The eﬀect of temperature on the phonon dispersion relation

The effect of temperature on the phonon

dispersion relation in grapheneMaster thesis by:

Inka L.M. Locht

Supervisor:

Prof.dr. A. Fasolino

External advisor:

Dr. P.C.M. Christianen

Department Theory of Condensed MatterInstitute for Molecules and MaterialsRadboud University of NijmegenHeyendaalseweg 1356525 AJ Nijmegenthe [email protected]

Abstract

We have studied the temperature dependence of the phonon dispersion relation in graphene.First, we calculated the dispersion relation in the harmonic approximation with an em-pirical potential lcbop [1]. Second, we calculated some thermodynamic quantities in thequasiharmonic approximation, using the same potential. Third, we adapted the scaild-method [2], to use lcbop and to calculate the temperature dependence of the phonondispersion relation and of the bending rigidity in graphene. We can conclude that thetemperature dependence of the bending rigidity is due to coupling in the whole phononspectrum. Calculation of this dependence was not yet possible, since we have found thatthe scaild-method is not directly applicable for two-dimensional materials. We discusspossible strategies to adapt the method to this situation.

Contents

1 Graphene 3

1.1 Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Reciprocal space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Lattice Dynamics 5

2.1 Theory of lattice dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Normal coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3 Properties for graphene . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.1 Symmetry properties . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.2 Bending rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Empirical potential 12

3.1 LCBOP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Harmonic approximation 14

4.1 Computational implementation . . . . . . . . . . . . . . . . . . . . . . . . 144.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.3 Flexural phonon mode under strain . . . . . . . . . . . . . . . . . . . . . . 16

5 Quasiharmonic approximation 20

5.1 Theory: Quasiharmonic Approximation . . . . . . . . . . . . . . . . . . . 205.2 Calculational implementation . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.2.1 Thermal expansion coefficient . . . . . . . . . . . . . . . . . . . . . 225.3 Calculation of the thermal expansion coefficient . . . . . . . . . . . . . . . 23

5.3.1 Direct minimisation of the free energy . . . . . . . . . . . . . . . . 235.3.2 Gruneisen formalism . . . . . . . . . . . . . . . . . . . . . . . . . . 245.3.3 Comparing different methods . . . . . . . . . . . . . . . . . . . . . 24

6 SCAILD 28

6.1 Supercell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286.1.1 Commensurate wavevectors . . . . . . . . . . . . . . . . . . . . . . 286.1.2 Folding of the Brillouin zone . . . . . . . . . . . . . . . . . . . . . 29

6.2 Theory: SCAILD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336.3 Computational implementation . . . . . . . . . . . . . . . . . . . . . . . . 35

i

6.4 SCAILD: Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.4.1 Effects of the variation of the size of the displacements . . . . . . . 386.4.2 Effect of variation of the lattice parameter . . . . . . . . . . . . . . 426.4.3 Effect of variation of the temperature . . . . . . . . . . . . . . . . 436.4.4 Effect of variation of the size of the supercell . . . . . . . . . . . . 456.4.5 Effect of averaging over previous frequencies . . . . . . . . . . . . . 476.4.6 Effect of scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

6.5 Long wavelength limit of membranes . . . . . . . . . . . . . . . . . . . . . 526.5.1 Theory: divergence for small wavevectors . . . . . . . . . . . . . . 526.5.2 Influence of a minimal wavevector . . . . . . . . . . . . . . . . . . 536.5.3 Different temperatures . . . . . . . . . . . . . . . . . . . . . . . . . 556.5.4 Localised modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.5.5 Size of the displacement . . . . . . . . . . . . . . . . . . . . . . . . 63

7 Summary and conclusions 65

A Additional calculations 68

A.1 Diamond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A.2 New torsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

References 71

ii

Introduction

This thesis focuses on graphene: a recently discovered and very promising crystal.Graphene is a two-dimensional structure consisting of carbon atoms closely packed ina honeycomb lattice. The material is recently discovered [3] and a lot of fundamentalresearch can still be done to unveil its properties and the physics behind it.

This thesis is a continuation and extension of the work done by Leendertjan Kars-semeijer for his master in our group[4]. He calculated, among other things, the phonondispersion relation in graphene in the harmonic approximation with an empirical poten-tial lcbop. In figure 1 his calculations are compared with experimental values. We seea remarkably good agreement, except for the out-of-plane acoustic mode. This mode(za-mode) is significantly softer than the experimental values. For two-dimensional ma-terials it is known that the acoustic out-of-plane mode has a quadratic dispersion, asshown by Lifshitz [5]. Since the mode is quadratic, it has low frequencies for a largerange of wavevectors. This means that this mode is easily populated even for low tem-peratures. Therefore L. Karssemeijer and A. Fasolino tried to include some anharmoniceffect by using the quasiharmonic approximation. The quasiharmonic approximationreproduced the negative thermal expansion coefficient of graphene, but at a certain tem-perature (about T = 300K) a problem occurred. At high enough temperatures, thelattice parameter that minimises the quasiharmonic free energy, becomes quite smalland the material becomes dynamically unstable, namely imaginary frequencies occur inthe harmonic approximation. The aim of this thesis is to include anharmonic effects inthe calculation of the phonon dispersion relation by a method named scaild developedby P. Souvatzis et al. [6], [2], [7]. This method will be explained in detail in section6.2, but the basic idea is to use a supercell (a sample significantly larger than one unitcell) and excite phonon modes in this supercell by displacing the atoms according tothe phonon dispersion relation and the polarisation of the modes. The distorted sam-ple gives rise to a new dispersion relation, which gives rise to new displacements. Thisprocedure is done in a self-consistent way. To understand this method and to interpretthe results, it is important to understand the harmonic approximation and the quasihar-monic approximation. Both are very powerful tools, but one also should realise the finiteapplicability of these approximations. Therefore this thesis begins with the harmonicand quasiharmonic approximations and is ordered in the following way:

Chapter 1 gives an introduction to some general properties of graphene as a crystal.

Chapter 2 contains the theory of lattice dynamics in the harmonic approximation.

2

0

200

400

600

800

1000

1200

1400

1600

Γ M K Γ

Fre

quen

cy (

cm-1

)

ZA

TA

LA

ZO

TO

LO LO

LA

TA

ZA

ZO

Figure 1: Phonon dispersion relation of graphene as calculated by L. Karssemeijer and A. Fasolino[4], reproduced with their permission.

Chapter 3 introduces the empirical potential lcbop that we used for all our calcula-tions.

Chapter 4 describes how we calculated the phonon dispersion relation in the harmonicapproximation and compares the results to experimental data.

Chapter 5 elaborates on the simplest extension of the harmonic approximation: thequasiharmonic approximation. With the quasiharmonic approximation some ther-modynamic quantities can be calculated. This chapter will give results for thethermal expansion coefficient of graphene, calculated in two different ways.

Chapter 6 introduces the scaild-method, the adjustments to this method and ourcomputational implementation. Furthermore this chapter demonstrates some ofthe difficulties and problems we encountered. In this chapter we investigate theeffect of several variables like the temperature and the initial lattice parameteron the behaviour of the acoustic out-of-plane mode and on the self-consistentprocedure in order to gain a better understanding. Finally it states a numberof results, questions and remarks.

Chapter 1

Graphene

This chapter gives a short introduction to the crystal structure of graphene in realspace and in reciprocal space. In reciprocal space we give the points and lines withhigh symmetry, since paths along these lines are frequently used to display the phonondispersion relation.

1.1 Crystal structure

Graphene is a 2-dimensional crystal structure in which the carbon atoms are arranged ina honeycomb lattice (see figure 1.1). We can also interpret this as two interpenetratingtriangular lattices A and B separated by a vector δ = (0, a/

√3). The basis vectors of

these lattices are given by:

a1 = a

(

10

)

a2 = a

(

cos 60 ◦

sin 60 ◦

)

= a

(

12

12

√3

)

where a is the length of the basis vectors and a =√3acc, where acc is the distance

between the nearest neighbours.

A

B

O

δa1

a2

Figure 1.1: Graphene lattice [4]

Γ

FBZ

b2

b1

K

M

K’

Figure 1.2: Brillouin zone[4]

1.2 Reciprocal space 4

1.2 Reciprocal space

For most of the calculations in solid state physics we use the reciprocal lattice of acrystal. The reciprocal lattice vectors are constructed with the following identity:

ai · bj = 2πδij

This gives the following reciprocal lattice vectors:

b1 =2π

a

(

1− 1√

3

)

b2 =2π

a

(

02√3

)

(1.1)

With these reciprocal vectors one can construct the first Brillouin zone which is shownin figure 1.2. In this figure some points with special symmetry (Γ, K, K ′ and M) aredenoted. The coordinates of these special points in reciprocal space are:

Γ =2π

a

(

00

)

K =2π

a

(

230

)

K ′ =2π

a

(

131√3

)

M =2π

a

(

121

2√3

)

These points are highly symmetrical and the lines among them also have special sym-metry. Phonon dispersion relations are often calculated and displayed along the linesthat connect these points.

Chapter 2

Lattice Dynamics

In crystals at finite temperatures the atoms are not exactly at their ideal position: theyoscillate around this position. One could describe these oscillations atom by atom, butthis would be very cumbersome for large systems. There is a very powerful theoremfor infinite systems, the Bloch theorem, that allows us to describe these oscillations ascollective waves. These waves can be seen as a kind of particles (quasiparticles) and arecalled phonons. These quasiparticles have a certain dispersion relation depending on thematerial: the phonon dispersion relation. In this thesis we will focus on the temperaturedependence of the phonon dispersion relation in graphene. Before discussing tempera-ture, in this chapter the harmonic theory of lattice dynamics will be explained. In theend of the chapter we will look at some symmetry properties for graphene. Moreoverwe will explain how one can obtain the bending rigidity for two-dimensional materialsfrom the phonon dispersion relation. The bending rigidity will play an important rolein this thesis, since it also describes the hardness or softness of the acoustic out-of-planephonon mode.

2.1 Theory of lattice dynamics

In a crystal all atoms have a certain equilibrium position. Together, these positions formthe crystal lattice. At temperatures larger than zero all atoms oscillate around theirequilibrium position. We describe these vibrations in the lattice as collective excitationspropagating through the lattice: phonons. Most of this paragraph follows the lecturesand the book of M.I. Katsnelson [8].

To understand the vibrational modes in a crystal we consider a potential V as afunction of the coordinates of the atoms in the crystal. We will use the following notation:

• The crystal consists of a number of unit cells labelled by the integer vector l =(lx, ly, lz). The position of the unit cell is given by Rl

• Each unit cell consists of n atoms labelled by k, with positions rk with respect tothe origin of the unit cell.

2.1 Theory of lattice dynamics 6

• The Cartesian coordinates are denoted with Greek indices α and β

For a first estimation of the dispersion relation of the phonons we use the harmonicapproximation. This approximation assumes that the crystal is described by a Bravaislattice, where the ions oscillate around their equilibrium position with deviations thatare small compared to the interatomic distance. With this assumption we can expandthe potential energy V as a function of the displacements of the atoms (uα(lk)) up tosecond order.

V ≈ V0+∑

lkα

(

∂V

∂uα(lk)

)

0

uα(lk)+1

2

∑

lkα,l′k′β

(

∂2V

∂uα(lk)∂uβ(l′k′)

)

0

uα(lk)uβ(l′k′) (2.1)

The first term is a constant. The second term is zero when the system is in equilibrium.The only important term contains the second derivatives of the potential energy, whichwe call the force constants:

φαβ(lk, l′k′) =

∂2V


With this approximation of the potential energy, we obtain a Gaussian Hamiltonianwhich can be solved. The force constant matrix describes the force on atom (lk) arisingfrom displacements of the other atoms.

Fα(lk) = − ∂V

∂uα(lk)≈ −

∑

l′k′β

(

∂2V


)

0

uβ(l′k′)

= −∑

l′k′β

φαβ(lk, l′k′)uβ(l

′k′)(2.2)

This force constant matrix has certain features. The force constant matrix is symmetric,since

∂2V


=∂2V

∂uβ(l′k′)∂uα(lk)

(2.3)

Furthermore we could displace the lattice rigidly by a vector v. This means replacinguβ(l

′k′) in eq.(2.2) with uβ(l′k′) + vβ. This operation cannot change the force on an

atom, therefore:∑

l′k′

φαβ(lk, l′k′) = 0 (2.4)

This is the condition of translational invariance.We can write the equation of motion for each atom:

mkuα(lk) = −∑

l′k′β

φαβ(lk, l′k′)uβ(l

′k′) (2.5)

where mk is the mass of the k-th atom in the unit cell. This set of 3n coupled lineardifferential equations can be solved with the ansatz:

2.1 Theory of lattice dynamics 7

uα(lk) =Aα,k(q)√

mk

ei(q·Rl−ω(q)t) (2.6)

where Aα,k(q) is the amplitude, q the wavevector with corresponding frequency ω(q)and t the time. When we substitute this ansatz into the equation of motion (2.5), wefind:

ω2(q)eiq·RlAα,k(q) =∑

l′k′β

1√mkmk′

φαβ(lk, l′k′)eiq·Rl′Aβ,k′(q) (2.7)

We can reformulate this linear relation in an eigenvalue problem:

ω2(q)Aα,k(q) =∑

βk′

Dαβ(k, k′, q)Aβ,k′(q) (2.8)

where the dynamical matrix Dαβ(k, k′, q) is given by:

Dαβ(k, k′, q) ≡ 1√

mkmk′

[

∑

l′

φαβ(lk, l′k′)eiq·(Rl′−Rl)

]

(2.9)

The dynamical matrix is Hermitian:

[

Dβα(k′, k, q)

]∗=

1√mkmk′

∑

l′

φβα(lk′, l′k)e−iq·(Rl′−Rl)

1=

1√mkmk′

∑

l′

φαβ(l′k, lk′)e−iq·(Rl′−Rl)

2=

1√mkmk′

∑

l′

φαβ(lk, l′k′)eiq·(Rl′−Rl)

= Dαβ(k, k′, q)

where we have used at 1 the symmetry of the force constant matrix. At 2 we statesthat each Bravais lattice contains the inversion centre. Since the dynamical matrix isHermitian it can be diagonalised and the eigenvalues are real. This system of 3n × 3nsolutions (eq.(2.7)) can be solved if the following determinant equals zero:

|Dαβ(k, k′, q)− δαβδkk′ω

2(q)| = 0 (2.10)

The square root of the eigenvalues ω2(q) give the phonon dispersion relation in thematerial. Notice that a negative value of ω2(q) signals a dynamical instability. In factif of ω2(q) < 0, ω(q) becomes imaginary and the displacement 2.6 grows exponentiallywith time instead of displaying oscillatory behaviour.

