Contents An overview Literature

The electronic structure of materials 1Quantum mechanics 2 - Lecture 8

Igor Lukacevic

UJJS, Dept. of Physics, Osijek

December 13, 2012

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

1 An overview

2 Literature

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Contents

1 An overview

2 Literature

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electronic ground state

Ground state

cohesive energy

equilibrium crystal structure

phase transitions betweenstructures

elastic constants

charge density

magnetic order

static dielectric susceptibility

static magnetic susceptibility

nuclear vibrations (in theadiabatic approximation)

...

Excited state

low-energy excitations

Pauli spin susceptibility

transport

electrical conductivity

optical properties

thermal excitation of electrons

spectra for adding electrons

spectra for removing electrons

...

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electronic ground state

Interplay:

electronic ground state ! spatial structure of the nuclei

⇓

bonding

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electronic ground state

1 closed-shell systemsrare gasesmolecular solidsvan der Waals interaction

2 ionic bonding

3 covalent bonding

4 metallic bonding

5 hydrogen bonding

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electronic ground state

1 closed-shell systems

2 ionic bondingelectronegativity differencecharge transferhcp, fcc or bccinsulators

3 covalent bonding

4 metallic bonding

5 hydrogen bonding

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electronic ground state

1 closed-shell systems

2 ionic bonding

3 covalent bondingcomplete change of theelectronic statesopen structures

4 metallic bonding

5 hydrogen bonding

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electronic ground state

1 closed-shell systems

2 ionic bonding

3 covalent bonding

4 metallic bondingpartially filled bandsclose-packed structures

5 hydrogen bonding

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electronic ground state

1 closed-shell systems

2 ionic bonding

3 covalent bonding

4 metallic bonding

5 hydrogen bondingp-e attractionno core repulsionintra- and inter- molecular

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electron density in the ground state

n(r) can be:measured experimentally

- x-ray scettering- high-energy electron scattering

calculated theoretically

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electron density in the ground state

n(r) can be:measured experimentally

- x-ray scettering- high-energy electron scattering

calculated theoretically

n(r) reveals:

core density

atomic-like

Debye-Waller factor

smearing of theaverage density due tothermal and zero-pointmotion

“outer” density

changes in density dueto bonding and chargetransfer

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electron density in the ground state

core density

atomic-like

Debye-Waller factor

smearing of theaverage density dueto thermal andzero-point motion

“outer” density

changes in densitydue to bonding andcharge transfer

A question

How would you reveal (calculate) the covalent bond density?

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Electron density in the ground state

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Volume or pressure as the fundamental variables

Equation of state: E = E(p,T )

E = E(V ,T = 0) very easy to calculate

99K one of the most important tests of the theory (e-e interaction)

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Volume or pressure as the fundamental variables

Fundamental quantities

E = E(V ) = Etotal(V ) ,

p = − dE

dV,

B = −V dp

dV= V

d2p

dV 2

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Volume or pressure as the fundamental variables

Fundamental quantities

E = E(V ) = Etotal(V ) ,

p = − dE

dV,

B = −V dp

dV= V

d2p

dV 2

How to test the theory using these variables:

1 equilibrium volume V0 =⇒ E0 , p = 0

2 equilibrium bulk modulus B0 =⇒ E0 , p = 0

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Volume or pressure as the fundamental variables

For example,

1 calc. E for several V

2 fit with analytic eq. of states

3dE

dVgives V0 and E0

4d2p

dV 2gives B

Accuracy within few percent ofexp.

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phase transitions under pressure

Experiments can now easily measure materials’ properties under pressure

“Bridgman” era: 1905. - 20th century40’s

DAC (diamond anvil cell) - C. E. Weir(1959)

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phase transitions under pressure

Pressure can change manymaterials’ properties

Band structure of K under pressure [2].

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phase transitions under pressure

Pressure can change manymaterials’ properties

The shift of optical absorption spectra under pressure [3].

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phase transitions under pressure

Pressure can change manymaterials’ properties

Existance of superconducting phases under pressure [4].

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phase transitions under pressure

Pressure can change manymaterials’ properties

E(V ) of various Si phases [5].

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phase transitions under pressure

How to calculate the pressureat which phase transitionoccurs?

1 G(T = 0) = H

stable structure enthalpy minimum

2 Gibbs construction oftangent lines betweenE(V ) curves

Enthalpies of various InP phases [6].

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phase transitions under pressure

How to calculate the pressureat which phase transitionoccurs?

1 G(T = 0) = H

stable structure enthalpy minimum

2 Gibbs construction oftangent lines betweenE(V ) curves

E(V ) of various Si phases [5].

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phase transitions under pressure

How to calculate the pressureat which phase transitionoccurs?

1 G(T = 0) = H

stable structure enthalpy minimum

2 Gibbs construction oftangent lines betweenE(V ) curves

So, how accurate are thesemethods?

