Solid State Theory

Solid State Theory

Spring Semester 2014

Manfred Sigrist Institut fu?r Theoretische Physik HIT K23.8

Tel.: 044-633-2584 Email: sigrist@itp.phys.ethz.ch Website:

Lecture Website:

Literature: ? N.W. Ashcroft and N.D. Mermin: Solid State Physics, HRW International Editions, 1976. ? C. Kittel: Einfu?hrung in die Festk?orperphysik, R. Oldenburg Verlag, 1983. ? C. Kittel: Quantentheorie der Festk?orper, R. Oldenburg, 1970. ? O. Madelung: Introduction to solid-state theory, Springer 1981; auch in Deutsch in drei B?anden: Festk?operphysik I-III, Springer. ? J.M. Ziman: Principles of the Theory of Solids, Cambridge University Press, London, 1972. ? M.P. Marder: Condensed Matter Physics, John Wiley & Sons, 2000. ? G. Grosso & G.P. Parravicini: Solid State Physics, Academic Press, 2000. ? G. Czychol: Theoretische Festk?orperphysik, Springer 2004. ? P.L. Taylor & O. Heinonen, A Quantum Approach to Condensed Matter Physics, Cambridge Press 2002. ? G.D. Mahan, Condensed Matter in a Nutshell, Princeton University Press 2011. ? numerous specialized books.

1

Contents

Introduction

5

1 Electrons in the periodic crystal - band structure

8

1.1 Symmetries of crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.1 Space groups of crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.1.2 Reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2 Bloch's theorem and Bloch functions . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.3 Nearly free electron approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.4 Tight-binding approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.4.1 Linear combination of atomic orbitals - LCAO . . . . . . . . . . . . . . . 16

1.4.2 Band structure of s-orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.3 Band structure of p-orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.4.4 Wannier functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4.5 Tight binding model in second quantization formulation . . . . . . . . . . 21

1.5 Symmetry properties of the band structure . . . . . . . . . . . . . . . . . . . . . 21

1.6 Band-filling and materials properties . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.6.1 Electron count and band filling . . . . . . . . . . . . . . . . . . . . . . . . 24

1.6.2 Metals, semiconductors and insulators . . . . . . . . . . . . . . . . . . . . 25

1.7 Semi-classical description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.7.2 Bloch oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

1.7.3 Current densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

1.8 Appendix: Approximative band structure calcuations . . . . . . . . . . . . . . . . 30

1.8.1 Pseudo-potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.8.2 Augmented plane wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2 Semiconductors

35

2.1 The band structure in group IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.1.1 Crystal and band structure . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.2 Elementary excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.2.1 Electron-hole excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2.2 Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2.3 Optical properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.3 Doping semiconductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.1 Impurity state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.3.2 Carrier concentration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4 Semiconductor devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4.1 pn-contacts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.4.2 Diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.4.3 MOSFET . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2

3 Metals

51

3.1 The Jellium model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.1 Theory of metals - Sommerfeld and Pauli . . . . . . . . . . . . . . . . . . 52

3.1.2 Stability of metals - a Hartree-Fock approach . . . . . . . . . . . . . . . . 54

3.2 Charge excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2.1 Dielectric response and Lindhard function . . . . . . . . . . . . . . . . . . 58

3.2.2 Electron-hole excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.3 Collective excitation - Plasmon . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.4 Screening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3.1 Vibration of a isotropic continuous medium . . . . . . . . . . . . . . . . . 66

3.3.2 Phonons in metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.3.3 Peierls instability in one dimension . . . . . . . . . . . . . . . . . . . . . . 69

3.3.4 Dynamics of phonons and the dielectric function . . . . . . . . . . . . . . 74

4 Itinerant electrons in a magnetic field

76

4.1 The de Haas-van Alphen effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.1.1 Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.1.2 Oscillatory behavior of the magnetization . . . . . . . . . . . . . . . . . . 78

4.1.3 Onsager equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.2.1 Hall effect of the two-dimensional electron gas . . . . . . . . . . . . . . . . 82

4.2.2 Integer Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.2.3 Fractional Quantum Hall Effect . . . . . . . . . . . . . . . . . . . . . . . . 89