At every q there are 3n solutions (the different branches), labelled by j. The eigen-vectors are not fully determined by eq.(2.8), so we can construct them in a way that theorthonormality relations hold:

∑

α,k

Aα,k(q, j)A∗α,k(q, j

′) = δj,j′

2.2 Normal coordinates 8

∑

j

Aα,k(q, j)A∗β,k′(q, j) = δk,k′δα,β

2.2 Normal coordinates

In order to recognise the wavelike behaviour, we will introduce normal coordinates. Wecan write the Hamiltonian of the crystal as:

H = T + V =1

2

∑

lkα

pα(lk)pα(lk)

mk

+1

2

∑

lkα

∑

l′k′β

φαβ(lk, l′k′)uα(lk)uβ(l

′k′)

(2.11)

In the harmonic approximation we can introduce normal coordinates Q(q, j) and P (q, j):

uα(lk) =∑

q,j

√

1

mkNAα,k(q, j)Q(q, j)eiq·Rl (2.12)

pα(lk) =∑

q,j

√

mk

NAα,k(q, j)P (q, j)eiq·Rl

When we substitute this in the harmonic part of the Hamiltonian eq.(2.11) it separatesinto 3n independent harmonic oscillators:

H =1

2

∑

q,j

(

P ∗(q, j)P (q, j) + ω2(q, j)Q∗(q, j)Q(q, j))

In this Hamiltonian we recognise the Hamiltonian of a set of harmonic oscillators, withfrequencies ω(q, j). We describe the vibrating crystal with this set of harmonic oscilla-tors. These collective waves are called phonons.

2.3 Properties for graphene

2.3.1 Symmetry properties

From the symmetry properties of the graphene crystal we can deduce certain features ofthe phonon dispersion relation. The mirror symmetry in the graphene plane implies:

φxz(lk, l′k′) = φyz(lk, l

′k′) = 0

This means that the out-of-plane modes are separated from the in-plane modes in thisapproximation. Taking into account that the two sublattices of graphene as shown infigure 1.1 are equivalent, we see that the two parts of the dynamical matrix belongingto each of the two should be the same:

Dαβ(1, 1, q) = Dαβ(2, 2, q)

2.3 Properties for graphene 9

From the condition for translational invariance (eq.(2.4)) and the definition of the dy-namical matrix (eq.(2.9)) we can see that at the zone centre (q = 0):

Dαβ(1, 1, q = 0) +Dαβ(1, 2, q = 0) = 0

These considerations lead to six phonon branches from which two are purely out-of-planeand four are in-plane modes. The three acoustical modes are for small q translationalmodes. The optical modes at q = 0 are modes where the atoms move in the unit celland the centre of mass of the unit cell does not move.The acoustic out-of-plane mode is given by:

ω2ZA(q) = Dzz(1, 1, q) +Dzz(1, 2, q) (2.13)

The optical out-of-plane mode is given by

ω2ZO(q) = Dzz(1, 1, q)−Dzz(1, 2, q)

The two acoustical in-plane modes are given by the eigenvalues of:

Dαβ(1, 1, q) +Dαβ(1, 2, q) α, β = x, y

The two optical in-plane modes are given by the eigenvalues of

Dαβ(1, 1, q)−Dαβ(1, 2, q) α, β = x, y

2.3.2 Bending rigidity

In this thesis we will focus on the flexural phonons. These are the phonons of the acousticout-of-plane mode. Waves propagating in thin plates, membranes or two-dimensionalcrystals are fundamentally different from waves in three-dimensional materials. Thedispersion of the longitudinal waves in particular shows a strange type of behaviour: itis quadratic in the wavevector near the zone centre. This paragraph contains two waysto calculate the dispersion of the out-of-plane waves near the zone centre. The firstway uses the argument of rotational invariance, which imposes a condition on the forceconstants that leads to a condition on the eigenvalues of the dynamical matrix. Thesecond way is by means of the theory of elasticity. An equation of equilibrium is derivedfrom the minimum of the free energy. This gives an equation of motion, which gives thedispersion relation near the zone centre. We give the dispersion relation for the acousticout-of-plane phonon mode in the most general form at the end of this paragraph.

Rotational invariance

Analogously to the condition of translational invariance in three-dimensional materials(eq.(2.4)), we can define a condition of rotational invariance for two-dimensional ma-terials in three-dimensional space. The lattice could be rotated by a uniform in-plane


rotation. The rotation may not cause any forces or torques. This condition of rotationalinvariance implies:

∑

l′k′

φzz(lk, l′k′)Rα

l′Rβ

l′= 0

Together with eq.(2.9) in the limit that q goes to zero and the condition for the acousticout-of-plane branch (eq.(2.13)) we obtain:

∂2

∂qα∂qβ[Dzz(1, 1, q) +Dzz(1, 2, q)]

∣

∣

∣

∣

q=0

= 0

This means that the right hand side of eq.(2.13) starts with terms of order q4, soωZA(q) ∝ q2 for q → 0. The rotational invariance of a two-dimensional crystal forin-plane rotations implies that the acoustic out-of-plane mode has a quadratic disper-sion near the zone centre.

Theory of elasticity for thin plates

Another way to calculate the behaviour of the dispersion relation for out-of-plane wavesis from the theory of elasticity for thin plates. Landau and Lifshitz [9] explained howthe theory of elasticity provides the free energy of a bent plate. This theory is valid forthin plates, where the thickness h is small compared to the dimensions in the other twodirections. In a thin plate, bent by external forces, as in figure 2.1, we denote the verticaldisplacement of a point on the neutral surface (the surface that lies midway through theplate) with ζ. We assume that the deformations are small. In [9] the free energy for a

6

-

z

x

6

?

h

?6ζ

Figure 2.1: A thin plate bent by external forces

thin plate is calculated as a function of the deformations. The equation of equilibriumcan be derived from the condition for the minimum of the free energy. The equation of


equilibrium for a thin plate bent by external forces P is:

Eh3

12(1 − σ2)∆2ζ − P = 0

where E is Young’s modulus and σ Poisson’s ratio. ∆ is the two-dimensional Laplacian.

If we replace the force by the acceleration −P → ρh∂2ζ∂t2

, where ρh is the mass perunit area, we find the equation of motion:

ρ∂2ζ

∂t2+

Eh3

12(1 − σ2)∆2ζ = 0 (2.14)

For waves we can take the ansatz ζ ∝ ei(q·r−ωt) and substitute it in the equation ofmotion eq.(2.14). This implies directly that ω2(q) ∝ q4 for the bending waves with longwavelengths in the thin plate.

Flexural phonons in a strained sample

Lifshitz [5] showed that in two-dimensional materials, ω2ZA(q) is quartic in q near the zone

centre, but gains a quadratic term when there is strain in the sample. The dispersionrelation ω(q) of the acoustical out-of-plane mode with strain can be written as:

ω2(q) =κ

ρ|q|4 + u

2(λ+ µ)

ρ|q|2 (2.15)

where κ is the bending rigidity, ρ is the density, u is the uniform dilatation of the crystal,λ and µ are the Lame coefficients. Without strain the second term would be zero andthe mode (ω(q)) would be purely quadratic. From the dispersion close to q → 0 of theacoustic out-of-plane mode one can calculate the bending rigidity κ.

Chapter 3

Empirical potential

Calculating the phonon dispersion relation can be done in different ways. There are twostarting points for computational methods. The first one is to start from elementaryequations like the Schrodinger equation and calculate the quantities needed. Thereforethis method is called ab initio or from first principles. The second one uses an empiricalpotential which gives the energy of a certain configuration of atoms. This potentialis based on a model and on certain parameters. The advantage of using an empiricalpotential is that it allows one to calculate much larger samples, since it is computationallyless time-consuming than ab initio calculations.

The empirical potential we use in this thesis is a long range bond order potential,namely lcbop developed by J.H. Los and A. Fasolino [1]. This chapter introduces thispotential.

3.1 LCBOP

lcbop is a bond order potential[10]. This type of potential can describe different bondingstates of the atom. It assumes that the strength of a chemical bond depends on thebonding environment of the atoms. This type of potential goes beyond the so calledforce fields that can describe the energy variations only around equilibrium. Bond orderpotentials allow changes of coordination and bond formation and breaking. Lchbop

describes all phases of carbon and transitions among them with good accuracy [11].The long range carbon bond order potential (lcbop) [1] has a short range and a

long range part (a later version also includes a medium range term [12]). The long rangepart is a Morse like potential. The short range part contains a number of modificationsas compared to the Brenner potential[13]. There are different models for deciding whenthe long range part should be switched off. In this potential the long range interactionsare only excluded for nearest neighbours.

The potential calculates the total binding energy of a configuration of atoms, givenby:

Eb =1

2

N∑

i,j

V totij =

1

2

N∑

i,j

(

fc(rij)VSRij + S(rij)V

LRij

)

3.1 LCBOP 13

The total pair interaction V totij is given as a sum of the sort range interaction V SR

ij and

the long range interaction V LRij , for different distances between the atoms:

Distance between atoms:

0 A 1 A 2 A 3 A 4 A 5 A 6 A

Short range

Middle range

Long range

The short range interaction describes the covalent bonds. In this term the bond ordercharacter of the potential is included. The long range interaction accounts for the non-bonded interactions. Later a middle range term is included which takes into accountthe rest of the attractive interactions in the middle range regime. These attractiveinteractions are also environment-dependent. The function fc(rij) is a smooth cutofffunction and S(rij) excludes only the nearest neighbours.

Chapter 4

Harmonic approximation

In this chapter I will explain briefly the methods used by Karssemeijer and Fasolino[4] for calculating the phonon dispersion relation in graphene. They used the theory oflattice dynamics as explained in section 2.1. This is a purely harmonic theory. For theircalculation they used the empirical potential lcbop, described in section 3.1.

4.1 Computational implementation

In section 2.1 the theory of lattice dynamics was explained. This theory provides aneasy method for calculating the phonon dispersion relation. In order to obtain thedispersion relation we need to calculate the dynamical matrix (eq.(2.9)) based on theforce constants (eq.(2.2)). If we know the potential energy for a certain configuration ofthe atoms, this is a straightforward task. With the potential energy we can calculate theforce constants that are second derivatives of the potential. To calculate the derivativeswe use the following approximation:

df(x)

dx≈ f(x+ ǫ)− f(x− ǫ)

2ǫ

for small epsilon. The error is of order O(ǫ2). The second derivative becomes:

d

dx

(

df(x)

dx

)

≈ d

dx

(

f(x+ ǫ)− f(x− ǫ)

2ǫ

)

≈

[

f(x+2ǫ)−f(x)2ǫ

]

−[

f(x)−f(x−2ǫ)2ǫ

]

2ǫ

≈ f(x+ ǫ′) + f(x− ǫ′)− 2f(x)

(ǫ′)2

with an error of O(ǫ4).In the program the force constants φαβ(lk, l

′k′) are calculated by first displacingatom (lk) in direction α and then displacing atom (l′k′) in direction β, both over adistance ǫ. For these situations the potential energy is calculated.

When all force constants are calculated the condition for translational invariance(eq.(2.4)) is not always completely fulfilled, due to computational errors. Since this

4.2 Results 15

condition immediately implies that there are three linear independent eigenvectors witheigenvalues ω(q = 0) = 0 at the zone centre, this condition must be exactly satis-fied. In order to correct the condition for translational invariance, we take the sum∑

l′k′ φαβ(lk, l′k′). This should be zero, but sometimes it is a small but finite number,

say δ. We subtract from each entry of the force constant matrix a fraction of δ whichis in proportion to the absolute value of the entry. As a result the sum of the row giveszero. In order to keep it symmetrical we impose φαβ(lk, l

′k′) = φβα(l′k′, lk), which will

change some other rows. Therefore we do this in a loop until the sums of all rows givezero up to a few digits.

After these checks of the translational invariance and the symmetry of the forceconstants, the dynamical matrix is calculated for a certain wavevector in reciprocalspace as in eq.(2.9). The dynamical matrix is diagonalised numerically by means of thezheev() routine from the lapack linear algebra package and we obtain the eigenvaluesdepending on the wavevector. We do this for a path in reciprocal space, for examplefrom the zone centre to a point with high symmetry. In this way we obtain the dispersionrelation ω(q). For a schematic overview of the program see figure 4.1.

Calculate force constants (eq.(2.2))Correct the translational invariance (eq.(2.4))Check the translational invarianceCheck the symmetry (eq.(2.3))

do for a set of wavevectors along a path in reciprocal spaceCalculate the dynamical matrix (eq.(2.9))

Check the HermiticityDiagonalise the dynamical matrixCalculate the eigenvalues

enddo

Figure 4.1: Schematic representation of the program for calculating phonon dispersion relation.

4.2 Results

Karssemeijer and Fasolino [14] calculated the phonon dispersion relation for different car-bon crystals, like nanotubes, graphene and graphite. I focused in my work on graphene.

Figure 4.2 shows the phonon dispersion relation in graphene together with someexperimental values for graphite of [15],[16],[17] and[18]. The calculations with lcbop

are in quite good agreement with the experimental data. There is a strong disagreementfor the acoustic out-of-plane branch (za-mode). For two-dimensional materials it isknown that the acoustic out-of-plane mode is a quadratic mode, as shown by Lifshitz[5]. Since the mode is quadratic, it has low frequencies for many wavevectors. Thismeans that this mode is easily populated even for low temperatures. Therefore this

4.3 Flexural phonon mode under strain 16

0

200

400

600

800

1000

1200

1400

1600

Γ M K Γ

Fre

quen

cy (

cm-1

)

ZA

TA

LA

ZO

TO

LO LO

LA

TA

ZA

ZO

Figure 4.2: Phonon dispersion relation in graphene calculated by L. Karssemeijer and A. Fasolino[14](solid lines). Experimental data for graphite are from inelastic x-ray scattering (squares from [15]and circles from [16]) and electron energy loss spectroscopy (diamonds from [17] and triangles from[18]).

mode could be strongly temperature dependent and we have to include anharmoniceffects. In the next chapter the simplest way of including anharmonic effects will beexplained: the quasiharmonic approximation. After that we will try a more elaboratemethod for including even more anharmonic effects: scaild. But first we will look whatthe effect of strain is on the za-branch in the harmonic approximation.

4.3 Flexural phonon mode under strain

In this section we will look how the phonon dispersion relation of the za-mode behavesunder strain. All calculations are done in the harmonic approximation. In section2.3.2 the relation between the acoustic out of plane mode and the bending rigidity wasexplained. In this section we will look in more detail at what this implies.

