezoelectric tensor components null, whereas

for another which crystallizes in trigonal system, class 32, it will have five piezoelec-

tric components (11,12,14,25,26) with only two independent components (11,14). Note

that, for any piezoelectric coefficient, eiv for instance, the subscript i gives the direction

of electrical vector (polarization or electric field), while v represents the component of

mechanical behaviour (strain or stress) where Voigt’s notation is applied.

42


Figure A.I.8: Form of piezoelectric tensor as related to the group symmetry, the image is

taken from Nye 1985.(47) Key of notation is also presented.

2.1.3 Piezoelectric Tensor and Dimensionality

If the dimensionality of material decreases, the number of components in the piezoelec-

tric tensor will be decreased as well. For instance, surfaces (2-dimensional materials)

where there is no periodicity through z-direction, will have ill-defined mechanical prop-

erties for orthogonal direction, and the piezoelectric third-rank tensor will have only 9

components instead of 18 for a 3D-material, see Table A.I.1. One-dimensional system

(such as polymer, that has repeating units through only one direction) will however

have only three components in its piezoelectric tensor. Molecules that have not any

periodicity through the three directions x, y, and z, should have zero components in its

piezoelectric tensor. Note that, molecule or 0D-system may be polarized through any

of the three directions, however they will have no any deformation components and so

no piezoelectric response, Table A.I.1.

Obviously, the symmetry of the material will then define the nonzero component

within the tensor. For example, if the material that crystallizes in trigonal system (class

32) is a 2D-material, the number of piezoelectric components will reduce to three with

only one independent component (11). This is because the deformation components 4

and 5 (yz and xz according to Voigt’s notation) will be ill-defined in this case.

43


Table A.I.1: Representation of piezoelectric tensor for different dimensionality systems,

where Voigt’s notation is applied for mechanical deformation. So, for instance, the compo-

nent 11 in the piezoelectric tensor refers to xxx, while 12 points out to xyy, and 36 to zxy.

Dash refers to polarized non-deformable 0D system.

Material Dimensionality No. of components Piezoelectric tensor

Bulk 3D 3 × 6

11 12 13 14 15 16

21 22 23 24 25 26

31 32 33 34 35 36

polarization × deformation

Surface 2D 3 × 3

11 12 16

21 22 26

31 32 36


Polymer 1D 3 × 1

11

21

31


Molecule 0D 3 × 0 −polarization × deformation

2.1.4 Quantum Chemical Point of View

Piezoelectricity can be defined as the second energy derivative with respect to both an

external electric field, E, and a mechanical deformation, η, (for strain S or stress T).

According to Born-Oppenheimer approximation, that separates both the electronic and

nuclear behaviours, total piezoelectricity “relaxed” can be obtained by the summation

of purely electronic “clamped-ion” and nuclear “vibrational” contributions. The purely

electronic contribution is calculated using clamped-ion conditions, where the fractional

coordinates are fixed after the deformation (without optimizing the geometry). While,

if the coordinates are left to relax by optimizing the deformed fractional coordinates,

the total piezoelectricity is actually obtained:

(piezo.)relax = (piezo.)clamp + (piezo.)vib (I.78)

44


Consquently, any piezoelectric constant can be theoretically decomposed into these two

contributions. For example, direct piezoelectric constant eiv can be written as:

erelaxiv = eclampiv + evibiv (I.79)

• The electronic contribution: The electronic contribution to direct piezoelectric

constant can thus be written as the second-order perturbation energy:

eeleiv ∝

n =0

(µi)0→n(− ∂H∂ηv

)0→n

(∆ε)0→n

(I.80)

µi = −∂H/∂Ei is the dipole moment, H is the energy of the system, (∆ε)0→n

is the energy difference between initial, Ψ0, and final, Ψn, spectroscopic states

(eigenvectors of the unperturbed Schrodinger equation). The numerator of the

above expression depends on the symmetry of the structure while the denomina-

tor is related to the electronic band gap.

• The vibrational contribution:

In the quantum theory of perturbation, the vibrational term of the piezoelec-

tricity has an expression similar to the vibrational contribution to the static

polarizability:

evibiv ∝

p

∂µi

∂Qp(− ∂2H

∂ηv∂Qp)

ω2p

(I.81)

where Qp is the harmonic normal mode with ωp frequency, µi the i-component of

the dipole moment, and ∂H∂ηv

the constraint related to the unit cell deformation

ηv. This latter expression shows the importance that a soft IR active frequency

mode could have on the piezoelectricity value, as for the static dielectric constant.

Considering the crystalline systems (applying the crystalline orbital basis, Bloch

functions, and Born-Von Karman boundary conditions), the dipole moment component

per unit cell can be obtained as follow:

µi = − 2

nk

k

jocc

jocc|ri + ı∇ki |jocc (I.82)

jocc is an occupied crystalline orbital at k-point, and the dipole moment operator

ri + ı∇ki acts in both direct and phases spaces(52; 53) of periodic systems. It is

important to mention here that, the calculation of the dipole moment using the last

equation I.82 is so tough since the dipole moment is ill-defined between the phases.

45


However, the derivative of the dipole moment can be well defined by applying Berry

phase-like scheme, as will be seen later.

According to the quantum mechanical perturbation theory, and as appeared from

equations I.80 and I.81, piezoelectricity can then be represented as a sum of fractional

terms with numerators and denominators.

For the electronic contribution: the numerators represent the products of allowed

transition moments (µ)0→n and (−∂H∂η

)0→n due to the electric and mechanical fields,

respectively. The two transition moment operators should eventually belong to the

same irreducible representation of symmetry group in order to exert a piezoelectric

effect. The denominators represent, however, the energy differences between initial,

Ψ0, and final, Ψn, spectroscopic states (generally direct electronic band gap).

For the vibrational contribution: the numerators represent the products of al-

lowed transition moments due to the electric and mechanical fields with however

respect to the harmonic normal mode Qp,∂µi

∂Qpand ∂2H

∂ηv∂Qp. While, the denominators

represent the square of frequency of IR active mode, ω2p. It is clear here that, if this

mode is soft (means a small frequency and a large polarization), this will lead to a

huge piezoelectricity induced from the vibrational contribution.

2.2 Associated Response Properties

2.2.1 Elasticity and Phonons

The modern theory of elasticity generalizes Hooke’s law to say that the strain (defor-

mation) of an elastic object or material is proportional to the stress applied to it.(54)

However, since general stresses and strains are described by second-rank tensors (which

may have multiple stretch and shear independent components), the “proportionality

factor” may no longer be just a single real number, but rather a tensor that can be

represented by a matrix of real numbers. The elasticity tensor is a fourth-rank tensor

with up to 21 independent constants that measure an object or substance’s resistance

to being deformed elastically (i.e., non-permanently) when a force is applied to it. A

stiffer material will have a higher elastic modulus.

Specifying how stress and strain are to be measured, including directions, allows

for many types of elastic moduli to be defined.

• Young’s modulus (Ys) describes tensile elasticity, or the tendency of an object

to deform along an axis when opposing forces are applied along that axis; it is

defined as the ratio of tensile stress to tensile strain. It is often referred to simply

as the elastic modulus.

46


• The shear modulus or modulus of rigidity (Gs) describes an object’s ten-

dency to shear (the deformation of shape at constant volume) when acted upon

by opposing forces; it is defined as shear stress over shear strain. The shear

modulus is part of the derivation of viscosity.

• The bulk modulus (Ks) describes volumetric elasticity, or the tendency of

an object to deform in all directions when uniformly loaded in all directions;

it is defined as volumetric stress over volumetric strain, and is the inverse of

compressibility. The bulk modulus is an extension of Young’s modulus to three

dimensions.

The elements of the fourth-rank elastic tensor C for 3D systems are usually defined

as second energy derivatives with respect to pairs of deformations:(47)

Cvu =1

V

∂2H

∂ηv∂ηu

0

(I.83)

where η is the symmetric second-rank pure strain tensor, V the equilibrium cell volume

and Voigt’s notation is used according to which v, u = 1, . . . , 6 (1 = xx, 2 = yy, 3 =

zz, 4 = yz, 5 = xz, 6 = xy). Since volume V is not uniquely defined for 1D and 2D

systems, it can be here omitted (length or surface could be respectively used). An

automated scheme for the calculation of C (and of s = C−1, the compliance tensor)

has been implemented in the Crystal program that exploits analytical gradients and

compute numerically the second derivatives.(55; 56; 57) The introducing of tensor order

is given in Figure A.I.9.

In crystalline solid structures, we assume that the atoms sit at lattice sites. How-

ever, this is not actually the case; since the atoms, even at lowest temperatures, perform

small vibrations (displacement) about their equilibrium positions. These displacement

motions can be explained upon the harmonic oscillator approximation, where each vi-

bration can be treated as though it corresponds to a spring and obeys Hook’s law:

the force required to extend the spring is proportional to the extension. In crystal,

the calculation of vibrational frequencies at Γ point (i.e., at the center of the Bril-

louin zone) are obtained by diagonalizing the mass-weighted Hessian matrix W (is

constructed by numerical differentiation of the analytical gradients with respect to the

atomic Cartesian coordinates)

W Γ

ai,bj =1√

MaMb

∂2H

∂rai∂rbj

(I.84)

Where Ma and Mb are the atomic masses of atoms a and b, and rai is the displacement

of atom a from its equilibrium position along the i-th Cartesian direction.

47


Figure A.I.9: Introducing of tensor order: second, third, and fourth-rank-tensors are

considered. Examples and total number of components are stated for each case. Note that,

zero-rank-tensor is corresponding to scalars, while first-rank-tensor is referred to vectors as

the dipole moment, for instance.

From equations I.83 and I.84, one can observe the connection between elasticity

and vibrational frequencies. By imposing equality between elastic and vibration strain

energies (the energy Evib of the mode should be equal to the energy required to induce

an equivalent deformation through a defined direction), the elastic constants can be

expressed as a function of phonon frequencies:

Evib = Eelast → 1

2ω2pQ

2p nCjjη

2j (I.85)

where, Qp is the harmonic normal coordinate of the mode p, ωp = 2πνp, and ηj

is the parallel component of the strain tensor. Note that, this equation is applied

for a special cylinder shape, here is BN-nanotube, then looking to the ring breathing

collective vibrational mode. Thus, the energy of the mode Evib of a (n, 0) tube should

be equal to the energy required to induce an equivalent deformation of the 2n BN

units in the monolayer Eelast. Details and some examples of application can be found

elsewhere.(58; 59)

48


2.2.2 Infrared Intensity

Molecular vibrational frequencies lie in the IR region of the electromagnetic spectrum,

and they can be measured using the IR technique. In IR, a polychromatic light (light

having different frequencies) is passed through a sample and the intensity of the trans-

mitted light is measured at each frequency. When molecules absorb IR radiation,

transitions occur from a ground vibrational state to an excited vibrational state. The

corresponding vibrational motion (phonon mode) will be IR-active, if a change in dipole

moment is occurred when IR radiation is absorbed. Dipole moment is a vector quantity

and depends on the orientation of the molecule and the photon electric vector. Dipole

moment is previously defined for the periodic solids, equation I.59, and the relation

between IR intensity I and dipole moment can be given as:

Ip =π

3

NA

c2dp

∂µ

∂Qp

2

, (I.86)

where NA is Avogadro’s number, c the speed of light, dp the degeneracy of the p-th

mode, µ the cell dipole moment and Qp the normal mode displacement coordinate.

So, the IR intensity is proportional to the square of the first-derivative of the dipole

moment µ with respect to the normal mode Qp.

In Crystal, the dipole moment is related to the Born effective charge tensor,

which is the first derivative of the polarization per unit cell with respect to the atomic

displacements when the applied electric field is zero:(60)

Z∗α,ij = Ω

∂P i

∂uαj

(I.87)

where Ω is here the cell volume. Born effective charge tensor can be also written for

an atom α in the form:

Z∗α,ij =

∂

∂uαj

∂H

∂Ei

=∂

∂uαj

µi (I.88)

where H is the total energy, Ei is the ith component of the applied electric field, and

uαj is the jth atomic displacement coordinate of atom α with respect to equilibrium.

Note that, it is possible to define a mass-weighted effective mode Born charge vector

Zp (in the normal mode basis), which is related to the atomic Born charge tensor Z∗α

(in the atomic basis) by means of:

Zp,i =

αj

tp,αj√Mα

Z∗α,ij (I.89)

One can simply deduce that the calculations of the IR intensity look extremely like

that of piezoelectricity, since the piezoelectricity is the derivative of the polarization

49


with respect to a deformation where the IR intensity is considered as the derivative of

polarization with respect to the atomic displacement or mode coordinate. More details

on the calculation of the infrared intensities in the Crystal program can be found in

Ref.(60).

2.2.3 Polarizability and Dielectric Constant

The polarizability α of an atom is defined in terms of the local electric field at the

atom:

µ = αElocal, (I.90)

where, the dielectric constant describes the interaction of dielectric materials and the

electrical field as be previously mentioned in equation I.62. From that equation, it is

defined in terms of the macroscopic field E as:

=E + 4πP

E= 1 + 4πχ (I.91)

Dielectric constant r is defined also as the relative permittivity of a material and it is

related to the refractive index n as: n = (r)1/2.

The polarizability is an atomic property, but the dielectric constant will depend on

the manner in which the atoms are assembled to form a crystal. Applying equation

I.62, the two properties could be related as follows:

= 1 +4πα

V(I.92)

where V represents here the volume of the cell that has the polarizability α.

It should be noted that, all of these response properties can be decomposed, as the

piezoelectricity, into two contributions: electronic and vibrational contributions. Then,

for example, the total static polarizability can be determined as follows:

α0 = αele + αvib = αele +

p

Z2

p

ν2p

(I.93)

where αele is the electronic (clamped ion) contribution. The vibrational (ionic) con-

tribution is given, in the double harmonic approximation, by the second term on the

right hand side. Z2

p is a mass weighted effective mode Born charge and νp is the vibra-

tional frequency of the mode p. Born charges are calculated in Crystal using a Berry

phase-like scheme.(61; 62)

The total static dielectric tensor can be written as:

0ij = eleij + vibij = eleij +4π

V

p

Zp,iZp,j

ν2p

(I.94)

50


where eleij and vibij are the electronic (clamped ion) and vibrational (ionic) contributions.

In the Crystal code, the electronic contribution to both polarizability and dielec-

tric tensor is evaluated through the Coupled Perturbed Hartree-Fock or Kohn-Sham

(CPHF/KS) scheme(63) as adapted to periodic systems.(64) This is a perturbative,

self-consistent method that describes the effect of an external electric field on the

relaxation of the crystalline orbitals. Additional details about the method and its im-

plementation in the Crystal program can be found elsewhere,(65; 66; 53) as well as

recent example of applications.(67; 68; 69; 70; 71; 72)

51

II. COMPUTATIONAL APPROACH

1 The Software Used: CRYSTAL Code

The crystal code is written in Fortran language (crystal14),(73) and its package is

used to perform ab initio calculations to obtain the ground state properties for periodic

systems. Systems periodic in 0 (molecules, 0D), 1 (polymers, 1D), 2 (slabs, 2D), and

3 dimensions (crystals, 3D) can be treated with the code on an equal level.(74) In

each case the fundamental approximation made is the expansion of the single particle

wave functions (Crystalline Orbital, CO) as a linear combination of Bloch functions

(BF) defined in terms of local functions (Atomic Orbitals, AOs). The crystal pack-

age permits the use of different Hamiltonains, Hartree-Fock as well as DFT-hybrids

(which adopt the Exchange-Correlation part following the Density-Functional theory

postulates).

The local functions are, in turn, linear combinations of Gaussian type functions

(GTF) whose exponents and coefficients are defined by input. Functions of symmetry

s, p, d and f , as well as sp shells (s and p shells, sharing the same set of exponents)

can be used. The use of sp shells can give rise to considerable savings in CPU time.

Input tools allow the generation of slabs (2D system) or clusters (0D system) from a

3D crystalline structure, the elastic distortion of the lattice, the creation of a super-cell

with a defect, and a large variety of structure editing.

crystal can use the following all electrons basis sets:

(a) General basis sets, including s, p, d, f functions (given in input);

(b) Standard Pople basis sets(75) (internally stored as in Gaussian 94).(76)

STOnG, Z=1 to 54

6-21G, Z=1 to 18

3-21G, Z=1 to 54

The standard basis sets (b) are stored as internal data in the crystal code.

The crystal09 input includes a title and three sections (referred to as “blocks”).

Each block consists of keywords (case insensitive, written left justified) and numerical

parameters (free format). Each block ends with the keyword END (mandatory: 3

characters only are interpreted, any ending is allowed, ENDgeom, ENDbas, etc) or

STOP. The latter will cause immediate termination of execution. In order to test

each block, the keyword TEST can be used. Input is processed up to that block, and

the program then stops. Optional keywords can be present in each section. Extended

information on input features are in “CRYSTAL User’s Manual”.(74)

52

1. The Software Used: CRYSTAL Code

1.1 Structure of Input File

The input deck has the following structure (mandatory data):

Title

input block 1 Geometry input

standard geometry input

optional geometry optimization and editing keywords

END

input block 2 Basis set input

standard basis set input

optional basis set related keywords

END

input block 3 Single particle Hamiltonian (default RHF)

and SCF control

SHRINK (number of k-points

sampling in reciprocal space (for 1D, 2D, 3D systems only)

optional general information and SCF related keywords

END

Where in the block 1, the geometry input is introduced as a starting point for the

geometry simulation. This block begins with the title of the input structure followed by

determining the type of periodicity of the structure by typing different keywords. To

present 3D systems, the keyword CRYSTAL is used, SLAB is used for 2D systems,

POLYMER for 1D one, and MOLECULE is used to simulate molecules (0D). Then,

the symmetry of the structure should be identified by defining the space group (sym-

metry group in the case of molecules). After defining the symmetry group, one needs to

specify the preliminary lattice parameters and fractional coordinates for the structure

under study. Moreover, one can replaced that by using the keyword EXTERNAL

to apply geometry from external file. At the end of this block, some keywords can be

typed to define the object of the calculation (optimizing the geometry, calculation of

the frequencies, calculation of polarizability, . . . . . . , etc), which will be introduced in

the next section.

The second block is used to introduce the basis set for each atom with different

conventional atomic number defined in the crystal structure input. Each atom defined

in the crystal structure links to the basis set by its conventional atomic number Z.

For the effective core pseudopotential (ECP) basis set, the atom be represented by a

53


conventional atomic number Z + 200 (Example: Si-conventional atomic number = 14,

all electrons; and = 214, ECP). The basis set block ends with the form:

99 0 conventional atomic number 99, 0 shell. A library of optimized basis

sets adopted for periodic systems are available at: http://www.crystal.unito.it.

The choice of the basis set is the most crucial point in performing ab initio calculations

of periodic systems. The optimization of the basis set will be discussed later.

Hamiltonian description, computational parameters, and SCF control are the third

blocks’ set of data which have to be present in this desk. Starting with the determina-

tion of the calculation type (RHF, UHF, DFT), and the description of the Hamiltonian

(LDA, B3LYP, . . . ); then defining the integrals of tolerance, and the shrinking factor

(to generate a commensurate grid of k-points in reciprocal space); and ending this

block by determining the SCF control parameters (max number of SCF cycles, and

the convergence on total energy). The effect of the computational parameters on the

calculation will be analyzed later with examples.

For each block, there is usualy optional keywords, more details can be found in

“CRYSTAL User’s Manual”.(74)

1.2 The Keywords Applied in This Study

Block 1

In this part, we will discuss the keywords which are applied in this study:

• For optimization of the geometry:

The keywords OPTGEOM and FULLOPTG have been used to perform a full

geometry optimization (atomic coordinates and lattice parameters). An optional

keyword FINALRUN, can be also applied to action after geometry optimiza-

tion. For instance, if FINALRUN = 0, the program will stops after optimiza-

tion. If FINALRUN = 1, a single-point energy calculation will be performed

after optimization. For FINALRUN = 2, the program performs a single-point

energy and gradient calculation after optimization. If FINALRUN = 3, the

code performs a single-point energy and gradient calculation and if convergence

criteria on gradients are not satisfied, optimization restarts. However for FI-

NALRUN = 4, step 3 will be iterated until full stable optimization achieves.

In most of our calculations, FINALRUN = 4 has been choosed.

• For Frequency calculations:

The calculation of the vibration harmonic frequencies at the Γ-point is invoked by

the keyword FREQCALC in input block 1. Eckart conditions(77) for cleaning

the Hessian matrix as regards translational and rotational vibration modes has

54


been applied by the keyword ECKART. INTENS keyword has been used to

perform the calculation of infrared (IR) intensities and the vibrational contribu-

tion to the polarizability. To scan the geometry along selected normal modes, one

can use the keyword SCANMODE. Reading the anisotropic dielectric tensor

for the calculation of the longitudinal optic (LO)/transverse optic (TO) splitting

can be obtained by the keyword DIELTENS (only to be used if INTENS is

active).

• For polarizability and dielectric constant (electronic contribution):

CPHF or CPKS, these keywords compute the polarizability and the first

and second order hyper-polarizabilities via the Coupled Perturbed Hartree

Fock/Kohn Sham (HF/KS) method.(63) Those keywords must be the last key-

word in the geometry input block. To improve the convergence of the Self-

Consistent Coupled-Perturbed (SC-CP) procedure, some keywords can be applied

such as: FMIXING (mixing of Fock/KS matrix derivatives from iterations n

and n−1 by a given percent), ANDERSON (mixing of Fock/Ks matrix deriva-

tives).

• For elastic and piezoelectric constants calculation:

The elastic and piezoelectric constants can be computed at once with the key-

word ELAPIEZO, to be inserted at the end of the Geometry input block. In

this case the total contribution (sum of electronic and vibrational parts) to these

response properties is calculated. One can calculate separately the electronic

contribution by the sub-keyword CLAMPION after ELAPIEZO, where the

relative positions of atoms induced by the strain are fixed without optimiza-

tion. The input geometry is assumed to be optimized; nevertheless, the user can

ask this module to perform a pre-optimization of the structure by means of the

PREOPTGEOM sub-keyword.

1.3 Piezoelectricity Computations in CRYSTAL

In crystal regime, direct e and converse d piezoelectric tensors describe the polariza-

tion P induced by deformation η and the strain induced by an external electric field

E, respectively:

direct effect Pi = eiv ηv at constant field (II.1)

converse effect ηv = divEi at constant stress (II.2)

The direct and converse piezoelectric tensors are connected to each other via the elas-

ticity: e = dC and d = e s. We recall that piezoelectric constants can be decomposed

55


into purely electronic “clamped-ion” and nuclear “internal-strain” contributions, as for

the dielectric tensor, erelaxiv = eclampiv +evibiv ; where it measures the piezoelectric effect due

to relaxation of the relative positions of atoms induced by the strain(78; 79) and can

be computed by optimizing the atomic positions within the strained cell. Voigt(47)

notation can be then applied in order to simplify the indices notation.

In the crystal code, two contributions to the piezoelectricity can be calculated as

follow:

The electronic contribution can be separately calculated by applying the equation

I.80. This can be obtained by the Berry phase method(61) already implemented in

crystal, the method consisting in the calculation of the first derivative of the dipole

moment component per unit cell (or polarization vector). The dipole moment can be

obtained by using the equation I.82.

While, the total (electronic plus vibrational) contribution to the piezoelectricity can

be also obtained using the Berry phase method. In that case, the dipole moment is

calculated after optimization of the fractional coordinates Qηvat each finite deforma-

tion ηv. In fact, it corresponds to the finite field nuclear relaxation (FFNR) method

proposed by Kirtman and Bishop(80) for calculating numerically vibrational contribu-

tion to the polarizability (or dielectric constant). In the piezoelectric case, it means

doing a Taylor development of the dipole moment µi of the unit cell with respect to

the deformation ηv:

µi(ηv, Qηv) = µi(0, Q0) +

∂µi(ηv, Qηv

)

∂ηv

ηv→0

ηv + ... (II.3)

When divided by the volume of the cell, the derivative of the dipole moment with

respect to the deformation (∂µi(ηv ,Qηv )

∂ηv) obtained by fit leads to the piezoelectricity

component eiv value including electronic and vibrational contributions. If the fractional

coordinates are fixed at the optimized geometry without deformation Q0, then the

previous development leads to the electronic contribution of the piezoelectricity only:∂µi(ηv ,Q0)

∂ηv

ηv→0, as mentioned above.

A very recent implementation(81) that allows the use of a quasi-analytical scheme

for the calculations of direct piezoelectric tensor is newly included in the crystal

program. This scheme, which is not applied in the present work, is adapted for both

the clamped ion electronic and nuclear relaxation contributions of the piezoelectricity,

and it gives a unique “proper”, instead of an approximated, value for the piezoelec-

tric tensor. The use of the analytical Coupled Perturbed Hartree Fock/Kohn-Sham

(CPHF/KS) procedure to obtain dipole derivatives with respect to lattice deformations

as well as internal coordinates, is the basis of such procedure. Hence, the required polar-

ization derivatives with respect to an external strain are evaluated through an extension

56


of the CPHF/KS treatment utilized for optic dielectric(53; 65) and Born charges,(73)

for both electronic and ionic contributions, respectively. For example, the analytical

calculation of ionic contribution to the piezoelectricity is composed of the analytical

calculations of Born charges in conjunction with a Hessian matrix that is obtained by

numerical differentiation of analytical gradients with respect to atomic coordinates and

lattice vector components. Thus, this treatment of the ionic contribution, for instance,

is considered as a quasi-analytical procedure.

1.4 Capability in Treating Systems of Different Dimensional-

ity: The Example of MgO Nanotubes

In this study, we investigate the accuracy of CRYSTAL code in treating systems of dif-

ferent dimensionality by examining the convergence of response properties from passing

on bulk (3D) to slab (2D) and then to nanotube (1D). The properties of two families

of MgO tubes, namely (n,0) and (m,m), are here explored. In both cases the coordi-

nation of cations and anions is 4, as in the monolayer, whereas it is 6 in the bulk. In

the (m,m) case however cation rings alternate with anion rings, whereas in (n,0) tubes

oxygen and magnesium atoms are present in the same ring (see Figure A.II.1). Tubes

in the range 6 ≤ n ≤ 140 and 3 ≤ m ≤ 70 have been considered, being n = 2 ∗ m

the number of MgO units in the unit cell (so, the maximum number of atoms is 280).

Tubes are built by rolling up the fully relaxed 2-dimensional conventional cell (2 MgO

units, with oxygen atoms protruding from the Mg plane alternately up and down by

0.38 A). Recent improvements in the CRYSTAL09 code permit full use of symmetry

(the point group contains 140 ∗ 4 = 560 symmetry operators for the (140, 0) tube) so

as to drastically reduce the computational cost.

The total energy and its difference with respect to the MgO monolayer (l-MgO), the

relative stability of the (n,0) with respect to the (m,m) family, the relaxation energy

and equilibrium geometry, the band gap, the IR vibrational frequencies and intensities,

and the electronic and ionic contributions to the polarizability are reported. All these

properties are shown to converge smoothly to the monolayer values as n → ∞, see

Table A.II.1. Absence of negative vibrational frequencies confirms that the tubes have

a stable structure. The parallel component of the polarizability α converges very

rapidly to the monolayer value, whereas α⊥ is still changing at n=140; however, when

extrapolated to very large n values, it coincides with the monolayer value to within

1%, Table A.II.1. Detailed informations can be found in our published article.(82)

57


Figure A.II.1: The optimized geometry of MgO systems of different dimensionality: bulk

((a), 3D); primitive planar (unbuckled) and conventional non planar (buckled) cell of the

monolayer ((b) and (c), 2D); (12,0) and (12,12) nanotubes, ((d) and (e), 1D).

2 The Optimization of The Basis Set

As mentioned before, the basis set choice is a critical issue in this kind of calculations.

Since, it determines the accuracy with which the calculated wave function and the de-

rived quantities may approximate the exact solution. For instance, Civalleri et al.(83)

have shown that the relative stability of silica polymorphs with tetrahedral Si coor-

dination is correctly described only if polarization d functions are included in the Si

and O sets. The d orbital in silicon play a fundamental role for the correct description

of the partially covalent character of the Si-O bonding, influencing directly the Si-O

equilibrium distance; those on oxygen are required to provide a correct description of

the dependence of total energy on the Si-O-Si bond angle.(84)

The optimization of the basis sets in this study has been performed manually by

changing the value of orbitals exponents (α in bohr−2) and searching the one cor-

responding to minimum energy. For example, one starts by looking the sp orbital

exponent value (αsp) for one atom, then proposes two other numerical values for this

58

2. The Optimization of The Basis Set

Table A.II.1: Calculated properties of the (n,0) series of MgO nanotubes and of the

monolayer (l-MgO). ∆E, δE and ∆E are the energy difference between the (n,0) tube and

the corresponding (n/2,n/2) one (they have the same number of atoms), the relaxation

energy for the rolled configuration and the energy difference between the relaxed tube and

l-MgO, respectively. Values reported are in µHa per MgO unit. Ru is the unrelaxed radius

(in A) of the Mg cations (the anions O are at Ru±0.38); RMg and RO are the same distances

after relaxation. The radius of the oxygen atom “outside” the tube (at Ru + 0.38 before

relaxation) remains essentially unaltered. BG is the band gap in eV. α and α⊥ are the

longitudinal and transverse components of the electronic αe, and static α0 polarizabilities

per MgO unit (in A3). The unrelaxed (sum over states) αSOS values are also reported. The

Layer∗ row gives the average of the monolayer perpendicular polarizabilities per MgO unit

using the relation α⊥n = 1

2(α⊥l + α

l ) (l stands for layer, n is the label of the tube) which

should be equal to the large radius limit (∞ row) of the transverse nanotube polarizability.

α

α⊥

n ∆E δE ∆E Ru RMg RO BG αeSOS α

eα0

αeSOS α

eα0

6 33601 4568 1.86 1.98 2.06 9.02 2.134 2.300 6.239 1.803 1.236 2.951

12 15664 4489 1646 3.71 3.82 3.86 9.55 2.088 2.259 6.873 1.746 1.272 2.905

24 3801 437 881 7.42 7.41 7.18 9.56 2.099 2.261 6.865 1.748 1.356 3.287

30 205 622 9.28 9.26 8.99 9.58 2.101 2.261 6.900 1.754 1.383 3.402

36 1684 109 459 11.13 11.11 10.81 9.59 2.102 2.261 6.940 1.758 1.403 3.493

40 1364 75 383 12.37 12.35 12.04 9.59 2.102 2.260 6.953 1.760 1.413 3.544

48 956 40 276 14.84 14.82 14.49 9.60 2.102 2.260 6.990 1.762 1.430 3.632

60 619 19 183 18.56 18.54 18.19 9.61 2.102 2.259 7.009 1.764 1.448 3.733

80 372 6 106 24.74 24.73 24.37 9.62 2.102 2.259 7.043 1.765 1.468 3.853

100 257 4 68 30.93 30.91 30.55 9.62 2.102 2.259 7.056 1.766 1.480 3.939

140 158 1 35 43.29 43.28 42.91 9.63 2.102 2.258 7.066 1.767 1.495 4.049

∞ 0 0 9.66 2.102 2.258 7.084 1.767 1.536 4.376

Layer∗ 1.767 1.535 4.422

Layer 0 0 9.65 2.101 2.258 7.082 1.432 0.811 1.761

exponent with changing factor ± 0.02. Then fitting the energies for these three expo-

nent values and finding out the value corresponding to the minimum energy; (fitting

function: f(α) = aα2 + bα + c, and αmin = −b2a). For the polarization orbitals (d, f),

the exponential changing factor was ± 0.01. It is known that, when the geometrical

and/or the basis set parameters are changed, the truncation criteria of the Coulomb

and exchange series, based on overlap can lead to the selection of different numbers

of bi-electronic integrals. This may be the origin of numerical noise in the optimiza-

tion curve. To avoid that, the indices of the integrals to be calculated can be selected

for a reference basis set (the basis with initial exponent value) by using the keyword

FIXINDEX. In this case, the crystal input ends with FIXINDEX followed by

END then BASE and the new basis set. This successive procedure repeated for the

other orbitals and the other atoms till the convergence of all exponents. A python

script has been used to generate the crystal input files with different exponent values

59


and submit the calculations.

Interestingly, one can improve as well the basis sets by adding additional sp or

polarization orbitals. Two examples will be introduced to elucidate the such strategy

and highlight the influence of the basis set on both structural and electronic properties.

EX.1: The basis set for α-quartz and GeO2 structuresWe started with the 86-311G* basis set (αsp=0.13, αd=0.6) for Si-atom(85) and

84-11G* (αsp=0.188, αd=0.6) for O-atom(86), then we optimized these external ex-

ponents of the sp and d shells of both atoms. The new obtained exponents were

(αsp=0.1534, αd=0.6804) for Si and (αsp=0.2105, αd=0.5664) for O-atom with a small

energy difference (4.1 mHa). Then, we improved Si-atom basis set by adding 1f func-

tion, with exponent value equals to 0.7, and then re-optimized the basis set. We gained

an interesting energy difference (-29.2 mHa) for exponent values as follow: (αsp=0.151,

αd=0.635, and αf=0.626) for Si and (αsp=0.198, αd=0.595) for O-atom. So, one can

conclude that, the addition of another function (orbital) is more effective on the expo-

nents value and the energy as well. Here, we will study solid solutions Si1−xGexO2 of

α-quartz where silicon atoms are progressively substituted with germanium atoms, to

different extent as a function of the substitutional fraction x. To have an extent range

of substitution, an elongated supercell (doubled along the c crystallographic axis) is

built with respect to the unit cell of pure α-quartz and a set of thirteen symmetry-

independent configurations is considered. To reduce the computational cost, the ef-

fective core Durand and Barthelat(87) pseudopotential (ECP) can be also used. An

effective Durand core is used to module the core electrons in silicon, germanium, and

oxygen atoms. Valence basis sets, (PS-211G* for silicon, PS-211G* for germanium,

and PS-41G*) which are already optimized in early studies(88) with respect to the de-

scription of SiO2 bulk structures of rutile and cristobalite, are adopted. The influence

of the basis set on the computed structural and electronic properties for α-SiO2 and

GeO2 hexagonal structures is shown in Table A.II.2.

From Table A.II.2, one can note that the addition of another d function to both

Si and Ge atoms, will improve the structural and electronic properties. However, the

addition of an f orbital overestimates the structural parameters as well as the band

gap and consequently underestimates the dielectric constant values (since the gap is a

denominator in the calculation of polarizability and dielectric constant). It can be also

noted that, the pseudopotential basis set describes very well the structural part in the

quartz case, with maximum error percentage 0.8% for the volume. However, it gives

a worth overestimation of the band gap value, 1.68 eV larger than the experimental

one. The situation is reversed for the GeO2 structure with a good estimation of the

gap compared to the other basis sets values (0.72 eV larger than the experimental

60


Table A.II.2: The influence of the basis set on the computed structural and electronic

properties for α-SiO2 and GeO2 hexagonal structures. Five optimized basis sets with dif-

ferent configuration of polarization orbitals are studied: the basis set B1 with only one d

polarization orbital for Si/Ge atoms; B2 basis with two d polarization orbitals, whereas the

basis sets B3 and B4 have one d, one f and two d, one f polarization orbitals, respectively;

the data obtained by the pseudopotential B5 is also reported. Si/Ge atoms are in 3a posi-

tion (u, 0, 13) and O atoms are in general 6c position (x, y, z) of space group P3221. θ is the

intertetrahedral angle (angle between SiO2 tetrahedrons). The percentage errors are given

in parenthesis with respect to the experimental data. Calculations are performed at the

PBE0 level.

B1 B2 B3 B4 B5 Exp.

α-quartz SiO2

a = b (A) 4.929 4.938 4.968 4.979 4.898 (0.4%) 4.916a

c (A) 5.432 5.438 5.464 5.473 5.399 (0.1%) 5.405a

V (A3) 114.289 114.839 116.794 117.539 112.177 (0.8%) 113.13a

ρ (g/cm3) 2.614 2.601 2.558 2.542 2.663 (0.3%) 2.655b

u 0.469 0.469 0.474 0.475 0.466 0.470a

Ox 0.413 0.413 0.416 0.416 0.411 0.413a

Oy 0.269 0.269 0.260 0.259 0.274 0.267a

Oz 0.216 0.215 0.208 0.207 0.220 0.214a

Si−O1 (A) 1.618 1.618 1.613 1.614 1.616 (0.6%) 1.607a

Si−O2 (A) 1.621 1.621 1.615 1.617 1.621 (0.4%) 1.614a

θ (deg) 142.808 143.200 146.171 146.537 141.189 (1.8%) 143.73a

Eindg (eV) 8.995 8.944 9.208 9.144 10.576 8.9c

∞xx = ∞yy 2.229 2.243 2.181 2.193 2.129 2.356d

∞zz 2.261 2.273 2.207 2.217 2.165 2.383d

0xx = 0yy 4.353 4.359 4.193 4.188 4.177 4.43d

0zz 4.559 4.575 4.404 4.405 4.381 4.64d

α-quartz GeO2

a = b (A) 5.045 5.039 5.062 5.059 5.101 (2%) 4.985e

c (A) 5.720 5.714 5.718 5.715 5.752 (1.9%) 5.646e

V (A3) 126.055 125.630 126.871 126.673 129.612 (6.7%) 121.50e

ρ (g/cm3) 4.186 4.200 4.159 4.165 4.071 (4.9%) 4.28b

u 0.454 0.454 0.455 0.455 0.454 0.451f

Ox 0.400 0.400 0.403 0.402 0.400 0.397f

Oy 0.299 0.299 0.297 0.297 0.298 0.302f

Oz 0.241 0.241 0.238 0.238 0.239 0.243f

Ge−O1 (A) 1.747 1.744 1.739 1.739 1.759 (1.3%) 1.736f

Ge−O2 (A) 1.752 1.749 1.743 1.744 1.764 (1.3%) 1.741f

θ (deg) 131.332 131.416 132.959 132.821 131.879 (1.4%) 130.05f

Eindg (eV) 6.633 6.665 6.770 6.760 6.440 5.94g

∞xx = ∞yy 2.518 2.550 2.487 2.519 2.478 2.89h

∞zz 2.617 2.646 2.576 2.609 2.559 2.99f

0xx = 0yy 5.339 5.297 5.130 5.149 5.124 6.65b

0zz 5.515 5.488 5.344 5.367 5.274 7.43*’ b

a Ref.(89), b Ref.(90), c Ref.(91), d Ref.(92),e Ref.(93), f Ref.(94), g Ref.(95), h Ref.(96)

*This value of dielectric constant has been calculated by solving system with linear

equations using the frequencies of the eight E(TO) modes, in which the author has

uncertainity with some frequencies, for more details see Ref.(97).

61


value) and an overestimation for the volume 6.7 % larger than experiments. In the

present study, we are interested in the piezoelectric properties of the Si1−xGexO2

solid solution, and we want to consider the whole range of compositions, where a

total of thirteen symmetry-independent configurations are considered. To reduce the

computational cost, we will use the optimized pseudopotential basis set, B5, for the

next part concerning the solid solutions of the quartz.

