Quantum Mechanic Perturbation Theory, Study notes of Quantum Mechanics

Download Quantum Mechanic Perturbation Theory and more Study notes Quantum Mechanics in PDF only on Docsity! G Quantum Mechanical Perturbation Theory Quantum mechanical perturbation theory is a widely used method in solid- state physics. Without the details of derivation, we shall list a number of basic formulas of time-independent (stationary) and time-dependent perturbation theory below. For simplicity, we shall use the Dirac notation for wavefunctions and matrix elements. G.1 Time-Independent Perturbation Theory Assume that the complete solution (eigenfunctions and eigenvalues) of the Schrödinger equation H0 ∣∣ψ(0) i 〉 = E(0) i ∣∣ψ(0) i 〉 (G.1.1) is known for a system described by a simple Hamiltonian H0. If the system is subject to a time-independent (stationary) perturbation described by the Hamiltonian H1 – which can be an external perturbation or the interaction between the components of the system –, the eigenvalues and eigenfunctions change. The method for determining the new ones depends on whether the unperturbed energy level in question is degenerate or not. G.1.1 Nondegenerate Perturbation Theory We now introduce a fictitious coupling constant λ, whose value will be treated as a parameter in the calculations and set equal to unity in the final result, and write the full Hamiltonian H = H0 + H1 as H = H0 + λH1 . (G.1.2) The parameter λ is purely a bookkeeping device to keep track of the relative order of magnitude of the various terms, since the energy eigenvalues and eigenfunctions will be sought in the form of an expansion in powers of λ: 580 G Quantum Mechanical Perturbation Theory∣∣ψi 〉 = ∣∣ψ(0) i 〉 + ∞∑ n=1 λn ∣∣ψ(n) i 〉 , Ei = E(0) i + ∞∑ n=1 λnE (n) i . (G.1.3) The series is convergent if the perturbation is weak, that is, in addition to the formally introduced parameter λ, the interaction Hamiltonian itself contains a small parameter, the physical coupling constant. By substituting this expansion into the Schrödinger equation and collect- ing the same powers of λ from both sides, we obtain H0 ∣∣ψ(0) i 〉 = E(0) i ∣∣ψ(0) i 〉 , H0 ∣∣ψ(1) i 〉 + H1 ∣∣ψ(0) i 〉 = E(0) i ∣∣ψ(1) i 〉 + E(1) i ∣∣ψ(0) i 〉 , H0 ∣∣ψ(2) i 〉 + H1 ∣∣ψ(1) i 〉 = E(0) i ∣∣ψ(2) i 〉 + E(1) i ∣∣ψ(1) i 〉 + E(2) i ∣∣ψ(0) i 〉 (G.1.4) and similar equations for higher-order corrections. The corrections to the en- ergy and wavefunction of any order are related to the lower-order ones by the recursion formula (H0 − E(0) i ) ∣∣ψ(n) i 〉 + (H1 − E(1) i ) ∣∣ψ(n−1) i 〉 − E(2) i ∣∣ψ(n−2) i 〉− . . .− E(n) i ∣∣ψ(0) i 〉 = 0 . (G.1.5) Multiplying the second equation in (G.1.4) (which comes from the terms that are linear in λ) by 〈 ψ (0) i ∣∣ from the left, the first-order correction to the energy is E (1) i = 〈 ψ (0) i ∣∣H1 ∣∣ψ(0) i 〉 . (G.1.6) To determine the correction to the wavefunction, the same equation is multi- plied by 〈 ψ (0) j ∣∣ (j = i): E (0) j 〈 ψ (0) j ∣∣ψ(1) i 〉 + 〈 ψ (0) j ∣∣H1 ∣∣ψ(0) i 〉 = E(0) i 〈 ψ (0) j ∣∣ψ(1) i 〉 . (G.1.7) Since the eigenfunctions of H0 make up a complete set, the functions ∣∣ψ(n) i 〉 can be expanded in terms of them:∣∣ψ(n) i 〉 = ∑ j C (n) ij ∣∣ψ(0) j 〉 . (G.1.8) The coefficients C(n) ii are not determined by the previous equations: their val- ues depend on the normalization of the perturbed wavefunction. Substituting the previous formula into (G.1.7), we have E (0) j C (n) ij + 〈 ψ (0) j ∣∣H1 ∣∣ψ(0) i 〉 = E(0) i C (n) ij , (G.1.9) G.1 Time-Independent Perturbation Theory 583 for the energy correction. In this method the energy denominator contains the perturbed energy Ei rather than the unperturbed one E(0) i . To first order in the interaction, Ei = E(0) i + 〈 ψ (0) i ∣∣H1 ∣∣ψ(0) i 〉 , (G.1.26) while to second order, Ei = E(0) i + 〈 ψ (0) i ∣∣H1 ∣∣ψ(0) i 〉 + ∑ j =i 〈 ψ (0) i ∣∣H1 ∣∣ψ(0) j 〉〈 ψ (0) j ∣∣H1 ∣∣ψ(0) i 〉 Ei − E(0) j + . . . . (G.1.27) It is easy to show that by rearranging the energy denominator and expanding it as 1 Ei −H0 = 1 E (0) i −H0 + ΔEi = 1 E (0) i −H0 ∞∑ n=0 ( −ΔEi E (0) i −H0 )n , (G.1.28) the results of the Rayleigh–Schrödinger perturbation theory are recovered. The formulas of time-dependent perturbation theory can also be used to determine the ground-state energy and wavefunction of the perturbed sys- tem, provided the interaction is assumed to be turned on adiabatically. The appropriate formulas are given in Section G.2. G.1.2 Degenerate Perturbation Theory In the previous subsection we studied the shift of nondegenerate energy levels due to the perturbation. For degenerate levels a slightly different method has to be used because the formal application of the previous formulas would yield vanishing energy denominators. Assuming that the ith energy level of the unperturbed system is p-fold degenerate – that is, the same energy E(0) i belongs to each of the states ∣∣ψ(0) i1 〉 ,∣∣ψ(0) i2 〉 , . . . , ∣∣ψ(0) ip 〉 –, any linear combination of these degenerate eigenstates is also an eigenstate of H0 with the same energy. We shall use such linear combinations to determine the perturbed states. We write the wavefunctions of the states of the perturbed system that arise from the degenerate states as∣∣ψ〉 = ∑ k cik ∣∣ψ(0) ik 〉 + ∑ n =i cn ∣∣ψ(0) n 〉 , (G.1.29) where the cik are of order unity, whereas the other coefficients cn that specify the mixing with the unperturbed eigenstates whose energy is different from E (0) i are small, proportional to the perturbation. By substituting this form into the Schrödinger equation, and multiplying both sides by 〈 ψ (0) ij ∣∣ from the left, 584 G Quantum Mechanical Perturbation Theory ΔEcij = ∑ k 〈 ψ (0) ij ∣∣H1 ∣∣ψ(0) ik 〉 cik + ∑ n =i 〈 ψ (0) ij ∣∣H1 ∣∣ψ(0) n 〉 cn (G.1.30) is obtained. Since the coefficients cn are small, the second term on the right- hand side can be neglected in calculating the leading-order energy correction, which is given by ∑ k [〈 ψ (0) ij ∣∣H1 ∣∣ψ(0) ik 〉− δjkΔE ] cik = 0 . (G.1.31) This homogeneous system of equations has nontrivial solutions if the deter- minant of the coefficient matrix vanishes: det (〈 ψ (0) ij ∣∣H1 ∣∣ψ(0) ik 〉− δjkΔE ) = 0 . (G.1.32) The solutions of this pth-order equation – that is, the eigenvalues of the matrix made up of the matrix elements 〈 ψ (0) ij ∣∣H1 ∣∣ψ(0) ik 〉 – specify the eventual splitting of the initially p-fold degenerate level, i.e., the shift of the perturbed levels with respect to the unperturbed one. Thus the interaction Hamiltonian needs to be diagonalized on the subspace of the degenerate states of H0. In general, the degeneracy is lifted at least partially by the perturbation. As discussed in Appendix D on group theory, the symmetry properties of the full Hamiltonian determine which irreducible representations appear, and what the degree of degeneracy is for each new level. G.2 Time-Dependent Perturbation Theory If the perturbation depends explicitly on time, no stationary states can arise. We may then be interested in the evolution of the system: What states can be reached at time t from an initial state ∣∣ψ(0) i 〉 if the perturbation is turned on suddenly at time t0? The answer lies in the solution of the time-dependent Schrödinger equation[H0 + λH1(t) ]∣∣ψi(t) 〉 = − i ∂ ∂t ∣∣ψi(t) 〉 . (G.2.1) The wavefunction ∣∣ψi(t) 〉 is sought in the form∣∣ψi(t) 〉 = ∑ j cij(t) ∣∣ψ(0) j 〉 e−iE (0) j t/ , (G.2.2) subject to the initial condition cij(t0) = δij . (G.2.3) Since the time dependence of the unperturbed state has been written out explicitly, the functions cij(t) are expected to vary slowly in time. Expanding the coefficients once again into powers of λ, G.2 Time-Dependent Perturbation Theory 585 cij(t) = c(0)ij (t) + ∞∑ r=1 λrc (r) ij (t) , (G.2.4) where, naturally, the zeroth-order term is a constant: c (0) ij (t) = δij . (G.2.5) Substituting this series expansion into the Schrödinger equation, we find −  i ∂ ∂t c (r) ij (t) = ∑ k ei(E (0) j −E (0) k )t/ 〈 ψ (0) j ∣∣H1(t) ∣∣ψ(0) k 〉 c (r−1) ik (t) . (G.2.6) The explicit formulas for the first two terms obtained through iteration are c (1) ij (t) = − i  t∫ t0 〈 ψ (0) j ∣∣H1(t1) ∣∣ψ(0) i 〉 ei(E (0) j −E (0) i )t1/dt1 , (G.2.7) and c (2) ij (t) = ( − i  )2 t∫ t0 dt1 t1∫ t0 dt2 ∑ k 〈 ψ (0) j ∣∣H1(t1) ∣∣ψ(0) k 〉 ei(E (0) j −E (0) k )t1/ ×〈ψ(0) k ∣∣H1(t2) ∣∣ψ(0) i 〉 ei(E (0) k −E (0) i )t2/ (G.2.8) In the interaction picture the time dependence of an arbitrary operator O is given by Ô(t) = eiH0t/Oe−iH0t/ . (G.2.9) Using this form for the Hamiltonian, which may have an intrinsic time de- pendence as well, the first two coefficients c(n) ij can be written in terms of the operators Ĥ1(t) = eiH0t/H1(t)e−iH0t/ (G.2.10) as c (1) ij (t) = − i  t∫ t0 〈 ψ (0) j ∣∣Ĥ1(t1) ∣∣ψ(0) i 〉 dt1 (G.2.11) and c (2) ij (t) = ( − i  )2 t∫ t0 dt1 t1∫ t0 dt2 ∑ k 〈 ψ (0) j ∣∣Ĥ1(t1) ∣∣ψ(0) k 〉 × 〈ψ(0) k ∣∣Ĥ1(t2) ∣∣ψ(0) i 〉 . (G.2.12) Since the intermediate states ∣∣ψ(0) k 〉 constitute a complete set, the previous formula simplifies to 588 G Quantum Mechanical Perturbation Theory in the interaction picture. Using the ground state ∣∣Ψ0 〉 of energy E0 of the unperturbed system, the energy correction due to the perturbation is ΔE = 〈 Ψ0 ∣∣H1S(0,−∞) ∣∣Ψ0 〉〈 Ψ0 ∣∣S(0,−∞) ∣∣Ψ0 〉 , (G.2.28) and the wavefunction is ∣∣Ψ〉 = S(0,−∞) ∣∣Ψ0 〉〈 Ψ0 ∣∣S(0,−∞) ∣∣Ψ0 〉 . (G.2.29) As J. Goldstone (1957) pointed out, the same result may be formu- lated in a slightly different way. Considering a many-particle system with a nondegenerate ground state, the contribution of each term in the perturba- tion expansion can be represented by time-ordered diagrams that show the intermediate states through which the system gets back to the ground state. This representation contains terms in which some of the particles participat- ing in the intermediate processes are in no way connected to the