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Introduction to Modern Solid State Physics Yuri M Galperin FYS 448 Department of Physics, P.O Box 1048 Blindern, 0316 Oslo, Room 427A Phone: +47 22 85 64 95, E-mail: iouri.galperinefys.uio.no Contents I Basic concepts 1 Geometry of Lattices 1.1 Periodicity: Crystal Structures 1.2 The Reciprocal Lattice 1.3 X-Ray Diffraction in Periodic Structures 1.4 Problems Lattice Vibrations: Phonons 2.1 Interactions Between Atoms 2.2 Lattice Vibrations 2.3 Quantum Mechanics of Atomic Vibrations 2.4 Phonon Dispersion Measurement 2.5 Problems Electrons in a Lattice 3.1 Electron in a Periodic Field 3.1.1 Electron in a Periodic Potential 3.2 Tight Binding Approximation 3.3 The Model of Near Free Electrons 3.4 Main Properties of Bloch Electrons 3.4.1 Effective Mass 3.4.2 Wannier Theorem → Effective Mass Approach 3.5 Electron Velocity 3.5.1 Electric current in a Bloch State Concept of Holes 3.6 Classification of Materials 3.7 Dynamics of Bloch Electrons 3.7.1 Classical Mechanics 3.7.2 Quantum Mechanics of Bloch Electron 3.8 Second Quantization of Bosons and Electrons 3.9 Problems i 3 10 18 21 21 23 38 43 44 45 45 46 47 50 52 52 53 54 54 55 57 57 63 65 67 ii II CONTENTS Normal metals and semiconductors Statistics and Thermodynamics 4.1 Specific Heat of Crystal Lattice 4.2 Statistics of Electrons in Solids 4.3 Specific Heat of the Electron System 4.4 Magnetic Properties of Electron Gas 4.5 Problems 69 Summary of basic concepts 93 Classical dc Transport 6.1 The Boltzmann Equation for Electrons 6.2 Conductivity and Thermoelectric Phenomena 6.3 Energy Transport 6.4 Neutral and Ionized Impurities 6.5 Electron-Electron Scattering 6.6 Scattering by Lattice Vibrations 6.7 Electron-Phonon Interaction in Semiconductors 6.8 Galvano- and Thermomagnetic 6.9 Shubnikov-de Haas effect 6.10 Response to “slow” perturbations 6.11 “Hot” electrons 6.12 Impact ionization 6.13 Few Words About Phonon Kinetics 6.14 Problems Electrodynamics of Metals 7.1 Skin Effect 7.2 Cyclotron Resonance 7.3 Time and Spatial Dispersion 7.4 Waves in a Magnetic Field 7.5 Problems 71 71 75 80 81 91 Acoustical Properties 8.1 Landau Attenuation 8.2 Geometric Oscillations 8.3 Giant Quantum Oscillations 8.4 Acoustical properties of semicondictors 8.5 Problems 97 97 101 106 109 112 114 125 130 140 142 145 148 150 152 155 155 158 165 168 169 171 171 173 174 175 180 CONTENTS iii Optical Properties of Semiconductors 9.1 Preliminary discussion 9.2 Photon-Material Interaction 9.3 Microscopic single-electron theory 9.4 Selection rules 9.5 Intraband Transitions 9.6 Problems 9.7 Excitons 9.7.1 Excitonic states in semiconductors 9.7.2 Excitonic effects in optical properties 9.7.3 Excitonic states in quantum wells 10 Doped semiconductors 10.1 Impurity states 10.2 Localization of electronic states 10.3 Impurity band for lightly doped semiconductors 10.4 AC conductance due to localized states 10.5 Interband light absorption III 181 181 182 189 191 198 202 202 203 205 206 211 211 215 219 225 232 Basics of quantum transport 237 11 Preliminary Concepts 11.1 Two-Dimensional Electron Gas 11.2 Basic Properties 11.3 Degenerate and non-degenerate electron gas 11.4 Relevant length