EX.2: The basis set adopted for SrTiO3 structuresIn this part of the study, we need to analyze the complete piezoelectric tensor

of ferroelectric strontium titanate SrTiO3 at low-temperature by ab initio theoretical

simulations. At room temperature, strontium titanate crystallizes in a simple cubic

structure of space group Pm3m where each Ti ion is octahedrally coordinated to six

O ions. On cooling, SrTiO3 undergoes a second-order antiferrodistortive (AFD) phase

transition at Ta = 105 K to a tetragonal phase with space group I4/mcm. By fur-

ther cooling below Ta, down to about 50 K, the ferroelectric instability leads to a

softening of the Ti-displacement phonon mode and anomalously to large values of the

static dielectric constants which grow according to a Curie-Weiss law. A ferroelectric

phase transition could be expected to occur at Tf ∼ 35 K; however, below a certain

temperature Tq = 37 K, these quantities saturate and the ferroelectric transition is

suppressed down to 0 K by strong zero-point quantum fluctuations.(98; 99) SrTiO3

then remains in a quantum coherent state (also called Muller state after its discover)

even at very low temperatures where it becomes a so-called quantum paraelectric.(100)

Figure A.II.2 represents graphically the considered four structures of SrTiO3, these

pictures have been prepared using the J-ICE online interface to Jmol.(101)

So, it was important to have a good representative basis set, here the influence of the

basis set has been also reported, see Table A.II.3. An atom-centered Gaussian-type-

orbital basis set is adopted which has been obtained by adding further polarization

functions to the optimized one used by Evarestov et al.(102) and is available on the

web:(103) an all-electron split-valence 8-411G(2d1f) for the Oxygen atoms, an all-

electron split-valence 86-411(2d1f) for Titanium atoms while the core of Strontium

atoms is described by a Hay-Wadt effective-core-pseudopotential(104) and the valence

by 211G(2d1f) functions. The optimization of the additional orbitals exponent is not

included in this part, since the used basis sets are very rich and as concluded from

the study on the quartz the more effective is the addition of polarization functions.

The structural and electronic properties for the four SrTiO3 models are reported in

Table A.II.3. The first model is the quadruple pseudocubic cell which introduced

with Pm3m symmetry and can be built by doubling the lattice parameter along z

(c = 2a0) and by doubling the cell in the xy plane so that a = b =√2a0. Such a

62


Figure A.II.2: Graphical representation of the structure of SrTiO3. In panel a), the cubic

Pm3m phase is represented in the xy plane; the conventional cubic cell (thick continuous

line) and the quadruple pseudo-cubic tetragonal cell (dashed line) are shown which contain 5

and 20 atoms, respectively. The same structure and cells are also represented in a different

view in panel b). Panel c) reports the structure of the I4/mcm tetragonal phase in the

xy plane; rotation of adjacent TiO6 octahedra along the z direction can be inferred from

comparison with panel a).

pseudo-cubic cell is represented in Figure A.II.2 in dashed lines. The second model

considering the tetragonal phase of I4/mcm symmetry, whose crystallographic cell

contains twenty atoms and almost coincides to the pseudo-cubic cell, apart from a

rotation of adjacent TiO6 octahedra along the z direction of an angle θ (as can be

inferred by comparing Figure A.II.2-a and Figure A.II.2-c) and a slight deviation from

the pseudo-cubic ratio c/2a0 = 1. A structural parameter u (0.25 in the pseudo-cubic

structure) is considered that corresponds to the fractional coordinate along z of the

oxygen atom in the 8h Wyckoff position; its value is related to the octahedra rotation

angle θ according to the relation θ = arctan (1− 4u). Starting from the I4/mcm

tetragonal phase, the symmetry has been lowered in order to describe a ferroelectric

phase to both a tetragonal I4cm and a orthorhombic Ima2 one. In both cases, a further

63


structural parameter appears, |δ|, which measures the displacement of Ti atoms from

their equilibrium positions in the Pm3m and I4/mcm phases.

From Table A.II.3, It appears that the addition of f orbital to the basis set with 2d

functions improves the structural paratmeters for both the pseudo-cubic and the high-

symmetry tetragonal (I4/mcm) phases. For the electronic level (band gap), the basis

B with 2d additional functions is the one which describes well the Highest Occupied

Molecular Orbital (HOMO) and the Lowest Unoccupied Molecular Orbital (LUMO)

band energies, 0.23 eV for the pseudo-cubic structure gap and 0.76 eV for the tetragonal

one, compared to experimental data. However, the richest basis in this study, basis C,

gives an overestimation with only 0.11 eV larger than the basis B values. The more

interesting phases in this part were the ferroelectric phases: low-symmetry (I4cm)

tetragonal and orthorhombic (Ima2) ones. Concerning these two phases, one can

note that, the obtained results have the same trend as those for cubic and tetragonal

structures: the structural lattice parameters increased for the basis B and improved

by applying the basis C, and the band gap is well-estimated with the basis B and

quite overestimation with the basis C; but always better than the basis with only

one additional d function (basis A). Piezoelectricity and elasticity are structure-related

properties, so basis set C that gives a good estimation for the structure parameters

will be used for the following calculations of SrTiO3 system.

3 The Choice of Computational Parameters

During our study, the structural changes among the different model sometimes are

quite small, though their effects on several properties turn out to be paramount. So,

the accuracy of ab initio simulations in reproducing basic structural and electronic

properties has to be carefully checked before computing more sophisticated quantities.

In this section, we discuss the effect of the computational parameters as: the DFT

integration grid, the shrinking factor, and the bi-electronic integrals tolerances. First,

we define those parameters, then we give some applications about the effect of these pa-

rameters on the electronic and structural results. These parameters are used to control

the precision of convergence in the calculations. In our application examples (SiO2-

GeO2 and SrTiO3), we are interesting in the convergence of the calculated parameters.

So, the results are compared to each other till convergence is found (comparison to

experimental data is not included).

64

3. The Choice of Computational Parameters

Table A.II.3: The influence of the basis set on the computed structural parameters and

band gap (Eg) of the four structures of SrTiO3 is here considered. The basis A corresponds

to the basis used by Evarestov et al with another polarization d (αd = 0.9 for Sr and Ti

atoms and 1.2 for O-atom). A more diffused polarization d orbital with α = 0.3 has been

added (all atoms) to basis A to give the basis set B. In the C basis, a polarization f function

has been added to all atoms with αf equals 0.9. Calculations are performed at the PBE0

level.

A B C Exp.

Pm3m

a = b (A) 5.517 5.521 5.506 5.501a

c (A) 7.802 7.808 7.785 7.780a

Eg (eV) 4.236 3.982 4.090 3.75b

I4/mcm

a = b (A) 5.513 5.517 5.502 5.507c

c (A) 7.827 7.817 7.793 7.796c

u 0.236 0.238 0.238 0.240d

θ (deg) 3.205 2.748 2.748 2.1d

Eg (eV) 4.265 4.010 4.117 3.246e

I4cm

a = b (A) 5.512 5.517 5.499 -

c (A) 7.827 7.817 7.808 -

c/2a0 1.003 1.001 1.003 -

u 0.238 0.239 0.238 -

θ (deg) 2.748 2.519 2.748 -

|δ| (A) 0.048 0.002 0.047 -

Eg (eV) 4.291 4.012 4.155 -

Ima2

a (A) 5.519 5.517 5.504 -

b (A) 5.517 5.517 5.504 -

c (A) 7.810 7.816 7.792 -

c/2a0 1.001 1.001 1.001 -

u 0.238 0.238 0.241 -

θ (deg) 2.748 2.748 2.062 -

|δ| (A) 0.002 0.001 0.001 -

Eg (eV) 4.298 4.015 4.129 -

a Ref.(105), b Ref.(106), c Ref.(107), d Ref.(108), e Ref.(109)

65


3.1 The DFT-Grid

In crystal, the DFT exchange-correlation density functional is integrated numerically

on a mesh of points in atomic domains over the cell volume: radial and angular points

of the atomic grid are generated through Gauss-Legendre and Lebedev quadrature

schemes, respectively. If one Lebedev accuracy level is associated with the whole

radial range, the atomic grid is called unpruned, or uniform. In order to reduce the

grid size and maintain its effectiveness, the atomic grids of spherical shape can be

partitioned into shells, each associated with a different angular grid. This procedure,

called grid pruning, is based on the assumption that core electron density is usually

almost spherically symmetric, and surface to be sampled is small. Also, points far

from the nuclei need lower point density, as associated with relatively small weights,

so that more accurate angular grids are mostly needed within the valence region than

out of it. The choice of a suitable grid is crucial both for numerical accuracy and

need of computer resources. So, the influence of this parameter “DFT-grid” on the

structural and electronic properties will be analyzed on our two examples: SiO2-GeO2

and SrTiO3.

3.2 The Number of k-Points

The numerical integration for periodic solid-state systems, 1D, 2D, 3D, is typically

carried out in reciprocal space where the first Brillouin Zone (BZ) is divided by a

finite number of k-points. The k-points describe the sampling of the electronic wave

function. In crystal, it’s mandatory to give information about the shrinking factor,

IS, which generates a commensurate grid of k-points in reciprocal space, according to

Pack-Monkhorst method,(27) where the Hamiltonian matrix computed in direct space

is Fourier transformed for each k value, and diagonalized, to obtain eigenvectors and

eigenvalues. A second shrinking factor, ISP, defines the sampling of k-points, “Gilat

net”,(110; 111) used for the calculation of the density matrix and the determination

of Fermi energy in the case of conductors (bands not fully occupied). The choice of

the reciprocal space integration parameters to compute the Fermi energy is a delicate

step for metals and semi-conductors. As will be shown later, this parameter is not so

effective for our two examples, which have quite large band gaps (see Tables A.II.2, and

A.II.3). However, for another case as graphene (the gap is nearly zero), the description

of the band gap area is deeply dependent on the number of k-points at the Dirac

fermions.

66

3. The Choice of Computational Parameters

3.3 The Bielectronic Integrals Tolerances

In crystal code, the evaluation of the Coulomb and HF exact exchange infinite series

is controlled by five bielectronic integrals tolerances parameters: (ITOL1, ITOL2,

ITOL3, ITOL4, ITOL5). ITOL1, or T1, represents the overlap threshold for Coulomb

integrals, ITOL2, or T2, corresponds to the penetration threshold for Coulomb

integrals, ITOL3 (T3) refers to the overlap threshold for HF exchange integrals, and

ITOL4 (T4), ITOL5 (T5) give the torelance limit for pseudo-overlap (HF exchange

series), for more details see “CRYSTAL User’s Manual”.(74) The values of those

parameters are set to T1 = T2 = T3 = T4 = 12T5. Selection is performed according to

overlap-like criteria: when the overlap between two Atomic Orbitals is smaller than

10−ITOL, the corresponding integral is disregarded or evaluated in a less precise way.

These five ITOL parameters control the accuracy of the calculation of the bielectronic

Coulomb and exchange series and their effect will be explained on the next examples.

EX.1: α-quartz and GeO2 structuresA number of structural and electronic parameters of the pure phases considered

(namely, α-SiO2 and α-GeO2) are reported in Table A.II.4. The lattice parameters,

bond angle and bond length, and the indirect band gap Eindg are reported. All the

values reported have been obtained by fully optimizing the structures with ECP basis

set B5 at PBE0 level, and shrinking factor 8. Table A.II.4 shows the effect of the

DFT integration grid and electronic integral tolerances on the computed structural

and electronic properties.

As mentioned above, each grid is represented by radial and angular points which are

generated through Gauss-Legendre and Lebedev schemes; each grid is labeled with the

symbol (nr,na) where nr represents the number of radial points and na the maximum

number of angular points. Three different grids have been used with G1 = (55,434), G2

= (75,974) and G3 = (99,1454). The truncation of infinite lattice sums for the integrals

evaluation is controlled by five thresholds and three different cases are used as well: T1

= (6 6 6 6 12), T2 = (8 8 8 8 16) and T3 = (10 10 10 10 20).

From inspection of Table A.II.4, the influence of both DFT-grid and electronic

integral tolerance appears to be not so effective, since the difference between considered

cases does not exceed 1%.

EX.2: SrTiO3 structuresA number of structural and electronic parameters of the four phases (namely,

pseudo-cubic Pm3m, I4/mcm, I4cm and Ima2) considered are reported in Table

67


Table A.II.4: The effect of the DFT integration grid and electronic integral tolerances

on computed structural and electronic properties. Three DFT grids (nr,na) with nr radial

points and a maximum of na angular points are used: G1 = (55,434), G2 = (75,974) and

G3 = (99,1454), with fixed integral tolerance T2. Three sets of integral tolerances are

considered: T1 = (6 6 6 6 12), T2 = (8 8 8 8 16) and T3 = (10 10 10 10 20), with fixed grid

G3. Calculations are performed with the pseudopotential, B5, basis sets and PBE0 hybrid

functional.

DFT Grid Integral Tolerances

G1 G2 G3 T1 T2 T3

α-quartz SiO2

a = b (A) 4.895 4.896 4.898 4.894 4.898 4.894

c (A) 5.397 5.398 5.400 5.396 5.400 5.397

c/a 1.103 1.103 1.102 1.103 1.102 1.103

V (A3) 112.019 112.067 112.177 111.942 112.177 111.975

ρ (g/cm3) 2.667 2.666 2.663 2.561 2.663 2.668

θ (deg) 141.040 141.087 141.189 140.982 141.189 140.996

Si−O1 (A) 1.616 1.616 1.616 1.616 1.616 1.616

Si−O2 (A) 1.621 1.621 1.621 1.621 1.621 1.621

Eindg (eV) 10.568 10.572 10.576 10.593 10.576 10.561

α-quartz GeO2

a = b (A) 5.099 5.099 5.101 5.099 5.101 5.099

c (A) 5.748 5.751 5.752 5.748 5.752 5.752

c/a 1.127 1.128 1.128 1.127 1.128 1.128

V (A3) 129.401 129.535 129.611 129.463 129.611 129.535

ρ (g/cm3) 4.077 4.073 4.071 4.075 4.071 4.073

θ (deg) 131.769 131.843 131.879 131.810 131.879 131.837

Ge−O1 (A) 1.759 1.759 1.759 1.759 1.759 1.759

Ge−O2 (A) 1.764 1.764 1.764 1.764 1.764 1.764

Eindg (eV) 6.439 6.439 6.440 6.427 6.440 6.518

68

4. The Choice of Hamiltonian

A.II.5. Along with the structural parameters, already defined at the beginning of Sec-

tion 2, the direct bang gap Eg and the difference between the cubic phase energy and

that of the other phases ∆Ec are reported as well. All the values reported have been

obtained by fully optimizing the structures at the PBE0 level.

Let us first consider the effect of the DFT grid; grid G1 = (55,434), which usually

performs rather well, is found to describe very poorly even the two simple structures of

cubic and tetragonal AFD phases. The structural and electronic parameters provided

by grid G2 = (75,974) are already quite converged, if compared with those obtained

with a richer grid G3 = (99,1454), especially for the first two structures in the table.

The only two (coupled) parameters that still vary when passing from G2 to G3 are

u and angle θ for the two ferroelectric structures I4cm and Ima2 thus revealing a

particularly flat potential energy surface in that region. However, the most relevant

structural parameter to piezoelectricity, that is the displacement |δ| of Ti atoms, is

already converged with G2 in both structures. The G2 grid will be used in the following.

In general, all the structural parameters are found to be more stable with respect

to the integral tolerances. A T2 = (10 10 10 12 24) set slightly improves upon T1 = (8

8 8 8 16) and is almost at convergence if compared with a richer set T3 = (12 12 12 15

30). Again, this is particularly so for the two simplest structures. In the following, we

will use the T2 set of integral tolerances.

Finally, by recalling that calculations refer to 0 K, let us note that, regardless of

the specific setting, the tetragonal AFD I4/mcm phase is always electronically more

stable that the cubic Pm3m one. The two ferroelectric phases are always electronically

more stable that the AFD one, with the orthorhombic Ima2 in turn more stable than

the tetragonal I4cm.

4 The Choice of Hamiltonian

In this section, we briefly discuss the influence of the adopted one-electron Hamilto-

nian on the structural and electronic properties of our two examples: α-SiO2; GeO2

systems, and SrTiO3 different phases. The description of the Hamiltonian has been

already discussed in Section 1.1, Chapter A and in this way we will directly introduce

into our examples.

EX.1: α-quartz and GeO2 structuresIn this example, four different one-electron Hamiltonians are considered: the lo-

cal density approximation (LDA) and a generalized gradient approximation (GGA),

namely Perdew-Burke-Ernzerhof (PBE)(14), to the density functional theory (DFT)

69


Table A.II.5: Influence of the DFT integration grid and electronic integral tolerances on

computed structural and electronic properties (as defined in the text) of the four structures

of SrTiO3 here considered. Three DFT grids (nr,na) with nr radial points and a maximum

of na angular points are used: G1 = (55,434), G2 = (75,974) and G3 = (99,1454). Three

sets of integral tolerances are considered: T1 = (8 8 8 8 16), T2 = (10 10 10 12 24) and T3

= (12 12 12 15 30). Calculations are performed with the PBE0 hybrid functional.

DFT Grid Integral Tolerances

G1 G2 G3 T1 T2 T3

Pm3m

a = b (A) 5.480 5.505 5.505 5.505 5.505 5.505

c (A) 7.750 7.785 7.785 7.786 7.785 7.785

Eg (eV) 4.141 4.110 4.110 4.088 4.110 4.126

I4/mcm

a = b (A) 5.481 5.501 5.501 5.502 5.501 5.501

c (A) 7.816 7.792 7.792 7.793 7.792 7.792

u 0.226 0.239 0.239 0.238 0.239 0.239

θ (deg) 5.484 2.519 2.519 2.748 2.519 2.519

Eg (eV) 4.225 4.135 4.135 4.177 4.135 4.149

∆Ec (mHa) 2.207 0.029 0.015 0.023 0.029 0.035

I4cm

a = b (A) 5.481 5.500 5.502 5.499 5.500 5.500

c (A) 7.816 7.798 7.796 7.808 7.798 7.798

u 0.226 0.240 0.242 0.238 0.240 0.240

θ (deg) 5.484 2.291 1.833 2.748 2.291 2.291

|δ| (A) 0.012 0.037 0.041 0.047 0.037 0.037

Eg (eV) 4.225 4.154 4.150 4.155 4.154 4.169

∆Ec (mHa) 2.208 0.031 0.019 0.037 0.031 0.040

Ima2

a (A) 5.481 5.504 5.506 5.504 5.504 5.502

b (A) 5.482 5.503 5.504 5.504 5.503 5.503

c (A) 7.816 7.789 7.787 7.792 7.789 7.791

u 0.226 0.239 0.241 0.241 0.239 0.239

θ (deg) 5.484 2.519 2.062 2.063 2.519 2.519

|δ| (A) 0.008 0.016 0.018 0.017 0.016 0.015

Eg (eV) 4.224 4.153 4.152 4.129 4.153 4.167

∆Ec (mHa) 2.208 0.041 0.031 0.043 0.041 0.042

70


and two hybrid schemes (namely B3LYP(9) and PBE0(26)) which include 20 and 25

% of exact HF exchange, respectively.

Table A.II.6: Effect of the adopted one-electron Hamiltonian on computed structural and

electronic properties for α-SiO2 and α-GeO2. See text for a definition of the quantities

reported.

This Study

LDA PBE B3LYP PBE0 LDA/PW(112) Exp.

α-SiO2

a = b (A) 4.777 4.921 4.939 4.898 4.883 4.916a

c (A) 5.343 5.429 5.435 5.399 5.371 5.405a

V (A3) 105.62 113.88 114.83 112.18 - 113.13a

Si-O1 (A) 1.629 1.629 1.619 1.616 1.613 1.607a

Si-O2 (A) 1.637 1.634 1.623 1.621 1.618 1.614a

O1-Si-O1 (deg) 107.11 108.45 108.67 108.48 108.47 108.93a

O1-Si-O2 (deg) 112.40 111.19 110.63 110.89 110.75 110.52a

θ (deg) 133.73 140.28 143.02 141.19 140.55 143.73a

Eg (eV) 7.0 7.4 10.1 10.6 5.785 8.9b

α-GeO2

a = b (A) 4.932 5.130 5.174 5.101 4.870 4.985c

c (A) 5.723 5.804 5.791 5.752 5.534 5.646c

V (A3) 120.57 132.30 134.24 129.61 - 121.50c

Ge-O1 (A) 1.779 1.779 1.760 1.759 1.693 1.736d

Ge-O2 (A) 1.788 1.784 1.765 1.764 1.699 1.741d

O1-Ge-O1 (deg) 104.57 106.61 107.27 106.70 106.16 107.72d

O1-Ge-O2 (deg) 115.03 112.98 111.96 112.55 113.03 110.49d

θ (deg) 124.56 130.69 134.38 131.88 130.56 130.05d

Eg (eV) 3.4 3.6 6.0 6.4 4.335 5.94e


Table A.II.6 shows the influence of the adopted one-electron Hamiltonian on com-

puted structural and electronic properties of the two structures α-SiO2 and α-GeO2.

71


Lattice parameters, cell volume, bond lengths and angles, intertetrahedral angle θ

and electronic band gap Eg are reported and compared with experimental values. As

regards lattice parameters, LDA underestimates them by 2.8 % for α-SiO2 and 1 %

for α-GeO2 while PBE overestimates them by 1.7 % and 2.9 %, respectively. Note

that the use of a pseudopotential plane wave basis(112) within LDA does not improve

significantly the theoretical results with respect to their experimental counterpart, in

particular as regards the band gap which affects the piezoelectric response properties

(see Table A.II.6 where results from Ref.(112) are also shown for comparison). The

B3LYP hybrid functional describes well the α-SiO2 structure, with a deviation of 0.5

%, but poorly the α-GeO2 structure (3.8 % of deviation from experiment). The PBE0

hybrid functional is providing the best overall description of the structural features of

the two structures with deviations of 0.4 % and 2 %, respectively. The description of

the θ intertetrahedral angle is also in fair agreement with experiment. Let us note that

the electronic band gap obtained with PBE0 is slightly overestimated with respect to

the experimental value, but remains satisfactory. The PBE0 hybrid functional is used

in the following of our calculations.

EX.2: SrTiO3 structuresHere, we briefly analyze the effect of the adopted one-electron Hamiltonian on

structural properties of the four SrTiO3 phases. The structural properties, as obtained

with the four Hamiltonians, are reported in Table A.II.7. To validate the accuracy

of the present calculations, we compare both with experiments and theoretical results

by El-Mellouhi et al.,(113) obtained with the screened hybrid HSE06 functional,(114)

which have been recently reported and declared to constitute one of the most accurate

ab initio datasets in the literature as concerns SrTiO3.

From the analysis of cubic Pm3m and tetragonal I4/mcm structures, for which

accurate experimental data are available, it can be noticed that the pure generalized-

gradient PBE functional overestimates the lattice parameters by 1.1 % and HF by

0.5 %, while a simple LDA functional underestimates them by 1.2 %. The global

hybrid PBE0 functional provides excellent lattice parameters for both structures with

an overall error of 0.06 %.

The HF description of the AFD phase is quite poor: along with a usual huge

electronic bang gap of 12.2 eV, with respect to an experimental value of 3.2 eV, it

describes a very small distortion with respect to the pseudo-cubic structure. The

rotation angle θ of the octahedra is very small, 0.2 with an experimental value of 2.1

at 4 K, the stretching of the cell is null (c/2a0 = 1.000) and the electronic relative

stability of the AFD phase with respect to the cubic is inverted. LDA overestimates

72


Table A.II.7: Influence of the adopted one-electron Hamiltonian on computed structural

and electronic properties (as defined in the text) of the four structures of SrTiO3 here

considered. The pseudo-cubic structure is considered for comparison with the others.

Present Study Ref.(113)

HF LDA PBE PBE0 HSE06 Exp.

Pm3m

a = b (A) 5.529 5.453 5.563 5.505 5.518 5.501a

c (A) 7.819 7.711 7.867 7.785 7.804 7.780a

c/2a0 1.000 1.000 1.000 1.000 1.000 1.000a

u 0.250 0.250 0.250 0.250 0.250 0.250a

θ (deg) 0.000 0.000 0.000 0.000 0.000 0.000a

|δ| (A) 0.000 0.000 0.000 0.000 0.000 0.000a

Eg (eV) 12.203 1.906 1.901 4.110 3.590 3.75b

I4/mcm

a = b (A) 5.528 5.441 5.554 5.501 5.515 5.507c

c (A) 7.816 7.730 7.881 7.792 7.809 7.796c

c/2a0 1.000 1.005 1.003 1.001 1.001 1.001c

u 0.249 0.228 0.231 0.239 0.241 0.240d

θ (deg) 0.229 5.029 4.346 2.519 2.010 2.1d

|δ| (A) 0.000 0.000 0.000 0.000 0.000 0.000c

Eg (eV) 12.204 2.014 1.974 4.135 3.227 3.246e

∆Ec (mHa) -0.001 0.246 0.155 0.029 0.013 -

I4cm

a = b (A) 5.529 5.442 5.551 5.500 - -

c (A) 7.818 7.729 7.900 7.798 - -

c/2a0 1.000 1.002 1.004 1.002 - -

u 0.249 0.229 0.231 0.239 - -

θ (deg) 0.229 4.802 4.346 2.519 - -

|δ| (A) 0.013 0.008 0.053 0.037 - -

Eg (eV) 12.203 2.010 2.014 4.154 - -

∆Ec (mHa) 0.000 0.243 0.169 0.031 - -

Ima2

a (A) 5.527 5.440 5.559 5.504 - -

b (A) 5.529 5.442 5.558 5.503 - -

c (A) 7.819 7.729 7.879 7.789 - -

c/2a0 1.000 1.002 1.002 1.001 - -

u 0.249 0.228 0.230 0.239 - -

θ (deg) 0.229 5.029 4.574 2.519 - -

|δ| (A) 0.009 0.007 0.031 0.016 - -

Eg (eV) 12.205 2.013 2.018 4.153 - -

∆Ec (mHa) 0.062 0.248 0.176 0.041 - -


73


the rotation angle θ, more than PBE and PBE0 which provides a reasonable agreement

with the experiment.

For each Hamiltonian, given the description of the first two structures, we expect a

similar description also for the two ferroelectric phases for which structural experimen-

tal data are not presently available. The hybrid PBE0 functional guarantees a good

description of the structural and electronic properties of SrTiO3 and it constitutes our

choice for the next calculations. The PBE0 results of the present work provides a good

agreement with experiments as that obtained by El-Mellouhi et al.(113) in their study

with a screened hybrid HSE06 (as can be inferred form the comparison of the two

corresponding columns in Table A.II.7).

5 Calibration of Piezoelectric Property Compu-

tations: The Case of α-Quartz Doped by Ge

(Si1−xGexO2)

Quartz, α-SiO2, is the well-known piezoelectric material. Due to its peculiar piezo-

electric properties, α-SiO2 is widely applied in the electronic industry. However, its

physical properties are severely reduced for applications requiring high thermal sta-

bility and high electromechanical coupling coefficients. These limitations are mainly

due to the α-SiO2 to β-SiO2 phase transition where the piezoelectric constant d11

vanishes and d14 remains the only non-zero component.(115) A series of solid solu-

tions for α-quartz homeotypes has been studied both experimentally and theoretically:

SiO2-GeO2,(116; 117; 118; 119; 120) SiO2-PON (phosphorus oxynitride),(121) SiO2-

AlPO4,(122) AlPO4-GaPO4,(123; 124; 125; 126; 127) AlPO4-AlAsO4,(123) AlPO4-

FePO4,(128) GaPO4-FePO4.(129) The structural parameters and the piezoelectric

properties of all these solid solutions are expected to vary as a function of the sub-

stitutional fraction x, thus representing an effective way for tuning such properties for

specific technological applications.

Among these solid solutions, the SiO2-GeO2 system is probably the most promising;

its phase diagram shows that the solubility of α-GeO2 into α-SiO2 can reach 31 % at

1000 K and 70 MPa, under hydrothermal conditions.(117) An attempt has been made

to synthesize a series of Si1−xGexO2 solid solutions of different chemical compositions,

as a function of x, by using hydrothermal and flux methods.(119; 130; 131) The first

measurement of the piezoelectric response of one of these compositions (Si0.93Ge0.07O2)

indicates that the main piezoelectric constants, d11 and d14, exceed those of α-quartz

74

5. Calibration of Piezoelectric Property Computations: The Case ofα-Quartz Doped by Ge (Si1−xGexO2)

Figure A.II.3: Graphical definition of (a) the intertetrahedral bridging angle θ and (b)

the tetrahedral tilting angle δ which is an order parameter for the α - β phase transition in

quartz.

by 20 - 30 %. Furthermore, its α - β phase transition temperature is found to be 1053

K, which is 207 K higher than that of pure quartz.(132)

In the present work, we apply first principles quantum mechanical techniques to

the simulation of several properties of the Si1−xGexO2 system with different chemical

compositions, where the data concerning the two end-members: α-SiO2 and α-GeO2,

will be used for the calibration of the method. An elongated supercell (doubled along

the c crystallographic axis) is built with respect to the unit cell of pure α-quartz where

Si atoms are progressively substituted with Ge atoms. The whole range of substitution

is here considered; beside the two end-members of the solid solution, α-SiO2 (x =

0) and α-GeO2 (x = 1), five intermediate compositions are taken into account: x =

0.16, 0.3, 0.5, 0.6 and 0.83. Some intermediate compositions require different atomic

configurations to be properly described; in some configurations, two Ge atoms directly

bridge one another by an O atom, while in others, Si atom(s) may lie between them.

All possible symmetry-independent configurations are simulated for each substitutional

fraction x; a total of thirteen symmetry-independent configurations are considered,

overall.

A complete set of piezoelectric constants (elements of direct and converse third-

rank piezoelectric tensors e and d, and electromechanical coupling coefficients k) and

elastic constants (elements of the fourth-rank elastic C and compliance s tensors) is

computed for each configuration. Electronic and nuclear contributions to the dielectric

tensor are also computed. The effect on the electromechanical coupling constants of

the chemical composition along the Si1−xGexO2 solid solution series is analyzed. As a

75


Table A.II.8: For any composition x, number Ntot of atomic configurations, number of Nirr

symmetry-irreducible configurations among them, multiplicity M and number of symmetry

operators Nops proper of each irreducible configuration. The elongated supercell has only

six Si-positions.

x 06

16

26

36

46

56

66

Ntot 1 6 15 20 15 6 1

Nirr 1 1 3 3 3 1 1

M 1 6 6 6 3 6 12 2 6 6 3 6 1

Nops 12 2 2 2 4 2 1 6 2 2 4 2 12

by-product, some insight can be gained about the influence of the chemical composition

on structural and electronic properties.

All the calculations are performed using the Crystal14 program(74; 73) with a

Gaussian-type-function, pseudopotential, basis set and the hybrid functional PBE0; all

geometries are fully optimized at this level of theory. For more information about the

choice of basis, Hamiltonian, and calculation parameters, see Sections: 2, 3, 4, Chapter

II.

To do Si1−xGexO2 solid solution, an elongated supercell (doubled along the c crys-

tallographic axis) is built with respect to the unit cell of pure α-quartz. A new feature

of the Crystal program is exploited for finding automatically all the possible atomic

configurations corresponding to any composition x.(133; 134) For any substitutional

fraction x, the program finds the total number Ntot of atomic configurations; a full

symmetry analysis is then performed to find Nirr symmetry-irreducible configurations

among them. To each irreducible atomic configuration, characterized by Nops symme-

try operators, a multiplicity M is associated.

A total of thirteen symmetry-independent configurations are considered; their prop-

erties are illustrated in Table A.II.8. For the two pure phases, silicon (germanium)

atoms are centered on equivalent 6a Wyckoff positions and only one atomic configura-

tion is obviously possible, that is characterized by 12 symmetry operators. For the two

76


compositions x = 16= 0.16 and x = 5

6= 0.83, there is one irreducible configuration out

of six possible atomic configurations, each one invariant under 2 symmetry operators.

Substitutional fractions x = 26= 0.3 and x = 4

6= 0.6 are described by a total of 15

atomic configurations, out of which 3 are found to be symmetry-independent: two of

them with a multiplicity M = 6 (2 symmetry operators each) and one with multiplic-

ity M = 3, with 4 symmetry operators. For the composition x = 36= 0.5, 20 atomic

configurations can be obtained; 3 are found to be symmetry-irreducible: the first one

has 2 symmetry operators and a multiplicity M = 6, the second one has no symmetry

at all and a multiplicity M = 12 while the third one is characterized by 6 symmetry

operators and a multiplicity M = 2.

Table A.II.9: Structural and energetic properties of the Si1−xGexO2 solid solution series,

as a function of the substitutional content x. Calculations are performed using the PBE0

hybrid functional. ∆E is the energy difference with respect to the two end-members; and is

obtained by the equation: ∆E = ESi1−xGexO2- (1-x)ESiO2

+ xEGeO2. All data reported

are per unit cell.

SiO2 Si0.83Ge0.17O2 Si0.67Ge0.33O2 Si0.5Ge0.5O2 Si0.33Ge0.67O2 Si0.17Ge0.83O2 GeO2

∆E (eV) 0 0.0197 0.0301 0.0340 0.0311 0.0202 0

a (A) 4.898 4.936 4.965 5.005 5.035 5.069 5.101

c (A) 5.400 5.459 5.516 5.575 5.633 5.692 5.752

c/a 1.102 1.106 1.111 1.114 1.119 1.123 1.128

V (A3) 112.2 115.2 118.1 120.9 123.8 126.6 129.6

θ (deg) 141.2 139.8 138.2 136.6 134.8 133.4 131.9

δ (deg) 18.6 19.7 20.9 22.1 23.4 24.5 25.6

ρ (g/cm3) 2.66 2.92 3.18 3.42 3.65 3.86 4.07

Si-O1 (A) 1.616 1.617 1.617 1.618 1.622 1.623 -

Si-O2 (A) 1.621 1.622 1.623 1.624 1.629 1.628 -

Ge-O1 (A) - 1.750 1.750 1.751 1.755 1.756 1.759

Ge-O2 (A) - 1.754 1.756 1.756 1.758 1.761 1.764

In the next tables, for each composition x, weighted average values will be reported

over all the corresponding irreducible atomic configurations. In general, we find that,

for a given substitutional fraction x, the variation of any considered property among

the different irreducible configurations is quite small (lower than 1 %). For instance,

the difference in the energy of the three irreducible configurations, in the case x = 26,

is less than 10−4 hartree and the difference in the lattice parameters and volume is

lower than 0.003 A and 0.3 A3, respectively. This is due to the similarities in bonding

between Ge and Si atoms belonging to the same group in the periodic table. Moreover,

it confirms that the use of a double cell supercell is sufficient to analyze the Ge-doped

quartz.

77


Structural and energetic properties of the Si1−xGexO2 solid solution series are re-

ported in Table A.II.9 as a function of the composition x. The energy difference,

∆E, with respect to the two end-members: α-SiO2 and α-GeO2, equilibrium lattice

parameters a and c, their ratio c/a, cell volume V , intertetrahedral bridging angle θ,

tetrahedral tilting angle δ, density ρ and several bond lengths and angles are reported.

As mentioned above, all the reported data, for each composition x, are the weighted av-

erage values over all the corresponding irreducible configurations, where the weights in

the averaging procedure are determined by the multiplicities M given in Table A.II.8.

Here for ∆E, we compared the total DFT-energy per formula unit of the mixed

crystal, ESi1−xGexO2, with respect to the DFT-energy of the two pure end-members

according to the following relation: ∆E = ESi1−xGexO2- (1-x)ESiO2

+ xEGeO2. The

obtained values (∆E, Table A.II.9) show that the energy of mixing is positive in all

cases, but if the entropic contribution T kB ln(Ntot) is considered to take into account

the compositional disorder at room temperature, the variation of the free enthalpy, ∆G,

becomes negative and the composition x = 0.5 becomes the most stable system. kBis Boltzmann constant and Ntot is the total number of atomic configurations given in

Table A.II.8. A numerical calculation of T kB ln(Ntot) for the considering supercell

(only 6SiO2 units within the supercell) shows that ∆G starts to be negative at 264 K

for the case x = 0.5.

Along the Si1−xGexO2 series, the cell volume V increases quite linearly with the

number of Ge atoms in the cell. Interestingly, also the distortion of the cell, that can be

quantified by the c/a ratio, increases linearly. The c/a ratio, as well as θ and δ angles,

reflects the intertetrahedral distortion in the helical chains; the ideal value of the c/a

ratio is 1.10 for pure α-SiO2 and 1.13 for pure α-GeO2 at room temperature.(135)

The c/a ratio, and θ and δ angles vary linearly with respect to the composition x, as

observed experimentally.(119; 130) Bond distances, X-O1 and X-O2 (with X = Si, Ge),

vary regularly as a function of the composition. For instance, this can be observed

in the systematic increase of the Si-O and Ge-O bond lengths, related to the internal

distortion of the tetrahedra, as the Ge content increases.

Before discussing into some detail the piezoelectric properties of the Si1−xGexO2

solid solution, let us recall that direct and converse piezoelectricity measure the varia-

tion of polarization under a finite strain and the strain induced by an applied electric

field, respectively. The two third-rank tensors associated with these properties, e and

d, are connected to each other via the elastic C and compliance s fourth-rank tensors,

according to the relations given in Section 1.3. For this reason we analyze first the

elastic properties.

In Table A.II.10 we report, for each considered composition x, elastic constants Cvu,

compliance constants svu, the intertetrahedral angle θ and the bulk modulus Ks of the

78


Table

A.II.10:Elastic

(inGPa)an

dcompliance

(inTPa−

1)constan

tsof

theSi 1−xGe xO

2solidsolution

asafunctionof

the

compositionx.Intertetrahedralbridgingan

gleθ(indeg)an

dbulk

modulusK

s(inGPa)

arealso

reported.Experim

entaldata

forx=

0arefrom

Ref.(135),

forx=0.07from

Ref.(132)an

dforx=

1from

Ref.(136).

Calculation

sperform

edat

PBE0level.

Substitutionfractionx

00.07

0.16

0.3

0.5

0.6

0.83

1

This

study

LDA/PW

Exp.

Exp.

This

study

This

study

LDA/PW

Exp.