incoming and outgoing particles. It can be demonstrated that the contributions of the disconnected parts are exactly canceled by the denominator in (G.2.28) and (G.2.29), so ΔE = ∞∑ n=0 〈 Ψ0 ∣∣∣H1 ( 1 E0 −H0 H1 )n ∣∣∣Ψ0 〉 con , ∣∣Ψ〉 = ∞∑ n=0 ( 1 E0 −H0 H1 )n ∣∣Ψ0 〉 con, (G.2.30) where the label “con” indicates that only the contribution of connected di- agrams need to be taken into account. It should be noted that instead of Goldstone’s time-ordered diagrams, the perturbation series for the ground- state energy can also be represented in terms of Feynman diagrams, which are more commonly used in the many-body problem. Only connected dia- grams need to be considered in that representation, too. Reference 1. C. Cohen-Tannoudji, B. Diu and F. Laloë, Quantum Mechanics, John Wiley & Sons, New York (1977). H Second Quantization The quantum mechanical wavefunction is most often considered as the func- tion of the space and time variables when the solutions of the Schrödinger equation are sought. In principle, this approach is applicable even when the system is made up of a large number of interacting particles. However, it is then much more convenient to use the occupation-number representation for the wavefunction. We shall introduce the creation and annihilation operators, and express the Hamiltonian in terms of them, too. H.1 Occupation-Number Representation It was mentioned in Chapter 12 on the quantum mechanical treatment of lattice vibrations that the eigenstates of the harmonic oscillator can be char- acterized by the quantum number n that can take nonnegative integer values. Using the linear combinations of the position variable x and its conjugate momentum, it is possible to construct operators a† and a that increase and decrease this quantum number. We may say that when these ladder operators are applied to an eigenstate, they create an additional quantum or annihi- late an existing one. Consequently, these operators are called the creation and annihilation operators of the elementary quantum or excitation. States can be characterized by the number of quanta they contain – that is, by the oc- cupation number. Using Dirac’s notation, the state ψn of quantum number n – which can be constructed from the ground state of the oscillator by the n-fold application of the creation operator a†, and thus contains n quanta – will henceforth be denoted by |n〉. The requirement that such states should also be normalized to unity leads to a|n〉 = √ n |n− 1〉 , a†|n〉 = √ n+ 1 |n+ 1〉 . (H.1.1) The operators a and a† of the quantum mechanical oscillator satisfy the bosonic commutation relation. 590 H Second Quantization This occupation-number representation can be equally applied to many- particle systems made up of fermions (e.g., electrons) or bosons (e.g., phonons, magnons). Any state of an interacting system consisting of N particles can be expanded in terms of the complete set of states of the noninteracting system. The eigenstates of the noninteracting many-particle system can, in turn, be expressed in terms of the one-particle eigenstates. When the one- particle problem is solved for the noninteracting system, the complete set φ1(ξ), φ2(ξ), . . . , φi(ξ), . . . of one-particle states is obtained, where the collec- tive notation ξ is used for the spatial variable r and spin s of the particles: ξ = (r, s). Such complete sets are the set of eigenfunctions for the harmonic oscillator, and the system of plane waves, Bloch functions, or Wannier functions for electrons. The construction of the complete set of many-particle functions from one-particle functions is different for bosons and fermions: for bosons, several particles may be in the same state, whereas this possibility is excluded by the Pauli principle for fermions. The two cases must therefore be treated separately. Bosons For noninteracting bosons the states of the many-particle system are described by means of those combinations of the one-particle states that are completely symmetric with respect to the interchange of the space and spin variables. If an N -particle system contains n1, n2, . . . , nk, . . . particles in the states φ1, φ2, . . . , φk, . . . , where ∑ k nk = N, (H.1.2) the wavefunction with the required symmetry properties is Φn1,n2,...,nk,... = ( n1!n2! . . . nk! . . . N ! )1/2∑ P φp1(ξ1)φp2(ξ2) . . . φpN(ξN) , (H.1.3) where 1, 2, . . . , k, . . . occurs among the indices pi exactly n1, n2, . . . , nk, . . . times, and summation is over all possible permutations of the indices. It turns out practical to introduce a more concise notation. If the wave- functions of the one-particles states are known, the wavefunction Φ is un- ambiguously characterized by the numbers n1, n2, . . . , nk, . . . that specify the occupation of each one-particle state, therefore the above-defined state can be concisely denoted by Φn1,n2,...,nk,... ≡ |n1, n2, . . . , nk, . . . 