The electronic structure of materials 1

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1 Quantum mechanics 2 - Lecture 9 December 18, 2013

2 1 An overview 2 Literature

3 Contents 1 An overview 2 Literature

4 Electronic ground state Ground state cohesive energy equilibrium crystal structure phase transitions between structures elastic constants charge density magnetic order static dielectric susceptibility static magnetic susceptibility nuclear vibrations (in the adiabatic 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.

5 Electronic ground state Interplay: electronic ground state spatial structure of the nuclei bonding

6 Electronic ground state 1 closed-shell systems rare gases molecular solids van der Waals interaction 2 ionic bonding 3 covalent bonding 4 metallic bonding 5 hydrogen bonding

7 Electronic ground state 1 closed-shell systems 2 ionic bonding electronegativity difference charge transfer hcp, fcc or bcc insulators 3 covalent bonding 4 metallic bonding 5 hydrogen bonding

8 Electronic ground state 1 closed-shell systems 2 ionic bonding 3 covalent bonding complete change of the electronic states open structures 4 metallic bonding 5 hydrogen bonding

9 Electronic ground state 1 closed-shell systems 2 ionic bonding 3 covalent bonding 4 metallic bonding partially filled bands close-packed structures 5 hydrogen bonding

10 Electronic ground state 1 closed-shell systems 2 ionic bonding 3 covalent bonding 4 metallic bonding 5 hydrogen bonding p-e attraction no core repulsion intra- and inter- molecular

11 Electron density in the ground state n(r) can be: measured experimentally - x-ray scettering - high-energy electron scattering calculated theoretically

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

12 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 the average density due to thermal and zero-point motion outer density changes in density due to bonding and charge transfer

Electron density in the ground state core density atomic-like Debye-Waller factor smearing of the average density due to thermal and zero-point

13 Electron density in the ground state core density atomic-like Debye-Waller factor smearing of the average density due to thermal and zero-point motion outer density changes in density due to bonding and charge transfer A question How would you reveal (calculate) the covalent bond density?

14 Electron density in the ground state

15 Volume or pressure as the fundamental variables Equation of state: E = E(p, T ) E = E(V, T = 0) very easy to calculate one of the most important tests of the theory (e-e interaction)

16 Volume or pressure as the fundamental variables Fundamental quantities E = E(V ) = E total (V ), p = de dv, B = V dp dv = V d2 p dv 2

17 Volume or pressure as the fundamental variables Fundamental quantities E = E(V ) = E total (V ), p = de dv, B = V dp dv = V d2 p dv 2 How to test the theory using these variables: 1 equilibrium volume V 0 = E 0, p = 0 2 equilibrium bulk modulus B 0 = E 0, p = 0

18 Volume or pressure as the fundamental variables For example, 1 calc. E for several V 2 fit with analytic eq. of states de 3 gives V0 and E0 dv d 2 p 4 dv gives B 2 Accuracy within few percent of exp.

properties under pressure Bridgman era: 1905.

19 Contents An overview Literature Phase transitions under pressure Experiments can now easily measure materials properties under pressure Bridgman era: th century 40 s Igor Lukaˇ cevi c DAC (diamond anvil cell) - C. E. Weir (1959)

20 Phase transitions under pressure Pressure can change many materials properties Band structure of K under pressure [2].

21 Phase transitions under pressure Pressure can change many materials properties The shift of optical absorption spectra under pressure [3].

22 Phase transitions under pressure Pressure can change many materials properties Existance of superconducting phases under pressure [4].

23 Phase transitions under pressure Pressure can change many materials properties E(V ) of various Si phases [5].

24 Phase transitions under pressure How to calculate the pressure at which phase transition occurs? 1 G(T = 0) = H stable structure enthalpy minimum 2 Gibbs construction of tangent lines between E(V ) curves Enthalpies of various InP phases [6].

25 Phase transitions under pressure How to calculate the pressure at which phase transition occurs? 1 G(T = 0) = H stable structure enthalpy minimum 2 Gibbs construction of tangent lines between E(V ) curves E(V ) of various Si phases [5].

