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MotivationCalculated Materials Properties

Computational ApproachesSummary

Computational Materials ScienceIntroduction

Volker Eyert

Institut für Physik, Universität Augsburg

Electronic Structure in a Nutshell

Volker Eyert Computational Materials Science



Organization of the Course

Mon Tue Wed Thu

General Comp. Mat. DFT DFT LDAScience Basics Basics Introduct.

ASW Introduction Standard Full. Pot. Miscell.Method History ASW ASW

Background

ASW Installation Distribution CTRL File Miscell.Program Data Files

Cu, Be FeS2 CrO2




Outline

1 Motivation

2 Calculated Materials Properties

3 Computational Approaches




Outline

1 Motivation






Outline

1 Motivation






Outline

1 Motivation






What Are We Aiming At?

Interpretation ofExperiments,

Understanding, and

Prediction ofMaterials Properties

-10

-8

-6

-4

-2

0

2

4

6

KWXΓLW

(E -

EF)

(eV

)






Understanding, and


-10

-8

-6

-4

-2

0

2

4

6

KWXΓLW

(E -

EF)

(eV

)





0 10 20 30 40 50 0 10

20 30

40 50

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

ρv / a.u.


Understanding, and


-578.060000

-578.055000

-578.050000

-578.045000

-578.040000

-578.035000

-578.030000

-578.025000

-578.020000

-578.015000

-578.010000

80 90 100 110 120 130 140 150 160 170

Si phase stability

fccbcc

scβ-tin

dia




Outline

1 Motivation






Structural and Mechanical Properties

Structural

geometries of molecules

crystal structures

electron densities

defect structures

interface structures

surface structures

adsorption





Structural


crystal structures

electron densities

defect structures


surface structures

adsorption





Structural


crystal structures

electron densities

defect structures


surface structures

adsorption

0 10 20 30 40 50 0 10

20 30

40 50

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

ρv / a.u.





Structural


crystal structures

electron densities

defect structures


surface structures

adsorption





Structural


crystal structures

electron densities

defect structures


surface structures

adsorption





Structural


crystal structures

electron densities

defect structures


surface structures

adsorption





Structural


crystal structures

electron densities

defect structures


surface structures

adsorption





-16532.950000

-16532.900000

-16532.850000

-16532.800000

-16532.750000

-16532.700000

-16532.650000

-16532.600000

850 900 950 1000 1050 1100 1150 1200

E (

Ryd

)

V (aB3)

V0 = 1056.15 aB3

E0 = -16532.904060 Ryd

B0 = 170.94 GPa

Mechanicalcompressibility

elastic moduli

thermal expansion

vibrational properties

hardness





Mechanical

compressibility

elastic moduli

thermal expansion


hardness





Mechanical

compressibility

elastic moduli

thermal expansion


hardness





0.000000

0.000500

0.001000

0.001500

0.002000

0.002500

0.003000

0.003500

0.004000

0.004500

0.005000

0.12 0.122 0.124 0.126 0.128 0.13

Si fcc, TO(Γ) phonon

Mechanicalcompressibility

elastic moduli

thermal expansion


hardness





Mechanical

compressibility

elastic moduli

thermal expansion


hardness




Electronic, Optical and Magnetic Properties

Electronic

band structuremetalsemiconductorinsulator

band gap

band offsets

density distributions

polarizabilities

ionization energies

electron affinities

electrostatic potential

work functions

-12

-10

-8

-6

-4

-2

0

2

4

6

KWXΓLW(E

- E

V)

(eV

)





Electronicband structure

band gap

band offsets


polarizabilities

ionization energies

electron affinities


work functions





Electronic

band structure

band gap

band offsetsdensity distributions

electrical momentselectric field gradients

polarizabilities

ionization energies

electron affinities


work functions

0 10 20 30 40 50 0 10

20 30

40 50

0.02 0.025 0.03

0.035 0.04

0.045 0.05

0.055

ρv / a.u.






band gap

band offsets


polarizabilities

ionization energies

electron affinities


work functions






band gap

band offsets


polarizabilities

ionization energies

electron affinities


work functions






band gap

band offsets


polarizabilities

ionization energies

electron affinities


work functions






band gap

band offsets


polarizabilities

ionization energies

electron affinities


work functions






band gap

band offsets


polarizabilities

ionization energies

electron affinities


work functions





-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 2 4 6 8 10 12 14

ε

E (eV)

Im εxxRe εxx

Optical

optical spectra

magneto-optical properties

dielectric response

luminescence

fluorescence





Optical

optical spectra


dielectric response

luminescence

fluorescence





Optical

optical spectra


dielectric response

luminescence

fluorescence





Optical

optical spectra


dielectric response

luminescence

fluorescence





Optical

optical spectra


dielectric response

luminescence

fluorescence





0 10 20 30 40 50 0 10

20 30

40 50

-0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

ρv / a.u.