〉 . (H.1.4) This is the occupation-number representation, while the vector space spanned by the set of all such basis states with nonnegative integers nk for bosons is called the Fock space. H.2 Second-Quantized Form of Operators 593 while for the reverse order of the operators the −1 factors due to antisym- metrization are different: a†k′ak|n1, n2, . . . , nk, . . . , nk′ , . . . 〉 (H.1.21) = (−1)Sk(−1)Sk′−1√nk √ 1 − nk′ |n1, n2, . . . , nk − 1, . . . , nk′ + 1, . . . 〉, and thus [ak, a † k′ ]+ ≡ aka † k′ + a†k′ak = δkk′ , (H.1.22) where [A,B]+ is the anticommutator of the two operators. Likewise, it can be shown that [ak, ak′ ]+ = 0 , [a†k, a † k′ ]+ = 0 . (H.1.23) The state Φ in which the one-particle states of index p1 < p2 < · · · < pN are filled can be written as Φ = a†p1 a†p2 . . . a†pN |0〉 (H.1.24) in terms of the creation operators, where |0〉 is the vacuum state. H.2 Second-Quantized Form of Operators In the discussion of many-particle systems we mostly encounter operators that are the sums of terms acting on individual particles or contain the variables of two particles. The kinetic energy of a system and the interaction with an applied field are examples for the first, while pair interaction between the particles is an example of the second. Below we shall show that the one- and two-particle operators can be expressed in simple forms in terms of the cre- ation and annihilation operators. Equivalence is based on the requirement that their action on the wavefunctions given in occupation-number representation lead to the same matrix elements as the usual representation. H.2.1 Second-Quantized Form of One-Particle Operators We shall first discuss one-particle operators. In complete generality, they can be written as F (1) = N∑ i=1 f(ξi) . (H.2.1) The operator f either leaves the particle in the same state or takes it into another. We shall first consider diagonal matrix elements. For bosons, each particle gives the same contribution because of symmetrization. By selecting a particle and assuming that it is in the state of label l, 594 H Second Quantization∫∫ . . . ∫ Φ∗ n1,n2,...,nk,... N∑ i=1 f(ξi)Φn1,n2,...,nk,... dξ1 dξ2 . . . dξN = N n1!n2! . . . nk! . . . N ! ∑ l ∫ φ∗l (ξ)f(ξ)φl (ξ) dξ (H.2.2) × ∑ P ′ ∫ . . . ∫ φ∗p1 (ξ2) . . . φ∗pN (ξN )φp1 (ξ2) . . . φpN (ξN ) dξ2 . . . dξN . To calculate the factor that remains after the separation of the matrix element of the state l, only those states need to be considered in the permutation P ′ that contain the state of label l only nl−1 times. Owing to the orthonormality of the one-particle states, the value of the previous formula is∑ l nl ∫ φ∗l (ξ)f(ξ)φl(ξ) dξ . (H.2.3) In the off-diagonal terms nonzero matrix elements are obtained between those states Φn1,n2,...,nk,...,nl,... and Φn1,n2,...,nk+1,...,nl−1,... for which the oc- cupation numbers of two one-particle states differ by one unit each. Then∫∫ . . . ∫ Φ∗ n1,n2,...,nk+1,...,nl−1,... ∑ i f(ξi)Φn1,n2,...,nk,...,nl,... dξ1 dξ2 . . . dξN . (H.2.4) Because of the normalization factors of the two wavefunctions the matrix element is proportional to I = ( n1!n2! . . . (nk + 1)! . . . (nl − 1)! N ! )1/2( n1!n2! . . . nk! . . . nl! N ! )1/2 . (H.2.5) Since each particle contributes by the same amount, the matrix element is∫∫ . . . ∫ Φ∗ n1,n2,...,nk+1,...,nl−1,... ∑ i f(ξi)Φn1,n2,...,nk,...,nl,... dξ1 dξ2 . . . dξN = NI ∑ kl ∫ φ∗k(ξ)f(ξ)φl (ξ) dξ (H.2.6) × ∑ P ′ ∫ . . . ∫ φ∗p1 (ξ2) . . . φ∗pN (ξN )φp1 (ξ2) . . . φpN (ξN ) dξ2 . . . dξN . After the separation of the integral for the selected particle, the remaining terms correspond to a state that contains N − 1 particles, with occupation numbers n1, n2, . . . , nk, . . . , nl − 1, . . . . Since there are (N − 1)! n1!n2! . . . nk! . . . (nl − 1)! . . . (H.2.7) H.2 Second-Quantized Form of Operators 595 such states, the separation of the ξ-integral leaves behind a factor √ nk + 1 √ nl, so the matrix element is √ nk + 1 √ nl ∫ φ∗k(ξ)f(ξ)φl (ξ) dξ . (H.2.8) The same expressions are obtained for the diagonal and off-diagonal matrix elements if the states are specified in occupation-number representation, the operator F (1) is chosen as F (1) = ∑ kl a†kfklal , (H.2.9) where fkl = ∫ φ∗k(ξ)f(ξ)φl (ξ) dξ , (H.2.10) and the previously obtained relations for the action of the creation and annihi- lation operators are used in the calculation of the matrix element. Therefore the operator given in (H.2.9), which acts in the Fock space, is the second- quantized form of one-particle operators for bosons. Note that while the sum is over N particles in the first-quantized formula (H.2.1) of the one-particle operator, it is over the quantum numbers of the one-particle states in the second-quantized formula. The intermediate steps are slightly different for fermions, since a Slater determinant wavefunction is specified in terms of the