θ141.2

140.6

143.7

142.6

139.8

138.2

136.7

134.8

133.4

131.9

130.6

130.0

Elastic

Constants

C11

93.20

76.2

86.79

85.5

83.62

76.06

69.80

64.42

59.45

54.73

66.7

64

C12

14.22

11.9

6.79

10.38

15.56

16.51

17.49

17.79

18.24

18.08

24.3

22

C13

20.38

11.2

12.01

-20.89

21.03

21.21

21.02

20.78

20.75

23.1

32

C14

14.97

17.0

18.12

16.3

12.23

9.80

7.79

5.99

4.45

3.15

32

C33

120.79

101.7

105.79

-114.06

108.57

104.24

100.48

97.42

94.68

118.7

118

C44

61.34

54.0

58.21

57.2

54.86

49.57

44.87

40.72

36.99

33.69

41.3

37

C66

39.07

32.1

40.00

37.5

33.82

29.61

26.09

23.14

20.59

18.31

21.2

21

Compliance

constants

s 11

11.95

-12.78

-13.49

14.98

16.52

18.06

19.79

21.71

-19.25

s 12

-2.00

--1.77

--2.55

-3.11

-3.77

-4.40

-5.24

-6.02

--4.68

s 13

-1.68

--1.25

--2.01

-2.32

-2.62

-2.89

-3.12

-3.44

--3.95

s 14

-3.40

--4.53

--3.57

-3.57

-3.54

-3.33

-3.02

-2.60

--1.29

s 33

8.85

-9.74

-9.57

10.21

10.78

11.27

11.67

12.07

-10.62

s 44

17.99

-20.00

-19.85

21.66

23.57

25.62

27.83

30.20

-27.17

s 66

28.29

-29.10

-32.32

36.41

40.71

45.19

50.14

55.54

-47.87

Bulk

modulus

Ks

45.4

35

38a/40b

-42.7

40.5

38.8

37.0

35.4

33.7

41

42

aRef.(136),

bRef.(137)

79


solid solution. Available experimental data are also reported, for pure end-members

and for the Si0.93Ge0.07O2 case. For the two end-members, results from Ref.(112),

as obtained at LDA level with a PW basis, are also reported. By comparing with

experiments, in these three cases, a good agreement is observed for most diagonal

elements such as C11, C44 and C66. Off-diagonal terms, such as C12 and C13, show a

larger discrepancy which, however, can be expected since they have very small values

and similar deviations are also found among different experiments.(90; 136) The C14 off-

diagonal constant, though small, is in relatively good agreement with the experimental

determinations. The pressure dependence of the elastic constants of α-SiO2 has recently

been investigated by means of theoretical LDA calculations: if zero pressure constants

were found in satisfactory agreement with experiments, the high pressure ones were

underestimated.(138)

If we look at the elastic properties of the Si1−xGexO2 solid solutions of intermediate

compositions, a very smooth connection is found between those of the end-members

as a function of x, for all the elastic constants. An overall index of this smoothness

is given by the bulk modulus Ks, reported in the last column of Table A.II.10, which

varies smoothly from 45.4 GPa for α-SiO2 to 33.7 GPa for α-GeO2. The good numerical

accuracy of our approach for computing such properties can be inferred from inspection

of the behavior of the very small C14 off-diagonal constant which regularly varies from

14.97 GPa (18.12 GPa in the experiment) to 3.15 GPa (2.2 GPa in the experiment).

The overall effect of the progressive substitution of Ge atoms for Si atoms, is that of

reducing the bulk modulus and, as a consequence, from our theoretical predictions, to

decrease the values of all the diagonal elastic constants (a behavior that looks consis-

tent). From the comparison with experiments, the trends of two elastic constants show

a discrepancy: i) the C13 constant is theoretically predicted to be almost independent

from the composition x, by passing from 20.38 GPa at x = 0 to 20.75 GPa at x = 1,

whereas experimentally it becomes three times larger (from 12 GPa to 32 GPa); ii) the

diagonal C33 elastic constant is theoretically predicted to decrease from 120.79 GPa

to 94.68 GPa while experimentally it grows in the opposite direction from 106 GPa

to 118 GPa. In this respect, we should mention that the experimental determination

of the elastic constants of the two end-members is not homogeneous since it has been

performed in two separate experiments; our theoretical predictions seem to be more

reliable in this case, at least as concerns the trend from α-SiO2 to α-GeO2 along the

solid solution series.

In Table A.II.11, we report direct and converse piezoelectric constants, electronic

and static dielectric constants (permittivity) and electromechanical coupling coeffi-

cients of the Si1−xGexO2 solid solution, as a function of the substitutional fraction

x. Experimental data, when available, are reported for the two pure end-members.

80


Table A.II.11: Direct and converse independent piezoelectric constants, electronic and

static dielectric constants (relative permittivity) and electromechanical coupling coefficients

of the Si1−xGexO2 solid solution, as a function of the composition x. Experimental data,

when available, are reported for end-members. Calculations performed at PBE0 level.

x 0 0.16 0.3 0.5 0.6 0.83 1

Calc. (Exp.) Calc. Calc. Calc. Calc. Calc. Calc. (Exp.)

Direct Piezoelectricity eiv (C/m2)

e11 0.179 (0.171)a 0.197 0.208 0.222 0.229 0.236 0.241 -

e14 -0.060 (-0.041)a -0.076 -0.092 -0.108 -0.122 -0.134 -0.145 -

e26 -0.184 (-0.171)a -0.196 -0.208 -0.218 -0.228 -0.234 -0.240 -

Converse Piezoelectricity div (pm/V)

d11 2.30 (2.31)a 2.89 3.46 4.08 4.70 5.49 6.31 (5.7)c

d14 0.18 (0.73)a 0.91 1.12 1.38 1.85 2.37 3.11 (3.82)d

d26 -4.78 (-4.62)a -5.79 -6.83 -8.15 -9.47 -10.87 -12.58 -

Electronic Permittivity ∞ij

∞11 2.13 (2.36)b 2.19 2.25 2.30 2.36 2.42 2.48 (2.89)e

∞33 2.16 (2.38)b 2.22 2.29 2.35 2.42 2.49 2.60 (2.99)e

Static Permittivity 0ij

011 4.21 (4.43)a 4.39 4.54 4.71 4.86 5.00 5.11 (6.65)f

033 4.42 (4.64)a 4.60 4.73 4.90 5.06 5.19 5.26 (7.43)†

Electromechanical Coupling kiv (%)

k11 9.61 (10.23) 10.90 11.90 13.04 13.78 14.50 15.32 -

k14 3.94 (2.63) 5.23 6.54 7.87 9.20 10.49 11.75 -

k26 15.26 - 17.10 19.07 20.92 22.86 24.52 26.38 -

a Ref.(139), b Ref.(92), c Ref.(140), d Ref.(90), e Ref.(96), f Ref.(97)† This value has been obtained from the oscillator strengths calculated from the eight

Raman LO and TO phonon modes of Ref.(97). This value has been computed by Zeng et al.(51), from experimental converse

piezoelectric strain, free stress dielectric and compliance constants of Ref.(139).

In particular, dielectric and piezoelectric properties of α-SiO2 have been extensively

measured and a quite complete set of these constants is experimentally known. The

agreement between our theoretical predictions and the experimental values is rather

good for almost all quantities in the table. The only significant discrepancy is observed

for the off-diagonal component, d14, of the converse piezoelectric tensor and is due to

81


the poor description of off-diagonal elastic and compliance constants (see again Table

A.II.10), probably affected by temperature effects, neglected in the calculations. Such a

discrepancy, however, is fairly acceptable considering that the value of d14 (0.18 pm/V)

is quite small if compared with the other converse piezoelectric constants (d11 = 2.30

pm/V and d26 = −2d11 = -4.60 pm/V). According to the symmetry space group of

the α-quartz structure (P3221), the piezoelectric constant e26 = −e11 and d26 = −2d11;

deviations from these relations in Table A.II.11 are due to numerical accuracy in the

supercell calculations with reduced symmetry.

The comparison with experiments is much more difficult for the second end-member,

α-GeO2, due to the fact that it does not exist in nature. Synthesizing and growing pure

α-GeO2 crystals artificially is a difficult task so that the determination of its response

properties is much less accurate than for α-SiO2.(141) For instance, the reported ex-

perimental value for the static dielectric constant 033 has been indirectly calculated

from longitudinal-optical, LO, and transverse-optical, TO, Raman frequencies, among

which some are not very accurately measured (due to the presence of shoulders in the

spectrum); the least-squares procedure used to obtain the static dielectric value is very

sensitive to the starting frequency values.(97)

The results of piezoelectric coefficients concerning the two pure structures of α-

SiO2 and α-GeO2 are used to calibrate our applied computational approach, since the

corresponding experimental results are available.

As regards the compositional effect on the piezoelectric response of the Si1−xGexO2

solid solution, Table A.II.11 clearly shows that both direct and converse piezoelectric

constants regularly increase passing from pure α-SiO2 to pure α-GeO2 by progressively

substituting Si atoms with Ge atoms. This behaviour is previously suggested from an

experimental study by Ranieri et al. couple years ago for, however, very small values

of x.(130)

For the dielectric response, it is seen from the table that the nuclear contribution

to the permittivity is as large as the electronic one. The latter is small compared to

the experimental one due to the overestimation of the electronic gap as mentioned

in Section 4. The static dielectric constant values, reported in the table, are used,

via expression (I.76), for computing the electromechanical coupling coefficients (kiv)

that express the efficiency of a piezoelectric material in converting mechanical into

electrical energy. Our computed values (via the second equality in equation I.76) for

α-quartz are compared with data reported by Zeng et al.,(51) who computed them

from experimental converse piezoelectric strain, free stress dielectric and compliance

constants of Ref.(139), following the first equality in equation (I.76). Again, also the

electromechanical coupling coefficients vary linearly along the Si1−xGexO2 solid solution

series and, in particular, they increase with x.

82


Figure A.II.4: Electromechanical coupling coefficients as a function of the intertetrahedral

bridging angle θ. The experimental values for pure end members (full black diamonds) are

also reported.(130) In the experimental work, Ranieri et al. also predicted the coupling

values for some intermediate compositions with small substitutional fraction x assuming a

linear behavior; these data are reported as empty black diamonds.

In Figure A.II.4, we report the computed electromechanical coupling coefficients,

k11, k14 and k26, as a function of the intertetrahedral angle θ that, as discussed in the be-

ginning of this section, is related to the structural deformation of the Si1−xGexO2 solid

solution. From the figure, a linear behavior is observed for both k14 and k26 coefficients

while k11 shows a slight deviation from linearity while increasing the content of Ge

atoms. The experimental values for pure end members are obtained from the so-called

AT-cut(142) which almost, but not exactly, corresponds to the k11 coefficient.(130) Our

theoretical calculations can describe in a quite accurate way both the absolute values

of these coefficients and their slope with respect to θ.

83

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92

Part B

Rationalization of Piezoelectric

Property

94

INTRODUCTION

In this part we will rationalize the parameters that affect the piezoelectric response

property of a material. This means, we will search the microscopic elements which allow

to modify the piezoelectricity. Upon Equations I.80 and I.81, we will determine the

microscopic properties that could influence and improve this macroscopic piezoelectric

property.

As previously mentioned, piezoelectricity can be theoretically expressed as the sum

of electronic and nuclear terms. Each term (contribution) is given as a sum of fractional

terms (numerators and denominators), since piezoelectricity is a second-order pertur-

bation response property as polarizability α. For instance, the electronic contribution

to the piezoelectricity is given as a fractional terms where the numerators are products

of allowed transition moments due to an external electric and mechanical fields, while

the denominators are the energy differences between ground and excited spectroscopic

states, Equation I.80.

Likewise, the numerators of the fractional terms that represent the vibrational con-

tribution to the piezoelectricity should be the products of allowed transition moments

induced by vibrational motion, with respect to the electric and mechanical fields as for

the electronic contribution. The denominators will be the energy differences between

initial and final vibrational states, and so can be given as the square of harmonic mode

wavenumber, Equation I.81.

So, the current part is classified into two chapters; one for the electronic and the

other for the vibrational contributions to the piezoelectricity. The outcomes obtained

from the analysis of the two equations describing the piezoelectricity contributions will

be confirmed by applying a case study for each contribution. For the electronic one, in-

duced in-plane piezoelectricity into exotic graphene doped by BN will be stated. While,

perovskite strontium titanate, SrTiO3, example will be considered in order to show the

microscopic effects that influence the vibrational contribution to the piezoelectricity.

For a reason of clarity, the vibrational contribution will be firstly discussed, then

the electronic one.

96

I. VIBRATIONAL CONTRIBUTION TO

THE PIEZOELECTRICITY

The vibrational contribution to the piezoelectricity has been earlier described by Equa-

tion I.81, mentioned at the bibliographic part, Part A. This equation can be written

with respect to vibrational mode as follow:

evibiv ≡ 2

p

υp = 0| ∂µi

∂QpQp|υp = 1υp = 1| ∂2H

∂ηv∂QpQp|υp = 0

ωp

≡ A× B

C,

Qp is the harmonic normal mode with ωp frequency, µi the i-component of the dipole

moment, and ∂H∂ηv

the constraint related to the unit cell deformation ηv. υp = 0 is

referred to the ground vibrational state (zero point energy) where the energy of the

system is equal to (1/2)ωp. The first excited vibrational state (υp = 1) is induced

when the system absorbs a quantum of energy according to E = ω, and so, its energy

being Evib = (3/2)ω. In the harmonic case, υp = 0|Qp|υp = 1 =

ω×

(υ+1)

2,

where = 1a.u. So, the above expression will lead to Equation I.81:

p

(∂µi∂Qp

)( ∂2H

∂ηv∂Qp)

ω2p

.

As the last expression is given with respect to the harmonic approximation, only the

transition υp = 0 → υp = 1, that can be represented by Γ-point, is considered.

• A × B, are two operators running over the vibrational phonon modes, one with

respect to the electric field ( ∂µi

∂Qp) and the other with respect to the mechanical

field ( ∂2H∂ηv∂Qp

). So, the non-zero product of the multiplication A × B can be

evaluated from group theory and the symmetry of the two operators, as it will

be mentioned below.

• C, is the wavenumber of the harmonic mode, i.e. the energy difference between

the two vibrational transition states; υp = 0 and υp = 1.

1 Symmetry Conditions for Non-Zero Piezoelec-

tricity

1.1 Symmetry of Operators Associated to Electric and Me-

chanical Fields

The numerator of the last expression, as well as in Equation I.81, is the product

of allowed transition moments induced by vibrational transition (vibrational motion)

97

I. VIBRATIONAL CONTRIBUTION TO THE PIEZOELECTRICITY

with respect to the electric and mechanical fields. The vibrational transition have to be

allowed by the two applied fields as distinctly appeared from the last expression. Upon

symmetry group theory the irreducible representation of each operator, one associated

with the electric field and the other to the mechanical one, should be identical for

this transition. This means, for the same normal mode Qp, that µi and ηv must

belong to the same irreducible representation of the symmetry group. This condition

will be exclusively achieved for non-centrosymmetric systems since the dipole moment

operator associated to the electric field is anti-symmetric while the one associated with

the mechanical field is totally symmetric.

1.2 Physical Aspect for Transition Moments

The allowed transition moments induced by vibrational motion with respect to the elec-

tric field can be experimentally seen from an infrared IR-spectra, while the vibrational

contribution to elasticity can give an indication about the magnitude of transition mo-

ments induced with respect to the mechanical field. Indeed, vibration is considered as

an intra-atomic (within or inside) displacement where the atoms in unchanged unit cell

are vibrating around their equilibrium positions. Mechanical deformation is, however,

an inter-atomic (between) displacement since the atoms are moving in a deformed unit

cell (cell is deformed with respect to neighbour cells). For the vibrational contribution

to the piezoelectricity expression, Equation I.81, the term appearing at the numera-

tor ( ∂2H∂ηv∂Qp

) represents a coupling between the harmonic normal mode coordinate Qp

and the deformation ηv of the cell. So, a material having an intense IR-peak at low

wavenumber should have a large induced moment due to this transition with respect

to the electric field. This material will have a large vibrational contribution to the

piezoelectricity if and only if this transition is also allowed with respect to the me-

chanical field, i.e. if the vibration-deformation coupling is nonzero. In Figure B.I.1,

we report polarized infrared reflectivity for compressively strained (001) SrTiO3 film

and the orthorhombic (110) NdGaO3 substrate measured at different temperatures and

taken from Ref.(1). A strong infrared anisotropy was observed at the 90 - 130 cm−1

range for both (001) SrTiO3 film and (110) NdGaO3 substrates at low temperature (see

red circle in bottom and top of the figure). These perovskite materials are expected to

show a huge vibrational contribution to the piezoelectricity.

2 IR-Active Soft Mode Contribution

The vibrational contribution to the piezoelectricity in a non-centrosymmetric material

will be largely affected by the existence of a soft infrared (IR)-active mode since the

98

2. IR-Active Soft Mode Contribution

Figure B.I.1: Polarized IR reflectance at different temperatures of compressively strained

(001) SrTiO3 film on the (110) NdGaO3 substrate, image is taken from Ref.(1).

denominators of Equation I.81 are the square of this harmonic mode frequency. Soft

modes correspond to atomic displacements that require a small energy (small frequency

ωp value) and induce a large polarization (large IR-intensity Ip). This mode type can

exist in systems that show many phase transitions such as perovskites. Perovskites

have a cubic structure with the general of ABO3 formula. In this structure, an A-site

ion on the corners of the lattice is usually an alkaline earth or rare earth element. B-site

99


ions, on the center of the lattice, are 3d, 4d, or 5d transition metal elements.(2) Here,

A cations are surrounded by 12-anions in the cudo-octahedral coordination and the B

cations are surrounded by 6-anions in the octahedral coordination, see Figure B.I.2.

Figure B.I.2: The structure of an ABO3 perovskite with the origin centered at (a) the

B-site ion and (b) the A-site ion, the image is taken from Ref.(2).

At room temperature, an ideal perovskite exhibits a centro-symmetric cubic space

group Pm3m, and so cannot allow the occurrence of piezoelectricity. However, decreas-

ing the temperature leads to a phase transition that initiates atomic displacement of

B-cation through x, y, and z-directions. Hence, the mode corresponding to this vibra-

tional motion of B-cation will induce a huge polarization. SrTiO3 perovskite structure

will be here considered, where its piezoelectric properties will be discussed into some

details.

3 Strontium Titanate Perovskite Example

3.1 Phase Transition

Strontium titanate SrTiO3 is probably the most studied complex oxide perovskite of

the ABO3 family due to its many technological applications in optoelectronics, macro-

electronics and ferroelectricity (see Ref.(3) and references therein). This material ex-

hibits an impressive variety of peculiar properties: a colossal magnetoresistance,(4)

anomalously large dynamical effective charges resulting in a giant LO-TO splitting,(5)

huge zero-point motion of Ti ions,(6) giant elastic softening (superelasticity) at low

100

3. Strontium Titanate Perovskite Example

temperature,(7) extremely large dielectric constants which increase when the temper-

ature decreases,(8; 9) superlattice high-Tc superconductivity,(10) anomalous ferroelas-

ticity, (11) etc.

At room temperature, SrTiO3 crystallizes in a simple cubic structure of space group

Pm3m where each Ti ion is octahedrally coordinated to six O ions. This arrangement of

atoms shows at least two types of structural instabilities, each connected to a particular

soft phonon mode of its first Brillouin zone (BZ): a structural R-point rotation of TiO6

octahedra and a Γ-point ferroelectric displacement of Ti ions from the center of the

octahedra. On cooling, SrTiO3 undergoes a second-order antiferrodistortive (AFD)

phase transition at Ta = 105 K to a tetragonal phase with space group I4/mcm.

The tetragonal phase is characterized by static rotations of TiO6 octahedra around

the tetragonal axis c and by a slight unit-cell stretching; the crystallographic axes of

the AFD phase are rotated by 45 around the c axis of the cubic phase. Two order-

parameters are associated with this phase transition: the octahedra-rotation angle θ

(reported to be 2.1 at 4.2 K)(12) and the tetragonality of the unit cell c/a (reported to

be 1.0009 at 10 K).(13) In recent years, many theoretical investigations have helped in

clarifying specific aspects of this transition(3; 14; 15; 16; 17; 18; 19) that was recently

found to be fully describable by classical Landau theory with terms up to the sixth

order of the free energy expansion.(20)

It is known for a long time that a ferroelectric transition to a lower (unknown)

symmetry phase can be induced by applying to SrTiO3 either an electric field(21)

along c or a stress perpendicularly to c.(22; 23) It has even been reported that room-

temperature ferroelectricity can be achieved by epitaxial strain.(24) More recently,

it has been found that the transition can also be driven by doping with Ca or Bi

atoms(25; 26) and by isotope substitution: when 16O atoms are fully replaced by their18O isotopes, a ferroelectric transition occurs at Tf = 24 K.(27; 28) Many optical and

spectroscopic measurements have been performed on this ferroelectric phase SrTi18O3:

Raman,(29; 30) hyper-Raman,(31) Brillouin scattering(32) and birefringence.(33) All

these analyses reported qualitative evidences of the ferroelectric transition, confirming

a reduction of the symmetry. Evidences of structural changes are rare: only a recent

neutron scattering experiment has revealed a lowering of the symmetry to a phase that

is most likely to be orthorhombic.(34) The orthorhombic symmetry of the ferroelectric

phase of SrTi18O3 has also been supported by ab initio theoretical phonon calculations

that suggest the Ima2 space group.(35)

Before presenting our results, let us briefly describe which are the structural features

of the four models we use for SrTiO3. At room temperature, strontium titanate exhibits

a simple cubic Pm3m structure (see Figure A.II.2) whose crystallographic cell contains

five atoms and is characterized by three identical lattice parameters a = b = c = a0. Ti

101


atoms are found at the vertices of the cube, O atoms at the midpoints of cube edges and

a Sr atom occupies the cube center. Let us introduce a quadruple pseudo-cubic cell,

which will prove useful in the subsequent discussion, by doubling the lattice parameter

along z (c = 2a0) and by doubling the cell in the xy plane so that a = b =√2a0. Such

a pseudo-cubic cell is represented in Figure A.II.2 in dashed lines.

Below Ta = 105 K, SrTiO3 undergoes a transition to a tetragonal phase of I4/mcm

symmetry, whose crystallographic cell contains 20 atoms and almost coincides to the

pseudo-cubic cell, apart from a rotation of adjacent TiO6 octahedra along the z direc-

tion of an angle θ (as can be inferred by comparing Figure A.II.2-a and Figure A.II.2-c)

and a slight deviation from the pseudo-cubic ratio c/2a0 = 1. A structural parameter

u (0.25 in the pseudo-cubic structure) is considered that corresponds to the fractional

coordinate along z of the oxygen atom in the 8h Wyckoff position; its value is related

to the octahedra rotation angle θ according to the relation θ = arctan (1− 4u).

Starting from the I4/mcm tetragonal phase, the symmetry has been lowered in

order to describe a ferroelectric phase to both a tetragonal I4cm and an orthorhombic

Ima2 one. In both cases, a further structural parameter appears, |δ|, which mea-

sures the displacement of Ti atoms from their equilibrium positions in the Pm3m

and I4/mcm phases. Note that, while in the tetragonal I4cm structure Ti atoms

are symmetry-constrained to move along the z direction, in the orthorhombic Ima2

structure they can move in the xy plane, much more along the x direction than along

y.

The structural geometry parameters of the four considered SrTiO3 phases are previ-

ously discussed and compared to their experimental counterparts in Sections 2, 3, and 4,

Chapter II, Part A. We have anticipated at the beginning of this section that, due to

an anomalously large zero-point motion,(6) many peculiar properties of SrTiO3, such

as giant LO-TO splitting, giant elastic softening, colossal magnetoresistance, strongly

depend on its soft phonon modes. In particular, its giant piezoelectricity at low tem-

perature is due to Ti atoms displacements from their equilibrium positions. So, we will

firstly discuss the vibration phonon frequencies of SrTiO3 and particularly their evolu-

tion when passing from the pseudo-cubic structure to AFD I4/mcm and ferroelectric

I4cm and Ima2 ones. The vibrational contribution to the piezoelectricity is connected

to the vibrational contribution to the dielectric tensor which can be computed analyti-

cally through equation (I.94). Phonon frequencies (expressed in wave-numbers ν = ν/c,

with c the speed of light), infrared intensities Ip and vibrational contributions to the

dielectric tensor vib are reported in Table B.I.1 for the four structures considered, as

obtained with the PBE0 hybrid functional. Experimental vibration frequencies are also

reported.

102


3.2 Phonon and Dielectric Properties

For the cubic Pm3m phase, accurate measurements of the phonon frequencies are

available(36; 37) both for Γ-point and R-point phonons, which correspond to the BZ

center frequencies of the pseudo-cubic structure reported in Table B.I.1. It is seen

that, when vibration frequencies greater than 100 cm−1 are considered, the overall

agreement between computed and observed values is definitely satisfactory with an

average discrepancy of 3 cm−1 and a maximum error of 16 cm−1 for the lowest frequency

of this set. The agreement is necessarily less satisfactory as regards low-frequency soft

modes. The first mode in the list, with a calculated imaginary frequency of i47 cm−1, is

the R-point mode corresponding to the octahedra rotation. At room temperature this

frequency is small and positive (see experimental value) and it decreases by lowering the

temperature until reaching zero at Ta = 105 K;(14) the computed value are “projected”

at 0 K and then imaginary. The second and third imaginary frequencies correspond

to the ferroelectric instability due to Ti atoms displacement. The corresponding soft

modes are expected to be rather anharmonic and thus difficult to be properly described

at harmonic level. The vibrational contributions to the dielectric tensor are small (2.57

and 2.48 for the parallel and perpendicular components, respectively), compared with

the experimental average value of 310 at room temperature,(38) because the second and

third modes contributions (that would be significant due to their strong IR activity)

have to be neglected due to their imaginary computed frequencies.

When passing from the cubic to the AFD tetragonal I4/mcm phase, the agree-

ment between computed and measured phonon frequencies slightly ameliorate; high-

frequency modes remain almost unchanged while the first mode in the list (TiO6 rota-

tion) that drives the transition at T = 105 K correctly becomes positive, 59 cm−1, and

comparable with the experimental value of 44 cm−1. The two soft modes connected

to the ferroelectric instability still show imaginary frequencies thus providing a small

value for vib, as for the cubic phase.

As concerns the ferroelectric phase, recent Raman measurements on SrTi18O3 have

revealed peaks at 11, 17 and 17.5 cm−1 that correspond to a large set of very soft

phonon modes which significantly affect many properties of the system.(29) While

the second mode in Table B.I.1 is still imaginary in the tetragonal I4cm structure,

it correctly becomes small and positive, 36 cm−1, in the orthorhombic Ima2 one. As

suggested by recent neutron scattering experiments and theoretical simulations, our

phonon calculations also support the orthorhombic symmetry of the low-temperature

ferroelectric phase of SrTiO3.(34; 35) Both ferroelectric phases show a large IR activity

and, consequently, a large vibration contribution to the dielectric tensor, particularly

due to the two ferroelectric soft phonon modes (when positive). However, even for

103


Table

B.I.1

:Phon

onwave-n

umbers

ν(cm

−1),

infrared

inten

sityIp(km/m

ol)andvibration

alcon

tribution

tothedielectric

tensor

vib(for

thefirst

three

structu

res,parallel

andperp

endicu

larcom

pon

ents

referto

thezdirection

)foreach

modepofthe

fourSrT

iO3stru

ctures

consid

ered.Dash

esindicate

nullvalu

es.IR

inten

sitiesandvibratio

nalcon

tribution

sto

thedielectric

tensors

arenot

reported

forim

aginary

phon

onfreq

uencies

(crosses).Thelast

tworow

srep

ort

thetotal

vibratio

nal

andelectro

nic

contrib

ution

sto

.Valu

esob

tained

with

thePBE0hybrid

function

al.Experim

ental

values

are

from

Refs.( 36

;37;39;29).

The

symmetry

labelin

gof

themodes

accordingto

theirrep

sof

thevariou

sstru

ctures

canbefoundin

Refs.(1

7;29;

35).

Cubic

Pm3m

Tetragon

alI4/m

cmTetragon

alI4cm

Orth

orhom

bic

Ima2

νca

lcνex

pIp

vib

vib

⊥νca

lcνex

pIp

vib

vib

⊥νca

lcνex

pIp

vib

vib

⊥νca

lcνex

pIp

vib

zz

vib

yy

vib

xx

i4752

××

×59

44-

--

5244

26842

-48

44-

--

-

i4152

××

×i27

11×

××

i2214

××

×36

145760

3906-

i2290

××

×i31

90×

××

9190

5206268

-58

905454

-689

-

2790

0.45-

-50

90-

--

5690

3034-

21071

905294

--

423

129145

--

-134

144-

--

134144

0.21-

-134

1440.22

--

-

144145

--

-149

144-

--

148144

--

-149

1440.23

--

-

158170

911.46

-161

17050

0.78-

162171

255-

2.10163

171554

-3.93

4.91

172170

176-

1.38174

170119

-0.92

177171

1201.65

-174

1713.35

0.05-

-

266265

0.52-

-271

2650.24

--

271265

4.63-

0.01271

2651.01

--

-

268265

--

-272

265-

--

272265

--

-273

265-

--

-

448446

--

-446

420-

--

425420

--

-440

4203.07

--

0.01

448446

--

-447

4207.21

-0.01

447420

10.65-

0.01443

420-

--

-

457450

--

-459

450-

--

460450

--

-460

4500.06

--

-

460450

--

-461

450-

--

461450

--

-461

450-

--

-

480474

--

-460

474-

--

460517

0.03-

-490

5170.90

--

-

483474

--

-487

474-

--

511517

--

-501

5170.45

--

-

544546

7731.11

-543

5461518

-1.09

544546

1489-

1.08545

546766

-1.09

548546

1529-

1.10547

546750

1.08-

550546

8081.14

-547

546757

1.081.09

1.10

867∼

800-

--

861∼

800-

--

862∼

800-

--

865∼

8000.03

--

-

vib

2.572.48

1.862.02

312.8213.2

3907695.1

429.2

ele

4.884.88

4.884.88

4.854.88

4.884.87

4.85

104


the orthorhombic phase, where all frequencies are positive, the value of vib (3907 for

the zz component) is still very small if compared with experimental values: about

104 for both parallel and perpendicular components of SrTiO3 in the low-temperature

regime.(8; 9) This underestimation is quite expected from the analysis of equation (I.94)

where the vibration contribution to is clearly shown to be inversely proportional to

ν2p ; the largest contribution for the low-temperature phases comes from ferroelectric

soft modes with experimental frequencies of 11 and 17 cm−1. In our calculations such

phonon frequencies are either imaginary or, in the orthorhombic phase, positive but

not small enough.

3.3 Elastic and Piezoelectric Behaviour

Since, the converse piezoelectric coefficients, d, are connected to the direct one, e,

via the elastic C and compliance s tensors, so we start at first by analysing such

quantities. In Table B.I.2, we report elastic and compliance constants of SrTiO3 for

the four structures here considered as computed with the PBE0 hybrid functional.

Electronic “clamped-ion” contribution is separated from the total “relaxed” constants

which include nuclear terms. The computed bulk modulus Ks is also reported. At

a first glance, it can be noticed that the “clamped-ion” contribution is essentially

the same for every structure, with a bulk modulus always about 199 GPa. While

the nuclear relaxation contribution is almost negligible for the cubic Pm3m and AFD

tetragonal I4/mcm phases, it becomes significant for the two ferroelectric phases where

the bulk modulus decreases from 199 to 185 and 171 GPa for the tetragonal I4cm and

the orthorhombic Ima2 phase, respectively. The orthorhombic phase thus provides a

larger elastic softening, again in better agreement with experimental observations.(7)

Let us enter into more detail in order to interpret such an elastic softening: in the

tetragonal phase, the softened constant is C33 ≡ Czzzz that passes from 365 to 282 GPa

whereas in the orthorhombic phase C11 ≡ Cxxxx from 370 to 298 GPa and C22 ≡ Cyyyy

from 371 to 330 GPa. This is due to the fact that while in the tetragonal structure Ti

atoms are symmetry-constrained to move along the z direction, in the orthorhombic

structure they can move in the xy plane, much more along x than along y. The same,

even if inverse, reasoning holds true for the compliance constants.

In Table B.I.3, we report direct and converse piezoelectric constants of the two

ferroelectric structures considered of SrTiO3. As for elastic and compliance tensors,

electronic “clamped-ion” and total “relaxed”, with nuclear contribution, constants are

reported. From inspection of that table, a large effect of nuclear relaxation can easily

be inferred. Direct piezoelectric constants are as large as 8.82 and 9.28 C/m2 for the

105


tetragonal and orthorhombic phases, respectively. Such a piezoelectric response is two-

orders of magnitude higher than that of α-quartz, a standard piezoelectric material,

whose largest constant e11 is 0.15 C/m2 at room temperature and 0.07 C/m2 down to

5 K.(40)

Let us consider, first, the tetragonal phase. A relatively small constant e31 ≡ ezxx =

0.20 C/m2 measures the polarization induced along z by a strain ηxx; nuclear relaxation

doubles its value. The largest constants is e33 ≡ ezzz which passes from -0.13 to 8.82

C/m2 after relaxation. This constant gives the variation of polarization along z when

the crystal is strained along the same direction; the effect of nuclear relaxation is huge

in this case (there is even a change of sign) because it directly involves the motion of

Ti atoms along z (ferroelectric soft phonon modes described just before). When the

crystal is deformed in the yz plane, a polarization appears along y which results in the

e24 ≡ eyyz = 4.86 C/m2 constant which is also quite affected by nuclear relaxation. As

anticipated in the introduction, one of the few piezoelectric quantities already reported

in the literature so far is e33 − e31 ∼ 6 C/m2, to be compared with our value of 8.62

C/m2.(41) Converse piezoelectric constants are the result of a coupling between direct

piezoelectric and compliance constants. In particular, the high relaxed value of d33 of

37.90 pm/V with respect to the purely electronic value of -0.55 pm/V is equally due,

on the one hand, to the relaxation effect on e33 and, on the other hand, to the softening

of the C33 elastic constant upon relaxation. The high value of d24 of 40.06 pm/V is

dominated by e24 and less affected by the small softening of C44.

In the orthorhombic Ima2 structure, Ti atoms can be displaced in the xy plane

and not along the z direction; as a consequence, a larger number of high piezoelectric

constants appears. The only relatively small constant e13 = 0.20 C/m2 measures the

variation of polarization along x when the crystal is strained along z and is the analogue

of e31 = 0.20 C/m2 for the tetragonal structure. The largest constant is e11 ≡ exxx =

9.28 C/m2 because Ti atoms can mainly be displaced along x. The two constants e12and e26, which measure the polarization induced along x and y by strains along y and

in the xy plane, respectively, have similar values of 6.70 and 6.04 C/m2. As regards the

converse piezoelectricity, a very large constant d24 ≡ dyyz = 68.10 pm/V is found which

describes the strain ηyz induced in the structure by an external electric field applied

along y. The effect of nuclear relaxation on this constant is huge (from -0.69 to 68.10

pm/V) and it is due to e24 and to the peculiar softening of the C66 elastic constant,

from 126 to 89 GPa (see Table B.I.2).

In conclusion, when a deformation is applied which involves a direction along which

Ti atoms can be displaced, a large piezoelectric response (dominated by vibrational

rather electronic term) is expected to arise in SrTiO3 at low temperatures. When the

corresponding ferroelectric phonon modes become very soft (as experimentally happens

106


at very low temperatures), a giant piezoelectric response is measured.(42) It can be

seen from this case study, how a soft IR-active mode could have on the piezoelectricity

value, more precisely on the vibrational contribution to the piezoelectricty as for the

static dielectric properties.

Barium Titanate Example

Another perovskite barium titanate (BaTiO3) structure is explicitly investigated, where

the ferroelectric rhombohedral phase that could present a huge piezoelectric response, is

herein found at higher temperature conditions (below 183 K) compared to orthorhombic

one of SrTiO3. Further information can be found in our corresponding published work,

Appdx 1: BaTiO3 Piezoelectricity.

107


Table

B.I.2

:Elastic

andcom

plian

cecon

stants

ofthefou

rstru

ctures

consid

eredofSrT

iO3 .

Electron

ic“clam

ped-ion

”andtotal

“relaxed”,

with

nuclear

contrib

ution

,con

stants

arerep

orted.Thecom

puted

bulk

modulusK

sis

also

reported

.Calcu

lations

perform

edat

PBE0lev

el.

Elastic

Tensor

C(G

Pa)

Com

plian

ceTensor

s(T

Pa−1)

Ks(G

Pa)

C11

C12

C13

C22

C23

C33

C44

C55

C66

s11

s12

s13

s22

s23

S33

s44

s55

s66

Pm3m

Relax

ed370

114114

370114

370133

133133

3.16-0.75

-0.753.16

-0.753.16

7.507.50

7.50199

Clam

ped

370114

114370

114370

133133

1333.16

-0.75-0.75

3.16-0.75

3.167.50

7.507.50

199

I4/m

cm

Relax

ed371

109116

371116

366132

132125

3.13-0.68

-0.783.13

-0.783.22

7.587.58

8.00199

Clam

ped

371109

116371

116366

132132

1253.13

-0.68-0.78

3.13-0.78

3.227.58

7.588.00

199

I4cm

Relax

ed371

108112

371112

282121

121124

3.19-0.63

-1.013.19

-1.014.34

8.248.24

8.07185

Clam

ped

371109

116371

116365

130130

1253.13

-0.68-0.78

3.13-0.78

3.227.63

7.637.99

199

Ima2

Relax

ed298

60115

330117

364131

12089

3.84-0.30

-1.123.43

-1.013.42

7.638.34

11.27171

Clam

ped

370110

116371

116367

131131

1263.15

-0.69-0.77

3.14-0.77

3.217.63

7.637.95

199

Table

B.I.3

:Direct

andcon

versepiezo

electriccon

stants

ofthetw

oferro

electricstru

ctures

consid

eredof

SrT

iO3 .

Electro

nic

“clamped-ion

”an

dtotal

“relaxed”,

with

nuclear

contrib

ution

,con

stants

arerep

orted.Calcu

lation

sperfo

rmed

atPBE0lev

el.

Direct

Piezo

electricitye(C

/m2)

Con

versePiezo

electricityd(pm/V

)

e11

e31

e12

e13

e33

e24

e35

e26

d11

d31

d12

d13

d33

d24

d35

d26

I4cm

Relax

ed-

0.20-

-8.82

4.86-

--

-8.41-

-37.90

40.06-

-

Clam

ped

-0.08

--

-0.130.05

--

-0.29

--

-0.550.37

--

Ima2

Relax

ed9.28

-6.70

0.20-

-4.61

6.0433.51

-20.08

-16.70-

-38.49

68.10

Clam

ped

0.02-

-0.060.06

--

0.04-0.09

0.04-

-0.260.24

--

0.31-0.69

108

II. ELECTRONIC CONTRIBUTION TO

THE PIEZOELECTRICITY

As for the vibrational contribution, the equation describing the electronic contribution

to the piezoelectricity is previously given in Part A, Equation I.80. In the case of

unique important transition, this equation can be written as follow:

eeleiv ≡ 20|ri|nn| ∂H∂ηv

|0εn − ε0

≡×

,

ri is the position or dipole moment component associated to the electric field

in i-direction, 0 represents the initial transition state (ground state) and n is

the final (excited) state. Note that 0 and n are not just electronic transition states

but rather spectroscopic states (eigenvectors of the unperturbed Schrodinger equation).

• × are two different transition moments running over the excited states. The

non-zero product of × can be estimated from the group theory and thus, from

the symmetry of operators associated to electric and mechanical fields.

• , is the energy difference between the initial and final spectroscopic eigenstates,

and so depends on the direct band gap if the corresponding transition is allowed.

1 Symmetry Conditions for Non-Zero Piezoelec-

tricity

1.1 Symmetry of Associated Operators

As appeared from the last expression and previously from Equation I.80, the numera-

tor is the product of allowed transition moments due to electric (polarization induced

by the transition 0 → n) and mechanical ((−∂H∂η

)0→n) fields. Both transition moments

must be allowed for the same transition 0 → n in order to supply a nonzero piezoelec-

tricity components. This means that, the two operators, ri and∂H∂ηv

, must belong to

the same irreducible representation of the symmetry group of the studied system. For

a centrosymmetric material, the irreducible representations of the position operator

(vector associated to the electric field) and the deformation (matrix associated to the

109

II. ELECTRONIC CONTRIBUTION TO THE PIEZOELECTRICITY

mechanical field) are never identical for the same transition, since r is anti-symmetric

and ηv is symmetric with respect to the center of inversion. Hence, a crystal with a

center of symmetry cannot be polarized under stress, and so cannot be piezoelectric,

as previously explained for the vibrational contribution at the beginning of this Part.

Indeed, the numerators of fractional terms that express vibrational (nuclear) contribu-

tion to the piezoelectricity are almost the same as for the electronic one except that a

vibrational rather than electronic transitions are considered.