We calculated the phonon dispersion relation for different lattice parameters or, inother words, under different strains. For easy reading we repeat the equation for theacoustic out of plane mode as was explained in 2.3.2:

ω2(q) =κ

ρ|q|4 + u

2(λ+ µ)

ρ|q|2 (4.1)

In figure 4.3 we show the acoustic out of plane mode for graphene with a latticeparameter a = 1.4195 A. This is slightly smaller than the equilibrium lattice parameter


at T = 0K: a(T=0K) = 1.4198 A. We see that this gives some imaginary frequencies,since the second term in equation (4.1) is negative and dominant over the first term forsmall q. In order to fit the phonon dispersion relation with equation (4.1), we plottedin graph 4.4(a) the frequency squared ω2, together with the fit. In graph 4.4(b) we tookthe square root. In figure 4.5 we show the behaviour of the fitting parameters. Withincreasing lattice parameter the bending rigidity decreases and seems to scale linearlywith the lattice parameter. The quadratic term (due to strain) increases with increasinglattice parameter and is zero for the equilibrium lattice parameter. From that we seethat at equilibrium lattice parameter and for T = 0K, the bending rigidity is κ = 0.66eV.

−50

0

50

100

150

200

250

300

Γ M

iω(q

)

ω(q

) in

cm

−1

Path in reciprocal space

N2, a=1.4195 ÅN128, a=1.4195 Å

(a) Dispersion relation ωza(q) from Γ to M .

−5

0

5

10

15

20

25

30

Γ 1/8 M

iω(q

)

ω(q

) in

cm

−1


N2, a=1.4195 ÅN128, a=1.4195 Å

(b) Detail: dispersion relation ωza(q) from Γ to14M .

Figure 4.3: In 4.3(b) the dispersion relation is plotted from Γ to M in the Brillouin zone of the unitcell with two atoms. For our supercell with 128 atoms the Brillouin zone folds in eight parts in thisdirection. From now on we will look at the path from Γ to 1

8M showed by the yellow (light grey)

points in the left graph and explicitly given in the right graph.


−50

0

50

100

150

200

Γ 1/16 M

ω2 (q

)


a=1.4170 Åa=1.4175 Åa=1.4180 Åa=1.4185 Åa=1.4190 Åa=1.4195 Åa=1.4200 Åa=1.4205 Åa=1.4210 Åa=1.4215 Åa=1.4220 Åa=1.4225 Åa=1.4230 Å

(a) ω2 fitted with f(q) = gq2 + hq4

−10

−5

0

5

10

15

20

25

30

35

Γ 1/16 M 1/8 M

iω(q

)

ω(q

) in

cm

−1


a=1.4170 Åa=1.4175 Åa=1.4180 Åa=1.4185 Åa=1.4190 Åa=1.4195 Åa=1.4200 Åa=1.4205 Åa=1.4210 Åa=1.4215 Åa=1.4220 Åa=1.4225 Åa=1.4230 Å

(b) ω plotted with the square root of f(q) = gq2 + hq4

Figure 4.4: In graph 4.4(a) ω2(q) and in graph 4.4(b) ω(q) of graphene are plotted for differentlattice parameters. The lines in graph 4.4(a) are a fit (f(q) = gq2 + hq4) to the data points. Ingraph 4.4(b) the square root of this fit is plotted.


−400

−300

−200

−100

0

100

200

300

400

500

1.417 1.418 1.419 1.42 1.421 1.422 1.423

Fitt

ed g

in f(

x) =

g x

2 + h

x4

Lattice parameter in Å(a) Parameter g of the fitf(q) = gq2 + hq4

700

705

710

715

720

725

730

735

740

745

1.417 1.418 1.419 1.42 1.421 1.422 1.423

Fitt

ed h

in f(

x) =

g x

2 + h

x4

Lattice parameter in Å(b) Parameter h of the fitf(q) = gq2 + hq4

0.645

0.65

0.655

0.66

0.665

0.67

0.675

1.417 1.418 1.419 1.42 1.421 1.422 1.423

κ ac

cord

ing

to h

of f

(x)

= g

x2 +

h x4 in

eV

Lattice parameter in Å

Kappa

(c) κ for the fitf(q) = gq2 + hq4

Figure 4.5: Behaviour of the fitting parameters as a function of the lattice parameter around theequilibrium value. In the last graph we converted the quartic term to the bending rigidity κ.

Chapter 5

Quasiharmonic approximation

In the theory of lattice dynamics that we derived in section 2.1, we used the harmonicapproximation. In the harmonic approximation the vibrations in the crystal are inde-pendent of the interatomic distance. This directly implies that the vibrational energydoes not depend on volume and so the equilibrium lattice parameter does not dependon temperature. Actually nothing depends on temperature, since temperature is notat all included in the harmonic approximation. A proper way to take the anharmoniceffects into account is to calculate all anharmonic terms, but this is not a reasonabletask. A simple way to take some anharmonic effects into account for calculations of thefree energy is the Quasiharmonic Approximation (qha). This chapter will give a shortintroduction to the theory of the quasiharmonic approximation. It provides two waysto calculate the thermal expansion coefficient. With these two ways the thermal expan-sion coefficient for graphene will be calculated and compared to the dft calculations ofMounet and Marzari [19].

5.1 Theory: Quasiharmonic Approximation

In the quasiharmonic approximation we use the harmonic partition function but weassume that the frequencies depend on a global static constraint X, usually the volume.The free energy can be calculated from the partition function. The partition functionconsists of a part due to the static lattice energy and a part due to the vibrations,which are in the harmonic approximation 3N independent oscillators. The (harmonic)partition function Z due to the vibrational contribution is given by:

Z =∏

qj

Zqj =∏

qj

[ ∞∑

n=0

e− ~ωqj

kBT(n+ 1

2)

]

=∏

qj

[

e− ~ωqj

2kBT

(

1− e− ~ωqj

kBT

)−1]

For the free energy in the quasiharmonic approximation one assumes that the frequenciesare dependent on the global constraint X, so ωqj → ωqj(X). Furthermore one has toinclude the zero temperature energy of the crystal U0(X) since it is volume dependent.This is the potential energy of the crystal if there were no vibrations (T = 0K). With

5.2 Calculational implementation 21

F = −kBT ln (Z), the free energy in the quasiharmonic approximation becomes:

F (X,T ) = U0(X) +1

2

∑

qj

~ωqj(X) + kBT∑

qj

ln

(

1− e− ~ωqj(X)

kBT

)

(5.1)

where q, j sums over all wavevectors q and their branches j in the Brillouin zone. Notethat X is usually the volume, but it can also contain anisotropic components of the straintensor, some external applied fields or distortions of the crystal lattice. When we takefor X the volume we can calculate many thermodynamic quantities like the Gruneisenparameters.

Whether the quasiharmonic approximation is enough to describe the system dependson the system and can only be checked by calculations and comparison to the experiment.

The quasiharmonic approximation can be useful to calculate the temperature depen-dence of thermodynamic quantities like the thermal expansion coefficient, the Gruneisenparameters and the heat capacity. In this thesis we will look especially at the thermalexpansion coefficient since the thermal expansion coefficient of graphene is strongly tem-perature dependent [19], [20]. Moreover the thermal expansion coefficient of grapheneis, unlike for most materials, negative at temperatures below about 800K. This is a fea-ture of membranes or layered materials (like graphite). One can intuitively understandthe negative thermal expansion coefficient in the following way: for finite temperaturesmembranes start to ripple. The deformations in the z-direction cause in-plane strain,which causes the area to decrease. It was proposed [20] that the lattice parameter ofgraphene first decreases and than above ∼ 1000K increases. This illustrates why thethermal expansion coefficient of graphene is very unusual.

5.2 Calculational implementation

To calculate the free energy (eq.(5.1)) for different lattice parameters we need to calculatethe phonon dispersion relation for these lattice parameters. We do this in the same wayas was explained in section 4.1, but now we start with a sample that is compressed orstretched.

The expression for the free energy includes a sum over all wavevectors in the Bril-louin zone. To perform this summation numerically a suitable mesh is needed. Theconvergence of the sum is highly dependent on the mesh. A common and proven meshis the Monkhorst-Pack mesh [21]. The Monkhorst-Pack mesh is a homogeneous gridof q-points in the Brillouin zone. The rows and columns of the grid lie parallel to thereciprocal lattice vectors.

qm1m2m−3 =m1

n1b1 +

m2

n2b2 +

m3

n3b3

where ni are the number of points along each reciprocal lattice vector bi, with i = 1, 2, 3.This grid includes a point exactly at the zone centre, which is often not desirable for theconvergence of the sum. Therefore a small offset ∆ is included:

qm1m2m−3 =m1

n1b1 +

m2

n2b2 +

m3

n3b3 +∆ (5.2)

5.2 Calculational implementation 22

The temperature dependence of the free energy is only included as a constant inexpression 5.1. A change in the lattice parameter however, influences the force constantsand therefore the phonon dispersion relation. An overview of the program to calculatethe free energy in the qha can be seen in figure 5.1.

do for several lattice parameters

Calculate a suitable mesh (eq.(5.2))


Calculate phonon dispersion relation for all qpoint in mesh

do for a mesh in reciprocal spaceCalculate the dynamical matrix (eq.(2.9))


enddo

Calculate free energy for different temperatures

do for several temperaturesCalculate the free energy (eq.(5.1))

enddo

enddo

Figure 5.1: Program for calculating the free energy due to the phonons in the quasi harmonicapproximation.

5.2.1 Thermal expansion coefficient

The thermal expansion coefficient can be calculated in two ways within the quasihar-monic approximation [19]. The first one is to minimise the quasiharmonic free energy(eq.(5.1)) for each temperature in the lattice parameter. This gives for each tempera-ture a lattice parameter a(T ) where the free energy has a minimum. The linear thermalexpansion coefficient α can be calculated by:

α(T ) =1

a(T )

da(T )

dT(5.3)

The second method is called the Gruneisen formalism [22], which assumes a linear depen-dence of the phonon frequencies on the three orthogonal cell dimensions. For graphene

5.3 Calculation of the thermal expansion coefficient 23

we have of course only two dimensions and moreover the lattice parameter depends onlyon one lattice parameter a. For a structure with only one lattice parameter the condition(∂F∂a

)T = 0 leads to (as explained for example in [23]):

α =1

a20∂2U0∂a2

∣

∣

∣

0

∑

qj

cv(qj)−a0

ω0,qj)(X)

∂ωqj(X)

∂a

∣

∣

∣

∣

0

(5.4)

where U denotes the potential energy and the subscript 0 indicates that the quantityis taken at the equilibrium lattice parameter. The quantity cv(qj) is the contributionto the specific heat from mode (q, j). The total heat capacity per unit cell at constantvolume can be calculated as:

Cv = −T

(

∂2Fvib

∂T 2

)

V

where Fvib is the vibrational part of the free energy and is given by the last term ofequation (5.1). By taking the second derivative with respect to temperature we obtain:

Cv =∑

q,j

cv(qj) = kB∑

q,j

(

~ωqj(X)

2kBT

)2 1

sinh2(

~ωqj(X)2kBT

)

5.3 Calculation of the thermal expansion coefficient

In this section we will calculate the thermal expansion coefficient in the two ways de-scribed in section 5.2.1 and compare our results with similar calculations from Mounetand Marzari [19] and with Monte Carlo simulations from Zakharchenko [24].

5.3.1 Direct minimisation of the free energy

In this subsection we calculate the thermal expansion coefficient by minimising the freeenergy for every temperature with respect to the lattice parameter. We do this in thefollowing way: for each temperature we calculate the quasiharmonic free energy forseveral lattice parameters, as is shown for T = 20K in figure 5.2. We performed thecalculations for 67 temperatures from 5K in steps of 5K and for 50 different latticeparameters between 1.4198 A and 1.4298 A in equal steps.

In order to find the minimum of the free energy for a given temperature with respectto the lattice parameter, we fitted a polynomial to the free energy and calculated theminimum of the polynomial. The order of the polynomial and the fitting range influencethe behaviour of the lattice parameter as is shown in graph 5.3(a). The thermal expan-sion coefficient was calculated by fitting a polynomial (up to 5th order) to the data for

acc(T ). From this fit we calculated α(T ) = 1a(T )

da(T )dT

.For low temperatures there is a good agreement between the different fitting methods

for the nearest neighbour distance. For higher temperatures the different fitting meth-ods give different results and we cannot say anything about the shape of the thermalexpansion coefficient α.


−7.1874

−7.1872

−7.187

−7.1868

−7.1866

−7.1864

−7.1862

−7.186

1.42 1.422 1.424 1.426 1.428

F p

er p

artic

le (

eV)

acc Å

T=20 K

Figure 5.2: Quasiharmonic free energy for graphene at T = 20K as a function of the lattice parameter.

5.3.2 Gruneisen formalism

In this subsection we calculated the thermal expansion coefficient with the Gruneisenformalism as explained in section 5.2.1. The calculation of the phonon dispersion rela-tion, its derivatives with respect to the lattice parameter and the second derivative ofthe static lattice energy with respect to the lattice parameter are calculated with lcbop,via the method explained in section 2.1.

There are two possibilities for the equilibrium lattice parameter that only differ bythe zero point motion: the lattice parameter that minimises the energy or the one thatminimises the free energy at T = 0K, namely including zero point motion. The latticeparameter that minimises the energy is aUmin

cc = 1.4198 A and the lattice parameterthat minimises the free energy is aFmin

cc = 1.4257 A. We cannot perform the calculationsat exactly aUmin

cc , since for the derivatives we need to calculate the dispersion relationin acc = aUmin

cc − ǫ. For lattice parameters smaller than aUmincc the dispersion relation

has imaginary frequencies, which is unphysical. Therefore we calculated the thermalexpansion coefficient with the Gruneisen formalism for an equilibrium lattice parameterof aUmin

cc = 1.4198 + 0.0002 = 1.4200 A. This is nearly the lattice parameter that min-imises the energy. Moreover we calculate the thermal expansion coefficient for a latticeparameter that minimises the free energy:aFmin

cc = 1.4257 A.When the thermal expansion coefficient is calculated with equation eq.(5.4) and one

knows the initial lattice parameter one can obtain the lattice parameter depending onthe temperature (acc(T )) by integrating eq.(5.3). The thermal expansion coefficient andthe lattice parameter are plotted in figure 5.4.

