vili Solid-State Physics for Electronics 4.4.1. Required functions 4.4.2. Dealing with overlapping energy bands 4.4.3. Permitted band populations 4.5. Semi-free electrons in the particular case of super lattices 4.6. Problems l16 4.6. 1. Problem 1: horizontal tangent at the zone limit (k t/a) of the dispersion curve 4.6.2. Problem 2: scale of m* in the neighborhood of energy discontinuities 4.6.3. Problem 3: study of EF T) Chapter 5. Crystalline Structure, Reciprocal Lattices and Brillouin Zones 123 5.1. Periodic lattices 5.1.1. Definitions: direct lattice 5.1.2. Wigner-Seitz cell 5.2. Locating reciprocal planes 125 5.2.1. Reciprocal planes: definitions and properties 5.2.2. Reciprocal planes: location using Miller indices 5.3. Conditions for maximum diffusion by a crystal (Laue conditions).. 128 5.3.1. Problem parameters 5.3.2. Wave diffused by a node located by pm, n,p=ma+nb+Pc 129 5.4. Reciprocal lattice 5.4. 1. Definition and properties of a reciprocal lattice 133 5. 4.2. Application: Ewald construction of a beam diffracted by a reciprocal lattice 5.5. Brillouin zones 5.5.1. Definition 135 5.5.2. Physical significance of Brillouin zone limits 553. Successive Brillouin zones 5.6. Particular properties 5.6.1. Properties of Gh, k I and relation to the direct lattice 137 5.6.2. A crystallographic definition of reciprocal lattice 139 5.6.3. Equivalence between the condition for maximum diffusion and braggs law 5.7. Example determinations of Brillouin zones and reduced zones 141 5.7.1. Example 1: 3D lattice 5.7.2. Example 2: 2D lattice 5.7.3. Example 3: ID lattice with lattice repeat unit(a) such that the base vector in the direct lattice is a 145 5.8. Importance of the reciprocal lattice and electron filling of Brillouin zones by electrons in insulators, semiconductors and metals.. 1 5.8.1. Benefits of considering electrons in reciprocal lattices
viii Solid-State Physics for Electronics 4.4.1. Required functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.4.2. Dealing with overlapping energy bands . . . . . . . . . . . . . . . . . 97 4.4.3. Permitted band populations . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5. Semi-free electrons in the particular case of super lattices . . . . . . . . 107 4.6. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.6.1. Problem 1: horizontal tangent at the zone limit (k | S/a) of the dispersion curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 4.6.2. Problem 2: scale of m* in the neighborhood of energy discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.6.3. Problem 3: study of EF(T) . . . . . . . . . . . . . . . . . . . . . . . . . 122 Chapter 5. Crystalline Structure, Reciprocal Lattices and Brillouin Zones 123 5.1. Periodic lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1.1. Definitions: direct lattice . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1.2. Wigner-Seitz cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2. Locating reciprocal planes . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 5.2.1. Reciprocal planes: definitions and properties . . . . . . . . . . . . . 125 5.2.2. Reciprocal planes: location using Miller indices . . . . . . . . . . . 125 5.3. Conditions for maximum diffusion by a crystal (Laue conditions) . . . 128 5.3.1. Problem parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 5.3.2. Wave diffused by a node located by U mnp , , ma nb pc GGG G 129 5.4. Reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 5.4.1. Definition and properties of a reciprocal lattice . . . . . . . . . . . . 133 5.4.2. Application: Ewald construction of a beam diffracted by a reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 5.5. Brillouin zones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.5.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 5.5.2. Physical significance of Brillouin zone limits . . . . . . . . . . . . . 135 5.5.3. Successive Brillouin zones . . . . . . . . . . . . . . . . . . . . . . . . 137 5.6. Particular properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 5.6.1. Properties of Ghkl , , G and relation to the direct lattice . . . . . . . . 137 5.6.2. A crystallographic definition of reciprocal lattice . . . . . . . . . . . 139 5.6.3. Equivalence between the condition for maximum diffusion and Bragg’s law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.7. Example determinations of Brillouin zones and reduced zones . . . . . 141 5.7.1. Example 1: 3D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 5.7.2. Example 2: 2D lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.7.3. Example 3: 1D lattice with lattice repeat unit (a) such that the base vector in the direct lattice is G a . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.8. Importance of the reciprocal lattice and electron filling of Brillouin zones by electrons in insulators, semiconductors and metals . . 146 5.8.1. Benefits of considering electrons in reciprocal lattices . . . . . . . 146