1.2 Physical interpretation for Transition Moments

Ultraviolet-visible (UV-vis) absorbption spectra and electronic contribution to elastic-

ity can give an indication about the allowed transition moments due to the electric

and mechanical fields, respectively. A material showing a high UV-vis peak (λmax) at

low wavenumber will show a large electronic transition moments due to the electric

field. However, this does not mean it will have a large electronic contribution to the

piezoelectricity, except if this transition is also allowed with respect to the mechani-

cal deformation field, and if this transition contributes considerably to the elasticity

of the material. For instance, in Figure B.II.1 we report the UV-visible spectra of

pure graphene and h-BN monolayers, as the variation of the imaginary part of po-

larizability with respect to the wavenumber hν. Graphene (CC) has a high peak for

the periodic αXX at very low wavenumber, this means that a huge polarization can

be induced in graphene due to this high polarized low energy difference (zero gap)

transition. For h-BN, this peak appears at larger frequency and its intensity is less

compared to graphene. From this figure, one can say that graphene could have a large

electronic contribution to the piezoelectricity. Actually, graphene is not piezoelectric

since it has an inversion symmetry center and then, this transition that would lead to a

huge electronic transition moments due to the electric field is not allowed with respect

to the mechanical deformation field. h-BN transition is however allowed with respect

to the mechanical field since it is non-centrosymmetric, but its electronic contribution

to the piezoelectricity is expected to be weak.

2 Transition Energy Condition for Large Piezoelec-

tricity

The denominators in the last expression, as well as in Equation I.80, are energy differ-

ences (∆ε)0→n between initial, Ψ0, and final, Ψn, spectroscopic states, and so related

to the electronic direct band gap (if the corresponding transition is allowed). Hence,

110

2. Transition Energy Condition for Large Piezoelectricity

0

100

0 2 4 6 8 10 12 14 16 18 20 22

Im(α

)

hν (eV)

BN: αxxBN: αzzCC: αxxCC: αzz

Figure B.II.1: Uv-visible spectra of pure graphene and h-BN monolayers, as the variation

of the imaginary part of the electronic uncoupled contribution (Sum Over States, SOS) of

polarizability, through periodic αXX and non-periodic αZZ directions, with respect to the

wavenumber hν.

in order to have a significant electronic contribution to the piezoelectricity, one have

to consider a non-centrosymmetric material that has a small direct band gap. Con-

ductors and semi-conductors are ideal candidates for such study. In the continuous

quest for the fabrication of nanoelectromechanical systems (NEMS) and nanoscale de-

vices, a great attention has been devoted in recent years to low-dimensional materials

due to their peculiar, highly-tunable, physico-chemical properties.(43; 44; 45) Among

other low-dimensional systems, such as nanoparticles, nanotubes, nanoribbons and

fullerenes, graphene-based(46) materials have been playing a paramount role in the

fabrication of innovative devices for electronics, optoelectronics, photonics and spin-

tronics, (47; 48; 49; 50; 51; 52; 53) due to the many extraordinary properties of the 2D

carbon allotrope: high electron-mobility, hardness and flexibility, anomalous quantum-

hall effect, zero band gap semi-metallic character, etc.(54; 55; 56) However, graphene

lacks any intrinsic piezoelectricity due to its symmetry inversion center that leads to a

zero value of the numerator as well as the denominator. Interestingly, breaking the in-

version center of graphene and keeping a small gap value should lead to a huge in-plane

piezoelectricity.

111


3 BN-Doped Graphene Example

The induction of piezoelectricity into graphene sheets would lead to a new branch of

possible applications in NEMS devices requiring high electromechanical coupling. This

can be achieved by breaking graphene symmetry center via the adsorption of atoms

on the surface of graphene,(57) hole formation,(58) application of nonhomogeneous

strain,(59) and chemical doping.(60; 61; 62) Among these strategies, chemical doping

seems the most promising as it already represents an effective experimental mean for

tuning structural and electronic properties (such as band gap and work function) of

graphene.(60; 63; 64; 65; 61)

Boron nitride (BN) chemical doping of graphene has recently been successfully

achieved in different configurations and concentrations: semiconducting atomic lay-

ers of hybrid h-BN and graphene domains have been synthesized,(63) low-pressure

chemical-vapor-deposition (CVD) synthesis of large-area few-layer BN doped graphene

(BNG) has been presented, leading to BN concentrations as high as 10%; the BN con-

tent in BNG layers has been discussed to be related to the heating temperature of the

precursor, as confirmed by X-ray photoelectron spectroscopy measurements.(64) The

synthesis of a quasi-freestanding BNG monolayer heterostructure, with preferred zigzag

type boundary, on a weakly coupled Ir-surface has also been recently reported.(65)

So, we will here show how, by doping graphene with BN inclusions arranged ac-

cording to different patterns and exploring different substitutional fractions x, a piezo-

electricity can be induced in 2D graphene which is found to be 3 to 4 times larger

than pure 2D BN monolayer and one order of magnitude larger than previously re-

ported on graphene. Carbon pairs are substituted with BN pairs so as to reduce point

symmetry from the centrosymmetric D6h to the non-centrosymmetric D3h group. The

full set of piezoelectric constants (elements of the third-order direct, e, and converse,

d, piezoelectric tensors) and elastic constants (elements of the fourth-rank elastic, C,

and compliance, s, tensors) of all configurations is determined. Both electronic and

nuclear-relaxation contributions to the piezoelectric and elastic response of BNG are

explicitly taken into account.

We consider graphene embedded with periodic arrangements of zigzag-edged

hexagonal (BN)3 which have recently been shown to be more stable than other

arrangements.(66) Different configurations are explored where the size of the BN rings

and their separation are changed. In order to unambiguously label each configura-

tion, we adopt a notation first introduced to study graphene antidot lattices and then

extended to BNG structures.(67) Each configuration is labeled by a pair of integer

indices within brackets, (R,W ). R represents the “radius” of the BN rings (measured

in units as the number of hexagonal carbon chains substituted with BN atoms) or,

112

3. BN-Doped Graphene Example

Figure B.II.2: BN-doped piezoelectric graphene in different configurations (R,W ). The

radius R of BN rings and the wall width W separating them are graphically defined. The

unit cell of each configuration is shown as thick black lines.

equivalently, the number of BN substituted atoms along each of the six sides of the BN

hexagons. The separation, or “wall width”, between neighboring BNs is represented by

the integer W corresponding to the number of non-substituted carbon chains between

neighboring BN hexagons. These two indices are graphically defined in Figure B.II.2,

where the (1, 5) and (2, 3) configurations are taken as representatives of two classes of

structures with different size. We anticipate that by increasing R (i.e. by increasing

the substitutional fraction of BN), the piezoelectric response of BNG decreases. The

structural parameter that effectively allows for fine-tuning BNG response properties is,

indeed, W , as already noticed for the electronic band gap,(67) and for this reason it

will be explicitly investigated in the following discussion.

3.1 Band Gap of BN-doped Graphene

Before illustrating the dependence on W of the elastic and piezoelectric response of

BNG structures, we discuss its effect on energetic and electronic properties, such as the

cohesive energy ∆E, the energy of mixing ∆Em and the electronic band gap Eg. All

these quantities are reported, for a series of (1,W ) BNG structures, in the two panels of

Figure B.II.3 as a function of an integer index p = 2R+W which measures the length of

the lattice parameters (kept equal to each other) in units of the total number of atomic

C-C chains contained in the unit cell (see Figure B.II.2 for a graphical interpretation

113


of this index). Note that p is related to the BN substitutional fraction x according to

x = 3/p2 for (1,W ) BNG series. This relation can easily be understood by referring,

for instance, to the two panels of Figure B.II.2, where it is seen that the unit cell

contains a total of 2p2 atoms, out of which 6 × R2 are BN-substituted. This gives a

substitutional fraction x = 6 × R2/2p2, which reduces to x = 3/p2 when R = 1. The

inset of the lower panel of the figure shows the effect of increasing the “wall width”

W on the structure of BNG and on the size of the corresponding unit cell for R=1.

The cohesive energy of each BNG structure (solid red squares in the upper panel of

the figure) is defined as: ∆E = (EBNG − NBEB − NNEN − NCEC)/(NB + NN + NC),

where EBNG is the energy per unit cell of the BNG configuration, EB, EN and EC

the energies of free B, N and C atoms, and NB, NN and NC the number of B, N and

C atoms per unit cell. The cohesive energy of pristine h-BN is -7.66 eV/atom and

remains negative for all BNG structures as a function of p, thus reflecting their strong

stability. As p increases (i.e. as the BN concentration decreases), ∆E regularly tends

to the pristine graphene limit of -6.29 eV/atom (represented as a horizontal dashed line

in the figure). The energy of mixing of a (BN)xG1−x structure, with x substitutional

fraction of BN for carbon pairs, ∆Em = E(BN)xG1−x− [xEBN+(1− x)EG], is a measure

of how favorable the formation of a BNG structure is with respect to isolated pristine

h-BN and graphene. Static computed values are reported, as empty blue squares in

the upper panel of Figure B.II.3. Given that entropic thermal terms are not explicitly

accounted, they are always positive but rather small, the maximum value being 0.44 eV

for the (1,1) case (i.e. p=3, x=0.33). ∆Em then progressively decreases as a function

of p; according to this merely electronic picture, BN-doped graphene is predicted to be

more stable than carbon-doped h-BN with respect to the separate phases of h-BN and

graphene.

The computed value of the band gap of pristine h-BN is here 6.73 eV with exper-

imental values usually found in the relatively wide range 4.6 < Eg ≤ 7 eV.(68; 69)

Electronic energy gaps of (1,W ) BNG structures are reported as a function of p in the

lower panel of Figure B.II.3. Different symbols are used for cases where p is a multi-

ple of 3 (empty blue circles) and where it is not (full red circles). Overall, the band

gap decreases as the concentration of BN decreases, as expected, but with different

steepness in the two cases so that the dependence of Eg on p appears to be oscillating,

where it appears to be linear with respect to BN concentration x by separating the

two p cases (see Figure B.II.4). The different behavior of these two classes (multiples

of 3 or not of the primitive cell of the pristine system) of graphene superlattices is now

well-understood in terms of the energy band-folding model.(70; 71; 72) Indeed, even

in pure graphene, when p is a multiple of 3, the two Dirac points K and K in the

primitive cell are folded to the Γ point of the hexagonal first-Brillouin zone (BZ) of the

114


Figure B.II.3: Energetic and electronic properties of (1,W ) BNG structures as a function

of p = 2R + W . Upper panel: cohesive energy ∆E (the graphene, p → ∞, limit is given

by the horizontal line) and energy of mixing ∆Em (zero by definition at p=0 and p=∞).

Lower panel: evolution of the electronic band gap Eg; red and blue lines are just meant as

eye-guides. The inset shows the structure of (1,W ) BNG configurations as p increases.

superlattice, giving rise to a fourfold degeneracy that can be broken, opening a band

gap, by a periodic arrangement of defects. In the other case, the twofold degenerate

Dirac points do not fold into Γ and a band gap opening can be induced by breaking

the inversion symmetry.(70) We will discuss below how these two cases can be clearly

discriminated also from the analysis of the elastic and piezoelectric response of BNG

structures. Band structures of pristine graphene and h-BN, and (1,W ) BNG struc-

tures with different BN concentrations (i.e. different p values) are reported in Figure

B.II.5. Panels (a) and (b) clearly show the occurrence of a zero and non-zero band gap

in graphene and h-BN, respectively, at point K of the BZ (schematically represented

115


0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.05 0.10 0.15

Eg (

eV

)

x

Multiple of 3Non multiple of 3

a1 = 8.257 +/- 0.136

a0 = 0.00692 +/- 0.00631

b1 = 4.109 +/- 0.025

b0 = -0.00048 +/- 0.00154

Figure B.II.4: Convergence of the band gap Eg (in eV) of BNG patterns to the graphene

limit (x = 0) as a function of the BN concentration x (or 3/p2). Values for the two cases of p,

multiple or not of 3, are represented by empty and full circles, respectively. The parameters

given in the insets are obtained with the fitting functions Eg = a0 + a1x, and Eg = b0 +

b1x, for the two cases of p, where the highest concentration points (x=0.33, and 0.19) are

excluded.

in blue lines). The area of Dirac conical intersection between valence and conduction

bands in graphene is highlighted in yellow whereas the highest occupied (HOCO) and

lowest unoccupied (LUCO) crystalline orbital energy levels are represented as dashed

green and red lines, respectively. Panels (c) and (d) clearly show the opening of a

direct band gap Eg at the Γ point for BNG configurations with p multiple of 3 (p=3

corresponding to a (1,1) and p=6 to a (1,4) structure). On the contrary, panels (e)

and (f) show how the direct band gap Eg for cases where p is not a multiple of 3

(p=4 and p=5 corresponding to (1,2) and (1,3) structures) is opened at the K point

of the BZ. Again, it is clearly seen that, as p increases (concentration decreases), Eg

systematically decreases and the band gap oscillation for the two classes (multiples of

3 or not) decreases as well leading to a linear behaviour between the gap value and

116


Figure B.II.5: Energy band structure of (a) pristine graphene, (b) pristine h-BN, (c-d)

BNG (1,W ) structures with p multiple of 3, (e-f) BNG (1,W ) structures with p not multiple

of 3. HOCO and LUCO energy levels are drawn as dashed green and red lines, respectively.

117


the concentration, as previously seen from Figure B.II.4. To avoid separating the ob-

tained results, the evaluation of piezoelectricity as a function of concentration x will

be considered.

3.2 Elastic and Piezoelectric Properties

Direct and converse piezoelectric tensors are connected via elastic stiffness and com-

pliance tensors; for this reason, before illustrating the piezoelectric response of BNG

structures, let us discuss their elastic behavior. According to Voigt’s notation,(73) the

elastic tensor C of a 2D system can be represented in terms of a 3×3 matrix whose

elements (i.e. the elastic stiffness constants) are defined as Cvu = 1/S[∂2H/(∂ηv∂ηu)]

where S is the area of the 2D cell, H the total energy per cell, η is the strain tensor

and v, u = 1, 2, 6 (1=xx, 2=yy and 6=xy). The compliance tensor is simply obtained

by inverting the stiffness tensor: s = C−1. Given their symmetry invariance with re-

spect to all the operators of the point group of the system, both tensors exhibit just

two symmetry-independent constants: C11, C12 and s11, s12. From the knowledge of

the full elastic tensor, a number of elastic properties can be derived, such as bulk,

shear and Young’s moduli, Poisson’s ratio, seismic wave velocities, etc.(73; 74; 75; 76)

For 2D systems, the Young’s modulus is given by YS = (C211 − C2

12)/C11, Poisson’s

ratio by ν = C12/C11 and bulk modulus by KS = (C11 + C12)/2 and corresponds to

KS = S(∂2H/∂S2).

All these quantities are given in Table B.II.1 for pristine graphene and h-BN, and

for (BN)xG1−x structures of intermediate compositions: from (1,1) where x=0.33 to

(1,8) where x=0.03. Total values are reported along with nuclear relaxation contribu-

tions (in parentheses). Few elastic properties have been experimentally determined for

graphene by nanoindentation in an atomic force microscope: its C11 stiffness constant

and its Young’s modulus.(77) Computed counterparts do agree with these determina-

tions, particularly so as regards their relative values, C11 being slightly larger than

YS. So far, no experimental determinations of such properties have been reported

for free-standing monolayer h-BN. Previous DFT calculations report KS values of 160

N/m with local-density(78) (LDA) and 179 N/m with generalized-gradient(76) (PBE)

approximations. Our computed value, at B3LYP hybrid level, is 185.3 N/m with a

Poisson’s ratio ν=0.210 which measures the induced deformation orthogonally to the

applied strain and nicely compares with a previous determination of 0.218.(76) As re-

gards intermediate compositions, a rather regular trend is observed in passing from

h-BN to pure graphene for all elastic properties, again with some oscillations corre-

sponding to BNG structures with p multiple of 3, such as (1,1), (1,4) and (1,7). The

118


Table B.II.1: Elastic properties of (BN)xG1−x structures in the whole composition range.

Elastic stiffness constants, Cvu, Young’s modulus, YS , and bulk modulus, KS , are given

in N/m. Compliance constants, svu, are given in 10−3 m/N while Poisson’s ratio, ν, is

dimensionless. Total values are reported along with purely nuclear relaxation effects (in

parentheses) using the B3LYP level.

C11 C12 s11 s12 YS KS ν

GExp.(77) 340±50 335±33

G 372.34 69.47 2.78 -0.52 359.38 220.91 0.187

(-6.56) (6.01) (0.06) (-0.07) (-8.89) (-0.28) (0.019)

(1,8) 371.30 67.06 2.78 -0.50 359.19 219.18 0.181

(-7.22) (6.43) (0.07) (-0.07) (-9.62) (-0.39) (0.020)

(1,7) 371.40 66.17 2.78 -0.50 359.61 218.79 0.178

(-7.69) (6.79) (0.08) (-0.08) (-10.18) (-0.45) (0.022)

(1,6) 369.60 66.97 2.80 -0.51 357.47 218.29 0.181

(-7.49) (6.52) (0.08) (-0.07) (-9.93) (-0.48) (0.021)

(1,5) 368.22 66.82 2.81 -0.51 356.09 217.52 0.181

(-7.70) (6.58) (0.08) (-0.07) (-10.17) (-0.56) (0.021)

(1,4) 367.52 64.91 2.81 -0.50 356.06 216.22 0.177

(-9.40) (7.47) (0.09) (-0.09) (-11.74) (-0.79) (0.024)

(1,3) 362.40 66.39 2.86 -0.52 350.24 214.40 0.183

(-8.41) (6.80) (0.09) (-0.08) (-11.00) (-0.81) (0.022)

(1,2) 355.89 65.77 2.91 -0.54 343.74 210.83 0.185

(-9.20) (6.94) (0.10) (-0.09) (-11.87) (-1.13) (0.024)

(1,1) 343.10 61.00 3.01 -0.54 332.25 202.04 0.178

(-13.29) (8.15) (0.14) (-0.11) (-16.30) (-2.57) (0.029)

h-BN 306.39 64.26 3.41 -0.72 292.91 185.33 0.210

(-9.92) (9.65) (0.15) (-0.16) (-13.97) (-0.14) (0.037)

nuclear relaxation effect is seen to be quite small in all cases if compared with the elec-

tronic one: it never counts more than 3.8% for the dominant C11 constant, which occurs

119


Figure B.II.6: Upper panel: direct piezoelectric constant e11 of BNG structures as a

function of -ln(x) where x is the BN substitution fraction; total (full symbols) and purely

electronic (empty symbols) values are reported for (1,W ) BNGs (circles), (2,W ) BNGs

(triangles) and pure h-BN (squares). Purely electronic and total data are fitted with a

function a + b/(−ln(x)). Lower panel: nuclear relaxation effect on the e11 piezoelectric

constant, ∆e11 = etot11 − eele11 as a function of -ln(x).

for the (1,1) BNG structure of highest BN concentration. Then, it rapidly decreases

to about 2% for BN concentrations below 5% and it remains almost constant down

to pristine graphene where it amounts to 1.8%. An analogous behavior is observed

also for the compliance constants. The vibrational contribution in pure graphene can

be interpreted by considering the elastic deformation as a movement of carbon atoms

in collective modes. This movement would lead to a polarization change which, how-

ever, is canceled by the inversion symmetry center of D6h graphene, as we will mention

below.

Pristine graphene, belonging to the D6h symmetry point group, exhibits an inver-

sion center which prevents it to be piezoelectric. BNG structures are here considered

with periodic arrangements of BN domains which break such an inversion and reduce

the symmetry to D3h, a point group that enables a piezoelectric response. A Berry-

120


phase approach is adopted that computes the direct piezoelectric constants eiv where

i = x, y represents the in-plane Cartesian component of the polarization and v is an

index representing the applied strain in Voigt’s notation. Given the D3h symmetry,

a single constant is enough for describing the whole in-plane anisotropic piezoelectric

response of BNG structures: e11 = −e12 = −e26, the other constants being null by

symmetry. The converse piezoelectric effect (represented by the d11 constant) is then

described by coupling the direct one with the compliance response. For this reason,

we discuss the effect of different BN patterns and concentrations, and of the nuclear

relaxation, on the direct e11 constant (see Figure B.II.6). In the upper panel, the direct

piezoelectric constant is reported as a function of BN doping concentration x, as ob-

tained by including (full symbols) or not (empty symbols) the nuclear relaxation effect,

for pure h-BN (squares), (2,W ) BNGs (triangles) and (1,W ) BNGs (circles). We no-

tice that: (i) the computed value of e11 for pure h-BN, 1.20×10−10 C/m, is remarkably

close to a previous theoretical determination: 1.19×10−10 C/m;(79) (ii) as discussed

for the electronic band gap Eg, at relatively high BN concentrations, the electronic

term of the piezoelectric response shows two distinct behaviors depending on p being

or not a multiple of 3; (iii) the two different BN patterns provide an almost identical

piezoelectric response as a function of BN concentration x (see the green arrows in the

figure that mark compositions where (1,W ) and (2,W ) give exactly the same piezo-

electric response). In the lower panel of the figure, the nuclear relaxation contribution

is reported. We see that: (i) it systematically reduces the purely electronic piezoelectric

response; (ii) it is rather large for pristine h-BN (-2.1×10−10 C/m, corresponding to

60% of the electronic term), it progressively reduces as the BN concentration decreases

and it becomes practically constant as x becomes lower than 10% (about -1.0×10−10

C/m, corresponding to just 20% and 16% of the electronic contribution for p multiple

or not of 3, respectively). The same trend has already been discussed for elasticity. An

explanation for such a behavior can be demonstrated as due to soft (low-frequency) col-

lective modes of small infrared (IR) intensity, see Figure B.II.7 that represnsts the IR

spectrum of large BNG supercells. The vibrational contribution to the elastic constant

of symmetry-broken graphene (D3h) is principally due to smooth collective E -modes

that can be observed in the infrared spectrum of a large supercell. In the same way,

these smooth modes are responsible of the non vanishing vibrational correction to the

piezoelectricity of the (symmetry-broken) BN-doped graphene when the concentration

x of BN goes to zero. In Figure B.II.7, we can see that the number of smooth modes

increases from pure BN (x = 1 or p = 1) to pure graphene (x = 0 or p → ∞) while

the mean IR intensity over the whole spectrum generally decreases. The frequency and

intensity (x,y) for the more and less intense modes are displayed in that figure in order

to show the decreasing in intensity for whole structures.

121


0

1

2

0 200 400 600 800 1000 1200 1400 1600

hν (cm-1

)

(265,0.2)

(294,0.1)(237,0.4)(549,0.2)

2

22

42

62

82

102

(902,20)

(1625,87)

100

200

300

400

500

600

700

800

900

1000

1100

IR inte

nsity (

km

.mol.

-1)

(1371,1056)

(1610,432)

(1599,376)

(1630,144)

BN(1,4)(1,5)(1,6)(1,7)

G

Figure B.II.7: Infrared (IR) spectrum, as the plot of IR-intensity in km/mol per atom

as a function of frequency for the IR-active modes. The modes with intensity less than 0.1

km/mol are excluded as a reason of clarity. Data for pristine h-BN and graphene (unit cell),

and (1,4), (1,5), (1,6), and (1,7) BNG are included. The frequency and intensity (x,y) for

the more and less intense modes for each structure are also displayed.

Another important finding, represented in Figure B.II.6, is that, for BN concentra-

tions below about 33%, the piezoelectric response of BN-doped graphene is found to

be essentially constant for p not multiple of 3, rapidly converging to the limit value of

about 4.5×10−10 C/m as x decreases. For p multiple of 3, the convergence is slower

and still not completely reached at p = 12 (the purely electronic contribution to e11

still increases to 4.45×10−10 C/m). However, the same piezoelectricity value as the

converged one for the p not multiple of 3 series should be obtained at the p → ∞ limit

(i.e. the piezoelectricity value of inversion symmetry-broken graphene). A fit of purely

122


electronic contributions leads to a limit value of about 5.5×10−10 C/m at infinite BN

dilution which would imply a corresponding value of approximatively 4.5×10−10 C/m

for the total piezoelectricity. The finite and constant piezoelectric response of D3h

graphene at infinite defect dilution can be explained as follows: as the BN concentra-

tion decreases, on the one hand the narrowing of the electronic band gap Eg would

lead to an infinite piezoelectric response whereas, on the other hand the reduction of

the degree of symmetry inversion breaking with respect to pure graphene would lead

to a zero piezoelectricity. As a result of the balance between these two limits, BNG

structures are found to exhibit a non-null, non-infinite, constant piezoelectric response.

Both compensating effects (i.e. vanishing band gap and recovering of inversion symme-

try) are essentially intrinsic of graphene and do not depend on the particular physical

or chemical nature of the inversion symmetry breaking defects.

Figure B.II.8: The variation of in-plane piezoelectricity as a function of electronic band

gap − ln(Eg) for the series (1,W ) of BN-doped graphene

As shown previously in Figure B.II.4, the band gap varies linearly as a function of

the concentration x, especially when separating the two classes: p multiple of 3 or not,

for higher concentrations. The evaluation of the piezoelectricity as a function of the

band gap is additionally introduced in Figure B.II.8 for the series (1,W ) BN-doped

graphene. At higher concentrations (p < 9), the band gap is affected by the configu-

ration of BN-dopant in graphene. However, at lower concentrations an evident similar

123


behaviour is obtained as for the variation of piezoelectricity with the concentration x

that is given in Figure B.II.6.

3.3 Independence of Piezoelectricity from DFT Functional

Commonly, the band gap is sensitive to the choice of Hamiltonian, however we show

here the piezoelectricity will not. In Table B.II.2, we report the numerical values of both

direct and converse piezoelectric constants of graphene, h-BN, and some of (1,W ) BNG

series, as a function of band gap Eg. Pure DFT/LDA and B3LYP Hamiltonains are

considered. Piezoelectric total values are reported along with purely nuclear relaxation

effects (in parentheses). Although LDA underestimates the band gap by, at least 40%

for the structure (1,2), the piezoelectric constants are practically equivalent to that

of B3LYP, particularly for the case where p is not a multiple of 3. This means that

the Hamiltonian affects equally both compensating effects, and impacts the numerator

evenly as the denominator in the calculation of piezoelectricity of graphene.

124


Table B.II.2: Direct and converse piezoelectric coefficients, e11 and d11, as calculated by

LDA/DFT and B3LYP levels of theory, and as a function of band gap Eg (in eV). Values

reported are in 10−10C/m for piezoelectric stress coefficient e11 and pm/V for piezoelectric

strain coefficient d11. Total values are reported along with purely nuclear relaxation effects

(in parentheses).

B3LYP LDA

system Eg e11 d11 Eg e11 d11

h-BN 6.73 1.20 0.50 4.86 0.92 0.39

(-2.07) (-0.75) (-2.57) (-0.95)

(1,0) 3.95 1.82 0.73 2.65 1.50 0.62

(-1.82) (-0.60) (-2.24) (-0.75)

(1,1) 2.52 2.31 0.82 2.65 2.30 0.85

(-1.33) (-0.38) (-1.69) (-0.48)

(1,2) 0.71 4.44 1.53 0.42 3.94 1.40

(-1.32) (-0.35) (-1.84) (-0.49)

(1,3) 0.49 4.52 1.53 0.29 4.18 1.45

(-1.22) (-0.32) (-1.67) (-0.44)

(1,4) 0.69 2.31 0.76 0.38 2.52 0.86

(-0.84) (-0.23) (-1.23) (-0.33)

(1,5) 0.25 4.72 1.57 0.15 4.42 1.51

(-1.08) (-0.27) (-1.45) (-0.40)

125


Conclusion

In this part, we have rationalized the microscopic parameters that affect the macro-

scopic piezoelectric property. The two contributions to the piezoelectricity, electronic

and vibrational, are explicitly analysed. In order to have a large vibrational contribu-

tion to the piezoelectricity, a non-centrosymmetric material that has a good IR-spectra

(an IR-active soft phonon mode) with a nonzero vibration-deformation coupling should

be considered. SrTiO3 and BaTiO3 perovskites are considered as perfect candidates for

examining such behaviour due to their phase transitions induced by the anomalously

large zero-point motion of Ti atoms at low temperature. This ferroelectric phase of

those perovskite has a giant piezoelectric effect at however very low temperature.

On the other hand, a non-centrosymmetric material that has a good UV-vis spectra

(an intense peak at low wavenumber), a nonzero transition moments with respect

to mechanical deformation (good electronic contribution to elasticity), and a small

electronic band gap, will show a large electronic contribution to the piezoelectricity.

In this respect, graphene semiconducting materials are considered to investigate this

attitude. Since graphene exerts an inversion symmetry center, it is intrinsically non-

piezoelectric, and so, the doping with different patterns and concentrations of BN is

attempted in order to break the inversion symmetry center in graphene. We have shown

that an in-plane piezoelectricity can be induced in this non-piezoelectric graphene by

breaking its inversion center, where the piezoelectric response (direct and converse) is

dominated by the electronic rather than nuclear term.

126

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131

Part C

Design of Materials for Large

Piezoelectricity: Application to

Graphene

132

INTRODUCTION

After calibration of the computational method, Section 5, Part A, and upon rational-

ization of the piezoelectric property, Part B, we will in this part design some materials

that can exert a significant piezoelectric effect. As shown in the last part, decreasing

the band gap is the way to obtain large electronic contribution to the piezoelec-

tricity, while increasing the mode softness (decreasing the frequency and increasing

IR-intensity of the mode) is the way to get major vibrational contribution to the

piezoelectricity.

We have recorded that the vibrational contribution leads always to an overall huge

piezoelectric coefficients rather than the electronic one. For example: the relaxed

piezoelectric response for the ferroelectric SrTiO3 phase, that is here predominated by

vibrational contribution, is very large compared to that induced in symmetry-breaking-

graphene and predominated by electronic contribution, regardless that different com-

ponents are considered. However, SrTiO3 ferroelectric phase that exerts an infinite

piezoelectric effect due to the existence of a soft mode, occurs only at very low tem-

perature (below 24 K) as previously shown. Even for the BaTiO3 ferroelectric phase

that may occur at quite higher temperature (below 187 K), it is still a challenge for the

practical synthesis. So, we will here search for a material with a piezoelectric response

predominated by the nuclear terms and experimentally accessible. For this purpose,

functionalized graphene by some defects that could break its planarity and realize the

ill-defined out-of-plane graphene macroscopic properties will be studied. Several simple

substitutions are considered where some C atoms in graphene are replaced by heav-

ier group-IV elements (Si, Ge and Sn). A more complex functionalization (namely,

pyrrolic N-doped graphene) is also investigated where different functional groups, such

as F, Cl, H3C and H2N, are studied. The piezoelectric effect in that case is found to

be dominated by the vibrational contribution.

Before discussing into details the out-of-plane piezoelectricity induced in graphene,

we will firstly emphasize our main outcomes from BN-doped graphene study of the

independence of in-plane piezoelectricity from the nature of defect that breaks graphene

inversion symmetry center, as well as the finite limit of in-plane piezoelectricity in

graphene that can be obtained with any in-plane defect.

134

I. INDEPENDANCE OF IN-PLANE

PIEZOELECTRICITY FROM DEFECT

NATURE IN GRAPHENE

In order to confirm that the induced in-plane piezoelectricity in graphene will not de-

pend on the particular physical or chemical nature of the inversion symmetry breaking

defects, and will tend towards a finite limit at infinite defect dilution, as shown from

BNG (BN-doped graphene), we here introduce different other defects.

1 Holes in Graphene

Two types of graphene holes are considered: D3h and C2v. A graphical representation

of the resulting structure of the perturbed graphene sheet in the two cases is given in

Figure C.I.1.

Figure C.I.1: Graphical representation of the structure of two defects, one of D3h and one

of C2v point-symmetry, of graphene (holes in this case) that break its inversion symmetry.

The unit cell of the p=4 case is sketched in both cases with black continuous lines.

For each type of holes, a series of structures corresponding to different defect con-

centrations are considered for which the purely electronic contribution of the in-plane

piezoelectric response, eele11 constant, is evaluated. These results are reported in Figure

135

I. INDEPENDANCE OF IN-PLANE PIEZOELECTRICITY FROMDEFECT NATURE IN GRAPHENE

Figure C.I.2: Dependence of the in-plane direct piezoelectric constant of graphene, eele11

(purely electronic contribution), on defect concentration x. Four different defects are con-

sidered: (1,W ) BNGs (full red circles), (2,W ) BNGs (empty black circles), D3h holes (full

blue triangles) and C2v holes (empty magenta triangles). For (1,W ) BNGs and D3h holes

the fitting a+ b/(−ln(x)) is also reported. All results are obtained at the B3LYP level. For

(1,W ) BNGs, a LDA result is also shown (red empty circle) at lower defect concentration.

C.I.2 along with the corresponding ones of the two series of (1,W ) and (2,W ) BN-

doped structures. From the analysis of the figure, it turns out that, although with

different steepness and behaviors, all defects induce a large piezoelectric response in

graphene essentially converging to the same value as the dilution of the defects in-

creases. For the two most regularly converging series ((1,W ) BNGs and D3h holes), a

fitting a + b/(−ln(x)) is reported which highlights the common piezoelectric response

in the limit of low defect concentration. The piezoelectricity value of each largest

supercell system is found in the range: eele11=5.6±0.4×10−10 C/m.

2 Si, Ge, and Sn doped Graphene

Substituting carbon atoms in graphene by IV-group elements with a larger size as

Si, Ge, or Sn, will break graphene center of symmetry as well as graphene planarity,

since the bond length has to be increased by 18, 23, or 50%, respectively, see Fig-

136

2. Si, Ge, and Sn doped Graphene

Figure C.I.3: Graphical representation for Si, Ge, and Sn doped graphene is given in

panel (a). In panel (b), the variation of substitutional fraction x of dopant in graphene

sheet is shown for Si-doped graphene, for instance. The unit cell used in the calculations is

highlighted.

ure C.I.3. This will induce both in-plane and out-of-plane piezoelectricity in graphene.

Before discussing the induced out-of-plane response, that will be considered for the

next chapter, we will introduce the results of in-plane response in order to confirm our

prior “universal” behaviour of finite limit at low defect concentration.

Si, Ge, and Sn doped graphenes have Cs symmetry, where the piezoelectricity ten-

sor is described by 5 components. However, the anisotropy is quite small and the

difference between the component values is less than 1% for almost cases. Thus, we

will here consider e11 −e12 −e26, and e31 e32 for those graphene structures. The

induced in-plane piezoelectric response in all of our simulated structures tends toward

a unique value of ∼ 5 × 10−10 C/m, in the limit of pure symmetry-broken graphene, as

reported in our previous study.(1) Direct and converse in-plane piezoelectric constants

for SiG (Si-doped graphene) as a function of Si-substitutional fraction x, are reported

in Table C.I.1.The variation of Si-concentration, x, in graphene sheet is represented

in Figure C.I.3, where x = 1/(2×p2) and p is an integer that measures the length of

137


Table C.I.1: In-plane direct and converse piezoelectric constants of SiG as a function

of substitutional fraction x. Electronic “clamped-ion” and total “relaxed”, with nuclear

contribution, constants are reported. e11 is given in 2-dimensional unit of 10−10C/m, and

d11 in pm/V. Calculations performed at B3LYP level.

clamped relaxed

x e11 d11 e11 d11

0.125 5.24 2.09 4.01 1.94

0.056 6.45 2.30 5.45 1.90

0.031 6.27 2.12 4.37 1.67

0.020 6.16 2.03 4.97 1.92

the lattice parameters (kept almost equal for both lattice parameters a and b) in units

of the total number of atomic C-C chains contained in the unit cell. The vibrational

contribution (relaxed - clamped, Table C.I.1) to the direct in-plane piezoelectric effect

is approximately constant, −1 × 10−10 C/m, as for BN-doped graphene (BNG). The

converse piezoelectric value is around 1.9 pm/V for x larger than 0.02, which is close

to the value of the infinite diluted BN-doped graphene, 1.5 pm/V. This can be ex-

plained from the elastic behaviour, see Table C.I.2. SiG is more rigid (smaller elastic,

larger compliance constants) than BNG, which increases the d11 component by more

than 25%. However, at the infinite Si-dilution, the in-plane response property will

depend on graphene only, and the same infinite diluted BN-doped graphene value will

be obtained.

In Table C.I.3, we report the in-plane piezoelectric constants for Ge- and Sn- doped

graphene as a function of the substitutional fraction x, where the same limit is obtained

for small x value. As mentioned before, the in-plane piezoelectric effect in graphene is

dominated by the electronic contribution, and a common limit value of ∼ 5 × 10−10

C/m (for low defect concentration) is obtained regardless of the particular chemical or

physical nature of the defects.

3 Finite Limit of In-Plane Piezoelectricity in

Graphene

Here, we explain with respect to Equation I.80 the convergence of in-plane piezoelec-

tricity induced in graphene to a finite limit. The numerator of the equation depends on

138

3. Finite Limit of In-Plane Piezoelectricity in Graphene

Table C.I.2: Elastic behaviour as a function of concentration fraction x for Si-doped

graphene sheets. Elastic and compliance constants are given in N/m and 10−3 m/N, respec-

tively.

Substitution fraction x

0.125 0.056 0.031 0.020

ele. relax. ele. relax. ele. relax. ele. relax.

Elastic constants

C11 311 251 341 252 356 274 365 278

C22 311 252 341 252 357 274 364 281

C12 60 44 61 -0.8 60 12 61 19

C66 125 104 140 126 148 131 152 131

Compliance constants

s11 3.34 4.11 3.03 3.97 2.89 3.66 2.82 3.61

s22 3.35 4.10 3.03 3.97 2.89 3.65 2.83 3.57

s12 -0.65 -0.73 -0.54 0.01 -0.49 -0.16 -0.47 -0.24

s66 8.02 9.63 7.14 7.91 6.76 7.62 6.60 7.63

the symmetry of the structure while the denominator is related to the electronic band

gap. For pristine graphene, both the numerator and the denominator would obviously

be null. Indeed, Ψ0 is symmetric (S) with respect to the inversion center while the

dipole moment operator is not: the dipole moment transition (µi)0→n would then be

allowed just for asymmetric (A) final states Ψn. However, since the − ∂H∂ηv

operator is

symmetric (S), (− ∂H∂ηj

)0→n = 0 would be zero in those cases. It follows that breaking

the inversion symmetry of graphene is necessary in order to generate a piezoelectric

effect (even a giant one, in principle, if only an infinitesimal energy band gap value was

kept).

Let us discuss what happens when an asymmetric (A) perturbation operator P is

included into the Hamiltonian operator of a system described by twofold degenerated

ΨS and ΨA wave functions of (S) and (A) symmetry, respectively. The asymmetric

perturbation can be considered as any physical or chemical modification of graphene

that breaks the inversion symmetry. After diagonalization of the Hamiltonian in this

(ΨS,ΨA) basis set, the gap between the two new wave functions: ΨS+ΨA and ΨS−ΨA

will now be equal to 2P , leading particularly to a P -independent polarization of the

lowest-energy ΨS + ΨA eigenvector, which will be equal to the non-zero unperturbed

transition moment µS→A, particularly so at the P → 0 limit, and which should vary

under deformation η. Indeed, the derivative of ΨS with respect to η can be described

139


Table C.I.3: In-plane direct piezoelectric constants as a function of substitutional fraction

x for Cs-symmetry structures, GeG and SnG. Total values are reported along with nuclear

relaxation contributions (in parentheses). Constant values are given in 10−10C/m.