Lowest transition pressures of several semisonductors [6].

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Elasticity: stress-strain relation

A question

What happens with the electronic ground state when strain is applied?

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Elasticity: stress-strain relation

A question

What happens with the electronicground state when strain is applied?

“Streckung des Grundgebietes”.

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Elasticity: stress-strain relation

Stress-strain relation

σαβ = − 1

V

∂Etot

∂uαβ

σαβ → stress tensor

uαβ → strain tensor

Stress-strain relation in Si [7].

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Magnetism and electron-electron interaction

What are magnetic systems?Ones in which the ground state has a broken symmetry with spin and/or orbitalmoments of the electrons.

Examples:

1 ferromagnets

2 antiferromagnets...

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Magnetism and electron-electron interaction

What are magnetic systems?Ones in which the ground state has a broken symmetry with spin and/or orbitalmoments of the electrons.

Explanation:

spin + orbital moment + Hund’s rules

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Magnetism and electron-electron interaction

Explanation:

spin + orbital moment + Hund’s rules

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Magnetism and electron-electron interaction

Basic equations

E(Vm) = Etot(Vm) ,

m(r) = − dE

dVm(r),

χ(r, r′) = − dm(r)

dVm(r′)=

d2E

dVm(r)dVm(r′)

m = n↑ − n↓ - magnetization

Vm = µHZeeman - effective Zeeman field which replaces e-e interaction

A question

What is m if we ignore the e-e interaction? What is m in a ferromagnet or aantiferromagnet?

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Magnetism and electron-electron interaction

Stoner parameter:

I · N(0)→ 1⇒ χ→∞

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phonons and displacive phase transitions

Experiments

Vibrational spectra:

infrared absorptionspectroscopy

light scattering

inelastic neutron scattering

Theory

E({RI}) = Etot({RI}) ,

FI = − dE

dRI,

CIJ = − dFI

dRJ= − d2E

dRIdRJ

A question

These equations hold only in adiabatic orBorn-Oppenheimer approximation. Can youremember what they say?

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phonons and displacive phase transitions

Experiments

Vibrational spectra:

infrared absorptionspectroscopy

light scattering

inelastic neutron scattering

Great synergy betweenexperiments and theory!

Theory

E({RI}) = Etot({RI}) ,

FI = − dE

dRI,

CIJ = − dFI

dRJ= − d2E

dRIdRJ

From these we get:

interatomic force constants

static dielectric constants

piezoelectric constants

effective charges

electron-phonon interaction...

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phonons and displacive phase transitions

Two approaches:

1 “frozen phonon”method

2 response functionmethod

Left: Two optic mode displacements in MgB2. Right: Two optic mode displacements of Ti

atoms in BaTiO3.

A question

What does the left figure reminds you of?

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Phonons and displacive phase transitions

Two approaches:

1 “frozen phonon”method

2 response functionmethod

Phonon dispersion curves for GaAs.

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Thermal properties

QMD = Quantum MolecularDynamics (Car-Parrinello MD)

liquids as a function of T

solids as a function of T

melting

chemical reactions of moleculesin solution

adsorption processes...

Phase diagram of carbon.

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Thermal properties

Water tough test for the theory (hydrogen bonding)

Radial density distributions in water molecule.

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Thermal properties

Water tough test for the theory (hydrogen bonding)

Proton transfer in water under high-temeprature/high-pressure conditions.

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Surfaces, interfaces and defects

Experiments

Powerfull techniques:

STM

x-ray diffraction

electron diffraction...

STM image of GaN (000-1) surface.

Theory

Supercell method - repeat supercells,not unit cells

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Surfaces, interfaces and defects

Surfaces Defects Molecules

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Surfaces, interfaces and defects

Electrolyses of water on metal surfaces - Pt(111)

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Contents

1 An overview

2 Literature

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1

Contents An overview Literature

Literature

1 R. M. Martin, “Electronic Structure - Basic Theory and PracticalMethods”, Cambridge University Press, Cambridge, 2004.

2 M. Alouani et al., Phys. Rev. B 39, 8096 (1989).

3 M. Moakafi et al., Eur. Phys. J. B 64, 35 (2008).

4 V. V. Struyhkin et al., Nature 390, 382 (1997).

5 M. T. Yin et al., Phys. Rev. B 26, 5668 (1982).R. Biswas et al., Phys. Rev. B 30, 3210 (1984).

6 A. Mujica et al., Rev. Mod. Phys. 75, 863 (2003).

7 O. H. Nielsen et al., Phys. Rev. Lett. 50, 697 (1983).

8 http://www.youtube.com/watch?v=UjXvI2w0VAU

Igor Lukacevic UJJS, Dept. of Physics, Osijek

The electronic structure of materials 1