occupation numbers, and the normalization factors are also different – nevertheless the final result is the same: the one-particle operators for fermions can again be represented as (H.2.9) in terms of creation and annihilation operators. H.2.2 Second-Quantized Form of Two-Particle Operators This approach can be extended to the two-body interaction term in the Hamil- tonian and similar operators that are the sums of terms containing the coor- dinates of two particles: F (2) = ∑ ij f(ξi, ξj) . (H.2.11) Since the variables of two particles appear in each term, such an operator has a nonvanishing matrix element only between states for which the occupation numbers of at most four one-particle states change: two decrease and two others increase by one. The matrix element to be evaluated is thus∫∫ . . . ∫ Φ∗ n1,...,nk+1,...,nl+1,...,nm−1,...,nn−1,... ∑ ij f(ξi, ξj) Φn1,...,nk,...,nl,...,nm,...,nn,... dξ1 dξ2 . . . dξN . (H.2.12) 598 H Second Quantization H = ∑ kl Hklc † kcl + 1 2 ∑ klmn U (2) klmnc † kc † l cmcn , (H.2.25) where Hkl = ∫ φ∗k(ξ) ( −  2 2me ∇2 + U(r) ) φl (ξ) dξ , (H.2.26) and U (2) klmn = ∫∫ dξ dξ′ φ∗k(ξ)φ∗l (ξ ′)U (2)(r, r′)φm(ξ′)φn(ξ) . (H.2.27) In general, the states are chosen in such a way that the one-particle part be diagonal. This is the case when the Bloch functions determined in the presence of a periodic potential are used as a complete basis set. However, this is not the only option. In the Hubbard model the Wannier states are used, and so the one-particle term in the Hamiltonian that describes the hopping of electrons between lattice points is not diagonal. Using the field operators instead of the creation and annihilation operators, H = ∫ ψ̂†(ξ) ( −  2 2me ∇2 + U(r) ) ψ̂(ξ) dξ + ∫∫ dξ dξ′ ψ̂†(ξ)ψ̂†(ξ′)U (2)(r, r′)ψ̂(ξ′)ψ̂(ξ) . (H.2.28) Writing out the spin variable explicitly, the spin independence of the potential and of the interaction implies H = ∑ σ ∫ ψ̂† σ(r) ( −  2 2me ∇2 + U(r) ) ψ̂σ(r) dr + ∑ σσ′ ∫∫ dr dr′ ψ̂† σ(r)ψ̂† σ′(r′)U (2)(r, r′)ψ̂σ′(r′)ψ̂σ(r) . (H.2.29) The description is highly simplified by choosing the plane waves as the complete set. The one-particle states are then characterized by the wave vector k and the spin quantum number σ. The usual formula Hkin = − ∑ i  2 2me ∂2 ∂r2 i (H.2.30) for the kinetic energy can be rewritten in second-quantized form as Hkin = ∑ kk′σσ′ c†kσHσσ′(k,k′)ck′σ′ , (H.2.31) where H.2 Second-Quantized Form of Operators 599 Hσσ′(k,k′) = 1 V ∫ dr e−ik·r ( −  2 2me ) ∂2 ∂r2 eik′·rδσσ′ =  2k2 2me δkk′δσσ′ , (H.2.32) and so Hkin = ∑ kσ  2k2 2me c†kσckσ . (H.2.33) The second-quantized form of the one-particle potential U(r) contains the Fourier transform of the potential: HU = 1 V ∑ kk′σ U(k − k′)c†kσck′σ = 1 V ∑ kq U(q)c†k+qσckσ . (H.2.34) For a spin-independent two-particle interaction U (2)(ri − rj) the second- quantized form is Hint = 1 2 ∑ k1k2k3k4 σσ′ U (2)(k1,k2,k3,k4)c † k1σc † k2σ′ck3σ′ck4σ , (H.2.35) where U (2)(k1,k2,k3,k4) = 1 V 2 ∫ dr1 ∫ dr2e−ik1·r1e−ik2·r2 × U (2)(r1 − r2)eik3·r2eik4·r1 = 1 V 2 ∫ dr1 ∫ dr2e−ik1·r1e−ik2·r2 (H.2.36) × 1 V ∑ q U (2)(q)eiq·(r1−r2)eik3·r2eik4·r1 = 1 V ∑ q U (2)(q)δk1,k4+qδk2,k3−q . By renaming the indices, the interaction term can be written as Hint = 1 2V ∑ kk′q σσ′ U (2)(q)c†k+qσc † k′−qσ′ck′σ′ckσ . (H.2.37) If the one-particle periodic potential is taken into account by using Bloch states instead of plane waves, and the corresponding creation and annihilation operators c†nkσ and cnkσ, then the entire one-particle part of the Hamiltonian – the kinetic energy plus the one-particle potential – can be diagonalized. This leads to Hkin + HU = ∑ nkσ εnkc † nkσcnkσ , (H.2.38) 600 H Second Quantization where εnk is the energy of Bloch electrons in the presence of the periodic potential. The interaction is not restricted to electrons in the same band. Electrons from different bands can be scattered to other bands provided the quasimomentum is conserved to within an additive reciprocal-lattice vector. H.2.5 Number-Density and Spin-Density Operators In the first-quantized formulation the number density of spinless particles is given by n(r) = ∑ l δ(r − rl) . (H.2.39) Its Fourier transform is n(q) = ∫ dr n(r)e−iq·r = ∑ l ∫ dr δ(r − rl)e−iq·r = ∑ l e−iq·rl . (H.2.40) In terms of plane-wave-creation and -annihilation operators, the general rule for one-particle operators implies n(q) = ∑ k,k′ c†kn(k,k ′)ck′ , (H.2.41) where n(k,k′) = 1 V ∫ dr e−ik·re−iq·reik′·r = δk′,k+q . (H.2.42) A part of the sum can then be evaluated; it yields n(q) = ∑ k c†kck+q . (H.2.43) Using an inverse Fourier transform it can be shown that the density operator in real space can be expressed particularly simply in terms of the field operator: n(r) = ψ̂†(r)ψ̂(r) . (H.2.44) For particles with spin, an additional sum over the spin quantum number appears: n(q) = ∑ kσ c†kσck+qσ , (H.2.45) while the number-density operator is