Substitution fraction x

0.125 0.056 0.031 0.020

GeG

e11 4.22 7.63 4.86 4.60

(-0.81) (-0.71) (-1.35) (-1.71)

e12 -4.15 -7.54 -4.89 -4.67

(0.87) (0.73) (1.37) (1.64)

e26 -4.10 -7.05 -4.89 -4.68

(0.94) (0.79) (1.38) (1.64)

SnG

e11 29.15 11.78 4.32 4.56

(-8.81) (-0.54) (-1.36) (-2.00)

e12 -4.85 -11.68 -4.37 -4.64

(0.05) (0.48) (1.34) 1.91

e26 -5.82 -10.79 -4.35 -4.58

(0.21) (0.01) (1.38) (2.01)

at the first order of perturbation by a combination of Ψn (S)-excited states (except

ΨS) that can interact with ΨA through the dipole moment operator, as well as the

derivative of ΨA with respect to η is a combination of Ψn (A)-excited states (except

ΨA) which interact with ΨS. In other words, the piezoelectricity expression is the same

as Equation I.80 where the ground state is ΨS+ΨA and the sum over n includes all the

excited states except ΨS or ΨA, i.e. it no longer take into account the 0-gap transition

(or the contact between CB and VB). It follows that the derivative of the spontaneous

polarization with respect to the deformation (the piezoelectricity) tends to a finite

value at the P → 0 limit (or, equivalently, at the infinite dilution of BN in graphene;

x → 0) and for any kind of asymmetric perturbation P . When P decreases (i.e. the

defect concentration decreases), the inversion symmetry in the system is progressively

recovered and the numerator of Equation I.80 decreases as fast as the energy gap in

the denominator decreases, leading to a finite 0/0 limit.

140

II. OUT-OF-PLANE PIEZOELECTRIC-

ITY IN GRAPHENE

Graphene is the material of the 21st-century, it has many extraordinary properties, and

it is the base for various applications. As seen in the last part, Chapter II, Section 3,

pristine graphene lacks any intrinsic piezoelectricity due to its symmetry inversion cen-

ter, however, an in-plane symmetry-breaking defect can induce an in-plane piezoelectric

response in graphene, that tends toward a unique value, neither null nor infinite, in

the limit of pure symmetry-broken graphene. Indeed, if the orientation of graphene

symmetry-breaking-defects is somewhere reversed, the perturbed graphene will again

exhibit an inversion symmetry to a significant degree, and the in-plane piezoelectric-

ity will be cancelled. This is the reason why we thought to engineer an out-of-plane

piezoelectricity into graphene sheets. In that case, the defect will be already oriented

to only one side of the graphene surface. Hence, in order to improve the piezoelectric

response induced into graphene, an out-of-plane piezoelectricity induced by an out-of-

plane symmetry-breaking-defect will be here discussed. The out-of-plane defect should

break graphene planarity in order to create non-periodic out-of-plane properties.

Breaking Graphene Planarity

In its intrinsic state (single-atom-thickness), graphene exhibits a perfect flat layer, and

so, its out-of-plane properties are ill-defined. Thus, protruding atoms out of graphene

xy-plane is necessary in order to define its properties through non-periodic z-direction.

Adsorption of atoms on graphene surface is a valid way to achieve that, but the in-

duced piezoelectricity is excessively small.(2) Substituting carbon atoms in graphene

by IV-group elements with a larger size as Si, Ge, or Sn, breaks graphene planarity,

as previously mentioned (see Figure C.II.1). So, carbon atoms are here substituted

with Si, Ge, or Sn atoms in order to reduce the point symmetry from the centrosym-

metric D6h to the non-centrosymmetric C2v group, whereas the flat structure is not a

minimum due to the size limitation effect (Si/Ge/Sn-C bond is larger than C-C one).

Then atoms, in particularly Si, Ge, or Sn, are buckling up and down leading finally

to non-centrosymmetric Cs structure. In these cases, the out-of-plane piezoelectric

component, e31, is no more equal to zero and is very large, nearly three hundred times

larger than the previously reported value of Li-adatom graphene, as will be seen later.

Further, pyrrolic-N-doped graphene (H-NG) is another accessible way to define

out-of-plane graphene properties, since NH group in the more stressed pyrrolic five-

141

II. OUT-OF-PLANE PIEZOELECTRICITY IN GRAPHENE

membered ring gets out from graphene plane by a deformation angle δ of 15.3, Fig-

ure C.II.1. Note that, in order to obtain pyrrolic-ring in graphene, some carbon atoms

are missing leaving asymmetric hole in graphene sheet and leading to an overall C1 point

group symmetry. Substitution of H atom in H-NG by different functional groups, such

as halogen (F-, Cl-), methyl (H3C-), and amino (H2N-), will additionally be investi-

gated in order to improve the softness of the vibrational mode, and so the piezoelectric

response. Thus, in this respect, we will discuss our results in two sections: Carbon

substitution in graphene and H-substitution in pyrrolic-N-doped graphene.

Figure C.II.1: The optimized structure of Si, Ge, Sn, and pyrrolic-N, piezoelectric doped-

graphene is shown in panels a, b, c, and d respectively. For each configuration front and

lateral views are represented, where the deformation angle δ (in degree) and bond lengths

(in A) are reported. The unit cell used in the calculation is displayed for each configuration.

Recently, the fabrication of doped-graphene monolayer has largely progressed, for

example, free-standing BN-, N-, B-, and Si-doped graphene monolayers have been

142

1. Carbon Substitution in Graphene

newly synthesized in laboratories and are found to be chemically stable at ambient

conditions.(3; 4; 5; 6) Not only the practical synthesis of chemically doped graphene

has recently been successfully achieved, but also a controllable concentration and con-

figuration of the dopant in graphene can be fulfilled.(7; 8) For instance, nitrogen doped

graphene has notably three common bonding configurations: pyridinic N, pyrrolic N,

and graphitic N, which are fundamentally affected by chemical vapor deposition (CVD)

process parameters: the precursor, catalyst, flow rate, and growth temperature.(8)

In the following we are going to calculate the piezoelectricity of these compounds

using the hybrid B3LYP functional(9) of the density-functional-theory (DFT). An

atom-centered all-electron basis set of triple-zeta quality, augmented with polarization

functions for all atoms,(10) is chosen, except for Ti-atom where an effective Durand

pseudopotential (PS) basis is applied.

1 Carbon Substitution in Graphene

In this part, we are interested in the out-of-plane piezoelectric effect induced in the

perturbed graphene sheet. For this component, the electronic contribution is negli-

gible and the vibrational contribution becomes predominant contrary to the in-plane

response. This can be explained as follows: in the non-periodic z-direction orthog-

onal to the plane, the vertical transition moments are not allowed by symmetry if

the sheet is perfectly flat or are small in the real case of doped-graphene while the

corresponding smallest transition energy (the denominator in Equation I.80) in that

direction is not null, even in the case of infinite dilution of the defect. On the other

hand, the asymmetric up and down vibrational motion of hetero-atom with respect to

the graphene plane leads to a crucial variation in the polarization and, then, to a large

vibrational contribution to piezoelectricty, especially if this mode is soft; the denom-

inator of Equation I.81 is small. A similar disposal is previously reported for SrTiO3

perovskite, where the vibrational contribution predominates due to the existence of a

soft mode. However, the piezoelectric response of the perovskite, which is infinite for

the ferroelectric phase, occurs at a very low temperature while doped-graphenes are

found to be stable at room temperature.

Table C.II.1 introduces the numerical value of out-of-plane direct and converse

piezoelectric constants for SiG structures as a function of concentration fraction x.

Considering e31 e32, total values are reported along with nuclear relaxation contribu-

tions (in parentheses). The vibrational contribution corresponds to 99.8% of the total

response. The vibrational mode corresponding to the vertical out-of-plane movement is

explained in Figure C.II.2, where the phonon wavenumber (ν) and IR-intensity through

143


Table C.II.1: Out-of-plane direct (e31) and converse (d31) piezoelectric constants of SiG

as a function of x. Total values are reported along with nuclear relaxation contributions (in

parentheses).

x e31 d31

10−10 C/m pm/V

0.125 37.50 12.70

(37.48) (12.68)

0.056 20.51 8.16

(20.47) (8.15)

0.031 13.92 4.87

(13.89) (4.86)

0.020 11.08 3.74

(11.06) (3.70)

z-direction (Ip(z)) give an indication about the softness and polarization induced by

the mode, respectively. For x = 0.125 (the largest concentration studied in our work),

ν = 36 cm−1 and Ip(z) = 4 km/mol, the soft mode leads to a considerable vibrational

contribution to the out-of-plane piezoelectricity. e31 equals 38 × 10−10 C/m, the value

that is 70 times larger than Li-adatom-graphene (0.55 × 10−10 C/m) for the same con-

centration, and 8 times more than the in-plane symmetry-broken-graphene limit (e11 =

4.5 × 10−10 C/m). Converse piezoelectric constant (d31 = 12.70 pm/V) is interestingly

two orders of magnitude larger than Li-adatom-graphene (0.1 pm/V), and almost one

order of magnitude of BNG (d11 = 1.5 pm/V).

As x decreases, the deformation angle δ (due to the out-of-plane displacement of

the defect) increases, Figure C.II.2 panel (b), and so increasing the energy required to

displace vertically Si-atom and leading to the loss of the mode softness. Moreover, the

Born charges or IR intensities per unit cell decrease with the decrease of x such that

the out-of-plane piezoelectric response finally vanishes, Table C.II.1. The results for

p = 2, x = 0.125, will be exclusively discussed in the remainder of this paper.

In general, the same trend is obtained for Ge- (GeG) and Sn- (SnG) doped graphene:

the vibrational contribution predominates the out-of-plane response that dramatically

decreases with the concentration of defect in graphene, conversely to the in-plane

response that tends to a finite limit. Figure C.II.3 represents a comparison of ob-

tained out-of-plane direct (e31, blue) and converse (d31, red) piezoelectric constants for

x = 0.125 of different simulated structures. The variation of most related parameters

144

1. Carbon Substitution in Graphene

Figure C.II.2: Atomic displacement corresponding to the vibrational mode, Si-atom moves

up where C-atoms move down and vice versa, see red arrows in panel (a). Phonon wavenum-

bers, ν, and IR-intensity, Ip(z) that can be expressed as ( ∂µz

∂Qp)2, corresponding to this dis-

placement are given for each SiG structure, where the concentration x is indicated at left

of the panel. panel (b) represents the variation of deformation angle (δ) with respect to x,

where a graphical sketch is introduced in panel (c) to show how δ can be calculated.

to the induced piezoelectricity (deformation angle δ, wavenumber ν, and IR-intensity

Ip(z) induced by the mode through z-direction) is additionally stated. For GeG, the

induced out-of-plane piezoelectricty is enormously improved (e31 = 157 × 10−10 C/m,

d31 = 114 pm/V), although the mode is more soft for SiG, Figure C.II.3 (Bottom).

Note that small deformation angle for GeG, δ = 11.4, keeps to some extent the mode

softness, ν = 115 cm−1, and then consequently permits the approach of Ge towards

the graphene plane leading to a prompt polarization: Ip(z) = ( ∂µz

∂Qp)2 ∼ 2 km/mol.

According to Equation I.81, the numerator consists of two terms: one refers to the

variation of polarization (dipole moment) with respect to the mode ∂µi

∂Qp, i.e. the Born

charge the square of which gives the IR intensity; and the second corresponds to the

second derivative of energy with respect to the mode and the in-plane deformation∂2H

∂ηv∂Qp. If the IR intensity of the soft mode is larger for SiG than for GeG moreover

with a smaller frequency for SiG (see values in Figure C.II.3), we could expect that SiG

145


Figure C.II.3: (Top) Column representation of direct, e31, and converse, d31, piezoelectric

constants (relaxed-ion) for our simulated structure, x = 0.125 for all structures. (Bottom)

The variation of induced piezoelectricity related parameters; deformation angle, δ, where

the angle initiated by vibrational mode amplitude at absolute zero temperature (±∆) is

included, wavenumber ν, and IR-intensity through z-direction. The inset in the figure

represents the vibrational mode amplitude (at 0 K) that seems almost symmetric for SiG

with respect to graphene plane.

has a higher out-of-plane piezoelectricity than GeG, as it is the case for the vibrational

contribution to the polarizability (αvibii ∝

p

(∂µi∂Qp

)2

ω2p

). However, a separated prelimi-

nary calculation of the coupling energy, ∂2H∂ηv∂Qp

, between the in-plane deformation (η11,

for instance) and the out-of-plane soft mode (Q3) responsible of the high IR intensity

along z-direction shows on the contrary that the second term in the numerator of the

piezoelectricity expression is much larger for the more asymmetric Ge-doped graphene

than for the flat Si-doped graphene, the latter having a small or even zero value of∂2H

∂ηv∂Qpfor this smooth mode for symmetry reason. Obviously, Si-atoms are highly

crossing graphene plane with nearly the same extent, see the inset of Figure C.II.3,

(angle corresponding to the mode amplitude ∆ = 7 at half-quantum vibrational en-

ergy, while δ = 4.45), and so a huge polarization is induced (vibrational contribution

of this mode to the polarizability αvibzz = 84.88 a.u. from a total vibrational contribution

of 85.22 a.u.) in this case. This polarization seems to be cancelled in the calculation of

146

2. H-Substitution in Pyrrolic-N-Doped Graphene

the piezoelectricity due to the quasi symmetric motion of Si. However, the asymmetric

motion of the Ge atoms with respect to graphene plane (δ ± ∆ equals 11 ± 3), with

even small induced polarization due to this atomic displacement (αvibzz of this mode

equals 3.70 a.u. from a total contribution of 3.84 a.u.), will largely contribute to the

piezoelectricity. This can be seen as well from the mean elastic behaviour of SiG and

GeG over the whole set of modes: (Cvibvv ∝

p

( ∂2H

∂ηv∂Qp)2

ω2p

) although the electronic C11

(or C22) contribution to elasticity is equal for SiG and GeG, the vibrational one is

however more than two times larger for Ge-doped graphene, see Table C.II.2. It comes

that the sum of quotients in Equation I.81 finally leads to a larger value for GeG than

SiG. In the case of a unique important mode, the vibrational contribution to the piezo-

electric coefficient evibiv could be obtained by the square root of the product: elasticity

Cvibvv × polarizability αvib

ii . Neglecting the symmetric soft mode for Si-doped graphene

(responsible of the large value of the polarizability but to a very small contribution to

the piezolectricity),

αvibii Cvib

vv leads to a piezoelectricity five times larger for GeG with

respect to SiG.

The out-of-plane piezoelectricity of SnG is not improved (2.8 ×10−10 C/m and 1.5

pm/V for e31 and d31), because the deformation angle δ becomes too large (42.1).

When the deformation angle δ increases, the vertical motion of hetero-atom through z-

direction becomes more difficult (required higher energy), then the hetero-atom cannot

penetrate graphene plane and so, the induced polarization by the mode gets smaller.

For SnG, the induced polarization by the mode corresponding to the vertical motion of

Sn-atom is extremely small (Ip(z) = 0.52 km/mol), and the frequency is quite large, ν

= 255 cm−1. This eventually leads, with however improved elastic behaviour compared

to SiG, to a smallest piezoelectric response, Figure C.II.3. H-N doped graphene piezo-

electricity will be discussed into details in the next section to be compared to chemical

substitution of H-atom by other functional groups.

2 H-Substitution in Pyrrolic-N-Doped Graphene

Other p-block elements related to realistic pyrrolic-N-doped graphene will be discussed

in this section. We are especially interested in the role of N-doped graphene pyrrolic

form for which model compounds are considered in order to analyse the substitution

effect on the piezoelectricity. The pyrrolic-N-doped graphene (H-NG) piezoelectricity

is quite improved compared to the SnG one, mainly due to a decreasing frequency value

for the vibrational mode corresponding to the vertical motion of the dopant. This can

be related to the lower value of the deformation angle δ, see Figure C.II.3 (Bottom).

The most findings for the H-NG case is that the converse piezoelectric constant, d31,

147


Table C.II.2: Elastic properties of simulated structures of Si, Ge, Sn, and H-N doped

graphene, x is here equal to 0.125. Electronic and vibrational contributions are separately

reported, where total values are the sum of both. Dashes refer to null value of Cs-symmetry

structures. Elastic constants, Cuv, are given in N/m, and compliance constants, suv, are

given in 10−3 m/N.

Electronic Vibrational

Si Ge Sn NH Si Ge Sn H-N

Elastic constants

C11 311 301 238 247 −60 −139 −81 −186

C22 311 300 238 274 −59 −137 −57 −178

C12 60 58 64 52 −16 −82 −32 −27

C16 - - - −20 - - - 13

C66 125 121 63 95 −21 −27 −39 −69

Compliance constants

s11 3.34 3.46 4.51 4.28 0.77 2.87 2.05 14.43

s22 3.35 3.46 4.52 3.80 0.76 2.81 1.19 7.81

s12 −0.65 −0.67 −1.20 −0.79 −0.07 1.61 0.07 −3.90

s16 - - - 0.85 - - - 3.61

s66 8.02 8.26 15.76 10.67 1.62 2.36 24.47 28.53

148


is larger than the direct e31 one, contrary to all the other cases, see Figure C.II.3

(Top). This is due to significant improvement of the vibrational contribution to elastic

properties (H-N doped graphene has the highest vibrational contribution among the

others), see Table C.II.2. Since d = e s, hence increasing of vibrational contribution

to elasticity will decrease the total value (electronic + vibrational) of elastic constants

and so increase the compliance; s = C−1 (s11 = 14.43 10−3 m/N, for H-NG case).

Figure C.II.4: Graphical representation of optimized geometry of pyrrolic-N-doped

graphene, H-NG, where H is here substituted by other functional groups (F-,Cl-, H3C-,

and H2N-). The unit cell of each configuration is shown as thick lines. The inset demon-

strates the angle θ.

The vibrational mode corresponding to the vertical motion of NH group out of

graphene plane is not so soft, ν = 179 cm−1. Substitution of H-atom by other functional

groups such as halogen (F-, Cl-), methyl (H3C-), and amino (H2N-), has been proposed

in order to improve the vibrational mode properties (frequency and polarization). A

graphical representation of optimized geometry of pyrrolic-N-doped graphene (H-NG),

where H-atom is replaced by other functional groups, is given in Figure C.II.4. Here,

we are concerned with the effect of H-substitution by a different functional group on the

induced out-of-plane piezoelectricity rather than N-doping effect. Since N-atom is here

149


Figure C.II.5: (a) Atomic displacement corresponding to the vibrational mode, where

arrows refer to the movement direction for each atom. The variation of θ angle, that has

been introduced in the previous figure, and phonon wavenumber, ν, as a function of the

structure are given in panels (b) and (c), respectively.

just doubly bonded to graphene, another angle θ should be considered (see the inset

of Figure C.II.4). The angle θ (C−N−X, X is the substituted functional group) gives

the degree of inclination of the substituted group with respect to the horizontal plane

of graphene. As θ approaches the value 180, the substituted group can penetrate the

graphene plane by vibration and induces a major polarization. So from Figure C.II.4,

it is expected that Cl-NG has the lowest piezoelectric contribution since the angle θ is

equal to 92. This is due to the large atomic size of Cl that leads consequently to the

increase of the mode frequency, ν = 280 cm−1. From a point of view, the reason behind

engineering poorly piezoelectric response by considering Cl-substitution is related to

the position of the element with respect to graphene surface. For Cl-NG case, it is

top site position contrarily to the hollow site position for the other functional groups.

This is in consonance with the reported piezoelectric values for adsorbed light atoms

on graphene surface, in which hollow site bonding cases (Li, K, and LiF) have higher

piezoelectric constants compared to top site bonding cases (H, F, and HF).(2)

Replacing H by F decreases the frequency of the mode, ν = 151 cm−1, but also

the polarization due to higher electronegativity of F-atom. For H3C-NG and H2N-

150


Table C.II.3: Out-of-plane direct and converse piezoelectric constants, relaxed total values,

for H-substituted pyrrolic-N-doped graphene. e and d constants are given in 10−10 C/m,

and pm/V, respectively.

H-NG F-NG Cl-NG H3C-NG H2N-NG

Direct

e31 11.81 6.87 2.83 18.56 27.29

e32 12.83 10.38 3.04 15.48 12.25

e36 -5.73 -2.9 -0.31 -2.47 -6.64

Converse

d31 13.53 0.94 0.88 23.79 38.19

d32 9.09 11.20 2.14 5.88 0.59

d36 -16.63 -20.80 -2.36 -30.83 -53.33

NG, the mode wavenumber decreases by 13 and 20 cm−1, respectively, while θ angle

decreases by 20 for H3C-NG due to steric hindrance, and increases by 2 for H2N-NG,

compared to H-NG. Figure C.II.5 represents the atomic displacement corresponding

to the vibrational mode, the mode wavenumber ν, the variation of θ, and the induced

out-of-plane polarization for each structure. From this vibrational geometrical analysis,

H2N-NG structure should have the highest vibrational contribution to the out-of-plane

piezoelectricity.

In Table C.II.3, we report the computed absolute values of the three direct and

converse out-of-plane piezoelectric components. The amplitude of e31 and e32 constants

are improved compared to H-NG, especially for H3C-NG and H2N-NG structures. As

for H-NG, converse piezoelectric constants d31 and d36 of H3C-NG and H2N-NG are

interestingly larger than their direct counterparts. Substitution of H by amino (NH2)

group appears the most effective enhancement: the soft mode frequency is lowered by

10%.

Conclusion

Here, we summarize our results about the in-plane and out-of-plane piezoelectricity

induced in graphene. Firstly, unlike existing piezoelectric materials, this new form of

piezoelectricity is engineered into a non-piezoelectric material, made possible by the 2D

nanoscale nature of graphene. Additionally, the applied electric or mechanical fields and

chemical doping (in-plane and out-of-plane defects) required to observe these effects in

graphene are readily experimentally accessible. The in-plane piezoelectricity induced

151


in graphene is predominated by the electronic contribution and tends toward a common

finite limit at low defect concentration whatever the nature of the defect. While, the

out-of-plane piezoelectric effect is predominated by the vibrational contribution, it

vanishes as the concentration of the defect decreases, and it depends substantially on

the defect nature.

In Table C.II.4, we report the numerical values of direct and converse piezoelec-

tric effect induced in doped graphenes (in-plane and out-of-plane symmetry-breaking

defects) as computed in the present study and as compared to experimental and theo-

retical values of other 2D and 3D piezoelectric materials. It is seen that the piezoelec-

tricity (in-plane finite limit) induced in graphene by any in-plane symmetry-breaking

defect is almost 4 times larger than that of pristine h-BN, and 60% larger than the

experimentally measured one for the h-MoS2 monolayer.(11) Furthermore, the induced

out-of-plane piezoelectricity for Ge-doped graphene monolayer at the finite concentra-

tion x = 0.125, for instance, is largely improved by more than 30 times compared

to the in-plane e11 doped-graphene,(1) and by 50 times compared to experimentally

measured e11 of h-MoS2 monolayer.(11) Interestingly, e31 of GeG and H2N-NG are also

about 300 and 50 times larger than Li-adatom graphene, respectively.

When comparing with 3D systems, we notice that the converse in-plane piezo-

electric constant d11 (in pm/V) is of the same order of magnitude but systematically

smaller than those of bulk quartz, GaN and AlN, while the converse piezoelectric d31

coefficient is nearly one and two orders of magnitude, for H2N-NG and GeG respec-

tively. In order to compare the present values of the direct piezoelectric response of the

monolayers with those of standard 3D piezoelectric materials, we need to define a vol-

ume and to consider, as it is generally done in these cases, a graphite-like system with

interlayer distance(15) of 3.35 A. It follows that in-plane doped graphene structures

exhibit a direct macroscopic piezoelectricity that is about 10 times larger than that

of pure α-quartz(12) or Ge-doped α-quartz,(13), 5 times larger than that (0.27 C/m2)

of polyvinylidene fluoride (PVF) and its copolymers(16), and is comparable to that

of bulk GaN and AlN.(14). As for the out-of-plane direct macroscopic piezoelectricity

of GeG, two orders of magnitude larger than PVF and its copolymers is interestingly

obtained.

The most advantages for in-plane piezoelectricity induced into graphene is that a

common universal (unique value) of piezoelectricity will be obtained in the limit of

vanishing defect concentration. This means that the same value of in-plane piezoelec-

tricity will be obtained whatever is the particular chemical or physical nature of the

defect. This also means it will be sufficient to just positioning graphene with only one

defect to gain a piezoelectric graphene. This defect can be made by removing only one

atom in the infinite graphene sheet. However, the orientation of the defect should be

152


Table C.II.4: In-plane and out-of-plane piezoelectric (direct and converse) constants of

perturbed graphene as computed in the present study and as compared to experimental

and theoretical values of other 2D and 3D piezoelectric materials. Direct e constants are

reported in 10−10C/m for 2D system and in C/m2 for 3D system. Converse d constants are

expressed in pm/V.

System e d

In-Plane Finite Limit 5 (e11) 1.5 (d11)

SiG (x = 0.125) 38 (e31) 13 (d31)

GeG (x = 0.125) 157 (e31) 114 (d31)

SnG (x = 0.125) 3 (e31) 1.5 (d31)

H-NG (x = 0.125) 12 (e31) 14 (d31)

H2NG (x = 0.125) 27 (e31) 38 (d31)

h-BN (This study) 1.2 (e11) 0.5 (d11)

Graphene + Li(2) 0.5 (e31) 0.1 (d31)

h-MoS2(11) 2.9 (e11) -

Bulk α-Quartz(12) 0.2 (e11) 2.3 (d11)

Si0.83Ge0.16O2(13) 0.2 (e11) 5.5 (d11)

Bulk GaN(14) 1.1 (e33) 3.7 (d33)

Bulk AlN(14) 1.5 (e33) 5.6 (d33)

achieved to avoid the cancellation of overall piezoelectricity if the perturbed graphene

has more than one defect.

The extreme advantages of inducing an out-of-plane piezoelectricity in graphene is

that, the local protrusion of the defect from graphene surface is a prerequisite in order to

identify an out-of-plane property. This means the defect will be already oriented to only

one side of the graphene surface. The obtained out-of-plane piezoelectric effect is largely

improved compared to the in-plane finite limit. However, out-of-plane piezoelectric

response in graphene vanishes at low defect concentration, so large piezoelectricity can

be merely obtained at high concentration.

Nevertheless, in-plane and out-of-plane symmetry-breaking defects will engineer

a piezoelectricity into this non-piezoelectric graphene, and have both of them their

significant technological applications.

153

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155

Conclusions

156

CONCLUSIONS

In this thesis, we have theoretically analyzed the technologically important piezoelectric

phenomena. From a theoretical point of view, and upon Born-Oppenheimer approxima-

tion, the overall static piezoelectric effect “relaxed” can be written as the sum of purely

electronic “clamped-ion” and vibrational-nuclear “internal-strain” contributions. In

general, the electronic contribution arises from the electronic cloud deformation due

to mechanical and/or field perturbation. This contribution can be individually cal-

culated using clamped-ion conditions, where the fractional coordinates are fixed “not

optimized” after applying of the mechanical field. The nuclear (ionic) contribution is

initiated from the relative displacements of nuclei in the perturbed system; and it can

be included in the calculations if the coordinates are free to relax. Experimentally, the

piezoelectric effect (generally, converse one) is measured as the strain produced when a

static electric field is applied into a piezoelectric material, where the two contributions

cannot be separated as for theoretical calculations. Note that, the direct and converse

piezoelectric tensors are connected to each other via the elasticity: e = dC and d = e s,

for s = C−1.

So, theoretically the total piezoelectric effect can be expressed as:

(piezo.)relax = (piezo.)clamp + (piezo.)vib

In order to explore into some details the piezoelectric properties, three procedures

were stated during this thesis: Firstly, the calibration of the computational parameters

and method has been done. In this respect, the well-known piezoelectric quartz mate-

rials are simulated, where their response properties are calculated and then compared

to their experimental counterparts. Additionally, the evaluation of piezoelectricity as

a function of concentration for α-quartz doped by Ge solid solution has been studied.

Our theoretical calculations gave a linear behavior for the piezoelectric coupling coef-

ficient as a function of concentration as observed and expected from the experimental

study.

Secondly, to rationalize the piezoelectric properties, microscopic effects that in-

fluence these properties have been studied. The equations describing electronic and

vibrational contributions to the piezoelectricity were separately examined. Upon the

perturbation theory, the piezoelectricity as a second energy derivative can be described

as a sum of fractional terms with numerators and denominators that would be different

for each contribution. For the electronic part, the numerators are products of allowed

transition moments with respect to electric and mechanical fields. The allowed tran-

sition moments due to the electric field (dipole moment vector) can be experimentally

seen from the UV-visible spectra of the material, however, these transitions should

158

CONCLUSIONS

be allowed as well with respect to the mechanical field so as to obtain a significant

electronic contribution to the piezoelectricity. For a centro-symmetric material, the

irreducible representations of operators associated with electric (three row-column vec-

tor) and mechanical (matrix, six by six array) fields will never be identical for the same

transition. The denominators, in electronic contribution case, are the energy differences

between initial and final spectroscopic transition states, and depend on the electronic

direct band gap. Since graphene is a well-known semi-conducting material with a

zero band gap, it has been chosen in order to elucidate these outcomes. Additionally,

graphene has many astonishing electronic and mechanical properties that make it as a

shining star climbed on the path of the scientists searching for new materials for future

electronic and composite industry. However, this 2D nanocarbon material is not intrin-

sically piezoelectric due to the existence of inversion symmetry center. So, for pristine

graphene both the numerator and denominator are null. Doping with different patterns

and concentrations of BN is suggested to break the symmetry center and engineer the

piezoelectricity into graphene. The case in which a large piezoelectricity, dominated

by the electronic rather than nuclear contribution, is obtained at low concentrations of

the dopant.

For the nuclear vibrational contribution to the piezoelectricity, the sum of fractional

terms run over phonon modes, where the numerators represent essentially the allowed

transition moments induced by a vibrational motion, with respect to the electric and

mechanical fields as for the electronic contribution. An indication about the allowed

transition moments induced by vibrational motion and due to electric field, can be

experimentally inferred from an IR-spectra of the material. These transitions induced

by a vibrational motion should be also allowed, as for the electronic contribution case,

with respect to the mechanical field. Vibrational motion and mechanical deforma-

tion can be considered as an intra and inter atomic displacement, respectively, hence

a vibrational-mechanical coupling is taken here into account. The denominator is the

energy difference between initial and final transition states with respect to a vibrational

motion, and so is the frequency of the vibrational phonon mode. From this analysis,

the large contribution of a soft IR-active phonon mode to the piezoelectricity has been

obviously deduced. Perovskites are considered as good candidates in order to investi-

gate such influence due to their phase transition that leads to a large variation of the

dielectric properties. SrTiO3 phases are simulated by the displacement of Ti ion from

a standard TiO6 octahedra leading to a ferroelectric phase transition. This soft mode

motion is found to largely affect the piezoelectric response of SrTiO3 that will be giant

at very low temperature (below approximately 24 K). Another perovskite structure of

barium titanate, BaTiO3, has been considered, where the ferroelectric phase showing

159

CONCLUSIONS

a huge piezoelectric response, could be herein found at higher temperature conditions

(below 183 K).

After rationalization of the parameters that affect the piezoelectric property, the

design of materials exerting a high piezoelectric effect has been attempted. It has been

shown that the large in-plane piezoelectricity induced at low concentrations of BN

in graphene can be acquired by including any in-plane defect that breaks graphene

centrosymmetric D6h point group to any non-centrosymmetric ones. Moreover, in

the limit of vanishing defect concentration, the piezoelectric response tends toward a

unique value, neither null nor infinite, whatever the particular chemical or physical

nature of the defect. We have recorded that the vibrational contribution leads always

to an overall huge piezoelectric coefficients rather than the electronic one. Further, if

the orientation graphene symmetry-breaking-defects is somewhere reversed, perturbed

graphene will exhibit inversion symmetry to a significant degree, and the in-plane

piezoelectricity should be cancelled. So, inducing an out-of-plane piezoelectricity in

graphene by breaking its planarity through the non-periodic z-direction was investi-

gated, the case in which the defect will be already oriented to one graphene side. This

is achieved via substitution of C atoms by heavier group-IV elements (Si, Ge and Sn)

or more complex pyrrolic N-doped graphene (H-NG). In this case, the piezoelectric ef-

fect is largely improved compared to the finite in-plane piezoelectric limit, however, at

higher concentration of defects. Additionally, the induced out-of-plane piezoelectricity

in graphene depends substantially on the physical and chemical nature of the defect

and it vanishes at low defect concentrations.

As piezoelectricity is a revolutionary source for “GREEN ENERGY”, the present

thesis could have some significant insights about this technologically important prop-

erty.

160

CONCLUSION GENERALE

Ce travail s’est focalise sur l’etude du phenomene de la piezoelectricite. L’effet statique

global dit “relaxed” de la piezoelectricite peut etre ecrit comme la somme de deux

contributions dans l’approximation de Born-Oppenheimer: la contribution electronique

dite “ clamped-ion ” et la contribution nucleaire dite “internal-strain”. La contribution

electronique provient de la deformation du nuage electronique sous l’effet d’un champ

mecanique. La contribution nucleaire est issue des deplacements relatifs des noyaux

dans le systeme perturbe.

(piezo.)relax = (piezo.)clamp + (piezo.)vib

A partir de la piezoelectricite et des tenseurs elastiques et dielectriques calcules,

le coefficient de couplage electromecanique peut etre determine. Ce coefficient rend

compte de l’efficacite du materiau a convertir l’energie mecanique en energie electrique

et inversement.

L’exploration theorique de la propriete piezoelectrique a ete realisee en trois etapes

au cours de ce travail. La premiere etape concerne la calibration des parametres de

calcul a partir de l’exemple des oxydes de silicium et de germanium, pour lesquels

des donnees experimentales sont disponibles. L’evaluation theorique des coefficients

piezoelectriques s’est revelee en bon accord avec sa contrepatrie experimentale.

La deuxieme etape concerne la rationalisation de la propriete piezoelectrique par

l’etude des effets microscopiques sous-jacents a ce phenomene. Les equations decrivant

les contributions electroniques et vibrationelles a la piezoelectricite ont ete examinees

separement. La piezoelectricite est une derivee seconde de l’energie qui peut etre decrite

par une somme de termes de perturbation au second ordre.

Pour la contribution electronique, les numerateurs sont des produits des mo-

ments de transition dus aux champs electriques et mecaniques. Pour un materiau

centro-symetrique, les representations irreductibles des operateurs associes aux champs

electrique et mecanique ne peuvent jamais etre identiques pour la meme transition,

conduisant ainsi a une piezoelectricite nulle. Les denominateurs sont les differences

d’energie entre les etats de transition spectroscopiques initiaux et finaux, donc reliees

au gap electronique direct si la transition correspondante est permise. Quoi qu’il en

soit, l’amplitude de telles transitions est observable sur le spectre UV/VIS du materiau.

Le graphene est donc apparu comme un bon candidat pour l’elaboration de materiaux

a forte piezoelectricite en raison de son gap nul. En outre, le graphene possede de

nombreuses proprietes electroniques et mecaniques etonnantes qui font de ce materiau

le materiau du XXIeme siecle selon la presse scientifique. Pour faire advenir une

161

CONCLUSION GENERALE

piezoelectricite non-nulle, la centro-symetrie de ce materiau peut etre brisee par dopage.

Cet aspect a ete traite au travers de l’exemple du graphene dope par le nitrure de bore

dans lequel il a ete egalement demontre que la propriete est ici largement dominee par

la contribution electronique.

Concernant la contribution vibrationnelle a la piezoelectricite, les series de

numerateurs representent les moments de transition permise entre 2 etats vibra-

tionnels du materiau sous l’effet des champs electrique et mecanique. Le mouvement

vibrationnel et la deformation mecanique correspondent aux deplacement intra et inter

atomiques, respectivement. De la meme facon ces deux operateurs doivent posseder la

meme representation irreductible pour conduire a une contribution vibrationnelle non-

nulle, ce qui ne peut etre le cas pour un materiau centro-symetrique. Le denominateur

represente la difference d’energie entre les etats vibrationnels initiaux et finaux, c’est

a dire la frequence d’un mode de vibration. L’amplitude de telles transitions est

observable sur le spectre infrarouge IR du materiau. De cette analyse, il vient que les

plus grandes contributions vibrationnelles a la piezoelectricite proviennent des modes

les plus mous actifs et intenses en IR. Les exemples de SrTiO3 et BaTiO3 ont ete

utilises pour illustrer cet aspect.

La troisieme etape de ce travail a ete consacree a la conception de materiaux a

base du graphene pouvant presenter une forte piezoelectricite. Il a ete montre qu’un

dopage dans le plan du graphene conduit a une piezoelectricite induite qui tend vers

une valeur finie a dilution infinie quelle que soit la nature de l’element dopant. Pour

illuster ce propos, le dopage du graphene par Si, Ge, Sn, BN, ainsi que la presence

de trous a ete consideres. Le dopage doit etre ici controle pour eviter de recouvrer

macroscopiquement un centre de symetrie qui annihilerait la piezoelectricite. La brisure

de symetrie du graphene par un defaut hors du plan a alors ete proposee puisque le

dopage peut s’effectuer dans ce cas sur un seul cote de la surface. Dans ce cas, une

composante piezoelectrique induite apparaıt perpendiculairement au plan du graphene.

La substitution d’atomes de Carbone dans le graphene par les elements plus lourds (Si,

Ge et Sn) et par le groupement chimique NH a ete realisee. De plus, le greffage chimique

de l’atome d’hydrogene par d’autres goupes tels que les halogenes (F, et Cl) , H3C,

et NH2 a ete considere afin d’ameliorer la propriete piezoelectrique induite. Dans ces

cas, l’effet piezoelectrique induit hors du plan du graphene est largement augmente par

rapport a la limite finie de la piezoelectricite dans le plan. Sa valeur, principalement

gouvernee par la contribution vibrationelle dans ce cas, tend vers zero a dilution infinie

mais reste aux grandes concentrations jusqu’a 50 fois plus grande que celle observee

pour les materiaux traditionnellement utilises, et 300 fois plus grande par rapport aux

materiaux de type ”ad-atom” sur graphene.

162

Appendix

164

Appdx 1: BaTiO3 Piezoelectricity

PHYSICAL REVIEW B 89, 045103 (2014)

Low-temperature phase of BaTiO3: Piezoelectric, dielectric, elastic, and photoelastic propertiesfrom ab initio simulations

A. Mahmoud,1,* A. Erba,1 Kh. E. El-Kelany,2,3 M. Rerat,2 and R. Orlando1

1Dipartimento di Chimica and Centre of Excellence NIS (Nanostructured Interfaces and Surfaces), Universita di Torino, via Giuria 5,

IT-10125 Torino, Italy2Equipe de Chimie Physique, IPREM UMR 5254, Universite de Pau et des Pays de l’Adour, FR-64000 Pau, France

3Chemistry Department, Faculty of Science, Minia University, Minia 61519, Egypt

(Received 7 November 2013; published 6 January 2014)

A complete theoretical characterization of dielectric, elastic, photoelastic, and piezoelectric tensors of the

low-temperature rhombohedral phase of BaTiO3 was performed by accurate ab initio simulations within periodic

boundary conditions, using one-electron Hamiltonians and atom-centered Gaussian-type-function basis sets as in

the CRYSTAL program. Because this phase is stable only at very low temperature, experimental characterization is

difficult, and none of such tensorial properties have been measured. For this reason, we validated our method by

comparing structural, electronic, and vibrational properties of the other three phases of BaTiO3 (cubic, tetragonal,

and orthorhombic) with available experimental data. The effect of the adopted one-electron Hamiltonian on the

considered tensorial properties, beyond the simple local density approximation and the dependence on the electric

field frequency of dielectric and photoelastic constants, is explicitly investigated.

DOI: 10.1103/PhysRevB.89.045103 PACS number(s): 31.15.A−, 78.20.hb, 77.84.Cg

I. INTRODUCTION

Barium titanate, BaTiO3, is one of the most studied fer-

roelectric ceramics. This material is widely used in advanced

technological applications such as capacitors or as components

of nonlinear optical, piezoelectric, and energy/data-storage

devices.1–4 Since the discovery of its ferroelectric character,

much attention has been devoted to the peculiar dielectric and

piezoelectric properties of BaTiO3.5

From a structural point of view, BaTiO3 shows a cubic

ABO3-type perovskite crystal structure at high temperature

where A sites host divalent cations (Ba2+ in this case) and

B sites are occupied by tetravalent cations (Ti4+ in this

case). With reference to the conventional cubic cell at high

temperature, Ba2+ ions are placed at corners, O2− ions at the

center of faces, thus forming an octahedron, and a Ti4+ ion at

the body center of the cell.