5.3.3 Comparing different methods

In this subsection we compare our methods with each other and with calculations doneby Mounet and Marzari [19]. Remarkably there are considerable differences among the


1.422

1.4225

1.423

1.4235

1.424

1.4245

1.425

1.4255

1.426

0 50 100 150 200 250 300 350

a cc

(Å)

T (K)(a) Nearest neighbour distance

−16

−14

−12

−10

−8

−6

−4

−2

0

0 50 100 150 200 250 300 350

α (1

0−6 K

−1 )

T (K)(b) Thermal expansion coefficient

Figure 5.3: The minimum of the free energy is found by fitting a polynomial to the free energy andcalculating the minimum. For the red crosses (dot-dashed line) a quadratic polynomial was used andthe fit was performed over the whole range. For the green dots (line with squares and stripes) acubic polynomial was used and the fit was again performed over the whole range. For the purplesquares (fine dotted line) a quadratic polynomial was used and the fit was first performed over thewhole range which gave an initial minimum, but secondly the fit was redone over a range of 0.002 Aaround the initial minimum. We see that up to around 100K the three methods agree reasonable forthe nearest neighbour distance. Above 100K the different fitting methods give different results andwe cannot say anything about the shape of α


-5

-4.5

-4

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0 200 400 600 800 1000 1200 1400

α (1

0-6 K

-1)

T (K)

(a) Thermal expansion coefficient

1.41

1.412

1.414

1.416

1.418

1.42

1.422

1.424

1.426

0 200 400 600 800 1000 1200 1400

a cc

(Å)

T (K)

(b) Nearest neighbour distance A

Figure 5.4: The yellow (light grey) lines represent the calculation for aFmin

cc = 1.4257 A. The green(dark grey) lines the calculations for aUmin

cc = 1.4198 + 0.0002 A= 1.4200 A.

methods.In all cases the lattice parameter decreases with increasing temperature. The curve

for the thermal expansion coefficient that Mounet and Marzari obtained via direct min-imisation of the free energy mostly resembles our calculations via the Gruneisen formal-ism. We find a quite large and negative thermal expansion coefficient in our calculationsvia direct minimisation of the free energy.


1.416

1.418

1.42

1.422

1.424

1.426

0 200 400 600 800 1000 1200 1400 1600 1800

a cc

(Å)

T (K)

1.4254

1.4258

1.4262

0 50 100

a cc

(Å)

T (K)

(a) Nearest neighbour distance

−14

−12

−10

−8

−6

−4

−2

0

0 200 400 600 800 1000 1200 1400 1600

α (1

0−6 K

−1 )

T (K)

−4

−2

0

0 50 100

α (1

0−6 K

−1 )

T (K)

(b) Thermal expansion coefficient

Figure 5.5: The colour scheme of these diagrams are the same as for the previous figures in thissection: nearest neighbour distance calculated by minimising the free energy: red crosses/dot-dashedline (fitted with quadratic polynomial over the whole range), olive green dots/line with squares andstripes (fitted with cubic polynomial over the whole range), purple squares/fine dotted line (fittedwith quadratic polynomial around the minimum). Thermal expansion coefficient calculated via theGruneisen formalism: yellow crosses/dashed line (aFmin

cc ), dark-green pluses/solid line (aUmin

cc ). Theadditional light-purple round dots are an approximation of the computational data from Mounet andMarzari [19]. They used the minimisation of the free energy for their calculations of the nearestneighbour. They calculated the thermal expansion coefficient by differentiation.

Chapter 6

SCAILD

So far we have used a harmonic approximation to the Hamiltonian. Even in the quasi-harmonic approximation we have used the harmonic partition function. When the tem-perature increases, the thermally induced displacements uα(l, k) become large. Thisimplies that we can no longer approximate the Hamiltonian only up to second order.One way to include the anharmonic effects would be to calculate all higher order terms,but this is very cumbersome. In this chapter we will try a method named scaild[6], [2],[7] in order to include anharmonic effects. This method calculates the phonon disper-sion relation in a self consistent way by exciting phonon modes in a supercell. Thereforethis chapter will begin with an explanation of these supercells and the theory behindthe scaild-method. It will continue with how we implemented this method and whichproblem we encountered. I will conclude with a number of results, questions and criticalremarks.

6.1 Supercell

To include anharmonic effects in the calculations of the phonon dispersion, we use thescaild method which will be explained in section 6.2. For this method we need asupercell much bigger than the unit cell of the crystal. In this supercell we will excitethe phonon waves, by displacing the atoms. These excitations will be performed onlywith the wavevectors that are commensurate with the supercell (paragraph 6.1.1), inorder to conserve the periodic boundary conditions. The distorted supercell will serveas the new ‘unit cell’. Since this supercell is much larger than the original unit cell, theBrillouin zone will fold, which will be explained in paragraph 6.1.2.

6.1.1 Commensurate wavevectors

A wavevector commensurate to the sample (or supercell) is a wavevector that “fits” intothe sample (or supercell) with periodic boundary conditions. For a linear chain we cansee this easily in figure 6.1.

6.1 Supercell 29

Figure 6.1: In this figure a linear chain with four atoms and periodic boundary conditions is drawn.The solid circles represent the atoms, the last open circle is drawn as a guide and is the same atomas the first due to periodic boundary conditions. The lines represent the commensurate waves. Forthis chain there are exactly four commensurate wavelengths.

For a two-dimensional structure like graphene the procedure is slightly more difficult,but essentially the same. For a sample (or supercell) with width Lx and length Ly awavelength that is commensurate obeys the following conditions:

nxλx = Lx with nx an integer and 0 ≤ nx < Nx

nyλy = Ly with ny an integer and 0 ≤ ny < Ny

Here Nx and Ny are the number of primitive cells in the x- and y-direction. In figure6.2 two commensurate wavevectors in a graphene sample are illustrated. Note that for atwo-dimensional sample there are linear combinations of commensurate wavevectors inthe x- and y-direction.

Figure 6.2: A graphene sample of 128 atoms denoted with yellow solid circles. Here two commensuratewavevectors are shown as a wave in the sample. In the graph on the right you can see a linearcombination of a wavevector in the x and one in the y-direction.

6.1.2 Folding of the Brillouin zone

For a linear chain we know that the Brillouin zone folds when we make our unit cellbigger. In two or three dimensions the same happens, but it is a bit more difficult tovisualise. For a supercell which is twice as large as the original unit cell, the Brillouinzone becomes twice as small. For graphene we can clearly see this in figure 6.3. Herethe dispersion is drawn for a unit cell consisting of two atoms and a supercell consistingof four atoms.

6.1 Supercell 30

-200

0

200

400

600

800

1000

1200

1400

1600

Γ M Γ

ω in

cm

-1

N=2N=4

(a) Dispersion relation (b) Supercell

Figure 6.3: In graph 6.3(a) the dispersion relation for two atoms per unit cell is drawn in green. Thedispersion relation for four atoms per supercell is drawn in yellow. We see that the two dispersionrelations overlap (as we expect), but that there is a folding of the Brillouin zone for the supercellof four atoms compared to the unit cell of two atoms. How the unit cell (N = 2) is extended to asupercell with four atoms is drawn in graph 6.3(b).

There is however an essential difference to the linear chain. With the linear chainwe can only expand the unit cell in one direction, but for a graphene lattice we cando it in two directions. The difference in the dispersion relation can be clearly seenby comparing graph 6.4(a) and 6.4(c). For the dispersion relation in graph 6.4(a) weextended the graphene unit cell only in the y-direction as is shown in graph 6.4(b). Sinceour path in reciprocal space is also in the y-direction we see exactly the folding that weexpect. For the dispersion relation in graph 6.4(c) we extended the graphene unit cellpartly in the y- and partly in the x-direction, as is shown in graph 6.4(d). We see thatthe Brillouin zone in the y-direction is less often folded as for graph 6.4(a). Moreoverthere are some extra branches due to folding in the x-direction. This can be explainedby the fact that the basis vectors of our lattice make an angle of 60 ◦ whereas we workwith Cartesian coordinates where the basis vectors make an angle of 90 ◦.

In figure 6.5(a) the Brillouin zone for graphene is the diamond from Γ to Γ1 to Γ2

to Γ3 and back to Γ. When we extend the sample y-direction (as in figure 6.3(b)), theline M1-Γ1 is folded to M1-Γ and we get the green dispersion relation in figure 6.5(c).When we extend the sample once in the y-direction and once in the x-direction (as infigure 6.4(d)) folding also occurs in the x-direction, which is not in the direction of one ofthe reciprocal lattice vectors. Due to folding in the x-direction, the line M3-M2 will befolded on the line B-A and the line M2-M4 on A-C. In figure 6.5(c) the yellow lines arethe dispersion relation on the path M3-M2-M4 plotted on the line B-A-C. We see thatthe green and the yellow lines together indeed give exactly the same dispersion relationas for a sample of 8 atoms (figure 6.4(d)) as shown in figure 6.4(c), with the dispersionrelation shown in figure 6.4(c). In figure 6.5(b) we see the Brillouin zone for a samplewith 16 atoms (4 in the x- and 4 in the y-direction). From the figure one can see howthe different lines fold on the line Γ-Γ1. For example: Q-Q1 folds to D-F , M3-M2 foldsto B-A, Q3-Q4 folds to H-E and so on.

6.1 Supercell 31

-200

0

200

400

600

800

1000

1200

1400

1600

Γ M Γ

ω in

cm

-1

N=2N=4N=8

(a) Dispersion relation (b) Supercell

-200

0

200

400

600

800

1000

1200

1400

1600

Γ M Γ

ω in

cm

-1

N=2N=4N=8

(c) Dispersion relation (d) Supercell

Figure 6.4: In the two graphs on the left the difference in the dispersion relation for different extensionsof the unit cell to a supercell of N = 8 atoms is plotted. The two different extensions of the unitcell are plotted in the two graphs on the right, where the two upper graphs correspond to each other,as well as the lower two. If the unit cell is extended to a supercell by pasting two unit cells in they-direction together, a path in the y-direction in reciprocal space folds exactly one time. Pasting thetwo unit cells in the x-direction causes extra branches along a path in the y-direction in reciprocalspace.

6.1 Supercell 32

Γ

Γ1

Γ2

Γ3

M1

M2

M3

M4

M5

A

B

C

x1

(a) Folding of the Brillouin zone for N = 8

Γ

Γ1

Γ2

Γ3

M1

M2

M3

M4

M5x2 x1

A

B

C

Q

Q1

Q2

Q3

Q4

Q5

H

D

E

F

G

(b) Folding of the Brillouin zone for N = 16

0

200

400

600

800

1000

1200

1400

1600

Γ M

ω (

cm-1

)

(c) Dispersion relation when folded once in both directions. The green lines are due to folding in they-direction. The yellow lines appear due to folding in the x-direction.

Figure 6.5:

6.2 Theory: SCAILD 33

The sample that we will use primarily contains 128 atoms and is folded 8 times inthe y- and 8 times in the x-direction. To calculate the displacements of the atoms inthe scaild-method we need the commensurate wavevectors. Moreover the size of thesereciprocal vectors turns out to be important.

In figure 6.6 the sizes of the reciprocal vectors that correspond to the different waysof folding are shown for the acoustical out-of-plane mode. The wavevectors that fold tothe Γ-point are exactly the commensurate wavevectors of the sample.

0

20

40

60

80

100

Γ 1/8M

ω c

m −

1

q = 0 Å −1

q = 0.18 Å −1

q = 0.32 Å −1

q = 0.37 Å −1 q = 0.37 Å −1

q = 0.41 Å −1

q = 0.55 Å −1

q = 0.41 Å −1

q = 0.64 Å −1

q = 0.74 Å −1

Figure 6.6: The dispersion relation of a cell with N = 128 atoms, which is folded 8 times in the x-and 8 times in the y-direction. The green lines represent the folding of the dispersion relation in they-direction, the yellow lines appear due to folding in the x-direction as explained. At each turningpoint the size of the reciprocal wavevector from where it folds, is denoted.

6.2 Theory: SCAILD

In this section we will explain the method we used to include anharmonic effects inthe calculation of the phonon dispersion relation. Our method of including anharmoniceffects is based on the scaild-method developed by P. Souvatzis and others [6], [2], [7].

6.2 Theory: SCAILD 34

This method was developed for materials that are unstable in the harmonic approxi-mation (dynamically unstable), like the high temperature bcc phase of many elementalmetals (Ti, Hf, Zr, La, . . . ). These materials are in the bcc phase at low temperaturesunstable, but stable at high temperatures. Zener [25] suggested that these dynamicallyunstable materials became stabilised by entropy. In the harmonic approximation notemperature and therefore no entropy is taken into account. The potential energy withthe first anharmonic term is:

V ≈ V0 +∑

lkα

(

∂V

∂uα(lk)

)

0

uα(lk) +1

2

∑

lkα,l′k′β

(

∂2V


)

0

uα(lk)uβ(l′k′)

+1

3!

∑

lkα,l′k′β,l′′k′′γ

(

∂3V


∂uγ(l′′k′′)

)

0

uα(lk)uβ(l′k′)uγ(l

′′k′′) + . . .

The main difference between our calculations and the scaild-method is that thescaild-method is based on ab initio calculations of the forces between the atoms,whereas we use an empirical potential to calculate the force constants.

The basic idea is to use a supercell (a sample significantly larger than one unit cell)and excite phonon modes in this supercell by displacing the atoms. The steps of thescaild-method will be explained briefly in the following paragraphs.

First calculate the phonon dispersion relation in the harmonic approximation. Ac-cording to this relation excite all lattice waves with a wavevector commensurate to thesupercell by displacing the atoms in the following way:

uα(lk) =∑

q,j

√

1

NAα,k(q, j)R(q, j)eiq·Rl (6.1)

where the sum over q runs over all wavevectors commensurate with the supercell andN is the number of atoms. We replaced the operators Q(q, j) in eq.(2.12) with realnumbers. We can do this at high enough temperatures (in the classical limit):

1√mk

Q(q, j) → R(q, j) = ±√

< Q∗(q, j)Q(q, j) >

mk

where the ± sign is chosen randomly and the thermodynamic average of the operatorsQ(q, j) give the mean-square atomic displacement:

< Q∗(q, j)Q(q, j) >=~

ωj(q)

[

1

2+ n

(

~ωj(q)

kBT

)]

(6.2)

where n(x) = 1/(ex − 1) is the Planck function.The second step is to calculate the frequencies (eq.(2.10)) in the supercell with the

displaced atoms. When we average these frequencies with the frequencies of the previousiterations we will obtain a new set of frequencies:

ω2j,(i)(q) =

1

NI

i∑

k=i−NI

(ωj,(k)(q))2 (6.3)

6.3 Computational implementation 35

where i indicates the iteration, NE is the current iteration and NI is the number ofprevious iterations that are taken into account. It should not be necessary to take allprevious iterations into account since all choices should converge eventually. To closethe cycle, calculate new atomic displacements with these frequencies (eq.(6.1)), and soon. This cycle is continued until self-consistency is reached.

6.3 Computational implementation

In this section I will explain how we implemented this method in our calculations. Theimplementation is based on our harmonic calculations of the phonon dispersion relationas explained in section 4.1. A crucial difference between our method and scaild isthat the scaild method is based on ab initio calculations, whereas we work with anempirical potential: lcbop. More details on this potential can be found in section 3.1.A schematic overview of our implementation can be seen in figure 6.7

6.3 Computational implementation 36

do over self consistency loops


do for a set of wavevectors along a path in reciprocal spaceCalculate the dynamical matrix (eq.(2.9))


enddo

do for all wavevectors commensurate with the sampleCalculate the dynamical matrix


enddo

Calculate the displacements for all atoms and displace them (eq.(6.1))Check for self consistency

enddo

Figure 6.7: Schematic representation of the program for calculating phonon dispersion relations takinginto account the phonon-phonon interaction in a self-consistent way.