Table of Contents ix 5.8.2. Example of electron filling of Brillouin zones in simple structures determination of behaviors of insulators. semiconductors and metals 5.9. The Fermi surface: construction of surfaces and properties 149 5.9.1. Definition 592. Form of the free electron Fermi surface 593. Evolution of semi-free electron Fermi surfaces 4. Relation between Fermi surfaces and dispersion curves 5.10. Conclusion. Filling Fermi surfaces and the distinctions between insulators. semiconductors and metals 5.10.1. Distribution of semi-free electrons at absolute zero 5.10.2. Consequences for metals, insulators/semiconductors and semi-metals 5.11. Problems 5.11. 1 Problem 1: simple square lattice 5.11.2. Problem 2: linear chain and a square lattice 511.3 Problem 3: rectangular lattice 162 Chapter 6. Electronic Properties of Copper and silicon 61. Introduction 17 6. 2. Direct and reciprocal lattices of the fcc structure 6.2.1. Direct lattice 6.2.2. Reciprocal lattice 6.3. Brillouin zone for the fcc structure 178 63. 1. Geometrical form 632. Calculation of the volume of the Brillouin zone 6.3.3. Filling the Brillouin zone for a fcc structure 6. 4. Copper and alloy formation 6.4.1. Electronic properties of copper l81 6.4.2. Filling the Brillouin zone and solubility rules 6.4.3. Copper alloys 184 6.5. Silicon 6.5. 1. The silicon crystal 6.5.2. Conduction in silicon 6.5.3. The silicon band structure 6.5.4 Conclusion 6.6 Problems 6.6.1. Problem 1: the cubic centered(cc)structure 6.6.2. Problem 2: state density in the silicon conduction band 194 Chapter 7. Strong Bonds in One Dimension 71. Atomic and molecular orbitals 7.1. 1. S-and p-type orbitals 7.1.2. Molecular orbitals 204
Table of Contents ix 5.8.2. Example of electron filling of Brillouin zones in simple structures: determination of behaviors of insulators, semiconductors and metals . . . 146 5.9. The Fermi surface: construction of surfaces and properties . . . . . . . 149 5.9.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.9.2. Form of the free electron Fermi surface . . . . . . . . . . . . . . . . 149 5.9.3. Evolution of semi-free electron Fermi surfaces . . . . . . . . . . . . 150 5.9.4. Relation between Fermi surfaces and dispersion curves . . . . . . . 152 5.10. Conclusion. Filling Fermi surfaces and the distinctions between insulators, semiconductors and metals . . . . . . . . . . . . . . . . . 154 5.10.1. Distribution of semi-free electrons at absolute zero . . . . . . . . . 154 5.10.2. Consequences for metals, insulators/semiconductors and semi-metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.11. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.11.1. Problem 1: simple square lattice . . . . . . . . . . . . . . . . . . . . 156 5.11.2. Problem 2: linear chain and a square lattice . . . . . . . . . . . . . 157 5.11.3. Problem 3: rectangular lattice . . . . . . . . . . . . . . . . . . . . . . 162 Chapter 6. Electronic Properties of Copper and Silicon . . . . . . . . . . . . 173 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.2. Direct and reciprocal lattices of the fcc structure . . . . . . . . . . . . . . 173 6.2.1. Direct lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 6.2.2. Reciprocal lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 6.3. Brillouin zone for the fcc structure . . . . . . . . . . . . . . . . . . . . . . 178 6.3.1. Geometrical form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 6.3.2. Calculation of the volume of the Brillouin zone . . . . . . . . . . . . 179 6.3.3. Filling the Brillouin zone for a fcc structure . . . . . . . . . . . . . . 180 6.4. Copper and alloy formation . . . . . . . . . . . . . . . . . . . . . . . . . . 181 6.4.1. Electronic properties of copper . . . . . . . . . . . . . . . . . . . . . . 181 6.4.2. Filling the Brillouin zone and solubility rules . . . . . . . . . . . . . 181 6.4.3. Copper alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 6.5. Silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.5.1. The silicon crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.5.2. Conduction in silicon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.5.3. The silicon band structure . . . . . . . . . . . . . . . . . . . . . . . . . 186 6.5.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 6.6. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 6.6.1. Problem 1: the cubic centered (cc) structure . . . . . . . . . . . . . . 190 6.6.2. Problem 2: state density in the silicon conduction band . . . . . . . 194 Chapter 7. Strong Bonds in One Dimension . . . . . . . . . . . . . . . . . . . . 199 7.1. Atomic and molecular orbitals . . . . . . . . . . . . . . . . . . . . . . . . 199 7.1.1. s- and p-type orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 7.1.2. Molecular orbitals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