Upon cooling, three consecutive ferroelectric transitions

occur starting from the cubic structure, due to the displacement

of Ti ions along different crystallographic directions, the

resulting macroscopic polarization of the material being

always parallel to this displacement.6 At 393 K, BaTiO3

undergoes the first transition from a cubic paraelectric to a

tetragonal ferroelectric phase, which corresponds to a small

structural elongation along a [001] crystallographic direction.

Then, an orthorhombic ferroelectric phase occurs between 278

and 183 K that can be interpreted as a deformation along a

face diagonal [011] direction.7 Finally, below 183 K, BaTiO3

transforms into a low-temperature ferroelectric rhombohedral

phase characterized by an elongation along the cell body

diagonal [111] direction. Two models have been proposed for

phase transitions of BaTiO3: the displacive model8 is governed

by a Ŵ point soft phonon mode; the order-disorder model9–11

implies the coexistence of local configurations with lower

symmetry with respect to the macroscopic order parameter,

that is, the macroscopic polarization in this case.

The structure of the paraelectric cubic phase of BaTiO3

was refined experimentally a long time ago.12,13 Single-

crystal x-ray and neutron diffraction structural studies of the

tetragonal,14,16 orthorhombic,17 and rhombohedral14–16 ferro-

electric phases of BaTiO3 have been reported. Among others,

the neutron diffraction study of the three ferroelectric phases

performed by Kwei et al.6 in 1993 is here taken as a reference

as regards their structural properties. Several computational

studies of the structures of the different phases of BaTiO3 have

also been reported, either with an atom-centered Gaussian-type

function (GTF)20–23 or a plane-wave pseudopotential24–31

basis set approach. Many of the above-mentioned theoretical

studies have also investigated the phonon properties of the cu-

bic phase,20,21,25,26,28 whereas no experimental spectroscopic

measurements are known. Raman scattering experiments have

been performed for the tetragonal,32,33 orthorhombic,34 and

rhombohedral35 phases. As regards theoretical investigations

of vibrational frequencies, several studies have addressed

this topic for the tetragonal,20,36–38 orthorhombic,20 and

rhombohedral20,22,31,38,39 phases.

Elastic, piezoelectric, and dielectric constants of cubic,40

tetragonal,41–43 and orthorhombic44 phases of BaTiO3 have

been measured experimentally. In particular, a complete set of

dielectric, elastic, piezoelectric, electro-optic, and elasto-optic

constants has been determined for the ferroelectric tetragonal

phase by Zgonik et al.43 The temperature dependence of

different elasto-optic tensor components of the cubic phase

has been measured also by Cohen et al.45 None of these

tensorial properties have been measured experimentally for

the low-temperature rhombohedral phase yet.

A couple of computational studies have been reported on

such tensorial properties of the rhombohedral phase. Wang

et al.46 computed elastic, polarization, and electrostrictive

properties of the four different phases of BaTiO3 by using a

local density approximation (LDA), with projector-augmented

waves (PAW) and density-functional perturbation theory

(DFPT). Elastic, dielectric, and piezoelectric properties of

the low-temperature ferroelectric rhombohedral phase have

been reported by Wu et al.31 who used the same LDA-DFPT

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166

MAHMOUD, ERBA, EL-KELANY, RERAT, AND ORLANDO PHYSICAL REVIEW B 89, 045103 (2014)

approach. In both cases, cell volumes were taken at the

experimental values.

In this paper, besides structural, electronic, and vibrational

properties of the four phases, we focus on accurate abinitio simulation of tensorial properties, such as dielectric,

elastic, piezoelectric, and photoelastic, of the low-temperature

rhombohedral phase of BaTiO3. Both electronic and nuclear

contributions to these properties are computed and discussed.

Recent developments by some of us of fully automated

algorithms for the calculation of such tensorial properties in

the CRYSTAL program now make such a complete theoretical

investigation feasible at relatively low computational cost.47–50

The effect of the adopted one-electron Hamiltonian on

these tensorial properties is investigated systematically here

for the first time beyond the simple LDA approach. In

particular, four one-electron Hamiltonians are considered

which are representative of four different classes: the reference

Hartree-Fock (HF) method, LDA and generalized-gradient

approximation (GGA) to the density functional theory (DFT),

namely, Perdew-Burke-Ernzerhof (PBE),51 and a global hybrid

functional (PBE0) with 25 % of exact HF exchange.52 All the

calculations reported in this manuscript have been performed

with a development version of the CRYSTAL program.53,54 A

fairly similar computational approach has been successfully

adopted by Evarestov and Bandura for computing structural,

thermodynamical, and phonon properties of the rhombohedral

phase of BaTiO3.20–22

The structure of the paper is as follows: In Sec. II we briefly

illustrate the theoretical methods used for the calculation of

structural parameters, dielectric, elastic, piezoelectric, and

photoelastic constants, and we report the main computational

parameters adopted for these calculations; in Sec. III A the ma-

jor outcomes of structural parameters and phonon frequencies

for the four different phases of BaTiO3 are presented; in the

following subsections elastic, piezoelectric, and photoelastic

properties of the low-temperature rhombohedral phase are

presented and discussed. Conclusions are drawn in Sec. IV.

II. COMPUTATIONAL METHOD AND DETAILS

All the calculations reported in this paper are performed

with a development version of the CRYSTAL program for

ab initio quantum chemistry of solid state.53,54 An atom-

centered Gaussian-type-function (GTF) basis set is adopted

whose coefficients for valence electrons have been reopti-

mized for the cubic phase of BaTiO3, by minimizing the

HF energy using the LoptCG script.55 Oxygen atoms are

described by a split-valence 8-411G(2d1f) basis set, titanium

atoms by a 86-411G(3d1f) one, while core and valence

electrons of barium atoms are described by a Hay-Wadt

small-core pseudopotential56–58 and 311G(1d1f) functions,

respectively.59

In CRYSTAL, the truncation of infinite lattice sums is

controlled by five thresholds T1, . . . ,T5; here T1 = T2 = T3 =

10−10 a.u., T4 = 10−12 a.u., and T5 = 10−24 a.u. Reciprocal

space is sampled according to a sublattice with shrinking

factor 8, corresponding to 35, 75, 105, and 65 points in the

irreducible Brillouin zone for cubic, tetragonal, orthorhombic,

and rhombohedral phases, respectively. The DFT exchange-

correlation contribution is evaluated by numerical integration

over the cell volume: radial and angular points of the atomic

grid are generated through Gauss-Legendre and Lebedev

quadrature schemes, using an accurate predefined pruned grid:

the accuracy in the integration procedure can be estimated by

evaluating the error associated with the integrated electronic

charge density in the unit cell versus the total number of

electrons per cell: 3 × 10−5|e| out of a total number of

56 electrons per cell for the rhombohedral phase, for instance.

A. Dielectric tensor

The electronic contribution to the static dielectric tensor, at

infinite electric field wavelength λ → ∞, is evaluated through

a coupled-perturbed-HF/Kohn-Sham (CPHF/KS) scheme60

adapted for periodic systems.61 From an experimental view-

point, it corresponds to the dielectric response of the crystal

measured for sufficiently high frequencies of the applied

electric field to make nuclear contributions negligible, but not

high enough for generating electronic excitations. CPHF/KS

is a perturbative, self-consistent method that focuses on the de-

scription of relaxation of crystalline orbitals under the effect of

an external electric field. The perturbed wave function is then

used to calculate the dielectric properties as energy derivatives.

Further details about the method and its implementation in the

CRYSTAL program can be found elsewhere62 as well as some

recent examples of its application.48–50,63–65 The electronic

dielectric tensor of a three-dimensional (3D) crystal is obtained

from the polarizability α as

= 1 +4π

Vα, (1)

where V is the cell volume. With such a scheme, the explicit

dependence of the polarizability and dielectric tensors from

the electric field wavelength λ can be computed as well.

B. Elastic, piezoelectric, and photoelastic tensors

The elements of the fourth-rank elastic tensor C for

3D systems are usually defined as second energy density

derivatives with respect to pairs of deformations:66

Cvu =1

V

∂2E

∂ηv∂ηu

0

, (2)

where η is the symmetric second-rank pure strain tensor

and V the equilibrium cell volume, and Voigt’s notation is

used according to which v,u = 1, . . . ,6 (1 = xx, 2 = yy,

3 = zz, 4 = yz, 5 = xz, 6 = xy). An automated scheme

for the calculation of C (and of S = C−1, the compliance

tensor) has been implemented in the CRYSTAL program that

exploits analytical gradients and computes second derivatives

numerically.47,67

In the linear regime, direct e and converse d piezoelectric

tensors describe the polarization P induced by strain η and the

strain induced by an external electric field E, respectively:

direct effect P = e η at constant field (3)

converse effect η = dT E at constant stress. (4)

The direct and converse piezoelectric tensors are connected

to each other: e = d C and d = e S. Our approach consists

045103-2

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LOW-TEMPERATURE PHASE OF BaTiO3: . . . PHYSICAL REVIEW B 89, 045103 (2014)

in directly computing the intensity of polarization induced by

strain. In CRYSTAL, the polarization can be computed either

via localized Wannier functions or via the Berry phase (BP)

approach.68 The latter scheme is adopted in the automated

implementation exploited here.48

The elements of the Pockels’ elasto-optic fourth-rank tensor

P (i.e., elasto-optic constants pijkl) are defined by the relation

−1ij =

kl

pijklηkl . (5)

In the above expression, −1 is the difference between

the inverse dielectric tensor of a strained and the unstrained

equilibrium configuration; i,j,k,l = x,y,z represent Cartesian

directions. If Voigt’s notation is used, Pockels’ tensor becomes

a 6 × 6 matrix like C and S. An automated implementation

in the CRYSTAL program of the calculation of elasto-optic

constants is exploited here.49

We recall that elastic, piezoelectric, and elasto-optic con-

stants can be decomposed into purely electronic “clamped-

ion” and nuclear “internal-strain” contributions; the latter

measures the effect of relaxation of the relative positions of

atoms induced by the strain and can be computed simply by

optimizing atomic positions within the strained cell.69,70

III. RESULTS AND DISCUSSION

The main objective of this section is that of reporting and

discussing the results of accurate ab initio simulations of

piezoelectric, elastic, dielectric, and photoelastic properties of

the low-temperature rhombohedral phase of BaTiO3. Given

the predictive character of this study, due to the lack of

experimental determinations of these tensorial quantities for

the low-temperature ferroelectric phase, the accuracy of

our theoretical approach will be discussed with respect to

structural, electronic, and vibrational properties of the four

phases of BaTiO3, for which experimental data have been

reported. Section III A is devoted to the illustration of the

structural and vibrational description of the four phases.

A. Structural and vibrational features of the four phases

Before illustrating the accuracy of the present simulations

in describing structural features of the four phases, let us recall

that, at high temperature, BaTiO3 shows a centro-symmetric

cubic structure (space group Pm3m) that can be described by

three symmetry-irreducible atoms in the cell: a Ba2+ cation is

placed at the origin, an O2− anion at the center of a face, and a

Ti4+ cation at the body center of the crystallographic cubic cell.

Upon cooling, at 393 K, the paraelectric cubic structure

undergoes a ferroelectric phase transition to a tetragonal

phase, with space group P4mm, that contains four symmetry-

irreducible atoms in the cell; with respect to the cubic lattice,

Ti and O atoms are displaced along the z direction, as

indicated in Table I. Below 278 K, symmetry is further

reduced to orthorhombic (space group Amm2) with four

symmetry-irreducible atoms in the cell; with respect to the

tetragonal phase, the second O atom is displaced also along

the y direction. Eventually, below 183 K, a rhombohedral

ferroelectric phase with space group R3m, is found with three

symmetry-irreducible atoms per cell: whereas Ba remains

fixed at the origin of the cell, both Ti and O atoms are displaced

along the three Cartesian axes, as shown in Table I.

Structural and electronic properties of the four phases of

BaTiO3 are given in Table II, as theoretically determined

in this study at the PBE0 level and compared with existing

experimental data and a previous theoretical investigation, at

PBE0 level, by Evarestov and Bandura.20 Lattice parameters

a,b,c, volume V , bulk modulus B, electronic direct band gap

Eg , and atomic positions, as defined in Table I, are reported.

For the rhombohedral phase, the lattice angle α is also shown.

For each ferroelectric phase, its energy difference Ec with

respect to the cubic phase is given.

As previously observed for other perovskites,48,73 the PBE0

hybrid functional is providing a fairly reasonable description

of the volume of the four systems under investigation, the

largest deviation from experiment being the underestimation

by 1.27% in the cubic phase; a slight underestimation by 0.26%

is found for the tetragonal phase, while slight overestimations

by 0.22% and 0.47% are obtained for the orthorhombic and

rhombohedral phases. In this respect, underestimations of the

volume of the two high-temperature cubic and tetragonal

phases can be easily understood in terms of the lack of

any thermal lattice expansion in the calculations, which

refer to 0 K. The computed bulk modulus B of the cubic

phase, 194 GPa, compares with the experimental value at a

pressure of 1.6 GPa, 195 GPa, thus indirectly confirming the

underestimation of the volume. The theoretical bulk moduli

of the three ferroelectric phases are much smaller than that

of the cubic phase, as also partially confirmed by available

experimental data for the tetragonal phase, 134–141 GPa, with

respect to 162–195 GPa for the cubic phase. Our computed

bulk moduli are obtained by fitting energy-volume data points

to the universal exponential Vinet’s equation of state.74

Besides volume, also atomic displacements from their

cubic sites to form the ferroelectric phases are relatively well

described at this level of theory. Even if the absolute values of

TABLE I. Fractional atomic coordinates of the symmetry-irreducible atoms of the four different phases of BaTiO3: cubic, tetragonal,

orthorhombic, and rhombohedral (corresponding space group symbols are also given).

Phase Cubic Tetragonal Orthorhombic Rhombohedral

space group Pm3m P4mm Amm2 R3m

Atomic positions Ba (0, 0, 0) Ba (0, 0, 0) Ba (0, 0, 0) Ba (0, 0, 0)

Ti ( 1

2, 1

2, 1

2) Ti ( 1

2, 1

2, 1

2+ zT i) Ti ( 1

2, 0, 1

2+ zT i) Ti ( 1

2+ xT i ,

1

2+ xT i ,

1

2+ xT i)

O ( 1

2, 1

2, 0) O1 ( 1

2, 1

2, zO1) O1 (0, 0, 1

2+ zO1) O ( 1

2+ xO , 1

2+ xO , zO )

O2 ( 1

2, 0, 1

2+zO2) O2( 1

2, 1

4+yO2, 1

4+zO2)

045103-3

APPENDIX

168


TABLE II. Structural and electronic properties of the cubic,

tetragonal, orthorhombic, and rhombohedral phases of BaTiO3, as

computed at PBE0 level and compared with experimental data and

a previous theoretical determination at the same PBE0 level (see

Ref. 20): lattice parameters a,b,c, volume V , bulk modulus B,

band gap Eg , and atomic positions as defined in Table I. For the

rhombohedral phase, the lattice angle α is also reported. For each

ferroelectric phase, the energy difference Ec with respect to the

cubic phase is given.

PBE0

Calc. (Ref. 20) Exp.

Cubic (Pm3m)

a (A) 3.980 3.992 3.99612

V (A3) 63.03 63.62 63.8112

B (GPa) 194 189 162a–195b

Eg (eV) 4.0 4.0 3.271

Tetragonal (P4mm)

a (A) 3.960 3.968 3.9976

c (A) 4.097 4.137 4.0316

V (A3) 64.24 65.14 64.416

zT i 0.0202 0.0203 0.02036

zO1 −0.0391 −0.0431 −0.02586

zO2 −0.0202 −0.0226 −0.01236

B (GPa) 117 112 134c–14143

Eg (eV) 4.2 4.1 3.471

Ec (mHa) −1.24 −1.51 –

Orthorhombic (Amm2)

a (A) 3.951 3.958 3.9836

b (A) 5.696 5.728 5.6756

c (A) 5.729 5.770 5.6926

V (A3) 128.91 130.81 128.636

zT i 0.0171 0.0175 0.01706

zO1 −0.0189 −0.0186 −0.01106

yO2 0.0086 – 0.00616

zO2 −0.0185 – −0.01576

B (GPa) 115 109 –

Eg (eV) 4.6 4.7 –

Ec (mHa) −1.56 −2.06 –

Rhombohedral (R3m)

a (A) 4.010 4.029 4.0046

α (deg) 89.800 89.727 89.8396

V (A3) 64.47 – 64.176

xT i −0.0132 −0.0151 −0.01286

xO 0.0232 0.0129 0.01096

zO 0.0124 0.0242 0.01936

B (GPa) 121 114 –

Eg (eV) 4.8 4.9 –

Ec (mHa) −1.62 −2.24 –

aAmbient pressure, high temperature from Ref. 13.bRoom temperature, 1.6 GPa from Ref. 72.cComputed from the elastic constants of Ref. 41.

the direct band gap of the cubic and tetragonal phases deviate

from the experimental data by 0.8 eV, the 0.2 eV increase of

the band gap from the cubic to the tetragonal phase is perfectly

reproduced. As a further internal check for consistency, the en-

ergy difference Ec of the ferroelectric phases with respect to

the cubic one is found to regularly increase, in absolute value,

while going down the phase transition series from tetragonal

to rhombohedral, according to their relative stability at 0 K.

According to the displacive model, ferroelectric phase

transitions in BaTiO3 are driven by soft phonon modes.8

The structural features of the ferroelectric phases are then

very closely connected to their vibrational properties. In

Table III we report computed phonon frequencies of both

transverse-optical (TO) and longitudinal-optical (LO) modes,

as compared with available experimental data. To the best

of our knowledge, no experimental phonon frequency data

have been reported for the cubic phase while a number of

theoretical simulations have been performed.20,21,25,26,28 A

quite complete experimental characterization is reported for

the tetragonal phase spectrum32 while few phonon frequency

values are reported for the orthorhombic and rhombohedral

phases.34,35 All phonon modes in the table are labeled

according to the symmetry irreducible representation (irrep)

they belong to. Infrared (IR) intensities Icalc for the TO modes

are also reported, as computed for each mode p by means of

the mass-weighted effective mode Born charge vector Zp,75,76

evaluated through a Berry phase approach:68,77

Ip =π

3

NA

c2dp|Zp|2 with |Zp|2 = |

∂μ

∂Qp

|2 , (6)

where NA is Avogadro’s number, c the speed of light, dp the

degeneracy of the pth mode, μ the cell dipole moment, and

Qp the normal mode displacement coordinate. More details

on the calculation of the infrared intensities in the CRYSTAL

program can be found in Ref. 78.

Since our simulations are performed at 0 K, soft phonon

modes driving the transitions from the cubic to the tetragonal

and orthorhombic phases are projected down to absolute zero

and, thus, show imaginary frequencies. As expected, the low-

temperature rhombohedral phase is characterized by phonon

frequencies which are all positive, which implies its stability

at 0 K. In particular, for the cubic phase, we find an imaginary

frequency of i230 cm−1 that is comparable to i240 cm−1

and i257 cm−1 obtained by Evarestov and Bandura in their

recent ab initio periodic study at the PBE and PBE0 level

of theory, respectively.20 For the tetragonal phase, calculated

optical phonon frequencies are in good agreement with the rich

set of experimental values measured by Nakamura.32 The only

significant deviation refers to a TO phonon mode belonging to

the totally symmetric A irrep, for which the calculated value

is 373 cm−1 and the measured one 275 cm−1.

For the orthorhombic and rhombohedral ferroelectric

phases, few experimental phonon frequencies are reported,

in good agreement with our simulated values, with the

exception of a single TO mode of A1 symmetry of the

rhombohedral phase where the computed value of 320 cm−1

largely overestimates the experimental value of 242 cm−1.

Given the overall good agreement with the experimental data,

such few deviations are likely to be due to specific problems in

extracting the correct vibration frequencies from incomplete

experimental spectra.

B. The rhombohedral phase

In this section, dielectric, elastic, piezoelectric, and photoe-

lastic properties of the low-temperature rhombohedral phase

045103-4

APPENDIX

169


TABLE III. Computed vibration frequencies of the optical phonon modes for the four different phases of BaTiO3 as compared with

available experimental data. For the tetragonal, orthorhombic, and rhombohedral phases, the measured frequencies are taken from Refs. 32,

34 and 35, respectively. The longitudinal-optical/transverse-optical (LO/TO) splitting has been computed. Theoretical values of the infrared

(IR) intensities are also reported for TO modes. Phonon modes are labeled according to the irreducible representation (irrep) they belong to.

Imaginary frequencies are labeled with i.

Cubic Tetragonal Orthorhombic Rhombohedral

Mode νcalc Icalc Mode νcalc νexp Icalc Mode νcalc νexp Icalc Mode νcalc νexp Icalc

(irrep) (cm−1) (km/mol) (irrep) (cm−1) (cm−1) (km/mol) (irrep) (cm−1) (cm−1) (km/mol) (irrep) (cm−1) (cm−1) (km/mol)

TO TO TO TO

T1u i230 – E i128 Soft – B2 i71 – A1 176 173 186

T1u 193 453 A 168 175 237 B1 174 313 E 180 575

T2u 317 Not active E 182 181 247 A1 178 Not active E 245 4681

T1u 489 60 B 303 Silent Not active B2 180 193 115 A2 302 Silent Not active

LO E 316 306 30 B1 285 270 1872 E 318 Not active

T1u 180 A 373 275 1308 B1 302 301 A1 320 242 1539

T1u 486 E 487 487 105 A2 316 320 Not active E 491 347

T1u 694 A 566 516 648 A1 318 Not active A1 557 522 325

LO A1 345 16 LO

E 176 180 B2 487 490 59 E 188

A 199 191 B1 496 270 A1 193 187

E 315 305 A1 562 532 198 E 318 310

E 479 468 LO E 473

A 492 471 B2 176 A1 503 485

E 701 717 A1 178 A1 729 714

A 775 725 B1 190 E 739

B1 300

A1 318 320

A1 349

B1 472

B2 477

A1 594

B2 704

B2 752 720

of BaTiO3 are discussed, and the effect on these properties of

the adopted one-electron Hamiltonian is investigated.

1. Dielectric tensor

Given the relatively high symmetry of the system, there

are only two independent components in the optical (i.e.,

purely electronic) dielectric tensor: 11 and 33. They were

computed as a function of the electric field wavelength λ

with four different one-electron Hamiltonians: HF, PBE0,

LDA, and PBE (Fig. 1). From previous applications of the

CPHF/KS method, we know that the generalized-gradient

approximation to the DFT, such as the PBE functional, usually

provides the best agreement with experimental dielectric

tensors, even better than hybrid schemes when the crystal

structure, including cell parameters, is fully relaxed.62,79

All the Hamiltonians describe 11 > 33. The HF method

gives very small values of dielectric constants, relatively close

to each other and almost independent on the electric field

wavelength λ. Pure DFT functionals, PBE and LDA, predict

much larger values (LDA more so than PBE), with a larger

separation between them and with a strong dependence on

λ. As expected, the hybrid PBE0 functional provides an

intermediate description of the dielectric constants, in all

respects.

FIG. 1. For each considered Hamiltonian, the two dielectric

constants 11 and 33 (in units of 0) are reported as a function of

the electric field wavelength λ, for the rhombohedral phase; 11 > 33

in all cases.

045103-5

APPENDIX

170


As clearly seen from pure DFT and hybrid data, which are

known to accurately describe dielectric properties of solids,

the explicit account of the dependence of computed dielectric

response properties on the electric field wavelength λ is

found to be mandatory before any comparison with future

experiments, which, as usual, will be performed at finite (and

relatively small) values of λ (between 500 and 600 nm).

Two previous theoretical studies, one by Wu et al.31 in

2005 and one by Evarestov and Bandura20 in 2012, discussed

their results of the dielectric constants of the rhombohedral

phase of BaTiO3 comparing with what they assumed to be

the corresponding experimental counterpart, that is, data from

Ref. 80 (2001). As a matter of fact, the values reported in

Ref. 80 are in turn taken from a much earlier study by Wemple

et al.71 (1968), which, however, refers to the room-temperature

tetragonal phase of BaTiO3 (see Fig. 4 therein and the related

discussion). In that work, the authors also explicitly measured

the λ dependence of the dielectric constants, in the 400–700 nm

range, which was found to be rather strong.

2. Elastic and photoelastic properties

Predicted elastic and photoelastic properties of low-

temperature BaTiO3, with four different Hamiltonians, are

given in Table IV. In particular, elasto-optic constants here

refer to the λ → ∞ limit; see below for an explicit account

of electric field wavelength dependence. Unfortunately no

experimental data are currently available to compare with.

From previous studies, we expect the hybrid PBE0 scheme

to give the best description of elastic properties and the PBE

functional the best description of photoelastic properties.47,49

Electronic “clamped-ion” and total “nuclear-relaxed” values

are reported; their difference corresponds to the nuclear

relaxation term, which is found to be dramatic. Even with

a balanced Hamiltonian, such as PBE, it can be as large as

twice the electronic contribution (see, for instance, C13 and

C14).

Elastic properties of isotropic polycrystalline aggregates

can be computed from the elastic and compliance constants

defined in Sec. II B via the so-called Voigt-Reuss-Hill averag-

ing scheme.81 For a rhombohedral crystal, the adiabatic bulk

modulus is defined as the average B = 12[BV + BR] between

Voigt upper and Reuss lower bounds:

BV = 19(2C11 + C33 + 2C12 + 4C13),

BR = (2S11 + S33 + 2S12 + 4S13)−1 .

The shear modulus G = 12[GV + GR] is expressed as the

average between Voigt upper GV and Reuss lower GR bounds:

GV = 115

(2C11 + C33 − C12 − 2C13 + 6C44 + 3C66),

GR = 15(8S11 + 4S33 − 4S12 − 8S13 + 6S44 + 3S66) .

From the average bulk and shear moduli defined above,

Young’s modulus E and Poisson’s ratio σ are defined as

follows:

E =9B G

3B + Gand σ =

3B − 2G

2(3B + G). (7) T

AB

LE

IV.

Ela

stic

and

photo

elas

tic

const

ants

of

the

rhom

bohed

ral

phas

eof

BaT

iO3

asco

mpute

dw

ith

four

dif

fere

nt

one-

elec

tron

Ham

ilto

nia

ns.

The

den

sity

ρ,

along

wit

ha

num

ber

of

poly

cryst

alli

ne

aggre

gat

eel

asti

cpro

per

ties

,su

chas

bulk

modulu

sB

,sh

ear

modulu

sG

,Y

oung’s

modulu

sE

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ois

son’s

rati

oσ

and

tran

sver

se,v

s,

and

longit

udin

al,v

p,

seis

mic

wav

e

vel

oci

ties

,ar

eal

sore

port

ed.E

lect

ronic

“cla

mped

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and

tota

l“n

ucl

ear-

rela

xed

”val

ues

are

report

edfo

rea

chquan

tity

.

Ela

stic

const

ants

(GP

a)P

hoto

elas

tic

const

ants

ρB

GE

σv

sv

p

C1

1C

12

C1

3C

14

C3

3C

44

C6

6p

11

p1

2p

13

p1

4p

33

p4

4p

66

(g/cm

3)

(GP

a)(G

Pa)

(GP

a)(k

m/s)

(km

/s)

HF

Rel

axed

290

66

176

282

45

112

0.0

74

0.2

41

0.2

49

−0.0

97

0.4

67

0.1

25

−0.0

83

5.9

2110

41

109

0.3

32.6

5.3

Cla

mped

399

116

91

28

395

116

141

0.0

36

0.0

96

0.1

43

−0.0

46

0.0

43

0.0

01

−0.0

30

5.9

2199

131

322

0.2

34.7

7.9

LD

A

Rel

axed

327

96

64

34

319

83

116

0.2

28

0.1

69

0.3

14

−0.1

69

0.5

10

0.2

07

0.0

30

6.3

1158

99

245

0.2

44.0

6.8

Cla

mped

381

119

109

9377

122

131

−0.0

26

−0.0

10

0.0

32

−0.0

41

−0.0

34

0.0

22

−0.0

08

6.3

1202

128

317

0.2

44.5

7.7

PB

E

Rel

axed

250

70

31

43

226

39

90

0.2

17

0.1

85

0.3

41

−0.1

66

0.5

58

0.2

61

0.0

16

5.8

5109

52

134

0.2

93.0

5.5

Cla

mped

323

101

85

12

299

95

111

−0.0

53

0.0

02

0.0

69

−0.0

35

0.0

05

0.0

05

−0.0

27

5.8

5165

105

259

0.2

34.2

7.2

PB

E0

Rel

axed

282

73

31

47

258

44

104

0.1

61

0.1

93

0.2

99

−0.1

40

0.5

00

0.2

13

−0.0

16

6.0

2121

61

156

0.2

83.2

5.8

Cla

mped

365

113

94

14

340

108

126

−0.0

32

0.0

32

0.0

98

−0.0

41

0.0

18

0.0

02

−0.0

32

6.0

2185

118

293

0.2

44.8

7.5

045103-6

APPENDIX

171


FIG. 2. (Color online) Elasto-optic constants pvu of the rhombo-

hedral phase of BaTiO3 as computed at PBE level, as a function of

the electric field wavelength λ. Computed values at the λ → ∞ limit

are also reported as red circles.

According to the elastic continuum theory, the three acoustic

wave velocities of a crystal, along any general direction, are

related to the elastic constants by Christoffel’s equation.82,83

Within the Voigt-Reuss-Hill averaging scheme, the average

values of transverse (shear), vs , and longitudinal, vp, seismic

wave velocities, for an isotropic polycrystalline aggregate, can

be computed from the elastic properties defined above and the

density ρ of the crystal as84

vs =

G

ρand vp =

B + 4

3G

ρ. (8)

All the elastic properties introduced above have been com-

puted with different Hamiltonians for the rhombohedral low-

temperature phase of BaTiO3 and reported in Table IV. The

relevance of the nuclear relaxation effect, which systematically

reduces the rigidity of the system, can be clearly seen from all

these average properties.

As stated above, the elasto-optic constants reported in

Table IV refer to the λ → ∞ limit. Since Brillouin scattering

experiments are usually performed at finite electric field

wavelengths (500 nm < λ < 600 nm), in order to facilitate

the comparison with future experimental measurements, we

have computed the explicit dependence of all these constants

on λ at the PBE level: a wide range, from 500 to 1000 nm, is

considered. Results are shown in Fig. 2, along with simulated

values at λ → ∞. None of the elasto-optic constants changes

sign as a function of λ. All six symmetry-independent positive

components decrease with λ while the only negative one, p14,

increases with λ. Three constants, p12, p66 and p14, show a

relatively small dependence on the field wavelength, whereas

TABLE V. Direct and converse piezoelectric constants of the

rhombohedral phase of BaTiO3, as computed with four different

Hamiltonians. Electronic and total nuclear relaxed values are given.

Direct (C/m2) Converse (pm/V)

e15 e21 e31 e33 d15 d21 d31 d33

HF

Relaxed −7.52 3.24 −3.30 −4.41 1562a −511a −9.2 −15.6

Clamped 0.14 −0.19 0.06 −0.14 1.6 −0.8 0.2 −0.4

LDA

Relaxed −5.81 3.75 −4.77 −6.46 −95.0 30.3 −8.7 −16.8

Clamped 0.13 −0.15 0.04 −0.12 1.1 −0.6 0.2 −0.4

PBE

Relaxed −4.31 1.93 −2.11 −3.52 −290 80.6 −5.2 −14.1

Clamped 0.20 −0.28 0.05 −0.23 2.5 −1.4 0.3 −0.9

PBE0

Relaxed −4.67 1.99 −2.17 −3.45 −271 73.9 −5.0 −12.2

Clamped 0.21 −0.28 0.06 −0.22 2.3 −1.2 0.3 −0.8

aThese unusual large values are due to very large elements of the HF

compliance tensor S = C−1 in this case.

the remaining four more strongly depend on λ, particularly in

the region between 500 and 600 nm.

3. Piezoelectricity

In recent years the interest in the piezoelectric response of

simple perovskites at very low temperature has been raised

by the discovery, by Grupp and Goldman in 1997, of a giant

piezoelectric effect of strontium titanate SrTiO3 down to 1.6 K

where piezoelectricity is usually severely reduced.48,85 Such

findings opened the way to applications of these materials at

cryogenic temperatures as actuators for adaptive optics and

low-temperature capacitors.

We report a complete characterization of the direct e and

converse d third-order piezoelectric tensors of rhombohedral

BaTiO3, as introduced in Sec. II B. Piezoelectric constants

are reported in Table V, as computed with four one-electron

Hamiltonians. For this property, as for elastic constants,

hybrid functionals, as PBE0, usually provide a rather good

description. Total values are given along with purely electronic

“clamped-ion” contributions. Nuclear relaxation effect plays

here a fundamental role: three direct, symmetry-independent,

piezoelectric constants out of four change their sign when

including such an effect (e15, e21, and e31). In particular, the

pure electronic contribution to e31 is predicted to be very small,

0.06 C/m2. All these features agree with the outcomes of

a previous LDA theoretical investigation, performed at the

experimental volume.31

Converse piezoelectricity depends on both the direct

piezoelectric tensor and the elastic tensor (or its inverse,

the compliance tensor). The hybrid PBE0 approach with full

geometry optimization is found to provide results that are in

good agreement with those of a previous LDA simulation,

constrained at the experimental volume, where d15 = −243.2,

d21 = 70.1, d31 = −6.8, d33 = −14.7.31

045103-7

APPENDIX

172


IV. CONCLUSIONS

Accurate ab initio simulations have been applied to the

theoretical characterization of several tensorial properties of

the low-temperature rhombohedral phase of barium titanate,

BaTiO3. Dielectric, elastic, piezoelectric, and photoelastic

tensors have been computed with four different one-electron

Hamiltonians (including Hartree-Fock, here reported as a

benchmark), representatives of four different levels of approx-

imation. The explicit treatment of the dependence of dielectric

and photoelastic constants on the electric field frequency

provides with results that can be more directly compared with

future experimental ones.

The adopted computational approach has been discussed

by comparing computed with experimental structural and

vibrational properties of four phases of BaTiO3: cubic Pm3m,

tetragonal P4mm, orthorhombic Amm2, and rhombohedral

R3m. The agreement with available experimental data and

previous theoretical investigations is remarkable. In gen-

eral, we expect predictions by the PBE0 hybrid functional

to be the most reliable as regards elastic and piezoelec-

tric properties of the rhombohedral phase, whereas those

by PBE to describe accurately dielectric and photoelastic

properties.

In the analysis of available experimental data, we could

realize that some experimental measurements of the dielectric

constants of BaTiO3 have been formerly wrongly assigned, by

some authors of theoretical investigations, to the rhombohedral

phase, and not to the correct phase, that is, the tetragonal one.

ACKNOWLEDGMENTS

Gustavo Sophia is kindly acknowledged for his support in

the basis set optimization with the LoptCG script. CINECA

Award No. HP10BLSOR4-2012 is acknowledged for the

availability of high-performance computing resources and

support. Improvements in the massive-parallel version of

CRYSTAL09 were made possible thanks to the PRACE proposal

no. 2011050810.

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045103-9

APPENDIX

174

Appdx 2: Publications

13296 Phys. Chem. Chem. Phys., 2013, 15, 13296--13303 This journal is c the Owner Societies 2013

Cite this: Phys. Chem.Chem.Phys.,2013,

15, 13296

The electronic structure of MgO nanotubes.

An ab initio quantum mechanical investigation

Khaled E. El-Kelany,w*a Matteo Ferrabone,b Michel Rerat,a Philippe Carbonniere,a

Claudio M. Zicovich-Wilsonc and Roberto Dovesib

The structural, vibrational and response properties of the (n,0) and (m,m) MgO nanotubes are

computed by using a Gaussian type basis set, a hybrid functional (B3LYP) and the CRYSTAL09 code.

Tubes in the range 6 r n r 140 and 3 r m r 70 were considered, being n = 2 m the number of

MgO units in the unit cell (so, the maximum number of atoms is 280). Tubes are built by rolling up the

fully relaxed 2-D conventional cell (2 MgO units, with oxygen atoms protruding from the Mg plane

alternately up and down by 0.38 Å). The relative stability of the (n,0) with respect to the (m,m) family,

the relaxation energy and equilibrium geometry, the band gap, the IR vibrational frequencies and

intensities, and the electronic and ionic contributions to the polarizability are reported. All these

properties are shown to converge smoothly to the monolayer values. Absence of negative vibrational

frequencies confirms that the tubes have a stable structure. The parallel component of the polarizability

aJ converges very rapidly to the monolayer value, whereas a

> is still changing at n = 140; however,

when extrapolated to very large n values, it coincides with the monolayer value to within 1%. The

electronic contribution to a is in all cases (aJ and a>; 6 r n r 140) smaller than the vibrational

contribution by about a factor of three, at variance with respect to more covalent tubes such as the BN

ones, for which the ratio between the two contributions is reversed.

I. Introduction

Since their discovery,1 carbon nanotubes (CNT) have attracted

the attention of the scientific community for their unique

electrical, mechanical and thermal properties.2 The search for

noncarbon nanotubes started very soon, at first in the domain

of layered highly anisotropic phases such as hexagonal boron

nitride and transition metal disulfide compounds that can also

adopt cage-like structures such as fullerenes,3–5 and then exploring

isotropic inorganic compounds that can be ‘‘precursors’’ of

nanomaterials in a large variety of morphological forms. Nowa-

days, great attention is paid to the preparation of nanomaterials

based on NaCl-like compounds, such as MgO. Solid MgO is

known to be an inert material with a high melting point,

consistent with strong ionic bonding, and a wide band gap of

7.8 eV.6 Its substrate has been used for high-temperature

superconductor (HTSC) thin-film coating applications world-

wide. The properties of materials at a finite scale of length are

often different from the corresponding bulk properties. For

example, a pronounced covalent contribution to the ionic

bonding exists in small MgO nanoparticles, whereas almost

pure ionic bonds are typical of the bulk of this compound.7–17

The atomic structures of small-sized MgO clusters have been

investigated experimentally7,18 and theoretically.7–17 Mass

spectroscopy experiments18,19 indicate that small (MgO)3 sub-

units are relatively stable. ‘‘Magic (MgO)i clusters’’ for i = 2, 4, 6,

9, 12, and 15 were discovered by Ziemann and Castleman using

laser-ionization time-of-flight mass spectrometry.7

In the present paper we investigate the properties of two

families of MgO tubes, namely (n,0) and (n,n). In both cases the

coordination of cations and anions is 4, as in the monolayer,

whereas it is 6 in the bulk. In the (n,n) case however cation rings

alternate with anion rings, whereas in (n,0) tubes oxygen and

magnesium atoms are present in the same ring (see Fig. 1). The

same scheme has previously been used for investigating

(CNT)20,21 imogolite,22 chrysotile,23 BNNT.24 Recent improve-

ments in the CRYSTAL09 code permit full use of symmetry

a Equipe de Chimie physique, IPREM UMR 5254, Technopole Helioparc,

2 avenue du President Pierre Angot, 64053 Pau Cedex 09, France.

E-mail: [email protected] di Chimica IFM, Universita di Torino and NIS -Nanostructured

Interfaces and Surfaces – Centre of Excellence, Via P. Giuria 7, 10125 Torino,

Italy. Web: http://www.nis.unito.itc Facultad de Ciencias, Universidad Autnoma del Estado de Morelos,

Av. Universidad, 1001, Col. Chamilpa, 62209 Cuernavaca (Morelos), Mexico

† Chemistry Department, Faculty of Science, Minia University, Minia 61519, Egypt.