scales 239 239 240 250 251 12 Ballistic Transport 12.1 Landauer formula 12.2 Application of Landauer formula 12.3 Additional aspects of ballistic transport 12.4 e − e interaction in ballistic systems 255 255 260 265 266 13 Tunneling and Coulomb blockage 273 13.1 Tunneling 273 13.2 Coulomb blockade 277 14 Quantum Hall Effect 14.1 Ordinary Hall effect 14.2 Integer Quantum Hall effect - General Picture 14.3 Edge Channels and Adiabatic Transport 14.4 Fractional Quantum Hall Effect 285 285 285 289 294 iv IV CONTENTS Superconductivity 307 15 Fundamental Properties 309 15.1 General properties 309 16 Properties of Type I 16.1 Thermodynamics in a Magnetic Field 16.2 Penetration Depth 16.3 Arbitrary Shape 16.4 The Nature of the Surface Energy 16.5 Problems 313 313 314 318 328 329 17 Magnetic Properties -Type II 17.1 Magnetization Curve for a Long Cylinder 17.2 Microscopic Structure of the Mixed State 17.3 Magnetization curves 17.4 Non-Equilibrium Properties Pinning 17.5 Problems 331 331 335 343 347 352 18 Microscopic Theory 18.1 Phonon-Mediated Attraction 18.2 Cooper Pairs 18.3 Energy Spectrum 18.4 Temperature Dependence 18.5 Thermodynamics of a Superconductor 18.6 Electromagnetic Response 18.7 Kinetics of Superconductors 18.8 Problems 353 353 355 357 360 362 364 369 376 19 Ginzburg-Landau Theory 19.1 Ginzburg-Landau Equations 19.2 Applications of the GL Theory 19.3 N-S Boundary 377 377 382 388 20 Tunnel Junction Josephson Effect 20.1 One-Particle Tunnel Current 20.2 Josephson Effect 20.3 Josephson Effect in a Magnetic Field 20.4 Non-Stationary Josephson Effect 20.5 Wave in Josephson Junctions 20.6 Problems 391 391 395 397 402 405 407 CONTENTS 21 Mesoscopic Superconductivity 21.1 Introduction 21.2 Bogoliubov-de Gennes equation 21.3 N-S interface 21.4 Andreev levels and Josephson effect 21.5 Superconducting nanoparticles V Appendices 22 Solutions of the Problems v 409 409 410 412 421 425 431 433 A Band structure of semiconductors 451 A.1 Symmetry of the band edge states 456 A.2 Modifications in heterostructures 457 A.3 Impurity states 458 B Useful Relations 465 B.1 Trigonometry Relations 465 B.2 Application of the Poisson summation formula 465 C Vector and Matrix Relations 467 vi CONTENTS Part I Basic concepts 455 Then one can invert this set of equations to express |pi , s through the functions Φl,m For references, √ 1 px ↑ = √ −Φ3/2,3/2 + √ Φ3/2,−1/2 − √ Φ1/2,−1/2 3 √ px ↓ = √ +Φ3/2,−3/2 − √ Φ3/2,1/2 − √ Φ1/2,1/2 3 √ i py ↑ = √ Φ3/2,3/2 + √ Φ3/2,−1/2 − √ Φ1/2,−1/2 3 √ i py ↓ = √ +Φ3/2,−3/2 + √ Φ3/2,1/2 + √ Φ1/2,1/2 3 √ √ 2 pz ↑ = √ Φ3/2,1/2 − √ Φ1/2,1/2 3 √ √ 2 pz ↓ = √ Φ3/2,−1/2 − √ Φ1/2,−1/2 3 Finally, we can write the spin-orbit Hamiltonian as Hso = λ [j(j + 1) − l(l + 1) − s(s + 1)] For p-orbitals, l = 1, s = 1/2 while j is given by the first index of Φij One can prove that the only non-vanishing matrix elements in p representation are px ↑ |Hso |py ↑ = −i px ↑ |Hso |pz ↓ = py ↑ |Hso |pz ↓ px ↓ |Hso |py ↓ px ↓ |Hso |pz ↑ py ↓ |Hso |pz ↑ ∆ ∆ ∆ ∆ = i ∆ = − ∆ = −i = −i Here ∆ = 3λ /2 is the spin-orbit splitting As a result, we arrive at the following structure of valence band, Fig A.2 Spin-orbit splitting is given in the table A 456 APPENDIX A BAND STRUCTURE