given in real space by n(r) = ∑ σ ψ̂† σ(r)ψ̂σ(r) . (H.2.46) We can now show that the field operator ψ̂† σ(r) indeed adds a spin-σ particle to the system at r. To this end, we shall rewrite the operator n(r) as I Canonical Transformation Instead of tackling the quantum mechanical eigenvalue problem directly, it is often more practical to perform a unitary canonical transformation on the Hamiltonian that leaves the energy spectrum unaltered. This can be achieved either by transforming away some degrees of freedom, and generating an ef- fective interaction among the remaining ones, or by transforming the Hamil- tonian directly to a diagonal form. Below we shall present both approaches. I.1 Derivation of an Effective Hamiltonian It is a recurrent situation in solid-state physics that a system is made up of two distinct parts whose components interact but we are interested only in the properties of one subsystem. The effects of the other subsystem – i.e., its degrees of freedom (or at least some of them) – can then be transformed away by means of a canonical transformation. This is the case for an interacting system of electrons and phonons when the effective interaction between the electrons mediated by the phonons is studied, as discussed in Chapter 23. Below we shall first treat the method of canonical transformation generally, and then present some other applications as well. I.1.1 General Formulation of the Problem By separating an unperturbed part H0 – whose energy eigenstates can be calculated exactly – from the interaction part Hint of the Hamiltonian, and by formally introducing a coupling constant λ, the Hamiltonian can be written in the generic form H = H0 + λH1 . (I.1.1) We shall now demonstrate that by means of a canonical transformation H̃ = eSHe−S , (I.1.2) 604 I Canonical Transformation where S† = −S (I.1.3) because of the unitarity of the transformation, the effects of the perturbation on the space of eigenstates of H0 can be taken into account by an equivalent interaction term instead of H1. When an arbitrary operator O and its transform Õ are considered, the series expansion of the unitary operator exp(±S) and the subsequent rear- rangement of the terms of the same powers of S gives Õ = eSOe−S = O + [S,O] + 1 2 [S, [S,O]] + 1 3! [S, [S, [S,O]]] + . . . . (I.1.4) Applying this formula to the Hamiltonian (I.1.1), H̃ = H0+λH1+[S,H0]+λ [S,H1]+ 1 2 [S, [S,H0]]+ 1 2λ [S, [S,H1]]+. . . . (I.1.5) The direct interaction term H1 can be eliminated by a suitable choice of S by requiring that λH1 + [S,H0] = 0 . (I.1.6) The operator S is thus proportional to λ. Eliminating H1 from the transformed Hamiltonian (I.1.5) by means of this equation, H̃ = H0 − 1 2 [S, [S,H0]] − 1 3 [S, [S, [S,H0]]] + . . . (I.1.7) to third order in the coupling constant. The Hamiltonian of the effective in- teraction is thus Heff = − 1 2 [S, [S,H0]] − 1 3 [S, [S, [S,H0]]] + . . . . (I.1.8) Alternatively, it can be written as Heff = 1 2 [S, λH1] + 1 3 [S, [S, λH1]] + . . . . (I.1.9) In most cases only the first (leading) term is taken into account. As we shall see, the operator S generating the canonical transformation can sometimes be given explicitly. In other cases we shall content ourselves with specifying the matrix elements of the transformed Hamiltonian between any initial and final states (|i〉 and |f〉, of energy Ei and Ef ) of the unperturbed system. By keeping only the first term on the right-hand side of (I.1.9) and inserting a complete set of intermediate states by making use of the property∑ j |j〉〈j| = 1, 〈f |Heff|i〉 = 1 2 ∑ j [〈f |S|j〉〈j|λH1|i〉 − 〈f |λH1|j〉〈j|S|i〉] . (I.1.10) Taking the matrix elements of (I.1.6) between the intermediate states, 〈j|λH1|j′〉 + 〈j|SH0 −H0S|j′〉 = 0 . (I.1.11) I.1 Derivation of an Effective Hamiltonian 605 If these states are eigenstates of the unperturbed system with an energy Ej then the previous equation implies 〈j|S|j′〉 = 〈j|λH1|j′〉 Ej − Ej′ . (I.1.12) By substituting this form of the matrix element into (I.1.10), and taking into account that the initial and final states are also eigenstates of the unperturbed Hamiltonian, we find 〈f |Heff|i〉 = 1 2 ∑ j 〈f |λH1|j〉〈j|λH1|i〉 [ 1 Ef − Ej − 1 Ej − Ei ] . (I.1.13) When elastic transitions are considered, and the common energy Ei = Ef is denoted by E0, 〈f |Heff|i〉 = − ∑ j 〈f |λH1|j〉〈j|λH1|i〉 Ej − E0 . (I.1.14) It is often not necessary to know these matrix elements over the entire Hilbert space of the system’s states; using physical considerations it may be sufficient to know them over a subspace. It is then often possible to find explicitly an effective Hamiltonian that gives the same matrix elements in that subspace. In Chapter 23 we showed how the effective electron–electron interaction can be derived from the electron–phonon interaction. Below we shall first derive the effective interaction between magnetic moments in an