Magnetic

spin-density distribution

magnetic moments

NMR chemical shifts





Magnetic


magnetic moments

NMR chemical shifts





Magnetic


magnetic moments

NMR chemical shifts




Transport, Chemical, and Thermodynamic Properties

Transport

electrical conductivity

thermal conductivity

diffusion constants

permeability

Chemicalcatalytic properties

corrosion

surface reactivity

photochemical properties

Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams

-578.060000

-578.055000

-578.050000

-578.045000

-578.040000

-578.035000

-578.030000

-578.025000

-578.020000

-578.015000

-578.010000

80 90 100 110 120 130 140 150 160 170

Si phase stability

fccbcc

scβ-tin

dia





Transport



diffusion constants

permeability


corrosion

surface reactivity


Thermodynamic

binding energies

phase transitions

phase diagrams

-578.060000

-578.055000

-578.050000

-578.045000

-578.040000

-578.035000

-578.030000

-578.025000

-578.020000

-578.015000

-578.010000

80 90 100 110 120 130 140 150 160 170

Si phase stability

fccbcc

scβ-tin

dia




Calculated Electronic Properties of MetalsCohesive Energies

Moruzzi, Janak, Williams, 1978

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga

E/R

yd

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In

E/R

yd




Calculated Electronic Properties of MetalsWigner-Seitz Radii


2

2.5

3

3.5

4

4.5

5

5.5


WSR

/au

2

2.5

3

3.5

4

4.5

5

5.5


WSR

/au




Calculated Electronic Properties of MetalsBulk Moduli


0

500

1000

1500

2000

2500

3000

3500

4000


Bul

k m

odul

us/k

bar

0

500

1000

1500

2000

2500

3000

3500

4000


Bul

k m

odul

us/k

bar




Calculated Electronic PropertiesARPES Data and Calculated Band Structure of Td-WTe2

Augustin et al., 2000




Calculated Optical PropertiesDiagonal-Conductivity and Normal Incidence Reflectivity of CrO2

Brändle et al., 1993




Calculated Electronic PropertiesExperimental and Calculated Fermi Surface of Cu and TiTe2

Straub et al., 1997

continuous penetration of a lifetime broadened peak intoa typically Gaussian energy window, given by the instru-mental response function.

This is illustrated on Cu~001! as an example. Figure 3shows the corresponding intensity map, which agrees withthat of other workers.4 The observed pattern approximatelyreflects a cut through the FS of the sp band, which is occu-pied in the center of the BZ. However, a closer inspection

shows that the position of the intensity maximum along theG-M direction of the surface BZ does not coincide with anindependent determination of the Fermi vector from the mea-sured dispersion of the sp band ~Fig. 4!, as obtained fromenergy distribution curves on the same surface. Rather, itappears slightly shifted towards the BZ center. Also shownin Fig. 3 is the gradient u¹kw(k)u of the intensity distribu-tion. In line with our above discussion, we now find two setsof maxima, the outer and more intense of which is inter-

FIG. 3. ~Color! Top: Fermi-level intensity map of Cu~001!. The surface Brillouin zone is indicated. Bottom: Modulus of the gradient(u¹kw(k)u) of the intensity map, together with the theoretical FS contour ~black line!.

13 476 55TH. STRAUB et al.

nent features are the ‘‘cigar-shaped’’ intensity maxima cen-tered at the M points, which originate from the Ti 3d band.The observed intensity pattern reflects the threefold symme-try of the crystal structure that requires a distinction betweenM and M 8 points.7 Though the electronic band structure it-self does not deviate much from sixfold symmetry in thehexagonal 1T structure,7 the optical transition matrix ele-ments are strongly affected by photoelectron diffraction offthe three Te atoms sitting atop each Ti site.13 Near the centerof the Brillouin zone ~BZ! three Te 5p bands cross the Fermilevel very close to each other, giving rise to a broad intensity

spot at the G point. Note that we can also access parts of thesecond BZ ~in the periodic zone scheme!. The identificationof the intensity maxima becomes possible by comparison toconventional band mapping results7 and to a correspondingFS cut derived from our ASW calculations @Fig. 1~b!#. Ap-parently, the overall shape of the measured intensity map isalready in good agreement with the calculated FS topology.However, the contours of the Ti 3d-derived electron pocketsat the M point~s! are clearly not resolved. This is a conse-quence of the very narrow ~occupied! width of the Ti 3dband. Its highest binding energy accessible by He-I radiation

FIG. 1. ~Color! ~a! Photoemission intensity map w(k) for excitations from the Fermi level of TiTe2 (hn521.2 eV!. The Brillouin zoneis indicated. ~b! Corresponding Fermi-surface contours obtained from an ASW band calculation. ~c! Modulus of the two-dimensionalgradient u¹kw(k)u of the map in ~a!. ~d! Modulus of the logarithmic gradient u¹klnw(k)u

13 474 55TH. STRAUB et al.