6.4 SCAILD: Variables 37

6.4 SCAILD: Variables

Performing the procedure as described in section 6.2 for graphene turned out not to beas simple as we hoped. In the next section we will show that when we implement thescaild-method naively, the phonon modes become more and more unstable (fig.6.8):we obtain negative eigenvalues ω2 of the dynamical matrix. To explain why the methoddoes not work in our case, we tried several adjustments. We looked what the effect wasof the following things:

• We investigate the influence of a factor (smaller than 1) with which the displace-ments (eq.(6.1)) are multiplied: uxy(lk) → fxyuxy(lk) for the in-plane componentsof the displacement and uz(lk) → fzuz(lk) for the out-of-plane component of thedisplacement. It might be that the changes in the positions of the atoms andtherefore in the dispersion relation are too fast to remain at equilibrium when itis reached. Therefore we try in section 6.4.1 to use a small factor for a slowerprocedure.

• Since graphene has a negative thermal expansion coefficient (up until about T ≈800 K) we started with different lattice parameters smaller than the one for T =0 K. We investigate the influence of the initial lattice parameter in section 6.4.2.

• We looked at the influence of the temperature in section 6.4.3 . For higher tem-peratures we expect the anharmonic effects to be bigger.

• The effect of the size of the supercell will be examined in section 6.4.4.

• The influence of the number NI of ωi(q, s) in equation (6.3), which is taken intoaccount for averaging, is the subject of section 6.4.5.

We looked at the influence of these variables on several quantities:

• First of all, the dispersion relation of the phonons: what happens to the acousticout-of-plane phonon mode.

• The free energy for a given iteration, although this is not a physical quantity, untilequilibrium is reached.

• The size of the displacement in the z direction after several iterations.

Furthermore we have a slightly different problem than the problems Souvatzis andcoworkers handle in their papers [6], [2], [7]. We do not know the lattice parameter,since the lattice parameter of graphene is strongly temperature dependent.

By looking at the dispersion relation at T = 0 K for different lattice parametersbetween 1.4170 A and 1.4230 A, we noted that only for the equilibrium lattice parametera = 1.4198 A at T = 0 K the dispersion relation of the acoustic out of plane phonon modeis purely quadratic. For either smaller or bigger lattice parameters a linear term appears(see section 4.3). This linear term appears due to strain in the sample, which caused


us to try using the presence of this linear term as a scaling criterion, to estimate theequilibrium lattice parameter at each temperature. In section 6.4.6 we try to rescale thesample in between the scaild-iterations in order to eliminate any strain in the sample.

In the next paragraphs we will look at the effect of these different variables on theevolution of the phonon dispersion relation with consecutive iterations. Although not alladjustments are physical, we try to get a better understanding by evaluating their effecton the behaviour of the phonon dispersion relation within the self-consistent procedure.

6.4.1 Effects of the variation of the size of the displacements

In this part we will investigate the influence of the factors by which the displacementsare multiplied (fxy, fz) at each iteration. In figure 6.8 we see that the za-branch showsimaginary frequencies for a lattice parameter a = 1.4186 A which is smaller than a(T=0K)

= 1.4198 A as was explained in section 4.3. The line indicated with it = 0 represents thezeroth iteration, namely for the flat, ideal situation with the given lattice parameter. Inthis figure the displacements in all directions are multiplied by a factor fxy = fz = 1.We see that for higher iterations the mode becomes more unstable.

-10

-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 2it = 4it = 6it = 8

it = 10

Figure 6.8: The dispersion relation of the acoustic out-of-plane phonon mode ωza for T = 200 Kand a sample of N = 128 atoms, factors fxy = fz = 1, NI = 1 and lattice parameter a = 1.4186A. The different lines represent different iterations in the scaild-procedure. Note that the za-modegoes down with consecutive iterations as is indicated by the arrow.


-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 2it = 4it = 6it = 8

it = 10it = 12it = 14it = 16it = 18it = 20

Figure 6.9: The dispersion relation of the acoustic out-of-plane phonon mode ωza for T = 200 Kand a sample of N = 128 atoms. We started with a lattice parameter a = 1.4186 A. We ignoredthe displacements in the x and y direction (fxy = 0) and displaced only in the z-direction (fz = 1),NI = 1. Note that the za-mode rises with consecutive iterations as is indicated by the arrow.

In the original scaild papers, the eigenvectors of the dynamical matrix were notrecalculated for each iteration. Therefore the displacements (eq.(6.1)) were calculatedwith the eigenvectors of the flat, ideal sample. This means that modes which were purelyout-of-plane were kept purely out-of-plane during the whole procedure. In our case werecalculate the eigenvectors in each iteration. This means that eigenvectors which arepurely out-of-plane for the flat sample, gain in-plane parts during the iterations. Toobserve how these in-plane components affect the behaviour, we displaced the atomsonly in the z-direction (fxy = 0 and fz = 1). In figure 6.9 we see that displacing theatoms only in the z-direction stabilises the mode. The only problem is that when wecontinue iterating, the mode continues to rise and does not show convergence.

For both cases (fxy = fz = 1 and fxy = 0, fz = 1) we tried to multiply thedisplacements in all directions with a factor (f = 0.2), so the two cases became fxy =fz = 1 · 0.2 = 0.2 and fxy = 0, fz = 1 · 0.2 = 0.2. The effect of this small factor was thateverything went slower, but eventually exhibited the same behaviour as fxy = fz = 1and fxy = 0,fz = 1.

We have just shown that for displacements only in the z-direction the mode doesnot stop growing and the displacements in the z-direction continue increasing. We know


that two-dimensional materials are stabilised by the anharmonic coupling between thebending modes and the in-plane stretching modes [26]. Therefore it might be useful toinclude at least some in-plane displacements. Consequently we multiplied the displace-ments in z-direction by a factor fz = 1 and in the x and y-direction by different factors:fxy = 0.2, fxy = 0.3, fxy = 0.4 and fxy = 0.5. The results are shown in figure 6.10.In the first case (fxy = 0.2) the phonon mode stabilises, but again continues growing.For fxy = 0.3 and fxy = 0.4 the mode stabilises a bit, destabilises a bit, stabilises, andso on, but finally destabilises. For fxy = 0.5 the mode clearly becomes more and moreimaginary with successive iterations.

-4

-2

0

2

4

6

8

10

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 10it = 20it = 30it = 40it = 50it = 60it = 70it = 80it = 90

it = 100

(a) fxy = 0.2

-4

-2

0

2

4

6

8

10

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1



it = 100

(b) fxy = 0.3

-4

-2

0

2

4

6

8

10

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1



it = 100

(c) fxy = 0.4

-4

-2

0

2

4

6

8

10

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1



it = 100

(d) fxy = 0.5

Figure 6.10: Behaviour of the dispersion relation for different sizes of the displacements in the xand y-direction. This figures are for T = 200 K, different fxy, fz = 1, NI = 1, N = 128 atoms,a = 1.4186 A. Note that the mode seems to stabilise for fxy = 0.2 only. For all other factors theza-mode eventually destabilises.

In some cases we looked at what happened to the free energy for successive iterations,where we calculated the free energy in the quasiharmonic approximation (eq.(5.1)). Cu-riously the free energy increases, although the modes stabilise (see figure 6.11). Thebehaviour of the free energy is dominated by the static lattice contribution. The staticlattice contribution grows for successive iterations, which is explained by the fact that anideal flat graphene sheet has certainly a lower potential energy than a rippled sheet. The


thermal contribution is negative and grows with successive iterations. In formula 5.1 wecan indeed see that for increasing ω(q), the thermal contribution goes to zero. It shouldbe this term that could construct a minimum in the free energy due to entropy. Someimportant notes should be made. First of all, we cannot calculate the free energy in thequasiharmonic approximation when there are still imaginary frequencies. Secondly, it isnot very useful to calculate the free energy when the system is not converged. Beforeconvergence, the calculation of the free energy is not physical; nevertheless, it might givesome insight into the behaviour of the self-consistent procedure.

-7.191

-7.19

-7.189

-7.188

-7.187

-7.186

-7.185

-7.184

-7.183

0 50 100 150 200 250 300

Fre

e en

ergy

per

ato

m (

eV)

Iteration(a) Free energy

0.1644 0.16441 0.16442 0.16443 0.16444 0.16445 0.16446 0.16447 0.16448 0.16449

0.1645 0.16451

0 50 100 150 200 250 300Zer

o po

int m

otio

n pe

r at

om (

eV)

Iteration(b) Zero-point motion

-0.0049

-0.00485

-0.0048

-0.00475

-0.0047

-0.00465

0 50 100 150 200 250 300The

rmal

con

trib

utio

n pe

r at

om (

eV)

Iteration(c) Thermal part of the free energy

-7.351

-7.35

-7.349

-7.348

-7.347

-7.346

-7.345

-7.344

-7.343

0 50 100 150 200 250 300Sta

tic la

ttice

ene

rgy

per

atom

(eV

)

Iteration(d) Static lattice contribution

Figure 6.11: Behaviour of the free energy and the different contributions to it. This simulation isdone for T = 200 K, fxy = 0, fz = 0.5, NI = 10, N = 128 atoms, a = 1.4186 A. From iteration200 and onwards there are no imaginary frequencies any more. This is denoted by a vertical line.The values before iteration 200 are calculated by skipping all wavevectors which have a negativeeigenvalue (imaginary frequency); therefore they are not reliable.

The most important result of this paragraph is that displacing the atoms accordingto eq.(6.1) in all directions only destabilises the material further. By displacing theatoms only in the out-of-plane direction, the za-mode grows with successive iterations.Notice, however, that this mode continues growing. The self-consistent procedure doesnot converge.


6.4.2 Effect of variation of the lattice parameter

For temperatures below about T ≈ 800 K, graphene has a negative thermal expansioncoefficient [20], which means that the lattice parameter decreases with increasing tem-perature. Therefore we started with a flat sample that has a lattice parameter smallerthan the lattice parameter at T = 0K. In figure 6.12 some iterations are shown fordifferent lattice parameters. We see indeed that for lattice parameters smaller thana(T=0K) = 1.4198 A some frequencies of the acoustic out-of-plane phonon mode becomeimaginary. Moreover we note that for smaller lattice parameters there are more wavevec-tors with imaginary frequencies and that ω2 is more negative. Since we displaced theatoms only in the z-direction and with a factor fz = 0.5 and fxy = 0, we see that forevery lattice parameter the mode goes up with successive iterations.

-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 50

it = 100it = 150it = 200it = 250it = 300it = 350it = 399

(a) a = 1.4179 A

-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 50

it = 100it = 150it = 200it = 250it = 300it = 350it = 399

(b) a = 1.4182 A

-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 50

it = 100it = 150it = 200it = 250it = 300it = 350it = 399

(c) a = 1.4186 A

-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 50

it = 100it = 150it = 200it = 250it = 300it = 350it = 399

(d) a = 1.4198 A (equilibrium for T = 0 K)

Figure 6.12: The behaviour of the dispersion relation for different lattice parameters. The calculationswere done for T = 200, fxy = 0, fz = 0.5, NI = 10 and N = 128 atoms. For smaller latticeparameters there are more imaginary frequencies and ω2 is more negative. In all cases the za-modeincreases with consecutive iterations as is indicated with the arrows.

To summarise: for a smaller lattice parameter the za-branch has more imaginaryfrequencies. However the size of the lattice parameter does not influence the behaviourof the phonon dispersion relation with consecutive iterations.


6.4.3 Effect of variation of the temperature

To investigate the influence of the temperature, we performed the calculations (withfxy = 0 and fz = 0.5) for different temperatures. We see in figure 6.13 that for highertemperatures, the za-mode increases faster.

-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 50

it = 100it = 150it = 200it = 250it = 300it = 350it = 399

(a) T = 200 K

-5

0

5

10

15

Γ 1/8 Miω

(q)

ω(q

) in

cm

-1


it = 0it = 50

it = 100it = 150it = 200it = 250it = 300it = 350it = 399

(b) T = 400 K

-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 50

it = 100it = 150it = 200it = 250it = 300it = 350it = 399

(c) T = 800 K

-5

0

5

10

15

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 50

it = 100it = 150it = 200it = 250it = 300it = 350it = 399

(d) T = 1500 K

Figure 6.13: The behaviour of the dispersion relation for different temperatures. These calculationsare done with displacements only in the z-direction (fxy = 0 and fz = 0.5). We used a sample ofN = 128 atoms and a lattice parameter of a = 1.4186 A. We averaged over the last 10 iterations:NI = 10. The higher the temperature, the faster the za-mode increases as is indicated with thearrows.

We also show in figure 6.14 how large the displacements in the z-direction are fora given iteration at the different temperatures in. First of all we see that the distancebetween atoms in the z-direction generally increases with iterations. Furthermore we seethat for higher temperatures the displacements in the z-direction become larger. In thesecond graph the distance between nearest neighbours is plotted for iteration 400. Atthe zeroth iteration we should see a sharp peak at a = 1.4186 A for each temperature.With successive iterations this peak broadens. The peak of the last iteration is plottedand we see that the higher the temperature, the broader the peak. We can also see thatthe fluctuations are not too large: the nearest neighbour distance is shorter than therange of the short-range term of lcbop and the typical distance in diamond (1.54 A) or


in other configurations.We conclude that the influence of temperature is as we expect. For higher tempera-

tures more phonon modes can be populated, the displacements grow faster and thereforethe distance between the atoms becomes greater.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 50 100 150 200 250 300 350 400

Larg

est d

ista

nce

in th

e z

dire

ctio

n in

Å

Iteration

T=200 KT=400 KT=800 K

T=1500 K

(a) Maximal excursion in the z-direction

0

10

20

30

40

50

1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49

Num

ber

of a

tom

s

Nearest neighbor distance in Å

T=200 KT=400 KT=800 K

T=1500 K

(b) Radial distribution function last iteration (400)

Figure 6.14: In 6.14(a) we show the maximal excursion in the z-direction for several temperatures.For higher temperatures the distance between the atoms grows faster. 6.14(a) shows the radialdistribution function at different temperatures.


6.4.4 Effect of variation of the size of the supercell

There are two reasons to vary the size of the supercell. The first reason is that there aremore commensurate wavevectors for a larger supercell. The second reason is that theBrillouin zone folds more often for a larger supercell. This implies that at a given size ofthe supercell, there is a commensurate wavevector for which the frequency is imaginary(when we start with a lattice parameter smaller than a(T=0K)).