x Solid-State Physics for Electronics 7.1.3. 0-and T-bonds 09 7.14. Conclusion 7.2. Form of the wave function in strong bonds: Floquet's theorem 210 7. 1. Form of the resulting potenti 210 7.2.2. Form of the wave function 7.2.3. Effect of potential periodicity on the form of the wave function and Floquet theorem 7.3. Energy of a ID system 215 731. Mathematical resolution in Id where xe r 7.3.2. Calculation by integration of energy for a chain of n atoms 217 3.3. Note 1: physical significance in terms of (Eo-a)and B 3.4. Note 2: simplified calculation of the energy 7.3.5. Note 3: conditions for the appearance of permitted and forbidden bands 7.4. ID and distorted AB crystals 4. 1. AB crystal 224 42 Distorted ch 7.5. State density function and applications: the Peierls etal-insulator transition 7.5.1. Determination of the state density functions 7.5.2. Zone filling and the peierls metal-insulator transition 230 7.5.3. Principle of the calculation of Relax(for a distorted chain) 7.6. Practical example of a periodic atomic chain; concrete calculations of wave functions, energy levels, state density functions and band filling. 233 7.6.1. Range of variation in k 233 6.2. Representation of energy and state density function for N=8 234 7.6.3. The wave function for bonding and anti-bonding states 235 7.6.4. Generalization to any type of state in an atomic chain 7. 8. Problems 241 8. 1. Problem 1: complementary study of a chain of s-type atoms where n=8 241 7.8.2. Problem 2: general representation of the states of a chain of o-S-orbitals(s-orbitals giving g-overlap)and a chain of o-p-orbitals. 243 7.8.3. Problem 3: chains containing both o-s-and o-p-orbitals 246 8.4. Problem 4: atomic chain with T-type overlapping of p-type orbitals: T-p-and T'-p-orbitals 247 Chapter 8. Strong Bonds in Three Dimensions: Band Structure of diamond and silicon nding the permitted band from ID to 3D for a lattice of atoms associated with single s-orbital nodes(basic cubic system, centered cubic, etc.) 250
x Solid-State Physics for Electronics 7.1.3. V- and S-bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.1.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 7.2. Form of the wave function in strong bonds: Floquet’s theorem . . . . . 210 7.2.1. Form of the resulting potential . . . . . . . . . . . . . . . . . . . . . . 210 7.2.2. Form of the wave function . . . . . . . . . . . . . . . . . . . . . . . . 212 7.2.3. Effect of potential periodicity on the form of the wave function and Floquet’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.3. Energy of a 1D system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 7.3.1. Mathematical resolution in 1D where x { r. . . . . . . . . . . . . . . 215 7.3.2. Calculation by integration of energy for a chain of N atoms . . . . 217 7.3.3. Note 1: physical significance in terms of (E0 – D) and E . . . . . . 220 7.3.4. Note 2: simplified calculation of the energy . . . . . . . . . . . . . . 222 7.3.5. Note 3: conditions for the appearance of permitted and forbidden bands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 7.4. 1D and distorted AB crystals . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.4.1. AB crystal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.4.2. Distorted chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 7.5. State density function and applications: the Peierls metal-insulator transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 7.5.1. Determination of the state density functions . . . . . . . . . . . . . . 228 7.5.2. Zone filling and the Peierls metal–insulator transition . . . . . . . . 230 7.5.3. Principle of the calculation of Erelax (for a distorted chain). . . . . . 232 7.6. Practical example of a periodic atomic chain: concrete calculations of wave functions, energy levels, state density functions and band filling . 233 7.6.1. Range of variation in k. . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.6.2. Representation of energy and state density function for N = 8 . . . 234 7.6.3. The wave function for bonding and anti-bonding states . . . . . . . 235 7.6.4. Generalization to any type of state in an atomic chain . . . . . . . . 239 7.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 7.8. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.8.1. Problem 1: complementary study of a chain of s-type atoms where N = 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 7.8.2. Problem 2: general representation of the states of a chain of V–s-orbitals (s-orbitals giving V-overlap) and a chain of V–p-orbitals . 243 7.8.3. Problem 3: chains containing both V–s- and V–p-orbitals . . . . . . 246 7.8.4. Problem 4: atomic chain with S-type overlapping of p-type orbitals: S–p- and S*–p-orbitals . . . . . . . . . . . . . . . . . . . . . 247 Chapter 8. Strong Bonds in Three Dimensions: Band Structure of Diamond and Silicon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 8.1. Extending the permitted band from 1D to 3D for a lattice of atoms associated with single s-orbital nodes (basic cubic system, centered cubic, etc.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
8.1.1. Permitted energy in 3D: dispersion and equi-energy curves 250 8.1.2. Expression for the band width 8.1.3. Expressions for the effective mass and mobility 8.2. Structure of diamond: covalent bonds and their hybridization 8.2.1. The structure of diamond 8.2.2. Hybridization of atomic orbitals 8.2.3. sp'Hybridization 8.3. Molecular model of a 3D covalent crystal(atoms in sp-hybridization states at lattice nodes) 8.3.1. Conditions 268 8.3.2. Independent bonds: effect of single coupling between neighboring atoms and formation of molecular orbitals 8.3.3. Coupling of molecular orbitals: band formation 73 8.4. Complementary in-depth study: determination of the silicon band structure using the strong bond method 275 8. 41. Atomic wave functions and structures 275 8.4.2. Wave functions in crystals and equations with proper values for a strong bond approximation 843. Band structure 282 8. 4.4. Conclusion 8.5. Problems 287 8.5.1. Problem 1: strong bonds in a square 2L 8.5.2. Problem 2. strong bonds in a cubic cente face centered lattice Chapter 9 Limits to Classical Band Theory: Amorphous Media 9. 1. Evolution of the band scheme due to structural defects(vacancies dangling bonds and chain ends) and localized bands 301 9. 2. Hubbard bands and electronic repulsions. The Mott metal-insulator transition 303 9.21. Introduction 303 9.2.2. Model 9.2.3. The Mott metal-insulator transition: estimation of transition criteria 9.2.4. Additional material: examples of the existence and inexistence of mott-Hubbard transitions 9.3. Effect of geometric disorder and the Anderson localization 9.3.1. Introduction 311 9.3.2. Limits of band theory application and the loffe-Regel conditions. 312 9.33. Anderson localization 314 9.3.4. Localized states and conductivity. The Anderson metal-insulator transition 319 94. Conclusion 32
Table of Contents xi 8.1.1. Permitted energy in 3D: dispersion and equi-energy curves . . . . . 250 8.1.2. Expression for the band width . . . . . . . . . . . . . . . . . . . . . . 255 8.1.3. Expressions for the effective mass and mobility . . . . . . . . . . . . 257 8.2. Structure of diamond: covalent bonds and their hybridization . . . . . . 258 8.2.1. The structure of diamond . . . . . . . . . . . . . . . . . . . . . . . . . 258 8.2.2. Hybridization of atomic orbitals . . . . . . . . . . . . . . . . . . . . . 259 8.2.3. sp3 Hybridization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 8.3. Molecular model of a 3D covalent crystal (atoms in sp3 -hybridization states at lattice nodes) . . . . . . . . . . . . . . . . . . . . . 268 8.3.1. Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 8.3.2. Independent bonds: effect of single coupling between neighboring atoms and formation of molecular orbitals . . . . . . . . . . . 272 8.3.3. Coupling of molecular orbitals: band formation . . . . . . . . . . . . 273 8.4. Complementary in-depth study: determination of the silicon band structure using the strong bond method . . . . . . . . . . . . . . . . . . . 275 8.4.1. Atomic wave functions and structures . . . . . . . . . . . . . . . . . . 275 8.4.2. Wave functions in crystals and equations with proper values for a strong bond approximation . . . . . . . . . . . . . . . . . . . . . . . . 278 8.4.3. Band structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 8.4.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 8.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 8.5.1. Problem 1: strong bonds in a square 2D lattice . . . . . . . . . . . . 287 8.5.2. Problem 2: strong bonds in a cubic centered or face centered lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 Chapter 9. Limits to Classical Band Theory: Amorphous Media . . . . . . 301 9.1. Evolution of the band scheme due to structural defects (vacancies, dangling bonds and chain ends) and localized bands . . . . . . . . . . . . . . 