OF SEMICONDUCTORS E |3/2,-+3/2> |3/2,-+ 1/2> |1/2,-+ 1/2> k Figure A.2: The general form of the valence band including spin-orbit coupling Semiconductor Si Ge GaAs InAs InSb InP GaP ∆ (eV) 0.044 0.29 0.35 0.41 0.82 0.14 0.094 Table A.1: Spin-orbit splitting for different semiconductors A.1 Symmetry of the band edge states One has to discriminate between direct gap (GaAs, InAs) and indirect gap (Si, Ge) materials, Fig A.3 In direct gap materials the conduction band minimum occurs at Γ-point and have a spherically symmetric central cell function So they are made of s atomic states As one goes apart, admixture of p-states appears This implies important limitations for selection rules In the valence band, he have Heavy holes: Φ3/2,3/2 = − √ (|px + i|py ) ↑ Φ3/2,−3/2 = √ (|px − i|py ) ↓ A.2 MODIFICATIONS IN HETEROSTRUCTURES Direct (k = 0) s-type 457 Indirect (k = X point) s+p (longitudinal) p (transverse) Heavy holes Light holes Spin-split holes Figure A.3: Schematic description of the nature of central cell functions Light holes: Φ3/2,1/2 = − √ [(|px + i|py ) ↓ −2|pz ↑] Φ3/2,−1/2 = √ [(|px − i|py ) ↑ +2|pz ↓] Split-off hole states: Φ1/2,1/2 = − √ [(|px + i|py ) ↓ +|pz ↑] Φ1/2,−1/2 = − √ [(|px − i|py ) ↑ +|pz ↓] A Brillouin zone for silicon is shown in Fig A.2 A.2 Modifications in heterostructures An important feature is how the bands align Three type are usually studied, Fig A.5 The problem of band offsets is very complicated, and several theories exist Quantum wells Let us discuss the case where the well region is made of direct gap material, so conduction band states are of s-type while valence band states are p-type For simplicity we discuss square-box confinement, Fig- A.6 Within the eective mass approach the Schrădinger o 458 APPENDIX A BAND STRUCTURE OF SEMICONDUCTORS equation for the electron states is (in the following we not discriminate between m∗ and m) − 2m∗ + V (z) ψ = Eψ Looking for the solution as Ψ(r) = ei(ky y+kx x) f (z) we have the equation for f ∂2 + V (z) ψ = En ψ 2m∗ ∂z 2 − For an infinite barrier, its solution has the form πnz , W πnz , = sin W π 2 n2 = 2mW f (z) = cos En if n is even if n is odd Then the total energy is 2 E = En + k 2m The situation for the valence band is much more complicated because the heavy- and light-hole states mix away from k = A.3 Impurity states A typical energy level diagram is shown on Fig A.7 Shallow levels allow a universal description because the spread of wave function is large and the potential can be treated as from the point charge, U (r) = e2 / r To find impurity states one has to treat Schrădinger equation (SE) including periodic o potential + Coulomb potential of the defect Extremum at the center of BZ Then for small k we have 2 En (k) = k 2m We look for solution of the SE (H0 + U )ψ = Eψ A.3 IMPURITY STATES 459 in the form ψ= Bn (k )φn k (r) , nk where φn k (r) are Bloch states By a usual procedure (multiplication by φ∗ (r) and intenk gration over r) we get the equation nk Un k Bn (k) = [En (k) − E]Bn (k) + nk nk Un k = V u∗ un k ei(k −k)r U (r) dr nk Then, it is natural to assume that B(k) is nonvanishing only near the BZ center, and to replace central cell functions u by their values