In this section we look at two different sizes of the supercell: N = 128 and N = 1024atoms. Note that if we double the size of the supercell in one direction, the size of theBrillouin zone halves. In the direction we are looking, the Brillouin zone for N = 1024atoms is four times smaller than the Brillouin zone for N = 128 atoms. Moreoverthere are some extra branches due to folding in the other direction (since graphene is ahexagonal lattice and we use Cartesian coordinates).

In figure 6.15 we see how the dispersion relation for N = 128 atoms evolves andhow we should compare it to the zeroth iteration for a sample with N = 1024 atoms.In figure 6.16 the dispersion relation for N = 1024 atoms is plotted. Each iteration isplotted in a separate graph together with the zeroth iteration in order to keep it clear.It is difficult to compare the dispersion relation for N = 128 (fig. 6.15) with the onefor N = 1024 (fig. 6.16) since there are more avoided crossings in the latter (becausewe took more commensurate wavevectors into account). We can see that for N = 1024atoms the dispersion relation stabilises more slowly than for N = 128 atoms. At the175th iteration the modes are still unstable, whereas for the smaller sample the modesstabilised within about 100 iterations.

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0it = 25it = 50it = 75it = 99

(a) N = 128

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0

(b) N = 1024

Figure 6.15: The behaviour of the dispersion relation for T = 200K, fxy = 0.2, fz = 1 and NI = 10.The lattice parameter is a = 1.4186 A. For the left graph the supercell contained N = 128 atoms.In the right graph only the zeroth iteration is plotted for a sample of N = 1024 atoms. One can seethat the Brillouin zone folds four times in the given direction.


-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0

(a) 0th iteration

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0it = 25

(b) 25th iteration

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0it = 50

(c) 50th iteration

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0it = 75

(d) 75th iteration

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0it = 100

(e) 100th iteration

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0it = 125

(f) 125th iteration

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0it = 150

(g) 150th iteration

-3

-2

-1

0

1

2

3

4

5

Γ 1/8 M

iω(q

)

ω(q

) in

cm

-1


it = 0it = 175

(h) 175th iteration

Figure 6.16: The behaviour of the dispersion relation for T = 200K, fxy = 0.2, fz = 1 and NI = 10.The lattice parameter is a = 1.4186 A. For these calculations the supercell contained N = 1024atoms. The black crosses indicate the dispersion for the given iteration and the zeroth iteration fora supercell of N = 128 atoms.


6.4.5 Effect of averaging over previous frequencies

To understand the different aspects of the scaild-procedure, we investigate the influenceof the number of frequencies from previous iterations that were taken into account tocalculate the mean of eq.(6.3). The simulations were done for fxy = 0 and are shown infigure 6.17. At first sight one cannot see any crucial difference. It looks as if for NI = 3and NI = 100 the za-branch rises faster than NI = 10 and NI = 20. This behaviouris not monotonic. Furthermore the distances between two consecutive (note that weplotted each 25th iteration) iterations differ.

-2

0

2

4

6

8

10

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 25it = 50it = 75

it = 100it = 125it = 150it = 175it = 200it = 225it = 250it = 275it = 300it = 325it = 350it = 375it = 399

(a) 3 previous iterations

-2

0

2

4

6

8

10

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 25it = 50it = 75


(b) 10 previous iterations

-2

0

2

4

6

8

10

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 25it = 50it = 75


(c) 20 previous iterations

-2

0

2

4

6

8

10

Γ 1/8 M

iω(q

)

ω

(q)

in c

m-1


it = 0it = 25it = 50it = 75


(d) 100 previous iterations

Figure 6.17: The behaviour of the dispersion relation taking into account a different amount of thefrequencies of the previous iterations. The simulations are done for T = 200 K, fxy = 0, fz = 0.5,a = 1.4186 A and N = 128 atoms.


6.4.6 Effect of scaling

In section 4.3, we saw that for a lattice parameter slightly smaller or bigger than thelattice parameter at T = 0K (a0), the dispersion relation of za-mode has a linear term.This linear term is positive for lattice parameters bigger than a0 and negative for latticeparameters smaller than a0.

Furthermore we know that when we include temperature (for example with thescaild-method), the graphene sheet ripples, which causes strain. The sample should beable to expand or to contract due to thermal expansion or contraction of the material.One could try to rescale the sample in between the scaild-iterations. The criterion thatshould be used for this is not evident. The free energy (in the quasi harmonic approx-imation) could be a good candidate, except that it is an equilibrium property and thesystem is definitely not in equilibrium.

Therefore we decided to take as our criterion that the linear term (which is due tostrain) in the za-mode (which is quadratic if there is no strain) should be zero.

Intuitively we can view it as follows: when we excite the phonon modes with thescaild-method and with factors fxy = fz = 1, the acoustic out-of-plane branch be-comes softer with successive iterations. The quadratic mode receives a negative linearterm. Scaling to a bigger lattice parameter means that there will be more in-planetension, which means that it is harder to excite out-of-plane modes. This means thatthe dispersion relation of the out-of-plane mode becomes harder and for a certain lat-tice parameter purely quadratic. These two opposite effects could probably lead to anequilibrium.

On the other hand, if we excite the phonon modes with factors fxy = 0.2 and fz = 1,the out-of-plane branch becomes harder with successive iterations. The quadratic modereceives a positive linear term. Scaling to a smaller lattice parameter means that therewill be less in-plane tension, which means that it is easier to excite the out-of-planemodes. This means that the dispersion relation of the out-of-plane mode becomes softerand, for a certain lattice parameter, purely quadratic.

The question arises whether these opposite effects cause the dispersion relation toconverge, or the sign of the linear term keeps alternating without convergence.

We tried to implement the following idea:

1. Start with lattice parameter a0 for which the linear term is approximately zero.

2. Perform some scaild-iterations until the linear term becomes appreciable.

3. Rescale the sample in such a way that the linear term is again approximately zero.

4. Continue doing 2. and 3.

In figure 6.18 we did this for fxy = fz = 1, in figure 6.19 for fxy = 0.2 and fz = 1.Note that the linear term changes considerably due to rescaling, whereas the quadratic

term barely changes due to scaling. This is in accordance to our expectations, since thelinear term is the term which includes the strain in the sample. In terms of the bend-ing rigidity it would mean that rescaling the sample barely affects the rigidity. On the


-10

-8

-6

-4

-2

0

2

4

0 1 2 3 4 5 6 7

Line

ar te

rm

Iteration

1.4198

1.4200

1.420021.42030

1.42040 1.42064

(a) Linear term

0.62

0.63

0.64

0.65

0.66

0.67

0.68

0.69

0.7

0.71

0 1 2 3 4 5 6 7

κ in

eV

iteration

1.41981.4200

1.42002

1.42030

1.42040

1.42064

(b) Quadratic term

Figure 6.18: In these simulations we chose T = 200K, fxy = fz = 1 and NI = 1. The green crossesrepresent successive iterations. When a green cross is encircled in yellow, we rescaled the sample.The encircled cross at the same iteration is the rescaled sample (with just above it the new latticeparameter in A), followed by some iterations.


-2

-1

0

1

2

3

4

5

6

0 5 10 15 20 25 30 35

Line

ar te

rm

Iteration

1.4198

1.419751.419731.419711.41968

1.419501.41948

(a) Linear term

0.68

0.7

0.72

0.74

0.76

0.78

0.8

0.82

0 5 10 15 20 25 30 35

κ in

eV

iteration

1.4198

1.419751.419731.41971

1.41968

1.41950

1.41948

(b) Quadratic term

Figure 6.19: In these simulations we chose T = 200K, fxy = 0.2, fz = 1 and NI = 1. The greencrosses represent successive iterations. When a green cross is encircled in yellow, we rescaled thesample. The encircled cross at the same iteration is the rescaled sample (with just above it the newlattice parameter in A), followed by some iterations. After rescaling the corresponding κ, given in thelower panel, does not converge but keeps growing.


other hand exciting phonons in the sample, and therefore rippling the sample, affectsthe bending rigidity considerably. We do indeed see that when we use fxy = fz = 1 theiterations destabilise the material and the rescaling results in a larger lattice parameter.For fxy = 0.2 and fz = 1, the iterations generally make the dispersion relation harderand the rescaling causes a smaller lattice parameter. Furthermore we see that the effectof one iteration with fxy = fz = 1 is generally much larger than the effect of one iterationwith fxy = 0.2 and fz = 1.

To summarise, we see that performing scaild-iterations and rescaling the sampledo indeed have opposite effects on the linear term. However this does not cause theza-branch to stabilise and converge. The sign of the linear term keeps alternating.

In summary, in the last six sections we looked at the effect of several variables onthe behaviour of the za-mode and the self-consistent procedure itself. Thinking aboutthese results we noticed that for two-dimensional materials we have an intrinsic problem:divergence in the long wavelength limit. These peculiarities of two-dimensional systemswill be explained in the next section. In addition we will try a method avoiding thestrong anharmonicity in the long wavelength limit.

6.5 Long wavelength limit of membranes 52

6.5 Long wavelength limit of membranes

In two-dimensional materials there are some quantities that diverge for small wavevec-tors. In this section we will briefly explain these divergences and look at the influenceof small wavevectors on our calculations.

6.5.1 Theory: divergence for small wavevectors

All our calculations are based on the theory of lattice dynamics as explained in section2.1, where we made the assumption that the average atomic displacement is very smallcompared to the interatomic distance. The mean square displacement can be calculatedby using the expression for the atomic displacements in normal coordinates (eq.(2.12)).

〈uα(lk)uβ(lk)〉 =∑

q,j

1

mNA∗

α,k(q, j)Aβ,k(q, j)〈Q∗(q, j)Q(q, j)〉

where the thermodynamic average 〈Q∗(q, j)Q(q, j)〉 is given by eq.(6.2). So in the har-monic approximation the mean square displacement is given by:

〈uα(lk)uβ(lk)〉 =∑

q,j

~

2mNωj(q)A∗

α,k(q, j)Aβ,k(q, j) coth

(

~ωj(q)

2kBT

)

(6.4)

For in-plane deformations the dispersion relation ω(q) near the zone centre is pro-portional to ∝ cs|q|. For q → 0 we can approximate a

qcoth (bq) by ≈ a

bq2. Performing

the integral over dq (in polar coordinates), gives that the in-plane displacements arelogarithmically divergent. This divergence is cut off by a minimal wavevector qmin ∝ 1

L,

with L the typical size of the system. The upper bound for the wavevector is determinedby the distance between the atoms qmax ∝ 1

a. Therefore the in-plane displacements grow

logarithmically with the sample size:

〈x2nj〉 = 〈y2nj〉 ∝ ln

(

L

a

)

Landau [27] and Peierls [28], [29] concluded from this argument that two-dimensionalcrystals could not exist. Mermin and Wagner [30], [31] made this statement more preciseand stated that two-dimensional crystalline order is impossible at non-zero temperatures.

For the acoustic out-of-plane branch the divergence is even more serious, since thedispersion relation ω(q) near the zone centre is proportional to∝ |q|2. For small wavevec-tors we can expand a

q2coth (bq2) ≈ a

bq4. When we integrate this over dq, we find that

the displacements in the z-direction have a much stronger divergence:

〈z2nj〉 ∝ L2

In this case the fluctuations in the z-direction could be as large as the in-plane size ofthe sample.


The ways the in-plane and the out-of-plane displacements diverge are calculated inthe harmonic approximation. If we take into account anharmonic coupling between thein-plane and out-of-plane long-wavelength deformations, we see that we have a problemof interacting fluctuations for small wavevectors. In the short wavelength limit graphenebehaves harmonically, but for longer wavelengths coupling between the modes plays animportant role. The separation between the nearly harmonic regime (large q, shortwavelengths) and the anharmonic regime (small q, large wavelengths) is approximatelygiven by the Ginzburg criterion[26]:

q∗ =

√

3TK

8πκ2

where T is the temperature, K the 2D bulk modulus and κ the bending rigidity. Belowq∗ the effects of interactions between fluctuations dominate. For the parameters ofgraphene q∗ ≈ 0.2 A−1.

Moreover the wavevector at the zone centre would be appropriate for an infinite sys-tem, whereas we have a finite system with a finite amount of commensurate wavevectors.The acoustic branches, which are zero at the zone centre, make a very large contributionto the displacements as calculated in eq.(6.1), since this expression has a factor 1/ωj(q).For an infinite system the wavevector at the zone centre would only count for an in-finitely small amount, since the weight with which each wavevector is counted goes withone over the number of commensurate wavevectors. For a finite system, this number isfinite and no longer infinitely small.

These are arguments for skipping the largest wave lengths (or smallest q, namelyq < q∗) of the acoustic out-of-plane mode in the calculations of the displacements of theatoms (eq.(6.1)).

6.5.2 Influence of a minimal wavevector

In this subsection we will look at the effect of truncating the sum in the calculationsof the displacements (eq.(6.1)) for very small wavevectors. The displacements becomethen:

uα(lk) =∑

q,j 6=ZA

|q|>q∗,j=ZA

√

1

NAα,k(q, j)R(q, j)eiq·Rl

where the sum over q runs over all commensurate wavevectors. If j denotes the za-modeonly the commensurate q with |q| > q∗ are taken into account. For calculations it is notimmediately clear at which wavevector we have to start the sum in our calculations ofthe displacements. The Ginzburg criterion gives us only an estimate. From the Monte-Carlo simulations of [32], [33] the deviation from the harmonic regime starts for q . 0.3A−1. This is only an estimate, but from these results there is no reason to assume thatq∗ > 0.4 or 0.5 A−1.


0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 40 80 120 160-20

0

20

40

60

80

100

120

140κ

(eV

)

Line

ar te

rm

Iteration

(a) q∗ = 0 A−1

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

0 40 80 120 160-20

0

20

40

60

80

100

120

κ (e

V)

Line

ar te

rm

Iteration

(b) q∗ = 0.32 A−1

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

0 40 80 120 160-20

0

20

40

60

80

100

120

140

κ (e

V)

Line

ar te

rm

Iteration

(c) q∗ = 0.37 A−1

0.65

0.7

0.75

0.8

0.85

0.9

0.95

0 40 80 120 160 200-10

0

10

20

30

40

50

60

70

80

κ (e

V)

Line

ar te

rm

Iteration

(d) q∗ = 0.41 A−1

Figure 6.20: The bending rigidity κ (green/dark grey) and the linear term (yellow/light grey) areplotted for different values of q∗. The calculations are done for T = 200, fxy = 0.2, fz = 1,N = 128 atoms.


The commensurate wavevectors are exactly those wavevectors that fold to the Γ-point. In figure 6.6 the length of the wavevectors from where the branches are foldedtowards the zone centre are denoted. The first three branches belong of course to |q| = 0A−1. The next commensurate wavevector is the branch that folds from |q| = 0.32 A−1.