301 9.2. Hubbard bands and electronic repulsions. The Mott metal–insulator transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 9.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 9.2.2. Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 9.2.3. The Mott metal–insulator transition: estimation of transition criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307 9.2.4. Additional material: examples of the existence and inexistence of Mott–Hubbard transitions . . . . . . . . . . . . . . . . . . . . 309 9.3. Effect of geometric disorder and the Anderson localization . . . . . . . 311 9.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 9.3.2. Limits of band theory application and the Ioffe–Regel conditions . 312 9.3.3. Anderson localization . . . . . . . . . . . . . . . . . . . . . . . . . . . 314 9.3.4. Localized states and conductivity. The Anderson metal-insulator transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
Solid-State Physics for Electronics 9.5. Problems 9.5.1. Additional information and Problem 1 on the mott transition insulator-metal transition in phosphorus doped silicon 9.5.2. Problem 2: transport via states outside of permitted band n low mobility media 331 Chapter 10. The Principal Quasi-Particles in Material Physics 335 101. Introduction 335 10.2. Lattice vibrations: phonons 336 10.2.1 Introduction 336 10.22 Oscillations within a linear chain of atoms 337 10.23. Oscillations within a diatomic and Id chain 10.2. 4. Vibrations of a 3D crystal 347 10.2.5. Energy of a vibrational mode 10.26 Phonons 350 10.2.7. Conclusion 351 10.3. Polarons 10.3.1. Introduction: definition and origin 10.3.2. The various polarons 352 10.3.3. Dielectric polarons 10.3.4. Polarons in molecular crystals 357 10.3.5. Energy spectrum of the small polaron in molecular solids 10.4. Excitons 10.4.1. Physical origin 364 10.42. Wannier and charge transfer excitons 10.43. Frenkel excitons 367 10.5. Plasmons 10.5.1. Basic definition 368 10.5.2. Dielectric response of an electronic gas: optical plasma 10.53 Plasmons 372 10.6. Problems 10.6.1. Problem 1: enumeration of vibration modes(phonon modes) 10.6.2. Problem 2: polaritons 375 Bibliography 385 Index 387
xii Solid-State Physics for Electronics 9.5. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 9.5.1. Additional information and Problem 1 on the Mott transition: insulator–metal transition in phosphorus doped silicon . . . . . . . . . . . 324 9.5.2. Problem 2: transport via states outside of permitted bands in low mobility media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Chapter 10. The Principal Quasi-Particles in Material Physics . . . . . . . . 335 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 10.2. Lattice vibrations: phonons . . . . . . . . . . . . . . . . . . . . . . . . . . 336 10.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 10.2.2. Oscillations within a linear chain of atoms . . . . . . . . . . . . . . 337 10.2.3. Oscillations within a diatomic and 1D chain . . . . . . . . . . . . . 343 10.2.4. Vibrations of a 3D crystal . . . . . . . . . . . . . . . . . . . . . . . . 347 10.2.5. Energy of a vibrational mode . . . . . . . . . . . . . . . . . . . . . . 348 10.2.6. Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350 10.2.7. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 351 10.3. Polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 10.3.1. Introduction: definition and origin . . . . . . . . . . . . . . . . . . . 352 10.3.2. The various polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . 352 10.3.3. Dielectric polarons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 10.3.4. Polarons in molecular crystals . . . . . . . . . . . . . . . . . . . . . 357 10.3.5. Energy spectrum of the small polaron in molecular solids . . . . . 361 10.4. Excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 10.4.1. Physical origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 10.4.2. Wannier and charge transfer excitons . . . . . . . . . . . . . . . . . 365 10.4.3. Frenkel excitons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 10.5. Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 10.5.1. Basic definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 10.5.2. Dielectric response of an electronic gas: optical plasma . . . . . . 368 10.5.3. Plasmons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 10.6. Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373 10.6.1. Problem 1: enumeration of vibration modes (phonon modes) . . . 373 10.6.2. Problem 2: polaritons . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 Bibliography ....................................... 385 Index ............................................ 387