Calculated Electronic PropertiesCalculated Electron Density of Si

0 10 20 30 40 50 0 10

20 30

40 50

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

ρv / a.u.

0 10 20 30 40 50 0 10

20 30

40 50

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

ρv / a.u.




Calculated Electronic PropertiesCalculated Electron Density of Si

0 10 20 30 40 50 0 10

20 30

40 50

-0.1

-0.05

0

0.05

0.1

-∇ 2ρ / a.u.

0 10 20 30 40 50 0 10

20 30

40 50

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11

ρv / a.u.




Calculated Optical PropertiesDielectric Function of Al2O3

Hosseini et al., 2005Ahuja et al., 2004

present work

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30

ε

E (eV)

Im εxxIm εzz




Calculated Optical PropertiesDielectric Function of Al2O3

Hosseini et al., 2005Ahuja et al., 2004

present work

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 5 10 15 20 25 30

ε

E (eV)

Re εxxRe εzz




Outline

1 Motivation






Theoretical Foundation

Newton’s 2nd Law

F = −∇E = ma

Schrödinger’s Equation

Hψ = Eψ




Foundation of Density Functional Theory 1964/1965

Pierre C. Hohenberg Walter Kohn

Lu Jeu Sham




Density Functional Theory I

Guidelines =⇒ to remember

theory for the ground stateno excitations (!)

electron density as the central variableno need for the many-body wave functionmany-body wave function and potential from the density

variational principle with respect to the electron densityminimization of total energy

many-body problem ⇐⇒ single-particle problemexact mapping































Density Functional Theory IIMany-Body ⇐⇒ Single-Particle

Exact Mapping

many-body problemreal, unsolvable (!!!)density ρmb

single-particle problemfictitious, solvable (???)

density ρsp!= ρmb

effective single-particle potential

veff ,σ(r) = vext (r) + vH(r) + vxc,σ(r)

single-particle equations: Kohn-Sham equations[

−∇2 + veff ,σ(r) − εσ

]

ψσ(r) = 0





Exact Mapping



density ρsp!= ρmb




−∇2 + veff ,σ(r) − εσ

]

ψσ(r) = 0





Exact Mapping



density ρsp!= ρmb




−∇2 + veff ,σ(r) − εσ

]

ψσ(r) = 0




DFT Implementations I

Kohn-Sham Equation

[

−∇2 + vext(r) + vH(r) + vxc,σ(r) − εσ(k)]

ψσ(k, r) = 0





Kohn-Sham Equation

[


ψσ(k, r) = 0

Kinetic Energy

non-relativistic

scalar-relativistic

fully relativistic





Kohn-Sham Equation

[


ψσ(k, r) = 0

External Potential (Ions)

non-periodic

periodic





Kohn-Sham Equation

[


ψσ(k, r) = 0

Effective Potential

all-electron muffin-tin

all-electron full potential

all-electron PAW

pseudopotential





Kohn-Sham Equation

[


ψσ(k, r) = 0

Exchange-Correlation Potential

non-spin-polarized

spin-polarized





Kohn-Sham Equation

[


ψσ(k, r) = 0


local-density approximation (LDA)

generalized gradient approximation (GGA)

beyond LDA/GGA





Kohn-Sham Equation

[


ψσ(k, r) = 0



generalized gradient approximation (GGA)beyond LDA/GGA

self-interaction correction (SIC)LDA+U, LDA+DMFT





Kohn-Sham Equation

[


ψσ(k, r) = 0



generalized gradient approximation (GGA)beyond LDA/GGA

optimized effective potential (OEP)exact exchange (EXX)screened exchange (sX)





Kohn-Sham Equation

[


ψσ(k, r) = 0



generalized gradient approximation (GGA)beyond LDA/GGA, beyond DFT

model GWGWtime-dependent DFT (TDDFT)