electron sys- tem, and then demonstrate that even the interaction between the magnetic moment and the electron system can be considered as an effective interaction, and obtained from the Anderson model that describes the interaction between conduction electrons and d-electrons that are “bound” to the atom. I.1.2 RKKY Interaction It was mentioned in Chapter 14 that a localized spin S1 placed in a system of free electrons interacts with them through its magnetic moment, and – provided S1 is fixed – it can polarize the electron system around itself. If a second spin S2 is placed at a distance r from the first, its orientation will not be arbitrary but determined by the local value of the spin density generated by the first spin. Since in reality the first spin is not fixed, interactions mediated by the mobile electrons may eventually lead to processes in which the two localized spins of magnitude S mutually flip each other. By choosing the kinetic energy of the mobile electrons as the unperturbed Hamiltonian, and the interaction between the conduction electrons and the localized spins as a perturbation, the canonical transformation is chosen in such a way that this direct interaction is replaced by an effective interaction between the two spins. 608 I Canonical Transformation Allowing all possible orientations for the spin of the electron–hole pair (and thus for the localized spin), the total contribution is 〈f |Heff|i〉 = − ∑ kk′ f0(εk)[1 − f0(εk′)] εk′ − εk ( J V )2∑ ll′ ei(k−k′)·(Rl′−Rl) × ∑ {M ′′ l } [ 2〈{M ′ l}|Sz l |{M ′′ l }〉〈{M ′′ l }|Sz l′ |{Ml}〉 +〈{M ′ l}|S+ l |{M ′′ l }〉〈{M ′′ l }|S− l′ |{Ml}〉 (I.1.25) +〈{M ′ l}|S− l |{M ′′ l }〉〈{M ′′ l }|S+ l′ |{Ml}〉 ] . Evaluating the sum for the complete set of intermediate spin states, 〈f |Heff|i〉 = − ( J V )2∑ ll′ ∑ kk′ f0(εk)[1 − f0(εk′)] εk′ − εk ei(k−k′)·(Rl′−Rl) ×2〈{M ′ l}| [ Sz l S z l′ + 1 2 ( S+ l S − l′ + S− l S + l′ )] |{Ml}〉. (I.1.26) This can be considered as the matrix element of the operator H = − ∑ ll′ J(Rl − Rl′)Sl · Sl′ , (I.1.27) thus indirect exchange can be described in terms of an effective Hamiltonian that has the same form as the Hamiltonian of direct exchange. To determine its strength, the notation r = R1 − R2 is introduced, and the sum I = ( 1 V )2∑ kk′ f0(εk)[1 − f0(εk′)] εk′ − εk e−i(k−k′)·r (I.1.28) has to be evaluated. Replacing the sum by an integral, the angular integrals are readily calculated: I = 1 (2π)6 kF∫ 0 k2 dk ∞∫ kF k′2 dk′ 1 εk′ − εk (I.1.29) ×2π π∫ 0 sin θ dθ e−ikr cos θ 2π π∫ 0 sin θ′ dθ′eik′r cos θ′ = − 4 (2π)4 kF∫ 0 k2 dk ∞∫ kF k′2 dk′ 1 εk′ − εk sin kr kr sin k′r k′r . Using the quadratic dispersion relation valid for free electrons and the nota- tions κ = kr and κ′ = k′r, we have I.1 Derivation of an Effective Hamiltonian 609 I = − me 22π4 1 r4 kFr∫ 0 κ2 dκ ∞∫ kFr κ′2 dκ′ 1 κ′2 − κ2 sinκ κ sinκ′ κ′ . (I.1.30) The κ′-integral is not affected significantly by shifting the lower limit of inte- gration to κ′ = 0 but, in order to avoid the singularity arising from κ′ = κ, the principal value of the integral needs to be taken. By considering the integral K = P ∞∫ 0 κ′2 dκ′ 1 κ′2 − κ2 sinκ′ κ′ (I.1.31) separately, the even character of the integrand implies K = 1 2P ∞∫ −∞ κ′2 dκ′ 1 κ′2 − κ2 sinκ′ κ′ = 1 4i P ∞∫ −∞ dκ′ [ κ′eiκ′ κ′2 − κ2 − κ′e−iκ′ κ′2 − κ2 ] . (I.1.32) The principal-value integrals can be determined by using the complex variable κ′± iη instead of κ′ (where η is an infinitesimal quantity), and performing the integral in the complex plane, along the contour shown in Fig. I.1. By using the variable κ′ +iη in the first term, the poles are in the lower half-plane, and the integration contour is closed in the upper half-plane. The opposite is done in the second term. 1 'i 1 'i 1 'i 1 'i 1’ plane 1’ plane Fig. I.1. The integration contours used for the two terms in the integrand of K 610 I Canonical Transformation Making use of the relation 1 x± iη = P 1 x ∓ iπδ(x) , (I.1.33) we have K = 1 2π cosκ . (I.1.34) Substituting this back into (I.1.30), I = − me 42π3 1 r4 kFr∫ 0 dκκ sinκ cosκ = − m 162π3 1 r4 ( sin 2κ− 2κ cos 2κ )∣∣∣kFr 0 = −mek 4 F 2π3 sin 2kFr − 2kFr cos 2kFr( 2kFr )4 . (I.1.35) By collecting all factors, the effective interaction between two localized spins can be written as Heff = −2J(r)S1 · S2 , (I.1.36) with an effective exchange constant J(r) = meJ 2k4 F 2π3 F (2kFr) , (I.1.37) where the function F (x) is defined by F (x) = x cosx− sinx x4 . (I.1.38) This is the RKKY interaction. The same result is obtained when the integral I is evaluated by another method. Using the variable k′ = k+q but neglecting once again the restriction imposed on q by the Pauli exclusion principle, I = 2me 2 ( 1 V )2∑ q ∑ |k|<kF 1 |k + q|2 − k2 e−iq·r , (I.1.39) which is the Fourier transform of the formula given in (C.2.32). After per- forming the angular integral, we may change to a complex variable again. The