at k = These function rapidly oscillate within the cell while the rest varies slowly Then within each cell cell u∗ un dr = δnn n0 because Bloch functions are orthonormal Thus, [En (k) − E]Bn (k) + U (kk )Bn (k ) = n U (kk) = V ei(k−bk )r U (r) dr = − 4πe2 V|k − k |2 Finally we get 2 k 4πe2 − E Bn (k) − 2m V k Bn (k ) |k − k |2 where one can integrate over k in the infinite region (because Bn (k) decays rapidly) Coming back to the real space and introducing F (r) = √ V Bn (k)eikr k we come to the SE for a hydrogen atom, − 2m − e2 F (r) = EF (r) r Here e4 m , t = 1, t2 2 F (r) = (πa3 )−1/2 exp(−r/a), a = Et = − For the total wave function one can easily obtain ψ = un0 (r)F (r) The results are summarized in the table /me2 460 APPENDIX A BAND STRUCTURE OF SEMICONDUCTORS Material m/m0 GaAs 12.5 0.066 InP CdTe 12.6 0.08 10 0.1 E1s (th.) E1s (exp.) (meV) (meV) 5.67 Ge:6.1 Si: 5.8 Se: 5.9 S: 6.1 S: 5.9 6.8 7.28 13 13.* Table A.2: Characteristics of the impurity centers Several equivalent extrema Let us consider silicon for example The conduction band minimum is located at kz = 0.85(2π/a) in the [100] direction, the constant energy surfaces are ellipsoids of revolution around [100] There must be equivalent ellipsoids according to cubic symmetry For a given ellipsoid, 2 (k + ky ) 2m 2mt x Here m = 0.916m0 , mt = 0.19m0 According to the effective mass theory, the energy levels are N -fold degenerate, where n is the number of equivalent ellipsoids In real situation, these levels are split due to short-range corrections to the potential These corrections provide inter-extrema matrix elements The results for an arbitrary ration γ = mt /m can be obtained only numerically by a variational method (Kohn and Luttinger) The trial function was chosen in the form E= F = (πa (kz − kz )2 + a2 )−1/2 ⊥ x2 + y z exp − + a2 a ⊥ 1/2 , and the parameters were chosen to minimize the energy at given γ Excited states are calculated in a similar way The energies are listed in table A.3 Material Si (theor.) Si(P) Si(As) Si(Sb) Ge(theor/) Ge(P) Ge(As) Ge(Sb) E1s (meV) E2p0 (meV) 31.27 11.51 45.5 33.9 32.6 11.45 53.7 32.6 31.2 11.49 42.7 32.9 30.6 11.52 9.81 4.74 12.9 9.9 4.75 14.17 10.0 4.75 10.32 10.0 4.7 Table A.3: Donor ionization energies in Ge and Si Experimental values are different because of chemical shift A.3 IMPURITY STATES 461 Impurity levels near the point of degeneracy Degeneracy means that there are t > functions, φj , j = 1, t nk which satisfy Schrădinger equation without an impurity In this case (remember, k ≈ 0), o t Fj (r)φj (br) n0 ψ= j=1 The functions Fj satisfy matrix equation, t αβ Hjj pα pβ + U (r)δjj ˆ ˆ j =1 Fj = EFj (A.1) α,β=1 If we want to include spin-orbital interaction we have to add Hso = [σ × 4mc c2 ˆ V ] · p Here σ is the spin operator while V is periodic potential In general H-matrix is complicated Here we use the opportunity to introduce a simplified (the so-called invariant) method based just upon the symmetry For simplicity, let us start with the situation when spin-orbit interaction is very large, and split-off mode is very far Then we have 4-fold degenerate system Mathematically, it can be represented by a pseudo-spin 3/2 characterized by a