Kohn-Sham Equation

[


ψσ(k, r) = 0

Wave Function, Expanded in

atomic orbitals

plane waves

augmented plane waves

augmented spherical waves

fully numerical functions





Kohn-Sham Equation

[


ψσ(k, r) = 0


atomic orbitalsGaussians (GTO)Slater-type (STO)numerical (DMol, Siesta)

plane waves



fully numerical functionsVolker Eyert Computational Materials Science




Kohn-Sham Equation

[


ψσ(k, r) = 0


atomic orbitals

plane waves








Kohn-Sham Equation

[


ψσ(k, r) = 0


atomic orbitals

plane wavesaugmented plane waves

augmented plane waves (APW)linear augmented plane waves (LAPW)full-potential linear augmented plane waves (FLAPW)






Kohn-Sham Equation

[


ψσ(k, r) = 0


atomic orbitals

plane waves

augmented plane wavesaugmented spherical waves

Korringa Kohn Rostoker (KKR)linear muffin-tin orbital (LMTO)augmented spherical wave (ASW)





Kohn-Sham Equation

[


ψσ(k, r) = 0


atomic orbitals

plane waves




courtesy of E. WimmerVolker Eyert Computational Materials Science



DFT Implementations IIPotentials and Partial Waves

Full Potential

0 10 20 30 40 50 0 10

20 30

40 50

-4-3.5

-3-2.5

-2-1.5

-1-0.5

v / a.u.

Full Potentialspherical symmetric nearnuclei

flat outside the atomiccores





John C. Slater Full Potential

spherical symmetric nearnuclei

flat outside the atomiccores

Muffin-Tin Approximation

approximate the potential





John C. Slater Muffin-Tin Approximation

distinguish:atomic regions

muffin-tin spheresveff ,σ(r) = veff ,σ(|r|)

remainderinterstitial regionveff ,σ(r) = 0





Muffin-Tin Potential Muffin-Tin Approximation

distinguish:atomic regions

muffin-tin spheresveff ,σ(r) = veff ,σ(|r|)

remainderinterstitial regionveff ,σ(r) = 0





Muffin-Tin Potential Partial Waves

muffin-tin spheresveff ,σ(r) = veff ,σ(|r|)solve radial Schrödingerequation numerically

interstitial regionveff ,σ(r) = 0exact solutions

plane wavesspherical waves

match at sphere surface(„augment“)




DFT Implementations IIIPlane Waves ⇐⇒ Spherical Waves

Augmented Plane Waves

require periodic cell

good for solids and surfaces

good for metals, semiconductors, and insulators

not recommended for rare earth and actinides

inefficient for open structures

systematic but slow convergence of basis set

matrix elements easy to calculate

easy to implement













easy to implement













easy to implement













easy to implement













easy to implement













easy to implement













easy to implement













easy to implement




DFT Implementations IVPlane Waves ⇐⇒ Spherical Waves

Augmented Spherical Waves

do not require periodic cell



good for all atoms including transition metals, rare earth,and actinides

still efficient for open structures

minimal basis set

computationally efficient

difficult to implement











minimal basis set













minimal basis set













minimal basis set













minimal basis set













minimal basis set













minimal basis set













minimal basis set






Summary

Density Functional Theory . . .

allows for an accurate determination of a still growingnumber of materials properties

is a ground state theory

is in principle exact

has given rise to a large variety of implementations




Summary









Summary









Summary









Further Reading I

P. Hohenberg and W. KohnInhomogeneous Electron GasPhys. Rev. 136, B864 (1964)

W. Kohn and L. J. ShamQuantum Density Oscillations in an InhomogeneousElectron GasPhys. Rev. 140, A1133 (1965)

W. KohnAn Essay on Condensed Matter Physics in the TwentiethCenturyRev. Mod. Phys. 71, S59 (1999)




Further Reading II

R. M. Dreizler and E. K. U. GrossDensity Functional Theory (Springer, Berlin, 1990)

R. G. Parr and W. YangDensity-Functional Theory of Atoms and Molecules (OxfordUniversity Press, Oxford, 1989)

H. EschrigThe Fundamentals of Density-Functional Theory (Editionam Gutenbergplatz, Leipzig 2003)




Further Reading III

J. Kübler and V. EyertElectronic structure calculationsin: Electronic and Magnetic Properties of Metals andCeramicsed by K. H. J. Buschow (VCH, Weinheim, 1992), pp. 1-145Volume 3A of Materials Science and Technology

E. WimmerPrediction of Materials Propertiesin: Encyclopedia of Computational Chemistryed by P. von Ragué-Schleyer (Wiley, New York, 1998)