Received 5th March 2013,

Accepted 3rd April 2013

DOI: 10.1039/c3cp50979f

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View Article OnlineView Journal | View Issue

175

This journal is c the Owner Societies 2013 Phys. Chem. Chem. Phys., 2013, 15, 13296--13303 13297

(the point group contains 140 4 = 560 symmetry operators forthe (140,0) tube) so as to drastically reduce the computationalcost. The total energy and its difference with respect to the MgOmonolayer (l-MgO), the relaxation geometry and energy, the IRvibrational frequencies and intensities are investigated as afunction of n. The polarizability of the tube (both parallel andperpendicular components, electronic and ionic contribution)is also explored, and it is shown to tend to the monolayer valuesas n-N.

The structure of the paper is as follows: Section II is devotedto the description of the method, section III presents the resultswhereas a few conclusions are drawn in section IV.

II. Computational method and details

Calculations were performed by using the periodic ab initio

CRYSTAL09 code,25 the hybrid B3LYP functional of the densityfunctional theory (DFT)26 and a Gaussian type basis set(a 8-511G* contraction for Mg,27 and a 8-411G* set for O,28

where the exponents of the most diffuse valence shells wereoptimized). The DFT exchange–correlation contribution is eval-uated by numerical integration over the unit cell volume. Radialand angular points of the integration grid are generatedthrough Gauss–Legendre radial quadrature and Lebedev two-dimensional angular point distributions. A (99,1454) prunedgrid (XXLGRID keyword in the CRYSTAL09 manual),29 corre-sponding to 99 radial and 1454 angular points, was employed.The integration accuracy can be estimated by the error in theelectronic charge per unit cell, De = 1.0 103|e| (out of a total of2800 electrons for the (140,0) MgO nanotube). Other details on thegrid generation and its influence on the accuracy and cost can befound in ref. 30. Evaluation of the Coulomb and exact exchangeinfinite series is controlled by five parameters29 (T1, T2, T3, T4, T5),

whose values are set to T1 = T2 = T3 = T4 = 12T5 = TI. In this work

we used TI = 10 and a shrinking factor (the number of pointsalong each reciprocal lattice vector at which the Fock matrix isdiagonalized) IS = 8. The electronic polarizability is evaluatedthrough the Coupled Perturbed Kohn–Sham (CPKS) scheme.31,32

The convergence of the SCF zeroth-order energy and CPHF/CPKSiterations is controlled by the TE and TCP parameters, respectively.The SCF cycles are terminated when the difference between thevalues of the total energy (E) or polarizability (a) for two successivecycles is less than 10TE Hartree or 10TCP Bohr3, respectively:here TE = 11 and TCP = 4 are used. Sometimes iterations producelarge oscillations in the Fock/KS matrix, in which case thesematrices were damped by mixing at them andm 1 cycles withthe FMIXING parameter29 of 60%. The symmetry of the tubes(rototranslational and planes, both vertical and horizontal)is fully exploited in the calculation, so that, in spite of ahigh number of atoms in the tube unit cell (up to 280) andthe use of an all-electrons basis set and a hybrid functional,the computational cost is low. For example, a full structureoptimization (18 steps) on a local computing cluster with 12 cores(Intel Xeon X5660 2.8 GHZ) costs about two hours of elapsed timefor the n = 100 tube.

A. Geometry optimization

Fractional atomic coordinates and unit-cell parameters wereoptimized within a quasi-Newton scheme using analytic energygradients combined with the BFGS algorithm for Hessianupdating.33–36 The two kinds of tubes have been rolled up(see the ‘‘NANOTUBE’’ keyword in the Nanotubes’ tutorial atwww.crystal.unito.it) starting from the primitive 2D cell (1 MgOunits), and imposing the full rototranslational symmetry(560 operators for the (140,0) tube). After geometry optimiza-tion, the vibrational spectrum was computed. The presence of

Fig. 1 Optimized geometry of MgO systems of different dimensionality: bulk ((a), 3D); primitive planar (unbuckled) and conventional non planar (buckled) cell of themonolayer ((b) and (c), 2D); (12,0) and (12,12) nanotubes, ((d) and (e), 1D).

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imaginary frequencies in the spectrum indicates that theoptimized structure is not a true global minimum. The sym-metry was then reduced in order to eliminate constraints andlocate the correct equilibrium geometry. To this aim the normalcoordinates corresponding to the imaginary frequencies wereexplored looking for a new energy minimum. The vibrationalfrequencies obtained on the resulting structure are all positive.The very large relaxation taking place in the nanotubes is due tothe fact that the 2D cell containing 1 MgO unit is not the moststable structure for the monolayer. A much more physicalmodel for the MgO slab is obtained by considering a supercellof the 2D lattice containing 2 MgO units, as shown in Fig. 1. Ifthis double 2D cell is optimized, the same kind of bucklingobserved in the nanotubes is reproduced in the monolayer.When the tubes are rolled up starting from this larger nonplanar 2D cell, the number of symmetry operators is half thatstarting from the planar layer (280 for the (140,0) tube, as onlyhalf of the cations and of the anions are symmetry related). Allfrequencies are in this case positive, confirming that theequilibrium position is a real minimum. The effect of suchrelaxation is illustrated in Fig. 2. Open circles give the trend ofthe energy difference between the unbuckled nanotubes andthe unbuckled planar slab. The full circles provide the sameenergy difference when the buckled structures are considered.

B. Frequencies and polarizability

The total static polarizability is determined as follows:

a0 ¼ ae þX

j

Zj2

nj2

(1)

where ae is the electronic (clamped ion) contribution. Thevibrational (ionic) contribution is given, in the double harmonicapproximation, by the second term on the right hand side.Frequencies were obtained by diagonalizing the dynamicalmatrix, found by numerical differentiation of the analytical

energy gradients (see ref. 37 for details). %Zj2 is a mass weighted

effective mode Born charge and nj is the vibrational frequencyof the mode j. Born charges were calculated using a Berry phase-like scheme.38,39 For the largest tube (n = 140) there are more than20 modes at less than 10 cm1; the Eckart conditions40 areimposed in order to eliminate translational and rotationalspurious contributions to the dynamic matrix.

III. Results and discussion

A. Bulk and monolayer

For comparison we report here bulk MgO and (001) monolayerproperties. If planarity is imposed to the layer, the surfaceformation energy is 0.974 eV and the lattice parameter of theprimitive cell (containing one Mg and one O atom/cell) shrinksfrom a = 4.23 Å (the bulk value; the experiment41 is at 4.19 Å) to2.80 Å (corresponding to 3.96 Å in a double cell). However thisgeometry is a saddle point rather than a minimum, as thenegative value of one border-zone (

-

k = (0.5,0.5)) frequencyconfirms (129 cm1). This mode corresponds to the displace-ment of the two oxygen atoms in opposite directions along z.When a double cell is optimized the oxygen atoms movevertically by 0.38 Å, the lattice parameter further reduces to3.89 Å and the energy lowers by 0.035 eV per MgO. The bulkband gap (BG) is 7.4 eV (7.8 eV from experiment42); it increasesto 9.65 eV in the relaxed monolayer. The triple-degenerate F1ntransverse optical (TO) and longitudinal optical (LO) vibrationalmodes are at 384 and 759 cm1 (exp = 394 and 724 cm1,respectively41); they split into two modes (A and E) withn = 577 cm1 and 609 cm1 for the planar slab and 572 cm1

and 621 cm1 for the non planar one.The calculated value of the bulk dielectric constant ee

(electronic) is 2.48 to be compared to 2.38 from experiment.43

The value of the static a0 constant is 9.69 (9.83 from experi-ment44). The fully relaxed a

J and a> slab values are 2.258 and

Fig. 2 Energy difference DE in mHa (103 Hartree) between the (n,0) nanotubes and the planar (unbuckled, open circle) and non planar (buckled, filled circles)monolayer of MgO. The zero of the energy corresponds to the non planar (conventional, double) cell. The energy of the primitive planar cell, containing 1 MgO units is1.302 mHa (0.035 eV) higher. When all the oxygens are forced to be symmetry related (and then to have the same radial distance from the tube axis) the large radiuslimit is the planar slab. When oxygens are allowed to relax in opposite directions the tube energy tends to the relaxed monolayer, in which the oxygen atoms areprotruding by 0.38 Å.

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0.811 for ae and 7.082 and 1.761 Å3 for a0 (the correspondingnumbers for the planar slab are 2.241, 0.756, 7.091 and 1.832 Å3).

Mulliken net charges in the bulk are 1.81 |e| and the bondpopulation is as small as +0.004 |e|, confirming the fully ionicnature of the Mg–O bond. In the slab the Mulliken net chargesare slightly smaller (1.72 |e|) and the bond population slightlylarger (+0.027 |e|), indicating the appearance of some degree ofcovalent character. The nanotube charge and bond populationare very close to the values in the slab even at low n values(1.72 |e| and +0.026 |e|, respectively, at n = 12) and convergevery rapidly to the slab limiting value.

Born charges provide a measure of the polarizability of thesystem, and represent a more physical measure of the beha-viour of charge distribution. Their value (we report one third ofthe trace of the tensor) is 1.98 for the bulk, 1.63 for themonolayer and 1.51 for the (12,0) MgO nanotube, with avariation from the bulk to the (12,0) nanotube nearly five timeslarger than for the Mulliken charges.

B. Nanotubes: geometry and energy

Table 1, first column, shows (DE), the energy differencebetween the (n,0) and the (n/2,n/2) tubes (see also Fig. 4, top).DE is as large as 16 mHa (0.435 eV) per MgO at n = 12 (24 atoms)and decreases by two orders ofmagnitude to 0.158mHa (0.0043 eV)per MgO at n = 140 (280 atoms). There are two possible reasonsfor the larger stability of the (n,0) tubes with respect to the(n/2,n/2) ones: (a) a different pattern of the fourfold coordina-tion of both cations and anions (for the (n/2,n/2) tubes the fourneighbors of Mg belong to different rings), whereas for (n,0) twoof the neighbors are on the same ring, as shown in Fig. 1, (b) thelower radius (and the higher strain) of (n/2,n/2) tubes withrespect to (n,0), as shown in Fig. 3. In order to separate thesetwo effects, DE was also reported as a function of the tube

radius (we used RMg for the comparison); the results are shownin Fig. 4, bottom. It turns out that the separation of the twocurves reduces drastically; (n,0) tubes are, however, more stablethan the (m,m) tubes, the difference tending to zero for verylarge tubes.

The second column in Table 1 provides the relaxation energydE of the (n,0) tubes, i.e. the energy difference between theequilibrium structure of the tube and the value obtained byrigidly rolling up the equilibrium monolayer. For the smallesttube it is as large as 33.6 mHa (0.914 eV) per MgO unit,then decreases rapidly (see Fig. 5). At n = 100 dE is negligible(4 mHa or 0.0001 eV, about four orders of magnitude smallerthan for n = 6).

The third column provides DE, the energy differencebetween the tube and the monolayer. Also DE becomes negli-gible (35 mHa or 0.00095 eV) at n larger than 100. It should benoticed that the very regular behaviour of both curves, shown inFig. 5, documents the high numerical accuracy of the code; theDE trend also implies that the same accuracy is obtained whentreating systems of different dimensionality (1D and 2D).Columns Ru, RMg and RO provide a measure of geometricalrelaxation, that for the smallest tubes is very important, asshown in Fig. 3. There are two types of O atoms, ‘‘inside’’ and‘‘outside’’ the Mg ring (as was for the slab, with one oxygenabove and one below the Mg plane). For small radii all atomstend to move outwards, in order to reduce the strain; in (12,0),the Mg radius increases by 0.11 Å whereas one anion (O outside)moves farther away by as much as 0.53 Å, in order to reduceshort range repulsion, which is no more compensated by thestrong electrostatic field as in the bulk. The oxygen moving‘‘inside’’ reduces its radius by 0.23 Å. At larger radii (the figurereports data for (24,0) and (12,12)) relaxation is nearly negligible,and the two oxygen radii are already at 0.38 Å with respect

Table 1 Calculated properties of the (n,0) series of MgO nanotubes and of the monolayer (l-MgO). DE, dE and DE are the energy difference between the (n,0) tubeand the corresponding (n/2,n/2) one (they have the same number of atoms), the relaxation energy for the rolled configuration and the energy difference between therelaxed tube and l-MgO, respectively. Values reported are in mHa per MgO unit. Ru is the unrelaxed radius (in Å) of the Mg cations (the anions O are at Ru 0.38); RMg

and RO are the same distances after relaxation (RO refers to the ‘‘inner’’ oxygen, see text and Fig. 3). The radius of the oxygen atom ‘‘outside’’ the tube (at Ru +0.38before relaxation) remains essentially unaltered. BG is the band gap in eV. aJ and a

> are the longitudinal and transverse components of the electronic ae, and static a0polarizabilities per MgO unit (in Å3). The unrelaxed (sum over states) aSOS values are also reported. The Layer* row gives the average of the monolayer perpendicularpolarizabilities per MgO unit using the relation a

>

n = 12(a>

l + aJ

l ) (l stands for layer, n is the label of the tube) which should be equal to the large radius limit (N row) ofthe transverse nanotube polarizability

n DE dE DE Ru RMg RO BG

aJ

a>

aeSOS a

ea0

aeSOS a

ea0

6 33 601 4568 1.86 1.98 2.06 9.02 2.134 2.300 6.239 1.803 1.236 2.95112 15 664 4489 1646 3.71 3.82 3.86 9.55 2.088 2.259 6.873 1.746 1.272 2.90524 3801 437 881 7.42 7.41 7.18 9.56 2.099 2.261 6.865 1.748 1.356 3.28730 205 622 9.28 9.26 8.99 9.58 2.101 2.261 6.900 1.754 1.383 3.40236 1684 109 459 11.13 11.11 10.81 9.59 2.102 2.261 6.940 1.758 1.403 3.49340 1364 75 383 12.37 12.35 12.04 9.59 2.102 2.260 6.953 1.760 1.413 3.54448 956 40 276 14.84 14.82 14.49 9.60 2.102 2.260 6.990 1.762 1.430 3.63260 619 19 183 18.56 18.54 18.19 9.61 2.102 2.259 7.009 1.764 1.448 3.73380 372 6 106 24.74 24.73 24.37 9.62 2.102 2.259 7.043 1.765 1.468 3.853100 257 4 68 30.93 30.91 30.55 9.62 2.102 2.259 7.056 1.766 1.480 3.939140 158 1 35 43.29 43.28 42.91 9.63 2.102 2.258 7.066 1.767 1.495 4.049N 0 0 9.66 2.102 2.258 7.084 1.767 1.536 4.376Layer* 1.767 1.535 4.422Layer 0 0 9.65 2.101 2.258 7.082 1.432 0.811 1.761

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to the Mg radius, as in the slab they are at 0.38 Å from the Mgmonolayer.

C. Nanotubes: band gap and polarizability

The bandgap (BG) shows good convergence to the slab indirect gapvalue of 9.65 eV: it is 9.02 eV at n = 6 (this tube is very strained),

increases to 9.55 eV at n = 12 and reaches 9.60 eV at n = 48. Thelast six columns in Table 1 report the values of the parallel andperpendicular polarizability of the (n,0) tubes as a function of n(see also Fig. 6). The electronic uncoupled contribution (oftenindicated as Sum Over States –SOS–) is first reported; it corre-sponds to the situation in which the effect of the applied fieldon the charge distribution is not taken into account (in otherwords the unperturbed charge distribution interacts with thefield). Then, the fully coupled polarizability is given, as result-ing from the CPKS-SCF scheme.45 It is interesting to note thatthe coupling (resulting from the difference between ae and aSOS)increases by about 7% the parallel value, and decreases by morethan 31% the perpendicular value at n = 6. At the other extreme(n = 140) the correction remains essentially the same for a

J

(7%), and reduces to about 15% for a>. The two components

Fig. 4 Energy difference DE in mHa per MgO units with respect to the slab forthe (n,0) (open circles) and (n/2,n/2) (filled circles) tubes as a function of n (top).In the bottom figure the same energy difference is plotted as a function of thetube radius.

Fig. 5 Relaxation energy dE and energy difference DE between the relaxed (n,0)nanotube and the relaxed monolayer as a function of n. Energies are in mHa perMgO unit.

Fig. 3 The unrelaxed and relaxed distances (in Å) of Mg and O atoms from the tube axis for (12,0), (6,6), (24,0) and (12,12) tubes.

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(aJ and a>) have about the same value at the SOS level, whereas

for the fully coupled case aJ is about two times larger than a> at

n = 6; this ratio decreases to 1.5 at n = 140. In the last line of thetable the values for the monolayer are shown: the asymptoticvalue for the tubes, obtained by fitting the data from n = 24 ton = 140 are also reported. The asymptotic (N) and the mono-layer values differ by less than 1% in all cases. Finally, thevibrational contribution to the polarizability (avib) is also shown,as obtained from a frequency calculation (avib = a0 ae, Table 1).The vibrational contribution is larger by a factor of 1.4 than theelectronic one for a> at n = 6, but increases much faster than thelatter along the series: at n = 140 the electronic contribution isonly 37% of the total. For the parallel contribution the situationis slightly different: the vibrational contribution is about twotimes the electronic one at n = 6, and this ratio remains aboutconstant up to n = 140. The parallel component of a0 increasesalong the series by about 0.8 Å3, to be compared to 1.5 Å3 for theperpendicular component. Also for the vibrational contributionthe extrapolated value a(n - N) is extremely close to themonolayer limit.

It is interesting to analyze where the vibrational contributioncomes from. For all tubes there are only four IR active modes(whose symmetry is A, E1, E2, and E3). They are obviously theonly ones that contribute to the polarizability. The A modedescribes a motion that contributes to polarization along thetube axis (direction x in our orientation), whereas the twofolddegenerate modes E contribute to the polarization in the other

two directions (y and z, equivalent by symmetry). The wave-numbers and intensities are reported in Table 2 and shown inFig. 7 (in the latter only the results from n = 24 are shown, tobetter identify the limiting values). In order to be able toextrapolate also the individual contributions to polarizabilityto the infinite radius limit, we performed two additionalcalculations for the (160,0) and (180,0) tubes (the latter con-tains 360 atoms). Graphical animation of the vibrational modesis provided as supplementary information on CRYSTAL’s web-site (http://www.crystal.unito.it/prtfreq/jmol.html). The A modecorresponds to the opposite displacements of Mg and O atomsalong the x-direction (the periodic direction). The corre-sponding vibrational frequency tends, in the infinite radiuslimit, to the one of the degenerate E modes of the monolayer.This mode contributes to the parallel polarizability axx of thetube which, in turn, corresponds in the n - N limit to theparallel component of the monolayer aXX. Table 2 shows thatfor the largest tubes the frequency and intensity coincide withthe monolayer values. The twofold degenerate E1 mode issimilar to the A mode since Mg and O atoms move in oppositedirections. Whereas in the Amode the displacement takes placealong the periodic x-direction, in the E1 mode the displacementis parallel to the circumference of the tube. This mode corre-sponds to the other component of the degenerate mode E of themonolayer, as confirmed by the trends of frequencies andintensities provided in Table 2. This mode contributes to thetransverse polarizability of the nanotubes ayy = azz, which in then-N limit corresponds to half of the parallel polarizability ofthe slab aYY/2. The E2 mode involves opposite displacements of

Fig. 6 The longitudinal aJ and transverse a> polarizability of (n,0) MgO nano-

tubes as a function of the nanotube size n. The electronic Uncoupled (or SOS:Sum Over States) aSOS, the electronic Coupled ae and the vibrational avib

contributions are reported. Values are in Å3 per MgO unit. The fitting function

a ¼ aþb

nþ

c

n2þ

d

n3was used.

Table 2 The IR active vibrational frequencies and intensities for the (n,0) seriesof MgO nanotubes and double cell monolayer (l-MgO). The extrapolated valuesare obtained by fitting the data from n = 80 to n = 180 with the function

f ðnÞ ¼ aþb

nþ

c

n2. The last line reports the double cell monolayer values. The tube

modes A and E1 modes of the tube correspond to the two components of thedegenerated mode E of the slab, whereas the degenerate E2 and E3 modes tendto the nondegenerate A and the degenerate E0 modes of the monolayer,respectively. The intensity of the A and E1 modes of the tube tends toEslab

2¼

786

2¼ 393. The values reported are in cm1 for the frequency and

km mol1 for the intensity

n

Frequency (n) Intensity (I )

A E1 E2 E3 A E1 E2 E3

6 637 636 420 716 397 256 31 012 591 684 520 685 401 260 70 024 589 572 591 707 397 2 220 15030 586 580 594 699 396 10 233 14236 582 585 598 690 395 39 224 13340 581 587 600 686 395 61 215 12648 579 591 604 679 394 106 195 11260 577 593 607 672 394 165 168 9180 575 594 611 666 393 236 138 63100 574 593 613 661 393 282 121 42140 573 591 616 658 393 327 107 21160 573 590 616 657 393 340 103 16180 573 588 617 657 393 348 100 12N 572 574 621 656 393 401 90 0Layer 572 621 654 786 91 0

E A E0 E A E0

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Mg and O atoms along the direction which is perpendicular tothe circumference of the tube. This mode is equivalent to theout of plane non degenerate mode of the monolayer. At then-N limit the contribution of the E2 mode to the transverseazz (ayy) polarizability of the tube tends to halve the perpendi-cular component aZZ of the slab (azz = ayy = 0.53 Å3 for n = 180,while aZZ/2 = (1.761 0.811)/2 = 0.48 Å3). The E3 mode issimilar to the E1 mode since the displacements of Mg and Oatoms are opposite and parallel to the circumference of thetube. The only difference is that neighboring rows (rings) ofatoms are moving in phase in E1 while in E3 they are moving inantiphase. In the slab this mode corresponds to a phonon in apoint of the First Brillouin Zone different from G. For thisreason in the n-N limit the contribution of this mode to thepolarizability is vanishing. In all cases, as shown in Table 2 andFig. 7, the asymptotic values for frequencies, intensities andcontribution to the polarizability coincide with high accuracywith the monolayer values.

It is interesting to compare the total polarizability of theMgO tube with the one of the more covalent BN case, for tubesof about the same radius. At n = 60, the numbers to becompared with the 60 entry of our Table 1 are 4.59 Å3 (electroniccontribution) and 7.15 Å3 (total) for aJ of BN, and 2.32 Å3 and3.17 Å3 for a> of BN.24 The electronic contribution is then abouttwice larger for BN than for MgO, whereas the total a0 of BN isabout the same as in MgO, indicating that the vibrational

contribution is much larger for MgO that for BN. The reasonis simple: looking just at the largest contribution to polariz-ability and considering the limiting case of the slab, the BNfrequency is larger than the MgO one (836 and 621 cm1,respectively; remember that the frequency appears squared tothe denominator in the formula defining the polarizability) andthe intensity (that appears in the numerator) much smaller(13.06 km mol1 for BN and 91 for MgO).

IV. Conclusions

In the present study the structural, vibrational and responseproperties of the MgO (n,0) and (m,m) nanotube families wereinvestigated. The (n,0) tubes were shown to be more stable thanthe (m,m) ones; the energy difference decreases when the radiusincreases and both families tend to the monolayer case(described by a cell containing two MgO units, the two oxygenatoms being displaced above and below the Mg plane) regu-larly. The tubes are shown to be stable structures (all vibrationalfrequencies are positive). Both the electronic and the ionic(vibrational) contributions to the polarizability were evaluatedand compared to the ones of the monolayer. The componentsparallel to the tube axis converge very rapidly to the monolayerasymptotic value, whereas the convergence of the perpendicularcomponent is much slower. The electronic contribution topolarizability is smaller than the vibrational one, indicatingthat these tubes are characterized by IR active modes with verylow frequencies. Overall, this study shows that the properties oftubes as large as (140,0), that contains 280 atoms in the unit celland have a radius not too far from the ones of technologicalinterest (30–100 nm),46 can be accurately simulated with a richall electron basis set and adopting a sophisticated functional,such as the hybrid B3LYP.

Acknowledgements

The authors acknowledge the CINECA Award No. HP10BLSOR4-2012 for the availability of high performance computing resourcesand support, and Mexican CONACyT for financial supportthrough project CB-2012-1 178853. El-Kelany also acknowledgesthe Egyptian government for supporting a grant to do this work atthe Universite de Pau et des Pays de l’Adour in France.

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Fig. 7 Frequencies (n) and intensities (I ) for the IR active modes of the (n,0)nanotubes as a function of n. The mode symmetry is indicated. The right scale inthe frequency and intensity figures represents the data of the E3 mode, and theleft scale corresponds to the data for the other three modes A, E1, E2. Values arein cm1 for frequency n and km/mol for the intensity (I ).

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R. Dovesi, J. Comp. Chem., 2010, 31, 855.21 R. Demichelis, Y. Noel, P. D’Arco, M. Rerat, C. M. Zicovich-

Wilson and R. Dovesi, J. Phys. Chem., 2011, 115, 8876.22 R. Demichelis, Y. Noel, P. D’Arco, L. Maschion, R. Orlando

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APPENDIX

182

1 © 2014 IOP Publishing Ltd Printed in the UK

1. Introduction

The low temperature phase of silica, α-quartz, has a trigonal

crystalline structure of symmetry group P3221 (or P3121),

containing three SiO2 formula units per unit cell. Its structure

consists of sharing-corner SiO4 tetrahedra. Among silica poly-

morphs, α-quartz (α-SiO2 in the following) is the most stable

form at ambient conditions (up to 3 GPa) [1]. Upon heating

at atmospheric pressure, α-SiO2 undergoes a phase transition

at 846 K to β-SiO2 [2]; in the transition, silicon atoms are

displaced by 0.03 nm and the crystalline system passes from

trigonal to hexagonal with space group P6222 (or P6422) [3].

Due to its peculiar piezoelectric properties, α-SiO2 is

widely applied in the electronic industry. However, its

Journal of Physics: Condensed Matter

Piezoelectric, elastic, structural and

dielectric properties of the Si1−xGexO2

solid solution: a theoretical study

Kh E El-Kelany1, 2, A Erba3, P Carbonnière1 and M Rérat1

1 Equipe de Chimie Physique, IPREM UMR5254, Université de Pau et des Pays de l'Adour, 64000 Pau,

France2 Chemistry Department, Faculty of Science, Minia University, Minia 61519, Egypt3 Dipartimento di Chimica and Centre of Excellence NIS (Nanostructured Interfaces and Surfaces),

Università di Torino, via Giuria 5, IT-10125 Torino, Italy

E-mail: [email protected]

Received 17 February 2014, revised 7 March 2014

Accepted for publication 11 March 2014

Published 25 April 2014

Abstract

We apply irst principles quantum mechanical techniques to the study of the solid solution

Si1−xGexO2 of α-quartz where silicon atoms are progressively substituted with germanium

atoms, to different extents, as a function of the substitutional fraction x. For the irst time,

the whole range of the substitution (x = 0.0, 0.16, 0.3, 0.5, 0.6, 0.83, 1.0), including pure

end-members α-SiO2 and α-GeO2, is explored. An elongated supercell (doubled along the

c crystallographic axis) is built with respect to the unit cell of pure α-quartz and a set of 13

symmetry-independent conigurations is considered. Their structural, energetic, dielectric,

elastic and piezoelectric properties are computed and analyzed. All the calculations are

performed using the CRYSTAL14 program with a Gaussian-type function basis set with

pseudopotentials, and the hybrid functional PBE0; all geometries are fully optimized at

this level of theory. In particular, for each coniguration, fourth-rank elastic and compliance

tensors and third-rank direct and converse piezoelectric tensors are computed. It has already

been shown that the structural distortion of the solid solution increases, almost linearly, as the

substitutional fraction x increases. The piezoelectric properties of the Si1−xGexO2 solid solution

are found to increase with x, with a similar quasi-linear behavior. The electromechanical

coupling coeficients are enhanced as well and the linear trend recently predicted by Ranieri

et al (2011 Inorg. Chem. 50 4632) can be conirmed from irst principles calculations. These

doped crystals do represent good candidates for technological applications requiring high

piezoelectric coupling and high thermal stability.

Keywords: α-quartz, piezoelectricity, PBE0 hybrid functional, Gaussian basis sets,

CRYSTAL program

(Some igures may appear in colour only in the online journal)

0953-8984/14/205401+9$33.00

doi:10.1088/0953-8984/26/20/205401J. Phys.: Condens. Matter 26 (2014) 205401 (9pp)

APPENDIX

183

PHYSICAL REVIEW B 88, 035102 (2013)

Piezoelectricity of SrTiO3: An ab initio description

A. Erba,1,* Kh. E. El-Kelany,2,3 M. Ferrero,1,2 I. Baraille,2 and M. Rerat2

1Dipartimento di Chimica and Centre of Excellence Nanostructured Interfaces and Surfaces (NIS), Universita di Torino, via Giuria 5,

IT-10125, Torino, Italy2Equipe de Chimie Physique, IPREM UMR 5254, Universite de Pau et des Pays de l’Adour, FR-64000, Pau, France

3Chemistry Department, Faculty of Science, Minia University, Minia 61519, Egypt

(Received 2 April 2013; published 2 July 2013)

The complete piezoelectric tensor of ferroelectric SrTiO3 at low temperature is computed by ab initio theoretical

simulations. Both direct and converse—coupled with elastic compliance—piezoelectricity are computed and

interpreted in terms of electronic and nuclear contributions. The role of the ferroelectric soft phonon mode on this

property is found to be dramatic thus leading to a possible giant piezoelectric response at very low temperature.

Two possible space groups are considered for the ferroelectric phase of SrTiO3, both compatible with the available

experimental data: a tetragonal I4cm and an orthorhombic Ima2 one. The piezoelectric response of the two

symmetries is predicted to be rather different and could be experimentally detected to clarify the (still unknown)

structure of the ferroelectric phase of SrTiO3.

DOI: 10.1103/PhysRevB.88.035102 PACS number(s): 31.15.A−, 77.84.−s

I. INTRODUCTION

Standard piezoelectric ceramics, such as lead zirconate

titanate (PZT) based materials, are widely used as sensor

and actuator devices, hydrophones, multilayered capacitors,

ultrasonic motors, transformers, and medical ultrasonics de-

vices for acoustic radiation force impulse imaging.1,2 Such

materials could be used for several applications at cryogenic

temperatures such as actuators for adaptive optics (space tele-

scopes and low-temperature capacitors, for instance); however,

their piezoelectric response is significantly reduced at very low

temperatures. In 1997, Grupp and Goldman discovered a giant

piezoelectric effect of strontium titanate (SrTiO3) down to

1.6 K, where the sole converse piezoelectric coefficient d31 =

16 × 10−10 m/V was reported which is comparable to those

of PZT at room temperature. These findings opened the way

for applications of SrTiO3 in ultralow-temperature scanning

microscopies and magnetic field-insensitive thermometers.3

Till now, this remained the only experimental determination

of a piezoelectric constant of SrTiO3, whose complete direct

and converse third-order piezoelectric tensors still have to

be determined and discussed. Even theoretically, only a few

features of the direct piezoelectric tensor e have been reported:

Furuta and Miura4 computed two constants, e31 and e33, with

an in-plane compressive tetragonal structure while Naumov

and Fu5 computed the quantity e33 − e31—which corresponds

to the piezoelectric response to a tetragonal strain at fixed

volume—of cubic SrTiO3 under a finite electric field. In what

follows we shall briefly recall the main structural and electronic

features of SrTiO3.

SrTiO3 is probably the most studied complex oxide

perovskite of the ABO3 family due to its many technolog-

ical applications in optoelectronics, macroelectronics, and

ferroelectricity (see Ref. 6 and references therein). This

material exhibits an impressive variety of peculiar properties:

a colossal magnetoresistance,7 anomalously large dynamical

effective charges resulting in a giant longitudinal optical-

transverse optical (LO-TO) splitting,8 the huge zero-point

motion of Ti ions,9 giant elastic softening (superelasticity) at

low temperature,10 extremely large dielectric constants which

increase when the temperature decreases,11,12 superlattice

high-Tc superconductivity,13 anomalous ferroelasticity,14 and

so on.

At room temperature, SrTiO3 crystallizes in a simple

cubic structure of space group Pm3m where each Ti ion is

octahedrally coordinated to six O ions. This arrangement of

atoms shows at least two types of structural instabilities, each

connected to a particular soft phonon mode of its first Brillouin

zone (BZ): a structural R-point rotation of TiO6 octahedra and

a -point ferroelectric displacement of Ti ions from the center

of the octahedra. On cooling, SrTiO3 undergoes a second-order

antiferrodistortive (AFD) phase transition at Ta = 105 K to a

tetragonal phase with space group I4/mcm. The tetragonal

phase is characterized by static rotations of TiO6 octahedra

around the tetragonal axis c and by a slight unit-cell stretching;

the crystallographic axes of the AFD phase are rotated by 45

around the c axis of the cubic phase. Two order parameters are

associated with this phase transition: the octahedra-rotation

angle θ (reported to be 2.1 at 4.2 K) (Ref. 15) and the

tetragonality of the unit cell c/a (reported to be 1.0009 at 10 K)

(Ref. 16). In recent years, many theoretical investigations have

helped in clarifying the specific aspects of this transition,6,17–22

which was recently found to be fully describable by classical

Landau theory with terms up to the sixth order of the free

energy expansion.23

By further cooling below Ta , down to about 50 K, the ferro-

electric instability leads to a softening of the Ti-displacement

phonon mode and to anomalously large values of the static

dielectric constants which grow according to a Curie-Weiss

law. A ferroelectric phase transition could be expected to

occur at Tf ∼ 35 K; however, below a certain temperature

Tq = 37 K, these quantities saturate and the ferroelectric

transition is suppressed down to 0 K by strong zero-point

quantum fluctuations.12,24 SrTiO3 then remains in a quantum

coherent state (also called the Muller state after its discoverer)

even at very low temperatures where it becomes a so-called

quantum paraelectric.25

It has been known for a long time that a ferroelectric

transition to a lower (unknown) symmetry phase can be

035102-11098-0121/2013/88(3)/035102(10) ©2013 American Physical Society

APPENDIX

184

Inducing a Finite In-Plane Piezoelectricity in Graphene with LowConcentration of Inversion Symmetry-Breaking Defects

Kh. E. El-Kelany,*,†,‡ Ph. Carbonniere,† A. Erba,¶ and M. Rerat†

†Equipe de Chimie Physique, IPREM UMR5254, Universite de Pau et des Pays de l’Adour, 64000 Pau, France‡Chemistry Department, Faculty of Science, Minia University, Minia 61519, Egypt¶Dipartimento di Chimica and Centre of Excellence NIS (Nanostructured Interfaces and Surfaces), Universita di Torino, via Giuria 5,IT-10125 Torino (Italy)

*S Supporting Information

ABSTRACT: We show that a finite in-plane piezoelectricity can beinduced in graphene by breaking its inversion center with any in-plane defect, in the limit of vanishing defect concentration. We firstconsider different patterns of BN-doped graphene sheets of D3h

symmetry, whose electronic and piezoelectric (dominated by theelectronic rather than nuclear term) properties are characterized atthe ab initio level of theory. We then consider other in-planedefects, such as holes of D3h or C2v point symmetry, and confirmthat a common limit value (for low defect concentration) of thepiezoelectric response of graphene is obtained regardless of theparticular chemical or physical nature of the defects (e11 ≈ 4.5 ×

10−10 C/m and d11 ≈ 1.5 pm/V for direct and conversepiezoelectricity, respectively). This in-plane piezoelectric responseof graphene is one-order of magnitude larger than the out-of-planepreviously investigated one.

1. INTRODUCTION

In the continuous quest for the fabrication of nano-electromechanical systems (NEMS) and nanoscale devices,great attention has been devoted in recent years to low-dimensional materials due to their peculiar, highly tunable,physicochemical properties.1−3 A variety of NEMS devices havebeen successfully produced (nanosized switches, sensors,motors, energy harvesters, actuators, etc.),4−7 which essentiallyrely on quantum-size effects.8 Most of such devices requiresome sort of dynamical control of atomic displacements andnanoscale deformations. In this respect, piezo-electricity turnsout to be an extremely useful property in that it allows for fine-tuning the induced nanostrain by modulating an appliedelectric field (or vice versa). Among other low-dimensionalsystems, such as nanoparticles, nanotubes, nanoribbons, andfullerenes, graphene-based9 materials have been playing aparamount role in the fabrication of innovative devices forelectronics, optoelectronics, photonics, and spintronics,10−15

due to the many extraordinary properties of the two-dimensional (2D) carbon allotrope: high electron-mobility,hardness and flexibility, anomalous quantum-hall effect, zeroband gap semimetallic character, etc.16−18

Graphene lacks any intrinsic piezoelectricity due to itssymmetry inversion center. The induction of piezoelectricityinto graphene sheets would lead to a new branch of possibleapplications in NEMS devices requiring high electromechanicalcoupling. A recent theoretical study has highlighted the

possibility of engineering piezoelectricity in graphene byadsorbing light atoms (such as H, Li, K, and F) on one sideof its surface; a rather small out-of-plane piezoelectric responsehas been reported.19 Apart from atom adsorption, othertechniques can be used to break the inversion symmetry ofgraphene sheets, such as hole formation,20 stacking control ingraphene bilayers,21 application of nonhomogeneous strain,22

and chemical doping.23−25 Among these strategies, chemicaldoping seems the most promising as it already represents aneffective experimental mean for tuning structural and electronicproperties (such as band gap and work function) ofgraphene.23,24,26−28

Boron nitride (BN) chemical doping of graphene hasrecently been successfully achieved in different configurationsand concentrations: semiconducting atomic layers of hybrid h-BN and graphene domains have been synthesized,26 low-pressure chemical-vapor-deposition (CVD) synthesis of large-area few-layer BN-doped graphene (BNG) has been presented,leading to BN concentrations as high as 10%; the BN contentin BNG layers has been discussed to be related to the heatingtemperature of the precursor, as confirmed by X-ray photo-electron spectroscopy measurements.27 The synthesis of aquasi-freestanding BNG monolayer heterostructure, with

Received: February 12, 2015Revised: March 25, 2015Published: March 31, 2015

Article

pubs.acs.org/JPCC

© 2015 American Chemical Society 8966 DOI: 10.1021/acs.jpcc.5b01471J. Phys. Chem. C 2015, 119, 8966−8973

APPENDIX

185

APPENDIX

Submitted

Piezoelectricity of Functionalized Graphene: A Quantum-mechanical Rationalization

Khaled E. El-Kelany,1, 2, ∗ Philippe Carbonniere,1, † Alessandro Erba,3 Jean-Marc Sotiropoulos,1 and Michel Rerat1

1Equipe de Chimie Physique, IPREM UMR5254,Universite de Pau et des Pays de l’Adour, 64000 Pau, France

2Chemistry Department, Faculty of Science, Minia University, Minia 61519, Egypt3Dipartimento di Chimica and Centre of Excellence NIS (Nanostructured Interfaces and Surfaces),

Universita di Torino, via Giuria 5, IT-10125 Torino, Italy

A large out-of-plane piezoelectricity can be induced in graphene by carbon substitution. Severalsimple substitutions are considered where C atoms are replaced by heavier group-IV elements (Si, Geand Sn). A more complex functionalization (namely, pyrrolic N-doped graphene) is also investigatedwhere different functional groups, such as F, Cl, H3C and H2N, are studied. Piezoelectric and elasticresponse properties of all systems are determined quantum-mechanically at the ab initio level oftheory. A rationalization of the physical and chemical parameters which most affect the out-of-planepiezoelectricity of functionalized graphene is reported, which reveals the dominant character of thenuclear over electronic contribution. The combination of an out-of-plane symmetry-breaking defectand a soft infrared-active phonon mode, with a large cell-deformation coupling, is shown to constitutethe necessary prerequisite to induce a large out-of-plane piezoelectric response into functionalizedgraphene.