5 Landau's Theory of Fermi Liquids

93

5.1 Lifetime of quasiparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

5.2 Phenomenological Theory of Fermi Liquids . . . . . . . . . . . . . . . . . . . . . 96

5.2.1 Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2.2 Compressibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5.2.3 Spin susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.2.4 Galilei invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.2.5 Stability of the Fermi liquid . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.3 Microscopic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.3.1 Landau parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.3.2 Distribution function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.3.3 Fermi liquid in one dimension? . . . . . . . . . . . . . . . . . . . . . . . . 109

6 Transport properties of metals

111

6.1 Electrical conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

6.2 Transport equations and relaxation time . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.1 The Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

6.2.2 The Drude form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6.2.3 The relaxation time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.3 Impurity scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3.1 Potential scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

6.3.2 Kondo effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6.4 Electron-phonon interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.5 Electron-electron scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.6 Matthiessen's rule and the Ioffe-Regel limit . . . . . . . . . . . . . . . . . . . . . 127

6.7 General transport coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

6.7.1 Generalized Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . 128

6.7.2 Thermoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3

6.8 Anderson localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.8.1 Landauer Formula for a single impurity . . . . . . . . . . . . . . . . . . . 133 6.8.2 Scattering at two impurities . . . . . . . . . . . . . . . . . . . . . . . . . . 135 6.8.3 Anderson localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

7 Magnetism in metals

138

7.1 Stoner instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.1.1 Stoner model within the mean field approximation . . . . . . . . . . . . . 139

7.1.2 Stoner criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

7.1.3 Spin susceptibility for T > TC . . . . . . . . . . . . . . . . . . . . . . . . . 143 7.2 General spin susceptibility and magnetic instabilities . . . . . . . . . . . . . . . . 144

7.2.1 General dynamic spin susceptibility . . . . . . . . . . . . . . . . . . . . . 144

7.2.2 Instability with finite wave vector Q . . . . . . . . . . . . . . . . . . . . . 147

7.2.3 Influence of the band structure . . . . . . . . . . . . . . . . . . . . . . . . 148

7.3 Stoner excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8 Magnetism of localized moments

153

8.1 Mott transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.1.1 Hubbard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

8.1.2 Insulating state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

8.1.3 The metallic state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.1.4 Fermi liquid properties of the metallic state . . . . . . . . . . . . . . . . . 158

8.2 The Mott insulator as a quantum spin system . . . . . . . . . . . . . . . . . . . . 160

8.2.1 The effective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

8.2.2 Mean field approximation of the anti-ferromagnet . . . . . . . . . . . . . . 161

8.3 Collective modes ? spin wave excitations . . . . . . . . . . . . . . . . . . . . . . . 163

4

Introduction

Solid state physics (or condensed matter physics) is one of the most active and versatile branches of modern physics that have developed in the wake of the discovery of quantum mechanics. It deals with problems concerning the properties of materials and, more generally, systems with many degrees of freedom, ranging from fundamental questions to technological applications. This richness of topics has turned solid state physics into the largest subfield of physics; furthermore, it has arguably contributed most to technological development in industrialized countries.

Figure 1: Atom cores and the surrounding electrons.

Condensed matter (solid bodies) consists of atomic nuclei (ions), usually arranged in a regular (elastic) lattice, and of electrons (see Figure 1). As the macroscopic behavior of a solid is determined by the dynamics of these constituents, the description of the system requires the use of quantum mechanics. Thus, we introduce the Hamiltonian describing nuclei and electrons,

H = He + Hn + Hn-e,

(1)

with

He =

i

p2i 2m

+

1 2

i=i

|ri

e2 - ri

|

,

Hn =

j

P

2 j

2Mj

+

1 2

j=j

Zj Zj e2 |Rj - Rj

|,

(2)

Hn-e = -

i,j

Zj e2 |ri - Rj

|

,

where He (Hn) describes the dynamics of the electrons (nuclei) and their mutual interaction and Hn-e includes the interaction between ions and electrons. The parameters appearing are

m free electron mass

9.1094 ? 10-31kg

e elementary charge

1.6022 ? 10-19As

Mj mass of j-th nucleus

103 - 104?m

Zj atomic (charge) number of j-th nucleus

The characteristic scales known from atomic and molecular systems are

5

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