To look at the effect of truncating the sum in the calculations of the displacementsfor wavevectors near the zone centre, we tried different values of q∗. We skipped only thethree branches belonging to |q| = 0 A−1, which should not make too much difference,since these are modes that cause a rigid translation. Next we skipped all wavevectors upto q∗ = 0.32 A−1, q∗ = 0.37 A−1 and q∗ = 0.41 A−1. In figure 6.20 we plotted the resultsfor T = 200K and different q∗. We plotted the bending rigidity κ and the linear term inthe equation for the dispersion relation of the acoustic out-of-plane branch (eq.(2.15))for each iteration. We see that for q∗ = 0.41 A−1 there is a region where the bendingrigidity and the linear term seem to converge.

0.8

1

1.2

1.4

1.6

1.8

0 20 40 60 80 100 120 140 160 180 200

0

20

40

60

80

100

120

140

160

κ (e

V)

Line

ar te

rm

Iteration

(a) With all wavevectors.

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 20 40 60 80 100 120 140 160 180 200

0

20

40

60

80

100

120

κ (e

V)

Line

ar te

rm

Iteration

(b) Skipped q < q∗ = 0.41 A−1

Figure 6.21: Calculations for T = 200, fxy = 0, fz = 1, N = 128 atoms. For comparing theevolution of the quadratic and linear term with and without skipping the wavevectors near the zonecentre, note the difference in the ranges of the vertical axes. The green (dark grey) lines denote thebending rigidity κ and the yellow (light grey) lines denote the linear term.

In figure 6.21 we compared the calculations with and without skipping the wavevec-tors near the zone centre. We see clearly that the dispersion grows slower when we skipthe wavevectors near the zone centre. Moreover, the fluctuations in the bending rigidityand the linear term are smaller in this case.

6.5.3 Different temperatures

Since the results when we skip small wavevectors look promising, we decided to investi-gate what happens at different temperatures and what happens to the other branches.From the results in the previous section we chose for the minimal wavevector q∗ = 0.41A−1. We evaluated the convergence of the bending rigidity and the linear term in ωza(q)for different temperatures in figure 6.22. We see that, for higher temperatures, the bend-ing rigidity and the linear term increase more quickly. There are some suggestions ofplateaus in both terms, which hold for about 20 iterations, but afterwards the terms


0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 20 40 60 80 100 120 140 160 180 200

0

20

40

60

80

100

120κ

(eV

)

Line

ar te

rm

Iteration

(a) T = 100K

0.7

0.8

0.9

1

1.1

1.2

0 20 40 60 80 100 120 140 160 180 200

0

20

40

60

80

100

120

140

κ (e

V)

Line

ar te

rm

Iteration

(b) T = 300K

0.8

1

1.2

1.4

1.6

1.8

0 100 200 300 400

0

50

100

150

200

250

300

350

κ (e

V)

Line

ar te

rm

Iteration

(c) T = 700K

0.8

1

1.2

1.4

1.6

1.8

0 100 200 300 400 0

50

100

150

200

250

300

350

400

450

κ (e

V)

Line

ar te

rm

Iteration

(d) T = 1000K

Figure 6.22: Quadratic and linear term of the acoustic out-of-plane mode for different temperatures.The parameters are: fxy = 0, fz = 1, N = 128 atoms, a = 1.4198 A and q∗ < 0.41 A−1. Note thedifferent vertical axes.

grow again.To investigate the behaviour of the other branches we calculated the density of the

phonon states by calculating the phonon frequency for all wavevectors in a grid thatcovers the Brillouin zone, as explained for the free energy in section 5.2. The density ofstates g(ω) was calculated with the help of Gaussian functions with standard deviationσ around each ωj(q) on the grid:

g(ω) =∑

q,j

e−(ω−ωj (q))

2

2σ2

In the figures 6.23, 6.24, 6.25 and 6.26, the density of states is plotted for several itera-tions. Using vertical lines we have denoted the frequency of several branches at severalq-points with high symmetry for the zeroth iteration. The shoulder at ω = 0 cm−1 ismainly due to the za-branch at Γ, since this branch is nearly flat near this point. Thefirst vertical line around ω = 265 cm−1 indicates the position of the peak due to theza-mode at M . From these densities of states we can draw several conclusions. First ofall we see that the za-mode shifts to higher frequencies with consecutive iterations. Onecan see this from the shoulder at ω = 0 cm−1 which broadens slightly, from the peak at


ω = 265 cm−1 which broadens and shifts clearly to higher frequencies, and from the pitaround ω = 405 cm−1, which shifts to higher frequencies. The optical out-of-plane mode(zo-mode) also shifts to higher frequencies, although the shift is smaller. This shift canbe deduced from the shift of the peak around ω = 540 cm−1 and the shoulder aroundω = 797 cm−1. We conclude that the out-of-plane branches seem to shift towards higherfrequencies. The two in-plane optical branches, however, shift a little towards lowerfrequencies. These effects can partly be explained by the strain that is induced due tothe ripples. The ripples cause strain and this strain causes the in-plane force constantsto decrease. As a result the phonon frequencies go down. However, the strain causes anincrease in the out-of-plane force constants and therefore the out-of-plane phonon fre-quencies increase. However, the effect of pure strain is larger for the optical modes thanfor the acoustical modes, as L.J. Karssemeijer showed in his master’s thesis [4]. Thereforethe rest of the shifts are due to including temperature with the scaild-method.

If we compare the different temperatures, we see that the shifts and the broadeningincrease with increasing temperature. Moreover, for T = 300, 700 and 1000K peaksabove the spectrum appear from a certain iteration onwards, as we describe in the nextsection.

6.5.4 Localised modes

The appearance of peaks above the spectrum is especially clear when we look in detailat the highest part of the density of states. In figure 6.27 the density of states g(ω) forω > 1400 cm−1 is plotted on the left hand side. We see that around iteration 121 amode starts to appear above the continuum, and at iteration 161 it is clearly separatedfrom the spectrum. If we look at the eigenvector of the dynamical matrix at the Γ pointfor this mode, we see clearly that it changes with consecutive iterations. On the righthand side of figure 6.27 we displayed the eigenvector of the dynamical matrix (Aα,k(q))in eq.(2.8)) at q = Γ−point. This eigenvector has an x, y and z-component, but wedisplayed only the in-plane components since it is an in-plane mode. Each circle denotesan atom and the colour denotes the size of the entry of the eigenvector in the directionthat is denoted above the graph. From the lowest two graphs, which belong to iteration1, we can clearly see that the highest mode is a purely in-plane mode, where the twoatoms in the unit cell move in an opposite direction along the x-axis. If we look at higheriterations we see that the eigenvector gains some y-components and that the wavelikecharacter becomes less strict. Around iteration 121 we see clearly that a considerablenumber of atoms have almost vanishing x and y-components in the eigenvector and onlyabout one third of the atoms have non-zero components. At iteration 161 we see thatthe mode is localised: only three or four atoms contribute with a non-zero element tothe eigenvector.


0 200 400 600 800 1000 1200 1400 1600 1800

g(ω

)

ω (cm-1)

ZOat Γ

LO,TOat Γ

ZAat M

ZOat M

TAat M

LA,LOat M

TOat M

ZA,ZOat K

TAat K

LA,LOat K

TOat K

it=1

it=21

it=41

it=61

it=81

it=101

it=121

it=141

it=161

it=181

Figure 6.23: The density of states for the phonon modes, calculated with σ = 2. The differentiterations are separated by a shift in the z-direction. The vertical lines denote certain branches atspecial q-points. The simulation was done for T = 100K, fxy = 0, fz = 1, N = 128 atoms anda = 1.4198 A. We skipped q < q∗ = 0.41 A−1 for the za-branch.


0 200 400 600 800 1000 1200 1400 1600 1800

g(ω

)

ω (cm-1)

ZOat Γ

LO,TOat Γ

ZAat M

ZOat M

TAat M

LA,LOat M

TOat M

ZA,ZOat K

TAat K

LA,LOat K

TOat K

it=1

it=21

it=41

it=61

it=81

it=101

it=121

it=141

it=161

it=181



0 200 400 600 800 1000 1200 1400 1600 1800

g(ω

)

ω (cm-1)

ZOat Γ

LO,TOat Γ

ZAat M

ZOat M

TAat M

LA,LOat M

TOat M

ZA,ZOat K

TAat K

LA,LOat K

TOat K

it=1

it=21

it=41

it=61

it=81

it=101

it=121

it=141

it=161

it=181



0 200 400 600 800 1000 1200 1400 1600 1800

g(ω

)

ω (cm-1)

ZOat Γ

LO,TOat Γ

ZAat M

ZOat M

TAat M

LA,LOat M

TOat M

ZA,ZOat K

TAat K

LA,LOat K

TOat K

it=1

it=21

it=41

it=61

it=81

it=101

it=121

it=141

it=161

it=181



1400 1500 1600

g(ω

)

ω (cm-1

)

1

11

21

31

41

51

61

71

81

91

101

111

121

131

141

151

161

171

181

191

it = 1, x

-0.1-0.08-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1

y

-2.5e-05-2e-05-1.5e-05-1e-05-5e-06 0 5e-06 1e-05 1.5e-05 2e-05 2.5e-05

it = 41, x

-0.15

-0.1

-0.05

0

0.05

0.1

0.15y

-0.05-0.04-0.03-0.02-0.01 0 0.01 0.02 0.03 0.04 0.05

it = 81, x

-0.03

-0.02

-0.01

0

0.01

0.02

0.03y

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

it = 121, x

-0.15

-0.1

-0.05

0

0.05

0.1

0.15y

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

it = 161, x

Direction eigenvector:

-0.5-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4 0.5

y

-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25

Figure 6.27: Left: the density of states of the phonon modes with ω > 1400 cm−1, calculated withσ = 0.7. Right: x and y entries of the eigenvector of the dynamical matrix for the highest phononmode at Γ. Each circle denotes an atom, the colours denote the size of the entry of the eigenvectorin the direction that is written above the graph. The eigenvector of the highest mode is displayed forfive iterations separated by 40 iterations.


6.5.5 Size of the displacement

In the previous sections we examined the influence of skipping all the wavevectors nearthe zone centre for calculations where we displaced the atoms in the z-direction only(fxy = 0). In this section we will compare those results to the calculations where wedisplaced the atoms in all directions (fxy = fz = 1). In figure 6.28 we plotted bothresults in one graph. For fxy = fz = 1, the distortions seem to influence the density ofstates more than for fxy = 0,fz = 1. The localised modes above the continuum appearmuch earlier and imaginary frequencies appear ‘below’ the original modes. Moreover wesee that the shift to higher frequencies of the out-of-plane modes is less pronounced.

0 500 1000 1500 2000

g(ω

)

iω (cm-1) ω (cm-1)

ZOat Γ LO,TO

at Γ

ZAat M

ZOat M

TAat M

LA,LOat M

TOat M

ZA,ZOat K

TAat K

LA,LOat K

TOat K

it=1

it=11

it=21

it=31

it=41

it=51

Figure 6.28: The density of phonon states, calculated with σ = 2. The different iterations areseparated by a shift. The vertical lines denote certain branches at special q-points. The simulationwas done for T = 300K, N = 128 atoms and a = 1.4198 A. We skipped q < q∗ = 0.41 A−1 forthe za-branch. The coloured lines denote the calculations for fxy = fz = 1. The grey lines are thecorresponding iterations for fxy = 0, fz = 1.

In the original papers on scaild [6], [2], [7] the eigenvectors of the dynamical matrixwere not recalculated each iteration. This means that each time when the displacementsof the atoms (eq.(6.1)) were calculated, the eigenvector of the undistorted sample wasused. This approximation was done to make the whole procedure less time consuming,since all calculations are done from first principles. Taking the original eigenvectors didnot influence the procedure for the materials Souvatzis and others investigated.


In the last part of this section we will look what happens for graphene when wecalculate the eigenvectors of the dynamical matrix for the flat, undistorted sample onlyand use them during the whole calculation. The density of states of this calculationis shown in figure 6.29 and can be compared to the results where we recalculated theeigenvector each iteration (fxy = fz = 1) again, plotted in the same figure by a thin greyline. In the case where we used the original eigenvectors (coloured lines), the distortionsseem to influence the density of states even more than for the case where we recalculatedthe eigenvectors at each iteration (grey lines). From this density of states we can concludethat the za-mode destabilises with consecutive iterations, as it did when we used therecalculated eigenvectors. Imaginary frequencies and localised modes appear in bothcases. This seems not the right way to include anharmonic effects in the calculation ofthe phonon dispersion relation.

0 500 1000 1500 2000

g(ω

)

iω (cm-1) ω (cm-1)

ZOat Γ LO,TO

at Γ

ZAat M

ZOat M

TAat M

LA,LOat M

TOat M

ZA,ZOat K

TAat K

LA,LOat K

TOat K

it=1

it=11

it=21

it=31

it=41

it=51

it=61

Figure 6.29: The density of phonon states, calculated with σ = 2. The different iterations areseparated by a shift. The vertical lines denote certain branches at special q-points. The simulationwas done for T = 300K, N = 128 atoms and a = 1.4198 A. We skipped q < q∗ = 0.41 A−1 for theza-branch. The coloured lines denote the calculations for which we used the original eigenvectors ineach iteration. The thin grey lines are the corresponding iterations for fxy = 1 and fz = 1, so wherewe calculated the eigenvectors each iteration again.

Chapter 7

Summary and conclusions

We have studied the temperature dependence of the phonon dispersion relation ingraphene using scaild [6] with an empirical potential lcbop [1], that describes theinteractions between carbon atoms, also far away from equilibrium structures. The ba-sic idea of the scaild-procedure is to use a supercell and excite phonon modes in thissupercell by displacing the atoms according to the dispersion relation. These thermaldisplacements change the frequencies and induce new thermal displacements in the nextiteration. This self-consistent procedure should be carried out until the phonon modesconverge. Calculating the displacements can be done in various ways. In chapter 6we showed that the behaviour of the self-consistent procedure strongly depends on thechoice of the eigenvector to be used for the self-consistent loop: if we recalculate theeigenvector at each iteration (fxy = fz = 1) the za-mode becomes imaginary near Γ,signalling an instability. Whereas it grows if we displace the atoms in the z-directiononly (fxy = 0, fz = 1). Therefore a standard application of the method seems notsuitable for two-dimensional materials.

The strong divergence of the out-of-plane fluctuations in the long wavelength limit fortwo-dimensional materials in the harmonic approximation, has motivated us to adaptthe scaild-method for two-dimensional crystals. We neglect the long wavelengths inour calculations of the displacements. This adjustment seems to be an improvement: forfxy = 0, fz = 1 since the za-mode still rises, but more gradually. Moreover some plateausappear for about 20 iterations or more, but they disappear with consecutive iterations,so the self-consistent method is still not convergent. We have noticed that localisedmodes appear above the spectrum. We have studied whether the appearance of thesemodes was related to the appearance of the plateaus. A detailed study of the behaviourof the density of states does not lead to a clear relation between the convergence of theza-mode and the appearance of the localised states.