I. INTRODUCTION

Piezoelectricity consists in the mutual conversion ofmechanical and electrical forces in the material and, sinceits discovery in 1880, it is central to a wide variety oftechnological applications: next-generation energy har-vesters,1,2 artificial muscles,3 sensors and actuators, etc.4

One obvious limitation of such an important property isthat of being restricted to non-centrosymmetric crystals.In this respect, it has recently been shown that a reduc-tion of the dimensionality of bulk materials represents aneffective way of enhancing (or even creating) a piezoelec-tric response:5 for instance, single-layered 2D materialssuch as h-BN, h-MoS2 and h-WS2 do show a piezoelectriceffect while their 3D bulk analogs do not.6–8

When it comes to low-dimensional systems, graphenewould clearly be the most promising material to work within the fabrication of electronic, optoelectronic and spin-tronic nano-devices due to all of its well-known remarkableproperties, including extraordinarily high electron mobil-ity, mechanical stiffness and flexibility.9–15 The exploita-tion of piezoelectricity of graphene would indeed leadto a new branch of possible applications in nano-electro-mechanical systems (NEMS) devices requiring high elec-tromechanical coupling. Unfortunately, graphene pos-sesses an inversion symmetry center in its undistortedD6h equilibrium configuration, which prevents a piezoelec-tric response to take place in its pristine form. However,its inversion center can be broken and piezoelectricityengineered by several means including adsorption, holecreation, application of biaxial strain, chemical doping,etc.16–21 A large out-of-plane piezoelectric response hasrecently been measured for a graphene single layer asdeposited on a SiO2 substrate.20

Among other strategies to induce a piezoelectric re-sponse in graphene, chemical doping seems the mostpromising as it already represents an effective exper-imental mean for tuning its structural and electronic

properties.22,23 Free-standing BN-, N-, B-, and Si-dopedgraphene monolayers have recently been synthesized andfound to be chemically stable at ambient conditions.24–27

Furthermore, both dopant concentration and spatial con-figuration have recently been shown to be tunable, to someextent.28,29 For instance, N-doped graphene exhibits threecommon bonding configurations: pyridinic, pyrrolic andgraphitic, whose relative occurrence is systematically af-fected by several factors of the chemical-vapor-depositionprocess: precursor, catalyst, flow rate, and growth temper-ature.29 Hydrothermal reduction of colloidal dispersionsof graphite oxide in the presence of hydrazine is an al-ternative approach to selectively obtain pyrrolic N-dopedgraphene,30 while a solvo-thermal synthesis via the re-action of tetrachloromethane with lithium nitride undermild conditions leads to production in gram scale.31

In a recent study, we have systematically investigatedthe in-plane piezoelectric response of graphene as inducedby several inversion symmetry-breaking defects and founda peculiar “universal” behavior: a common finite in-planepiezoelectric response (characterized by a direct piezoelec-tric coefficient e11 of about 5×10−10 C/m) in the limitof vanishing defect concentration, thus highlighting anintrinsic nature of the piezoelectric activity of graphene.21

The present investigation aims at providing a completequantum-mechanical rationalization of the overall (in-plane and out-of-plane) piezoelectric effect as inducedin graphene by any inversion symmetry-breaking defect.While confirming the “universal” in-plane behavior, theatomistic mechanisms behind a possible giant out-of-planepiezoelectricity are here addressed and understood. Dif-ferent kinds of chemical doping are considered (Si, Ge, Sn,pyrrolic N), which can lead to an out-of-plane piezoelec-tric response up to 300 times larger than the largest onereported so far in the literature for free-standing graphene(which was obtained by adsorption of Li atoms on thegraphene surface).16

In an ideally planar structure, out-of-plane response

186

Appdx 3: Optimized Geometry

1) MgO system, DFT-B3LYP calculations1) MgO system, DFT-B3LYP calculations

TEST11 - MGO BULK CUBICTEST11 - MGO BULK CUBIC

CRYSTALCRYSTAL

0 0 00 0 0

225225

4.227487214.22748721

22

12 0.0000000000E+00 0.0000000000E+00 0.0000000000E+0012 0.0000000000E+00 0.0000000000E+00 0.0000000000E+00

8 -5.0000000000E-01 -5.0000000000E-01 -5.0000000000E-018 -5.0000000000E-01 -5.0000000000E-01 -5.0000000000E-01

TEST11 - MGO 001 SLABCUTTEST11 - MGO 001 SLABCUT

CRYSTALCRYSTAL

0 0 00 0 0

225225

4.227487214.22748721

22

12 0.0000000000E+00 0.0000000000E+00 0.0000000000E+0012 0.0000000000E+00 0.0000000000E+00 0.0000000000E+00

8 -5.0000000000E-01 -5.0000000000E-01 -5.0000000000E-018 -5.0000000000E-01 -5.0000000000E-01 -5.0000000000E-01

SLABSLAB

0 0 10 0 1

1 11 1

TEST11 - MGO (6,0) NANOTUBETEST11 - MGO (6,0) NANOTUBE

CRYSTALCRYSTAL

0 0 00 0 0

225225

4.227487214.22748721

22

12 0.0000000000E+00 0.0000000000E+00 0.0000000000E+0012 0.0000000000E+00 0.0000000000E+00 0.0000000000E+00

8 -5.0000000000E-01 -5.0000000000E-01 -5.0000000000E-018 -5.0000000000E-01 -5.0000000000E-01 -5.0000000000E-01

SLABSLAB

0 0 10 0 1

1 11 1

NANOTUBENANOTUBE

6 06 0

TEST11 - MGO (6,6) NANOTUBETEST11 - MGO (6,6) NANOTUBE

CRYSTALCRYSTAL

0 0 00 0 0

225225

4.227487214.22748721

22

12 0.0000000000E+00 0.0000000000E+00 0.0000000000E+0012 0.0000000000E+00 0.0000000000E+00 0.0000000000E+00

8 -5.0000000000E-01 -5.0000000000E-01 -5.0000000000E-018 -5.0000000000E-01 -5.0000000000E-01 -5.0000000000E-01

SLABSLAB

0 0 10 0 1

1 11 1

NANOTUBENANOTUBE

6 66 6

187

2) SrTiO2) SrTiO33 perovskiteperovskite, DFT-PBE0 calculations, DFT-PBE0 calculations

Simple Simple

CubicCubic

PmPm33mm

SrTiO3_cubicSrTiO3_cubic

CRYSTALCRYSTAL

0 0 00 0 0

221221

3.892412283.89241228

33

238 0.000000000E+00 0.000000000E+00 0.000000000E+00238 0.000000000E+00 0.000000000E+00 0.000000000E+00

22 5.000000000E-01 5.000000000E-01 5.000000000E-0122 5.000000000E-01 5.000000000E-01 5.000000000E-01

8 5.000000000E-01 5.000000000E-01 0.000000000E+008 5.000000000E-01 5.000000000E-01 0.000000000E+00

TetragonalTetragonal

II44/mcm/mcm

SrTiO3_tetragonal highsymmSrTiO3_tetragonal highsymm

CRYSTALCRYSTAL

0 0 00 0 0

140140

5.50087266 7.791850185.50087266 7.79185018

44

238 0.000000000E+00 -5.000000000E-01 2.500000000E-01238 0.000000000E+00 -5.000000000E-01 2.500000000E-01

22 0.000000000E+00 0.000000000E+00 0.000000000E+0022 0.000000000E+00 0.000000000E+00 0.000000000E+00

8 2.3917883591E-01 -2.6082116410E-01 -9.424987442E-188 2.3917883591E-01 -2.6082116410E-01 -9.424987442E-18

8 2.1360418867E-17 -2.1360418867E-17 2.500000000E-018 2.1360418867E-17 -2.1360418867E-17 2.500000000E-01


II44cmcm

SrTiO3_tetragonal distorduSrTiO3_tetragonal distordu

CRYSTALCRYSTAL

0 0 00 0 0

108108

5.50020902 7.798371205.50020902 7.79837120

238 5.000000000E-01 1.2817797694E-16 -2.532323321E-01238 5.000000000E-01 1.2817797694E-16 -2.532323321E-01

22 0.000000000E+00 0.0000000000E+00 -4.738998212E-0322 0.000000000E+00 0.0000000000E+00 -4.738998212E-03

8 2.396238167E-01 -2.6037618333E-01 -1.089183945E-048 2.396238167E-01 -2.6037618333E-01 -1.089183945E-04

8 0.000000000E+00 0.0000000000E+00 2.501891671E-018 0.000000000E+00 0.0000000000E+00 2.501891671E-01

OrthorhombicOrthorhombic

Ima2Ima2

SrTiO3 orthorombicSrTiO3 orthorombic

CRYSTALCRYSTAL

0 0 00 0 0

4646

7.78935980 5.50287032 5.504250877.78935980 5.50287032 5.50425087

55

238 -8.993096525E-18 -1.080901785E-17 -2.4905327401E-01238 -8.993096525E-18 -1.080901785E-17 -2.4905327401E-01

22 2.500000000E-01 2.295983048E-04 2.5288188939E-0122 2.500000000E-01 2.295983048E-04 2.5288188939E-01

8 5.000000000E-01 4.084698391E-17 2.4657719709E-018 5.000000000E-01 4.084698391E-17 2.4657719709E-01

8 2.500000000E-01 2.394447323E-01 -1.4491961196E-028 2.500000000E-01 2.394447323E-01 -1.4491961196E-02

8 2.500000000E-01 -2.610074566E-01 6.8261487101E-038 2.500000000E-01 -2.610074566E-01 6.8261487101E-03

APPENDIX

188

3) BaTiO3) BaTiO33 perovskiteperovskite, DFT-B3LYP calculations, DFT-B3LYP calculations

Simple Simple

CubicCubic

PmPm33mm

BaTiO3_CubicBaTiO3_Cubic

CRYSTALCRYSTAL

0 0 00 0 0

221221

3.97972498 3.97972498

33

256 0.000000000E+00 0.000000000E+00 0.000000000E+00256 0.000000000E+00 0.000000000E+00 0.000000000E+00

22 -5.000000000E-01 5.000000000E-01 5.000000000E-0122 -5.000000000E-01 5.000000000E-01 5.000000000E-01

8 -5.000000000E-01 5.000000000E-01 0.000000000E+008 -5.000000000E-01 5.000000000E-01 0.000000000E+00


PP44mmmm

BaTiO3_TetragonalBaTiO3_Tetragonal

CRYSTALCRYSTAL

0 0 00 0 0

9999

3.97490125 4.29043343 3.97490125 4.29043343

44

256 0.0000000000E+00 0.000000000E+00 4.451319666E-02256 0.0000000000E+00 0.000000000E+00 4.451319666E-02

TI -5.0000000000E-01 -5.000000000E-01 -4.381258256E-01TI -5.0000000000E-01 -5.000000000E-01 -4.381258256E-01

O -5.0000000000E-01 -5.000000000E-01 -2.134939359E-02O -5.0000000000E-01 -5.000000000E-01 -2.134939359E-02

O -5.0000000000E-01 0.000000000E+00 -4.917189887E-01O -5.0000000000E-01 0.000000000E+00 -4.917189887E-01


AmmAmm22BaTiO3_ORTHOROMBICBaTiO3_ORTHOROMBIC

CRYSTALCRYSTAL

0 0 00 0 0

3838

3.95077626 5.69591519 5.728609483.95077626 5.69591519 5.72860948

44

256 0.000000000E+00 0.000000000E+00 3.6807016715E-03256 0.000000000E+00 0.000000000E+00 3.6807016715E-03

22 -5.000000000E-01 0.000000000E+00 -4.8079241882E-0122 -5.000000000E-01 0.000000000E+00 -4.8079241882E-01

8 0.000000000E+00 0.000000000E+00 4.8643744942E-018 0.000000000E+00 0.000000000E+00 4.8643744942E-01

8 -5.000000000E-01 2.576815985E-01 2.3238713387E-018 -5.000000000E-01 2.576815985E-01 2.3238713387E-01


RR33mmBaTiO3_RhombohedralBaTiO3_Rhombohedral

CRYSTALCRYSTAL

0 1 00 1 0

160160

4.07745679 89.636249 4.07745679 89.636249

256 -2.891578506E-03 -2.891578506E-03 -2.8915785056E-03256 -2.891578506E-03 -2.891578506E-03 -2.8915785056E-03

22 4.815071351E-01 4.815071351E-01 4.8150713508E-0122 4.815071351E-01 4.815071351E-01 4.8150713508E-01

8 -4.861340896E-01 -4.861340896E-01 2.8652622621E-028 -4.861340896E-01 -4.861340896E-01 2.8652622621E-02

APPENDIX

189

4) BNG and Graphene Holes, 4) BNG and Graphene Holes, DFT-B3LYP calculationsDFT-B3LYP calculations

Pristine Pristine

GrapheneGraphene

DD6h6h

Pure GraphenePure Graphene

SLABSLAB

8080

2.451462692.45146269

11

6 3.3333333333E-01 -3.3333333333E-01 0.0000000000E+006 3.3333333333E-01 -3.3333333333E-01 0.0000000000E+00

h-h-BNBN

DD3h3h

BN-SlabBN-Slab

SLABSLAB

7878

2.503424772.50342477

22

5 -3.3333333333E-01 3.3333333333E-01 0.0000000000E+005 -3.3333333333E-01 3.3333333333E-01 0.0000000000E+00

7 3.3333333333E-01 -3.3333333333E-01 0.0000000000E+007 3.3333333333E-01 -3.3333333333E-01 0.0000000000E+00

BNGBNG

(1,5)(1,5)

DD3h3h

BNG_R=1, W=5, p = 2R + W BNG_R=1, W=5, p = 2R + W

SLABSLAB

7878

17.1900823517.19008235

2424

5 -4.8044753459E-02 -9.6089506918E-02 0.0000000000E+005 -4.8044753459E-02 -9.6089506918E-02 0.0000000000E+00

6 -4.8748971569E-02 1.9141327205E-01 0.0000000000E+006 -4.8748971569E-02 1.9141327205E-01 0.0000000000E+00

6 -4.7593518780E-02 3.3410601737E-01 0.0000000000E+006 -4.7593518780E-02 3.3410601737E-01 0.0000000000E+00

6 -4.7489849871E-02 4.7625507506E-01 0.0000000000E+006 -4.7489849871E-02 4.7625507506E-01 0.0000000000E+00

6 9.5549294706E-02 -9.5549294706E-02 0.0000000000E+006 9.5549294706E-02 -9.5549294706E-02 0.0000000000E+00

6 9.5353267110E-02 3.3324431294E-01 0.0000000000E+006 9.5353267110E-02 3.3324431294E-01 0.0000000000E+00

6 9.5309701022E-02 4.7618954097E-01 0.0000000000E+006 9.5309701022E-02 4.7618954097E-01 0.0000000000E+00

6 2.3817108666E-01 4.7634217331E-01 0.0000000000E+006 2.3817108666E-01 4.7634217331E-01 0.0000000000E+00

6 2.3814800387E-01 -3.8092599807E-01 0.0000000000E+006 2.3814800387E-01 -3.8092599807E-01 0.0000000000E+00

6 3.8176390814E-01 1.9088195407E-01 0.0000000000E+006 3.8176390814E-01 1.9088195407E-01 0.0000000000E+00

6 -4.7594897035E-01 1.9057832220E-01 0.0000000000E+006 -4.7594897035E-01 1.9057832220E-01 0.0000000000E+00

6 -3.3333333333E-01 3.3333333333E-01 0.0000000000E+006 -3.3333333333E-01 3.3333333333E-01 0.0000000000E+00

7 -9.6807494858E-02 -4.8403747429E-02 0.0000000000E+007 -9.6807494858E-02 -4.8403747429E-02 0.0000000000E+00

6 -9.7641500578E-02 9.7641500578E-02 0.0000000000E+006 -9.7641500578E-02 9.7641500578E-02 0.0000000000E+00

6 -9.5650397459E-02 2.3916865656E-01 0.0000000000E+006 -9.5650397459E-02 2.3916865656E-01 0.0000000000E+00

6 -9.5237329451E-02 3.8127039505E-01 0.0000000000E+006 -9.5237329451E-02 3.8127039505E-01 0.0000000000E+00

6 4.7514973121E-02 2.3800859641E-01 0.0000000000E+006 4.7514973121E-02 2.3800859641E-01 0.0000000000E+00

6 4.7665310532E-02 3.8111735517E-01 0.0000000000E+006 4.7665310532E-02 3.8111735517E-01 0.0000000000E+00

6 4.7678276922E-02 -4.7616086154E-01 0.0000000000E+006 4.7678276922E-02 -4.7616086154E-01 0.0000000000E+00

6 1.9047702209E-01 3.8095404419E-01 0.0000000000E+006 1.9047702209E-01 3.8095404419E-01 0.0000000000E+00

6 1.9059871648E-01 -4.7611855332E-01 0.0000000000E+006 1.9059871648E-01 -4.7611855332E-01 0.0000000000E+00

6 3.3333333333E-01 -3.3333333333E-01 0.0000000000E+006 3.3333333333E-01 -3.3333333333E-01 0.0000000000E+00

6 4.7683112247E-01 2.3841556123E-01 0.0000000000E+006 4.7683112247E-01 2.3841556123E-01 0.0000000000E+00

6 -3.8090849906E-01 2.3818300188E-01 0.0000000000E+006 -3.8090849906E-01 2.3818300188E-01 0.0000000000E+00

APPENDIX

190

BNGBNG

(2,3)(2,3)

DD3h3h

BNG_R=2, W=3, p = 2R + WBNG_R=2, W=3, p = 2R + W

SLABSLAB

7878

17.2651487217.26514872

2424

5 -4.7702974051E-02 -9.5405948102E-02 0.0000000000E+005 -4.7702974051E-02 -9.5405948102E-02 0.0000000000E+00

5 -4.7722977494E-02 1.9110698829E-01 0.0000000000E+005 -4.7722977494E-02 1.9110698829E-01 0.0000000000E+00

6 -4.9012160985E-02 3.3369561378E-01 0.0000000000E+006 -4.9012160985E-02 3.3369561378E-01 0.0000000000E+00

6 -4.7674669099E-02 4.7616266545E-01 0.0000000000E+006 -4.7674669099E-02 4.7616266545E-01 0.0000000000E+00

5 9.6497403487E-02 -9.6497403487E-02 0.0000000000E+005 9.6497403487E-02 -9.6497403487E-02 0.0000000000E+00

6 9.5566288606E-02 3.3518586066E-01 0.0000000000E+006 9.5566288606E-02 3.3518586066E-01 0.0000000000E+00

6 9.5544021394E-02 4.7657443725E-01 0.0000000000E+006 9.5544021394E-02 4.7657443725E-01 0.0000000000E+00

6 2.3831991674E-01 4.7663983348E-01 0.0000000000E+006 2.3831991674E-01 4.7663983348E-01 0.0000000000E+00

6 2.3826586661E-01 -3.8086706669E-01 0.0000000000E+006 2.3826586661E-01 -3.8086706669E-01 0.0000000000E+00

6 3.8503584826E-01 1.9251792413E-01 0.0000000000E+006 3.8503584826E-01 1.9251792413E-01 0.0000000000E+00

6 -4.7505768376E-01 1.9088610989E-01 0.0000000000E+006 -4.7505768376E-01 1.9088610989E-01 0.0000000000E+00

6 -3.3333333333E-01 3.3333333333E-01 0.0000000000E+006 -3.3333333333E-01 3.3333333333E-01 0.0000000000E+00

7 -9.6700782165E-02 -4.8350391083E-02 0.0000000000E+007 -9.6700782165E-02 -4.8350391083E-02 0.0000000000E+00

7 -9.5039948390E-02 9.5039948390E-02 0.0000000000E+007 -9.5039948390E-02 9.5039948390E-02 0.0000000000E+00

6 -9.7825806292E-02 2.4059095594E-01 0.0000000000E+006 -9.7825806292E-02 2.4059095594E-01 0.0000000000E+00

6 -9.5687172050E-02 3.8180674429E-01 0.0000000000E+006 -9.5687172050E-02 3.8180674429E-01 0.0000000000E+00

7 4.9049551051E-02 2.4152407974E-01 0.0000000000E+007 4.9049551051E-02 2.4152407974E-01 0.0000000000E+00

6 4.7617872925E-02 3.8124630213E-01 0.0000000000E+006 4.7617872925E-02 3.8124630213E-01 0.0000000000E+00

6 4.8032325958E-02 -4.7598383702E-01 0.0000000000E+006 4.8032325958E-02 -4.7598383702E-01 0.0000000000E+00

6 1.9078082469E-01 3.8156164938E-01 0.0000000000E+006 1.9078082469E-01 3.8156164938E-01 0.0000000000E+00

6 1.9065266551E-01 -4.7593124958E-01 0.0000000000E+006 1.9065266551E-01 -4.7593124958E-01 0.0000000000E+00

6 3.3333333333E-01 -3.3333333333E-01 0.0000000000E+006 3.3333333333E-01 -3.3333333333E-01 0.0000000000E+00

6 4.7881624056E-01 2.3940812028E-01 0.0000000000E+006 4.7881624056E-01 2.3940812028E-01 0.0000000000E+00

6 -3.8074477076E-01 2.3851045849E-01 0.0000000000E+006 -3.8074477076E-01 2.3851045849E-01 0.0000000000E+00

Graphene Graphene

HolesHoles

DD3h3h

Graphen_three_holes, p = 4Graphen_three_holes, p = 4

SLABSLAB

8080

2.462.46

11

6 0.3333333 -0.3333333 0.00006 0.3333333 -0.3333333 0.0000

SUPERCELLSUPERCELL

4 04 0

0 40 4

ATOMREMOATOMREMO

33

17 18 2217 18 22

Graphene Graphene

HolesHoles

CC2v2v

Graphen_one_hole, p = 4Graphen_one_hole, p = 4

SLABSLAB

8080

2.462.46

11

6 0.3333333 -0.3333333 0.00006 0.3333333 -0.3333333 0.0000

SUPERCELLSUPERCELL

4 04 0

0 40 4

ATOMREMOATOMREMO

11

1717

APPENDIX

191

Appdx 4: GT-Atomic Basis Set

1) MgO system

2) SiO2/GeO2 system (pseudopotential basis)

192

3) BNG (BN-doped graphene)

BB NN

5 8

0 0 6 2.0 1.0

8564.86606870 0.00022837198155

1284.15162630 0.00176825764470

292.278716040 0.00914070805160

82.7754691760 0.03634263898900

27.0179392690 0.11063458441000

9.81496196600 0.23367344321000

0 0 2 2.0 1.0

3.93185590590 0.41818777978000

1.65955997120 0.22325473798000

0 0 1 0.0 1.0

0.53318702000 1.00000000000000

0 0 1 0.0 1.0

0.26659351000 1.00000000000000

0 2 4 1.0 1.0

22.4538758030 0.00502655751790

5.10450583300 0.03280173896500

1.49860813440 0.13151230768000

0.50927831315 0.33197167769000

0 2 1 0.0 1.0

0.53857716000 1.00000000000000

0 2 1 0.0 1.0

0.26928858000 1.00000000000000

0 3 1 0.0 1.0

0.75005942000 1.00000000000000

7 8

0 0 6 2.0 1.0

19730.800647 0.00021887984991

2957.8958745 0.00169607088030

673.22133595 0.00879546035380

190.68249494 0.03535938260500

62.295441898 0.11095789217000

22.654161182 0.24982972552000

0 0 2 2.0 1.0

8.9791477428 0.4062389614800

3.6863002370 0.2433821717600

0 0 1 0.0 1.0

0.7865398200 1.0000000000000

0 0 1 0.0 1.0

0.2677997200 1.0000000000000

0 2 4 3.0 1.0

49.200380510 0.0055552416751

11.346790537 0.0380523797230

3.4273972411 0.1495367102900

1.1785525134 0.3494930523000

0 2 1 0.0 1.0

0.3780331700 1.0000000000000

0 2 1 0.0 1.0

0.1473661500 1.0000000000000

0 3 1 0.0 1.0

0.3612294900 1.0000000000000

CC

6 8

0 0 6 2.0 1.0

13575.349682 0.00022245814352

2035.2333680 0.00172327382520

463.22562359 0.00892557153140

131.20019598 0.03572798450200

42.853015891 0.11076259931000

15.584185766 0.24295627626000

0 0 2 2.0 1.0

6.2067138508 0.41440263448000

2.5764896527 0.23744968655000

0 0 1 0.0 1.0

0.4941102000 1.00000000000000

0 0 1 0.0 1.0

0.1644071000 1.00000000000000

0 2 4 2.0 1.0

34.697232244 0.00533336578050

7.9582622826 0.03586410909200

2.3780826883 0.14215873329000

0.8143320818 0.34270471845000

0 2 1 0.0 1.0

0.5662417100 1.00000000000000

0 2 1 0.0 1.0

0.2673545000 1.00000000000000

0 3 1 0.0 1.0

0.8791584200 1.00000000000000

APPENDIX

193

4) SrTiO3 perovskite

TiTi OO

22 9

0 0 8 2. 1.

225338 0.000228

32315 0.001929

6883.61 0.011100

1802.14 0.05

543.063 0.17010

187.549 0.369

73.2133 0.4033

30.3718 0.1445

0 1 6 8. 1.

554.042 -0.0059 0.0085

132.525 -0.0683 0.0603

43.6801 -0.1245 0.2124

17.2243 0.2532 0.3902

7.2248 0.6261 0.4097

2.4117 0.282 0.2181

0 1 4 8. 1.

24.4975 0.0175 -0.0207

11.4772 -0.2277 -0.0653

4.4653 -0.7946 0.1919

1.8904 1.0107 1.3778

0 1 1 2. 1.

0.807363556283 1. 1.

0 1 1 0. 1.

0.339249226927 1. 1.

0 3 3 2. 1.

8.84510207254 0.150823763226

2.73393052121 0.397241364052

1.11345677732 0.535341075348

0 3 1 0. 1.

0.9 1.

0 3 1 0. 1.

0.3 1.

0 4 1 0. 1.

0.9 1.

8 7

0 0 8 2. 1.

8020. 0.00108

1338. 0.00804

255.4 0.05324

69.22 0.1681

23.90 0.3581

9.264 0.3855

3.851 0.1468

1.212 0.0728

0 1 4 6. 1.

49.43 -0.011 0.0097

10.47 -0.091 0.069

3.235 -0.039 0.207

1.22 0.379 0.347

0 1 1 0. 1.

0.458752225991 1. 1.

0 1 1 0. 1.

0.171116621926 1. 1.

0 3 1 0. 1.

0.9 1.

0 3 1 0. 1.

0.3 1.

0 4 1 0. 1.

0.9 1.

SrSr

238 6

HAYWSC

0 1 2 8.0 1.0

3.2429 0.23210 -0.12996

2.4027 -0.70898 0.050457

0 1 1 2. 1.

0.694 1. 1.

0 1 1 0. 1.

0.258 1. 1.

0 3 1 0. 1.

1.2 1.

0 3 1 0. 1.

0.4 1.

0 4 1 0. 1.

0.9 1.

APPENDIX

194

5) BaTiO3 perovskite

TiTi OO

22 9

0 0 8 2. 1.

225338 0.000228

32315 0.001929

6883.61 0.011100

1802.14 0.05

543.063 0.17010

187.549 0.369

73.2133 0.4033

30.3718 0.1445

0 1 6 8. 1.

554.042 -0.0059 0.0085

132.525 -0.0683 0.0603

43.6801 -0.1245 0.2124

17.2243 0.2532 0.3902

7.2248 0.6261 0.4097

2.4117 0.282 0.2181

0 1 4 8. 1.

24.4975 0.0175 -0.0207

11.4772 -0.2277 -0.0653

4.4653 -0.7946 0.1919

1.8904 1.0107 1.3778

0 1 1 4. 1.

0.814451146126 1. 1.

0 1 1 0. 1.

0.336378101288 1. 1.

0 3 3 0. 1.

8.84788506429 0.159767501392

2.74895047211 0.41228313479

1.08500450732 0.522225358902

0 3 1 0. 1.

0.887193042249 1.

0 3 1 0. 1

0.383058673677 1.

0 4 1 0. 1

0.877460982061 1.

8 7

0 0 8 2. 1.

8020. 0.00108

1338. 0.00804

255.4 0.05324

69.22 0.1681

23.90 0.3581

9.264 0.3855

3.851 0.1468

1.212 0.0728

0 1 4 6. 1.

49.43 -0.011 0.0097

10.47 -0.091 0.069

3.235 -0.039 0.207

1.22 0.379 0.347

0 1 1 0. 1.

0.463604985443 1. 1.

0 1 1 0. 1.

0.168429994525 1. 1.

0 3 1 0. 1.

0.897886263616 1.

0 3 1 0. 1.

0.303192559578 1.

0 4 1 0. 1.

0.875815796753 1.

BaBa

256 5

HAYWSC

0 1 3 8. 1.

1.3144 -1.3797 -0.1776

0.5144 1.1476 0.6089

0.287 2.0729 -0.3133

0 1 1 2. 1.

0.208815648861 1. 1.

0 3 1 0. 1.

0.898742189735 1.

0 3 1 0. 1.

0.324340582524 1.

0 4 1 0. 1.

0.890604589584 1.

APPENDIX

195

6) Si, Ge, Sn doped graphene

CC SiSi

6 8

0 0 6 2.0 1.0

13575.349682 0.00022245814352

2035.2333680 0.00172327382520

463.22562359 0.00892557153140

131.20019598 0.03572798450200

42.853015891 0.11076259931000

15.584185766 0.24295627626000

0 0 2 2.0 1.0

6.2067138508 0.41440263448000

2.5764896527 0.23744968655000

0 0 1 0.0 1.0

0.4941102000 1.00000000000000

0 0 1 0.0 1.0

0.1644071000 1.00000000000000

0 2 4 2.0 1.0

34.697232244 0.00533336578050

7.9582622826 0.03586410909200

2.3780826883 0.14215873329000

0.8143320818 0.34270471845000

0 2 1 0.0 1.0

0.5662417100 1.00000000000000

0 2 1 0.0 1.0

0.2673545000 1.00000000000000

0 3 1 0.0 1.0

0.8791584200 1.00000000000000

14 10

0 0 7 2.0 1.0

44773.358078 0.00055914765868

6717.1992104 0.00432060401890

1528.8960325 0.02218709646000

432.54746585 0.08648924911600

140.61505226 0.24939889716000

49.857636724 0.46017197366000

18.434974885 0.34250236575000

0 0 3 2.0 1.0

86.533886111 0.02130006300700

26.624606846 0.09467613931800

4.4953057159 -0.32616264859000

0 0 2 2.0 1.0

2.1035045710 1.39808038500000

1.0106094922 0.63865786699000

0 0 1 0.0 1.0

0.7422443800 1.00000000000000

0 0 1 0.0 1.0

0.2160762500 1.00000000000000

0 2 5 6.0 1.0

394.47503628 0.00262856939590

93.137683104 0.02055625774900

29.519608742 0.09207026280100

10.781663791 0.25565889739000

4.1626574778 0.42111707185000

0 2 1 2.0 1.0

1.4499318500 1.00000000000000

0 2 1 0.0 1.0

0.5949286700 1.00000000000000

0 2 1 0.0 1.0

0.1146786100 1.00000000000000

0 3 1 0.0 1.0

0.5074090300 1.00000000000000

GeGe Sn (pseudopotential)Sn (pseudopotential)

32 15

0 0 8 2.0 1.0

466115.00592 0.0002248726466

69875.420762 0.0017435426729

15903.276716 0.0090691482206

4501.8233453 0.0369061746850

1466.0570924 0.1205016790700

527.07841728 0.2874864170300

205.00395074 0.4162232188500

81.251596065 0.2239784569500

0 0 4 2.0 1.0

505.74661282 -0.0251846092910

156.96593744 -0.1189892972100

25.761448176 0.5493013587000

11.106654687 0.5293930912900

0 0 2 2.0 1.0

250 3

DURAND

0 1 2 4.0 1.0

0.664045 -0.531536 -0.094875

0.236154 0.858798 0.474750

0 1 1 0.0 1.0

0.08 1.0 1.0

0 3 1 0.0 1.0

0.20 1.0

APPENDIX

196

17.272059104 -0.2285459572800

2.9438289048 0.6837793031700

0 0 1 2.0 1.0

1.2786569600 1.0000000000000

0 0 1 0.0 1.0

0.3368300500 1.0000000000000

0 0 1 0.0 1.0

0.16841502500 1.0000000000000

0 2 6 6.0 1.0

2633.9346241 0.0022143925310

624.00161628 0.0181408991410

200.58528404 0.0866321849220

75.097081525 0.2564902059200

30.214388474 0.4265861126200

12.440087567 0.2620052731300

0 2 3 6.0 1.0

45.981316002 -0.0203217676780

6.9945654416 0.3201374452700

2.9686001327 0.5905101455500

0 2 1 2.0 1.0

1.2044364000 1.0000000000000

0 2 1 0.0 1.0

0.4237841000 1.0000000000000

0 2 1 0.0 1.0

0.2118920500 1.0000000000000

0 3 5 10.0 1.0

119.44887581 0.0105865445210

35.062915293 0.0696012809450

12.636924529 0.2280703528700

4.8888672922 0.4030106722000

1.8453195392 0.4130484701500

0 3 1 0.0 1.0

1.9531065900 1.0000000000000

0 3 1 0.0 1.0

0.6401807200 1.0000000000000

0 3 1 0.0 1.0

0.1305796200 1.0000000000000

APPENDIX

197

7) N-pyrrolic dope graphene (HN-G)

NN CC

7 8

0 0 6 2.0 1.0

19730.800647 0.00021887984991

2957.8958745 0.00169607088030

673.22133595 0.00879546035380

190.68249494 0.03535938260500

62.295441898 0.11095789217000

22.654161182 0.24982972552000

0 0 2 2.0 1.0

8.9791477428 0.4062389614800

3.6863002370 0.2433821717600

0 0 1 0.0 1.0

0.7865398200 1.0000000000000

0 0 1 0.0 1.0

0.2677997200 1.0000000000000

0 2 4 3.0 1.0

49.200380510 0.0055552416751

11.346790537 0.0380523797230

3.4273972411 0.1495367102900

1.1785525134 0.3494930523000

0 2 1 0.0 1.0

0.3780331700 1.0000000000000

0 2 1 0.0 1.0

0.1473661500 1.0000000000000

0 3 1 0.0 1.0

0.3612294900 1.0000000000000

6 8

0 0 6 2.0 1.0

13575.349682 0.00022245814352

2035.2333680 0.00172327382520

463.22562359 0.00892557153140

131.20019598 0.03572798450200

42.853015891 0.11076259931000

15.584185766 0.24295627626000

0 0 2 2.0 1.0

6.2067138508 0.41440263448000

2.5764896527 0.23744968655000

0 0 1 0.0 1.0

0.4941102000 1.00000000000000

0 0 1 0.0 1.0

0.1644071000 1.00000000000000

0 2 4 2.0 1.0

34.697232244 0.00533336578050

7.9582622826 0.03586410909200

2.3780826883 0.14215873329000

0.8143320818 0.34270471845000

0 2 1 0.0 1.0

0.5662417100 1.00000000000000

0 2 1 0.0 1.0

0.2673545000 1.00000000000000

0 3 1 0.0 1.0

0.8791584200 1.00000000000000

HH

1 4

0 0 3 1.0 1.0

34.061341000 0.00602519780

5.1235746000 0.04502109400

1.1646626000 0.20189726000

0 0 1 0.0 1.0

0.4157455100 1.00000000000

0 0 1 0.0 1.0

0.1795111000 1.00000000000

0 2 1 0.0 1.0

0.8000000000 1.00000000000

APPENDIX

198

Appdx 5: Solid Solution of SiO2-GeO2

199

-216.2362727144-216.2362727144 -216.236439407-216.236439407 -216.2363301494-216.2363301494 -216.2364438427-216.2364438427 -216.236600945-216.236600945

xx = 3/6 (N = 3/6 (Ntottot = 20, N = 20, Nirrirr = 3) = 3)

-216.0137111381-216.0137111381 -216.0137172954-216.0137172954 -216.0139187892-216.0139187892 -216.0130272115-216.0130272115 -216.0138112775-216.0138112775

-216.0138104816-216.0138104816 -216.0138105112-216.0138105112 -216.0139175103-216.0139175103 -216.0139149036-216.0139149036 -216.0138105062-216.0138105062

-216.0138150336-216.0138150336 -216.0139148786-216.0139148786 -216.013917487-216.013917487 -216.0138149666-216.0138149666 -216.0138150065-216.0138150065

-216.0138113117-216.0138113117 -216.0131428115-216.0131428115 -216.0139188169-216.0139188169 -216.0138105582-216.0138105582 -216.013811323-216.013811323

APPENDIX

200

DESIGN OF ENHANCED PIEZOELECTRIC MATERIALS

FROM QUANTUM CHEMICAL CALCULATIONS

An exhaustive analysis of the technologically important piezoelectric phenomena is

here done by applying quantum chemical simulations. At first, the calibration of the

assumed computational scheme is examined by comparing our calculated piezoelectric

properties of the well-known piezoelectric quartz to their experimental counterparts.

Secondly, the microscopic parameters that influence each contribution of piezoelec-

tric macroscopic property are distinctly rationalized. After the rationalization of the

piezoelectric property, the design of materials that exhibiting a high piezoelectric effect

has been attempted. It has been shown that a large in-plane piezoelectricity induced

in graphene by doping can be acquired by including any in-plane defect(s). More-

over, in the limit of vanishing defect concentration, the piezoelectric response tends

toward a unique value, neither null nor infinite, regardless of the particular chemi-

cal or physical nature of the defect. The induction of an out-of-plane piezoelectricity

in graphene by breaking its planarity through the non-periodic z-direction is stated,

where the obtained piezoelectric response is largely improved compared to the finite

in-plane piezoelectric limit, at however higher concentration of the defect. Contrarily

to what has been discussed for the in-plane piezoelectric effect, the out-of-plane one

eventually vanishes as far as the limit of infinite defect dilution is reached, and so it

relies ultimately on the nature of the defect.

CONCEPTION PAR LA MODELISATION MOLECULAIRE DE

MATERIAUX A PROPRIETES PIEZOELECTRIQUES AUGMENTEES

Une analyse exhaustive de la piezoelectricite a ete realisee par la modelisation

moleculaire basee sur l’application des principes de la mecanique quantique. La calibra-

tion de la methode et des parametres du calcul est d’abord examinee en comparant les

resultats calcules concernant les oxydes de silicium et de germanium a leurs homologues

experimentaux. Ensuite, les parametres microscopiques qui influencent chaque contri-

bution de cette propriete macroscopique de reponse sont distinctement rationalises. En-

fin, apres la rationalisation de la propriete piezoelectrique, la conception de materiaux

montrant un effet piezoelectrique eleve a ete tentee. Nous avons montre que la grande

piezoelectricite induite par un dopage dans le plan du graphene tendra vers une valeur

unique, ni nulle ni infinie, et de facon independante de la nature physique ou chimique

particuliere du defaut. L’induction d’une piezoelectricite hors du plan du graphene en

brisant sa planeite selon la direction z est etudiee. La reponse piezoelectrique obtenue

est largement amelioree par rapport a la limite finie de la piezoelectricite dans le plan,

mais aux grandes concentrations du defaut. En effet, contrairement a la composante

dans le plan de la piezoelectricite induite dans le graphene, la composante hors du plan,

depend de la nature du defaut et diminue jusqu’a tendre vers zero a dilution infinie.

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