pseudo-vector J ˆ There are only invariants quadratic in p, namely p2 and (ˆ · J)2 Thus we have only p two independent parameters, and traditionally the Hamiltonian is written as H= ˆ p2 m0 γ1 + γ − γ(ˆ · J)2 p (A.2) That would be OK for spherical symmetry, while for cubic symmetry one has one more invariant, i p2 Ji2 As a result, the Hamiltonian is traditionally expressed as ˆi H= ˆ p2 m0 γ1 + γ2 − γ3 (ˆ · J)2 + (γ3 − γ2 ) p p2 Ji2 ˆi (A.3) i This is the famous Luttinger Hamiltonian Note that if the lattice has no inversion center there also linear in p terms Now we left with coupled Schrădinger equations (A.1) To check the situation, let us o first put U (r) = and look for solution in the form Fj = Aj (k/k)eikr , k ≡ |k| 462 APPENDIX A BAND STRUCTURE OF SEMICONDUCTORS The corresponding matrix lemenets can be obtained by substitution k instead of the operator p into Luttinger Hamiltonian The Hamiltonian (A.2) does not depend on the ˆ direction of k Thus let us direct k along z axis and use representation with diagonal Jz Thus the system is decoupled into independent equation with two different eigenvalues, E = γ1 + 2γ 2m0 2 k , E = γ1 − 2γ 2m0 2 k If γ1 ± 2γ > both energies are positive (here the energy is counted inside the valence band) and called the light and heavy holes The effective masses are m (h) = m0 /(γ1 ± γ) The calculations for the full Luttinger Hamiltonian (A.3) require explicit form of J-matrices It solutions lead to the anisotropic dispersion law E ,h = 2m0 γ1 k ± γ2 k 2 2 2 2 +12(γ3 − γ2 )(kx ky + ky kz + kz kx ) 1/2 The parameters of Ge and Si are given in the Table A.3 Material γ1 γ2 γ3 ∆ Ge 4.22 0.39 1.44 0.044 11.4 Si 13.35 4.25 5.69 0.29 15.4 Table A.4: Parameters of the Luttinger Hamiltonian for Ge and Si The usual way to calculate acceptor states is variational calculation under the spherical model (A.2) In this case the set of differential equations can be reduced to a system of differential equation containing only parameter, β = m /mh A.3 IMPURITY STATES 463 Figure A.4: Constant energy ellipsoids for Si conduction band There are equivalent valleys resulting in a very large density of states Type I Type III Type II Figure A.5: Various possible band lineups in heterostructures E=E2 +p2/2m E1 E=E1 +p2/2m E2 x-y plne, E vs k z Figure A.6: Subband levels in a quantum well 464 APPENDIX A BAND STRUCTURE OF SEMICONDUCTORS Ec ED EA Ev Figure A.7: band diagram of a semiconductor Appendix B Useful Relations B.1 Trigonometry Relations cos x sin x = cos2 x sin2 x B.2 ix e ± e−ix = (B.1) (1 ± cos x) (B.2) Application of the Poisson summation formula The Poisson formula reads ∞ ϕ(2πN + t) = N =−∞ 2π In our case ϕ(2πN + t) = ∞ eilt ∞ ϕ(τ )e−ilτ dτ −∞ l=−∞ (2πN + t)3/2 3/2 (2π) (B.3) if we replace N → −N and put t = 2πx − π The other difference is that the function ϕ is finite So, if we introduce the maximal integer in the quantity x as [x] we get the limits of the ϕ 1 from 2π x − − x − to t 2 The lower limit we replace by that is good for large x Finally we get the for Φ(x) Φ(x) = (2π)5/2 ∞ t τ 3/2 e−ilτ dτ ilt e l=−∞ 465 466 APPENDIX B USEFUL RELATIONS Then we take into account that for l = t τ 3/2 dτ = t5/2 , and combine the item with ±l As a result, we get (−1)l e2πilx t 3/2 +e−2πilx = (−1)l t τ t τ 3/2 e−ilτ dτ + τ 3/2 eilτ dτ = cos [2πl(x − τ ) dτ ] Then we integrate by parts twice to transform τ 3/2 → τ −1/2 and changing τ → (π/2l)z we get l Φ(x) = (2πx)5/2 + ∞ (−1) (2πlx)1/2− 5/2 l5/2 (2π) √ √ π − sin(2πlx)S 4lx + cos(2πlx)C 4lx Here we have introduced the Fresnel integrals S(u) = C(u) = u sin u cos π x π x dx dx These function oscillate with the period ≈ and with dumping amplitudes; the asymptotic behavior being 0.5 at u → ∞ The non-oscillating term can be easily summed over l: ∞ l=1 (−1)l (2πlx)1/2 = (2πx)1/2 5/2 l ∞ l=1 (−1)l π2 = − (2πx)1/2 l2 12 Appendix C Vector and Matrix Relations Vector A is a column Ai Usually, two kind of vector products are defined: the scalar product AB = A · B = Ai Bi i and the vector product i1 i2 i3 A1 A2 A3 B1 B2 B3 [AB] = [A × B] = det where ii are the unit vectors We have AB = BA, [A × B] = − [B × A] ˆ Matrix A is a table with the elements Aik If Aik = Ai δik ˆ the matrix is called the diagonal, the unit matrix has the elements δik We have ˆA = 1ˆ ˆˆ = A 0-matrix has all the elements equal to 0; (A + B)ik = Aik + Bik ˆ ˆ ˆ A1 Trace of the matrix defined as ˆ Tr A = ˆˆ ˆˆ Tr (AB) = Tr (B A) Aii ; i Matrix dot product is defined as ˆˆ (AB)ik = Ail Blk l ˆˆ ˆˆ ˆ ˆˆ ˆˆ ˆ It is important that AB = B A At the same time, C(B A) = (C B)A The inverse matrix is defined by the relation ˆˆ ˆ ˆ AA−1 = A−1 A = ˆ We have ˆˆ ˆ ˆ (AB)−1 = B −1 A−1 467 468 APPENDIX C VECTOR AND MATRIX RELATIONS The conjugate matrix is defined as ˆ A† ik = A∗ ki ˆ ˆ If A† = A the matrix is called the Hermitian The matrix is called unitary if ˆ ˆ ˆ ˆ A† = A−1 , or A† A = ˆ Sometimes it is useful to introduce a row A† with the elements Ai We get A† A = i A2 i ˆ Eigenvectors u and eigenvalues λ of the matrix A are defined as solutions of matrix equation: ˆ Au = λu or (Aik − λδik )uk = k It is a set of i equations The number of solutions is equal to the matrix’s range They exist only if ˆ det (A − λˆ = 0, 1) ˆ it is just the equation for the eigenvalues λi If the matrix A is Hermitian its eigenvalues are real Bibliography [1] N W Ashkroft, and N D Mermin, Solid State Physics (Holt, Rinehart and Winston, New York, 1976) [2] C Kittel, Quantum Theory of Solids (John Wiley and Sons, New York, 1987) [3] K Seeger “Semiconductor Physics”, Springer (1997) [4] Jasprit Singh, Physics of Semiconductors and their Heterostructures (McGraw-Hill, Inc., New York, 1993) [5] R B Leighton, Principles of Modern Physics (McGraw-Hill, Inc., New York, 1959) [6] H Haug and S W Koch Quantum theory of the optical and electronic properties of semiconductors, World Scintific, 1990 469 ... Problems 9.7 Excitons 9.7.1 Excitonic states in semiconductors 9.7.2 Excitonic effects in optical properties 9.7.3 Excitonic states in quantum wells ... semiconductors 10.1 Impurity states 10.2 Localization of electronic states 10.3 Impurity band for lightly doped semiconductors 10.4 AC conductance due to localized states... Quantization of Atomic Vibrations: Phonons The quantum mechanical prescription to obtain the Hamiltonian from the classical Hamilton function is to replace classical momenta by the operators: ˙ ˆ Qj