The za-mode is characterised by a peculiar quadratic dispersion, which is determinedby the bending rigidity κ. The bending rigidity has been found in mc-simulations todepend on the temperature, contrary to what is expected in the continuum limit, whichtakes into account only coupling between the in-plane and out-of-plane acoustic modes.The fact that the dispersion relation of the za-mode changes with temperature and with

66

iterations, clearly shows that the bending rigidity becomes temperature dependent dueto interactions with the whole phonon spectrum. This itself is an important conclusion.

The determination of the dependence of κ on the temperature, however, has notyet been calculated, because the method is not directly applicable to two-dimensionalcrystals. The fact that the procedure strongly depends on the choice of the eigenvec-tor that is used for the self-consistent procedure, leads us to the conclusion that thescaild-method is not applicable, without any further adjustments, for two-dimensionalmaterials.

An additional feature of scaild for two-dimensional materials is that displacingthe atoms out-of-plane induces strain. Moreover the lattice parameter of graphene isstrongly temperature dependent. Whether graphene is dynamically stable (has onlyreal frequencies in the harmonic approximation) depends on the lattice parameter. Theunknown lattice parameter when we include temperature, and the induced strain due todisplacing the atoms out-of-plane, may be a problem. One could think of a method todeal with strain and temperature at the same time.

In the last week of the period for my master research we had a fruitful discussionand made plans for future investigations with Olle Eriksson and Petros Souvatzis thathave developed the method for group IV elements. To prove that the two-dimensionalityof graphene is the reason that the scaild-method does not converge, we want to lookat the phonon dispersion relation of graphite (a very similar, but 3-dimensional crystal)and diamond as two three-dimensional carbon allotropes.

Aknowledgement

The process of writing my master research was like a landscape with hills and valleys.Sometimes I was merrily walking through the landscape learning all kind of new things.At other moments I was climbing an endless looking mountain, for instance when I triedto figure out why the self-consistent method I used did not lead to convergence. Duringthese times the landscape became darker and darker. But I was never alone, a lot ofpeople accompanied me during my journey and gave me new courage to continue. I wantto thank all the people that have helped me during my journey. In particular, I wantto thank Misha Katsnelson, Olle Erikson and Petros Souvatzis for our discussions aboutmy project and for all opportunities they saw. I want to thank my roommates for all(scientific) discussions, but also for the fun. I want to thank Margot Peters for all thegood cups of tea.

Annalisa Fasolino was the scientific supervisor of my journey, but her guidance wasnot only in a professional way. She also had eye for all the personal and emotional issuesI had during my master research: deciding what to do with my future, applying for aPhD in Sweden. . .When I needed her help, she was always there for me. She also taughtme to walk alone, to enjoy the environment and to keep in mind that every cloud has asilver lining. I want to thank her for her great supervision and support.

My parents have always guided me, also during the time before I started workingon my thesis. I want to thank them for their continuous support. Last, but not least, Iwant to thank Jord Boeijink for everything he does for me and means to me.

Appendix A

Additional calculations

The aim of my master research was to include anharmonic effects in the calculation ofthe phonon dispersion relation of graphene. This quest I described in my thesis. Apartfrom this I have calculated some other things and the results I present in the appendix.This is only to keep in mind that we have once calculated this and to have the resultsand pictures clustered, so it will be really short.

A.1 Diamond

We calculated the dispersion relation for diamond in the harmonic approximation withlcbop. The diamond lattice consists of two face centred cubic (fcc) lattices one of whichis shifted by (1/4, 1/4, 1/4). A set of primitive vectors is:

a1 =a

2

001

a2 =a

2

101

a3 =a

2

110

The Brillouin zone with the special q-points is shown in figure A.1. The coordinates ofthe symmetry points in reciprocal space for the given choice of primitive vectors is:

Γ =2π

a

000

K =2π

a

34034

U =2π

a

1−1

414

W =2π

a

1012

L =2π

a

12−1

212

X =2π

a

100

The phonon dispersion relation of diamond is shown in figure A.2 together withexperimental data from [34].

It is very strange that the highest mode at L has a higher frequency than the highestmode at Γ. This is very unusual and we need to check the reliability.

A.1 Diamond 69

(010)

(100)

(001)

ΓU

X

KL

W

Figure A.1: Brillouin zone of a face centred cubic lattice, with the most important symmetry points.

0

200

400

600

800

1000

1200

1400

Γ K/U X Γ L X W L

ω(q

) in

cm

-1

LCBOP

Figure A.2: The phonon dispersion relation calculated in the harmonic approximation with lcbop.The experimental data (red triangles, squares and circles) are measured with inelastic neutron scat-tering by Warren and Yarnell [34].

A.2 New torsion 70

A.2 New torsion

Harmonic approximation

Recent studies of graphitic materials have us come to recognise that the torsion in lcbop

is underestimated. The improvements of the potential are in progress [35]. In this sectionwe calculated the phonon dispersion relation for graphene in the harmonic approximationwith the original torsion and the improved torsion. The results are shown in A.3.

0

200

400

600

800

1000

1200

1400

1600

Γ M K Γ

ω(q

) in

cm

-1

Old torsionNew torsion

Figure A.3: The dispersion relation for graphene calculated in the harmonic approximation withlcbop. We compared the original torsion (old torsion) with the improved torsion (new torsion).Experimental data are from inelastic x-ray scattering (squares from [15] and circles from [16]) andelectron energy loss spectroscopy (diamonds from [17] and triangles from [18]).

The calculations with the improved torsion have a better agreement with the exper-iments. However the za- and zo-modes are still too soft.

SCAILD

Since the za-mode is still too soft in the harmonic approximation, we ran some calcu-lations in the scaild-method with the improved torsion. In figure A.4 we calculatedthe bending rigidity per iteration. The behaviour of the linear term is similar for bothtorsions. The behaviour of the bending rigidity is quite different. For the improvedtorsion κ of the undistorted sample is significantly larger, but barely increases.

In figure A.5 we plotted the density of phonon states each 20th iteration. If we

A.2 New torsion 71

0.8

1

1.2

1.4

1.6

0 100 200 300 400 500 600 700 800

0

50

100

150

200

250

300

350

κ (e

V)

Line

ar te

rm

Iteration

Figure A.4: The bending rigidity κ (green) and the linear term (yellow) in the phonon dispersion rela-tion of the acoustic out-of-plane mode. The solid lines are calculated with the improved torsion. Forcomparison we plotted the calculations with the original torsion with dashed lines. Both calculationswere done for T = 300K, fxy = 0, fz = 1, N = 128 atoms and a = 1.4198 A. We averaged overthe previous 10 frequencies and we skipped q < q∗ = 0.41 A−1 for the za-branch.

compare this with the density of states calculated with the original torsion in figure6.24, we see that the peak belonging to the za-mode at M and the pit belonging to theza and zo-modes at K are at much higher frequencies for the improved torsion. Howeverthe shift of this peak and pit with consecutive iterations is much smaller, especially forthe peak of the za-mode at M.

A.2 New torsion 72

0 200 400 600 800 1000 1200 1400 1600 1800

g(ω

)

ω (cm-1)

ZAat Morig

ZA,ZOat Korig

ZOat Γ

LO,TOat Γ

ZAat M

ZOat M

TAat M

LA,LOat M

TOat M

ZA,ZOat K

TAat K

LA,LOat K

TOat K

it=1

it=41

it=81

it=121

it=161

it=201

it=241

it=281

it=321

it=361

it=401

it=441

it=481

it=521

it=561

it=601

it=641

it=681

it=721

it=761

it=801

it=841

Figure A.5: Density of phonon states for calculations with the improved torsion. The calculationswere done for T = 300K, fxy = 0, fz = 1, N = 128 atoms and a = 1.4198 A. We averaged overthe previous 10 frequencies and we skipped q < q∗ = 0.41 A−1 for the za-branch. The colouredvertical lines denote the frequencies of certain branches at special q-points. The frequencies of theza-mode at M and the za and zo-modes at K are very different for the improved torsion. The thinvertical black lines denote these frequencies calculated with the original torsion.

Bibliography

[1] J. H. Los and A. Fasolino, “Intrinsic long-range bond-order potential for carbon:Performance in monte calrlo simulations of graphitization,” Phys. Rev. B, vol. 68,no. 024107, 2003.

[2] P. Souvatzis, O. Eriksson, S. P. Rudin, and M. I. Katsnelson, “The self-consistentab initio lattice dynamical method,” Comp. Mat. Sc., vol. 44, pp. 888–894, 2009.

[3] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos,I. V. Grigorieva, and A. A. Firsov, “Electric field effect in atomically thin carbonfilms,” Science, vol. 306, no. 5696, pp. 666–669, 2004.

[4] L. J. Karssemeijer, “Thermal expansion of carbon structures,” Master Thesis,http://www.ru.nl/tcm/education 0/bachelor-master/, 2010.

[5] I. M. Lifshitz, Zh. Eksp. Teor. Fiz, vol. 22, p. 475, 1952.

[6] P. Souvatzis, O. Eriksson, S. P. Rudin, and M. I. Katsnelson, “Entropy driven sta-bilization of energetically unstable crystal structures explained from first principlestheory,” Phys. Rev. Lett., vol. 100, no. 095901, 2008.

[7] P. Souvatzis, D. Legut, O. Eriksson, and M. I. Katsnelson, “Ab initio study of latticevibrations and stabilization of the β phase in ni-ti shape-memory alloy,” Phys. Rev.B, vol. 81, p. 092201, 2010.

[8] M. Katsnelson, Graphene, Carbon in two dimensions. cambridge university

press, 2012.

[9] L. Landau and E. Lifshitz, Theory of elasticity. Pergamon Press, 1970.

[10] J. Tersoff, “Empirical interatomic potential for carbon, with applications to amor-phous carbon,” Phys. Rev. Lett., vol. 61, pp. 2879–2882, 1988.

[11] L. M. Ghiringhelli, J. H. Los, E. J. Meijer, A. Fasolino, and D. Frenkel, “Modelingthe phase diagram of carbon,” Phys. Rev. Lett., vol. 94, p. 145701, 2005.

[12] J. H. Los, L. M. Ghiringhelli, E. J. Meijer, and A. Fasolino, “Improved long-rangereactive bond-order potential for carbon. i. construction,” Phys. Rev. B, vol. 72, p.214102, 2005.

BIBLIOGRAPHY 74

[13] D. W. Brenner, “Empirical potential for hydrocarbons for use in simulating thechemical vapor deposition of diamond films,” Phys. Rev. B, vol. 42, pp. 9458–9471,1990.

[14] L. J. Karssemeijer and A. Fasolino, “Phonons of graphene and graphitic materialsderived from the empirical potential lcbopii,” Surf.sc., vol. 605, no. 17-18, pp. 1611–1615, 2011.

[15] J. Maultzsch, S. Reich, C. Thomsen, H.requardt, and P. Ordejon, “Phonon disper-sion in graphite,” Phys. Rev. Lett, vol. 92, no. 7, p. 075501, 2004.

[16] M. Mohr, J. Mailtzsch, E. Dobardzic, S. Reich, I. Milosevic, M. Damnjanovic,A. Bosak, M. krisch, and C. Thomsen, “Phonon dispersion of graphite by inelasticx-ray scattering.” Phys. Rev. B, vol. 76, no. 3, p. 035439, 2007.

[17] C. Oshima, T. Aizawa, R. Souda, Y. Ishizawa, and Y. Sumiyoshi, “Surface phonondispersion curves of graphite (0001) over the entire energy region.” Solid StateCommun., vol. 65, no. 12, pp. 1601–1604, 1988.

[18] S. Siebentritt, R. Pues, K. Rieder, and A. Shikin, “Surface phonon dispersionin graphite and in a lanthanum graphite intercalation compound,” Phys. Rev. B,vol. 55, no. 12, pp. 7927–7934, 1997.

[19] N. Mounet and N. Marzari, “First-principles determination of the structural, vibra-tional and thermodynamic properties of diamond, graphite, and derivatives,” Phys.Rev. B, vol. 71, no. 205214, 2005.

[20] K. Zakharchenko, M. I. Katsnelson, and A. Fasolino, “Finite temperature latticeproperties of graphene beyond the quasiharmonic approximation,” Phys. Rev. Lett.,vol. 102, no. 4, p. 046808, 2009.

[21] H. J. Monkhorst and J. D. Pack, “Special points for brillouin-zone integrations,”Phys. Rev. B, vol. 13, no. 12, p. 51885192, 1976.

[22] T. H. K. Barron, J. G. Collins, and G. K. White, “Thermal-expansion of solids atlow-temperatures,” Adv. Phys., vol. 29, no. 4, pp. 609–730, 1980.

[23] Ashcroft and Mermin, Solid State Physics. W.B. Saunders Company, 1976.

[24] K. Zakharchenko, Temperature effects on graphene from a flat crystal to a 3D liquid.Ipskamp Drukkers, Enschede, 2011.

[25] C. Zener, Influence of Entropy on Phase Stabilization. Edited by P.S. Rudman,J. Stringer and R.I. Jaffe, Phase Stability in Metals and Alloys, McGraw-Hill,

New York , 1967.

[26] D. Nelson, T. Piran, and S. Weinberg, Styatistical Mechanics of Membranes andSurfaces. World Scientific, Singapore , 2004.

BIBLIOGRAPHY 75

[27] L. Landau, Phys. Z. Sowjetunion, vol. 11, p. 25, 1937.

[28] R. Peierls, Helv. Phys. Acta, vol. 7, p. 81, 1934.

[29] ——, Ann. Inst. Henri Poincare, vol. 5, p. 177, 1935.

[30] N. D. Mermin, “Crystalline order in two dimensions,” Phys. Rev., vol. 176, pp.250–254, 1968.

[31] N. D. Mermin and H. Wagner, “Absence of ferromagnetism or antiferromagnetismin one- or two-dimensional isotropic heisenberg models,” Phys. Rev. Lett., vol. 17,pp. 1133–1136, 1966.

[32] A. Fasolino, J. Los, and M. Katsnelson, “Ripples in graphene,” Nat. Mater., vol. 6,no. 11, pp. 858–861, 2007.

[33] J. H. Los, M. I. Katsnelson, O. V. Yazyev, K. V. Zakharchenko, and A. Fasolino,“Scaling properties of flexible membranes from atomistic simulations: Applicationto graphene,” Phys. Rev. B, vol. 80, p. 121405, 2009.

[34] J. L. Warren, J. L. Yarnell, G. Dolling, and R. A. Cowley, “Lattice dynamics ofdiamond,” Phys. Rev., vol. 158, pp. 805–808, 1967.

[35] J. H. Los, A. Fasolino, and M. Akhukov, “In progress.”

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The eﬀect of temperature on the phonon dispersion relation