26 Phase transitions under pressure How to calculate the pressure at which phase transition occurs? 1 G(T = 0) = H stable structure enthalpy minimum 2 Gibbs construction of tangent lines between E(V ) curves So, how accurate are these methods? Lowest transition pressures of several semisonductors [6].

27 Elasticity: stress-strain relation A question What happens with the electronic ground state when strain is applied?

28 Elasticity: stress-strain relation A question What happens with the electronic ground state when strain is applied? Streckung des Grundgebietes.

29 Elasticity: stress-strain relation Stress-strain relation σ αβ = 1 V σ αβ stress tensor u αβ strain tensor E tot u αβ Stress-strain relation in Si [7].

30 Magnetism and electron-electron interaction What are magnetic systems? Ones in which the ground state has a broken symmetry with spin and/or orbital moments of the electrons. Examples: 1 ferromagnets 2 antiferromagnets.

31 Magnetism and electron-electron interaction What are magnetic systems? Ones in which the ground state has a broken symmetry with spin and/or orbital moments of the electrons. Explanation: spin + orbital moment + Hund s rules

32 Magnetism and electron-electron interaction Explanation: spin + orbital moment + Hund s rules

33 Magnetism and electron-electron interaction Basic equations E(V m) = E tot(v m), m(r) = de dv m(r), χ(r, r ) = dm(r) dv m(r ) = d 2 E dv m(r)dv m(r ) m = n n V m = µh Zeeman - magnetization - 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 a antiferromagnet?

34 Magnetism and electron-electron interaction Stoner parameter: I N(0) 1 χ

35 Phonons and displacive phase transitions Theory Experiments Vibrational spectra: infrared absorption spectroscopy light scattering inelastic neutron scattering A question E({R I }) = E tot({r I }), F I = de, dr I C IJ = df I dr J = d2 E dr I dr J These equations hold only in adiabatic or Born-Oppenheimer approximation. Can you remember what they say?

36 Phonons and displacive phase transitions Theory Experiments Vibrational spectra: infrared absorption spectroscopy light scattering inelastic neutron scattering Great synergy between experiments and theory! E({R I }) = E tot({r I }), F I = de, dr I From these we get: C IJ = df I dr J = d2 E dr I dr J interatomic force constants static dielectric constants piezoelectric constants effective charges electron-phonon interaction.

37 Phonons and displacive phase transitions Two approaches: 1 frozen phonon method 2 response function method Left: Two optic mode displacements in MgB 2. Right: Two optic mode displacements of Ti atoms in BaTiO 3. A question What does the left figure reminds you of?

38 Phonons and displacive phase transitions Two approaches: 1 frozen phonon method 2 response function method Phonon dispersion curves for GaAs.

39 Thermal properties QMD = Quantum Molecular Dynamics (Car-Parrinello MD) liquids as a function of T solids as a function of T melting chemical reactions of molecules in solution adsorption processes. Phase diagram of carbon.

40 Thermal properties Water tough test for the theory (hydrogen bonding) Radial density distributions in water molecule.

41 Thermal properties Water tough test for the theory (hydrogen bonding) Proton transfer in water under high-temeprature/high-pressure conditions.

$Surfaces, interfaces and defects Experiments Powerfull techniques: STM x-ray diffraction electron$

42 Surfaces, interfaces and defects Experiments Powerfull techniques: STM x-ray diffraction electron diffraction. Theory Supercell method - repeat supercells, not unit cells STM image of GaN (000-1) surface.

43 Surfaces, interfaces and defects Surfaces Defects Molecules

44 Surfaces, interfaces and defects Electrolyses of water on metal surfaces - Pt(111)

45 Contents 1 An overview 2 Literature

46 Literature 1 R. M. Martin, Electronic Structure - Basic Theory and Practical Methods, Cambridge University Press, Cambridge, 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

The Oxford Solid State Basics

The Oxford Solid State Basics Steven H. Simon University of Oxford OXFORD UNIVERSITY PRESS Contents 1 About Condensed Matter Physics 1 1.1 What Is Condensed Matter Physics 1 1.2 Why Do We Study Condensed