previous result is then recovered through integration along the cuts of the logarithmic function. It should be noted that the generator S of the transformation can be determined explicitly in this case. Since[ c†k′αckβ , ∑ k′′σ εk′′c†k′′σck′′σ ] = ( εk′ − εk ) c†k′αckβ , (I.1.40) it is straightforward to show that I.2 Diagonalization of the Hamiltonian 613 where the operators ak, a†k, b−k, and b†−k satisfy bosonic commutation re- lations. A similar situation is encountered in the Bogoliubov treatment of superfluidity2 (which we shall not discuss), and in Chapter 32, where the excitations of the one-dimensional Luttinger liquid is studied by means of bosonic density fluctuations. A very similar but fermionic problem is encountered in Chapter 34 on the BCS theory of superconductivity, where the eigenstates of the Hamiltonian HBCS = E0 + ∑ k ξk ( c†k↑ck↑ + c†−k↓c−k↓ ) − ∑ k ( Δkc † k↑c † −k↓ + Δ∗ kc−k↓ck↑ ) (I.2.2) are sought. For bosonic and fermionic systems alike, we shall use the Hamil- tonian H = E0 + ∑ k [ εk ( a†kak + b†−kb−k ) + γk ( akb−k + b†−ka † k )] , (I.2.3) which is bilinear in the creation and annihilation operators, and demonstrate how it can be diagonalized by means of a canonical transformation H̃ = eSHe−S . (I.2.4) Since the canonical transformation does not change the eigenvalues, the energy spectrum can be read off immediately from the diagonal form. I.2.1 Bosonic Systems We shall first consider a bosonic system, and show that diagonalization can be achieved by the choice S = ∑ k θk ( b†−ka † k − akb−k ) , (I.2.5) where θk is real. Performing the canonical transformation for each term of the Hamiltonian (I.2.3), H̃ = E0 + ∑ k [ εk ( ã†kãk + b̃†−kb̃−k ) + γk ( ãkb̃−k + ã†kb̃ † −k )] , (I.2.6) where ã†k = eSa†ke−S , ãk = eSake−S , (I.2.7) and b̃†−k and b̃−k are defined by similar formulas. 2 N. N. Bogoliubov, 1947. 614 I Canonical Transformation Applying the expansion (I.1.4) to ã†k, and using the relations [S, a†k] = −θkb−k , [S, ak] = −θkb†−k , [S, b†−k] = −θkak , [S, b−k] = −θka†k (I.2.8) that follow from the explicit form of the generator S and the bosonic commu- tation relations, the repeated application of the commutators yields ã†k = a†k − θkb−k + 1 2θ 2 ka † k − 1 3!θ 3 kb−k + . . . = cosh θk a † k − sinh θk b−k . (I.2.9) Likewise, it can be proved that ãk = cosh θk ak − sinh θk b † −k , b̃†−k = cosh θk b † −k − sinh θk ak , b̃−k = cosh θk b−k − sinh θk a † k . (I.2.10) Inserting these formulas into the canonically transformed Hamiltonian, H̃ = E0 + ∑ k { εk [ (cosh θka † k − sinh θk b−k)(cosh θk ak − sinh θk b † −k) + (cosh θk b † −k − sinh θk ak)(cosh θk b−k − sinh θk a † k) ] (I.2.11) + γk [ (cosh θk ak − sinh θk b † −k)(cosh θk b−k − sinh θk a † k) + (cosh θk a † k − sinh θk b−k)(cosh θk b † −k − sinh θk ak) ]} . The off-diagonal terms vanish if − 2εk sinh θk cosh θk + γk ( cosh2 θk + sinh2 θk ) = 0 . (I.2.12) The solution of this equation is cosh2 θk = 1 2 ( εk√ ε2k − γ2 k +1 ) , sinh2 θk = 1 2 ( εk√ ε2k − γ2 k −1 ) . (I.2.13) The remaining diagonal Hamiltonian reads H̃ = E0 + ∑ k ωk ( a†kak + b†−kb−k + 1 ) , (I.2.14) where ωk = εk ( cosh2 θk + sinh2 θk )− 2γk sinh θk cosh θk = √ ε2k − γ2 k . (I.2.15) I.2 Diagonalization of the Hamiltonian 615 Instead of this method, the reverse approach is usually applied. An inverse canonical transformation is performed on the operators, by introducing αk = e−SakeS = cosh θk ak + sinh θk b † −k , α† k = e−Sa†keS = cosh θk a † k + sinh θk b−k , βk = e−Sb−keS = cosh θk b−k + sinh θk a † k , β†k = e−Sb†−keS = cosh θk b † −k + sinh θk ak . (I.2.16) In terms of them, the original Hamiltonian becomes diagonal, H = E0 + ∑ k ωk ( α† kαk + β†−kβ−k + 1 ) . (I.2.17) I.2.2 Fermionic Systems The same procedure can be applied to fermions – moreover, the formula (I.2.5) for the generator S of the transformation can be used without any modifica- tions. The anticommutation relations for fermions then yield [S, a†k] = θkb−k , [S, ak] = θkb † −k , [S, b†−k] = −θkak , [S, b−k] = −θka†k , (I.2.18) hence ã†k = a†k + θkb−k − 1 2θ 2 ka † k − 1 3!θ 3 kb−k + . . . = cos θk a † k + sin θk b−k , (I.2.19) and ãk = cos θk ak + sin θk b † −k , b̃†−k = cos θk b † −k − sin θk ak , b̃−k = cos θk b−k − sin θk a † k . (I.2.20) The Hamiltonian can be diagonalized if 2εk sin θk cos θk − γk ( cos2 θk − sin2 θk ) = 0 , (I.2.21) which implies cos2 θk = 1 2 ( 1 + εk√ ε2k + γ2 k ) , sin2 θk = 1 2 ( 1 − εk√ ε2k + γ2 k ) , (I.2.22) and the new eigenvalues are given by Ek = εk ( cos2 θk − sin2 θk ) + 2γk sin θk cos θk = √ ε2k + γ2 k . (I.2.23) Just like for bosons, the inverse procedure is usually followed for fermions, too, as in Chapter 34 on superconductivity: the Hamiltonian is diagonalized in terms of the new creation and operation operators that are linear combinations of the original operators. Figure Credits 619 Fig. 25.8 W. J. Turner and W. E. Reese, Phys. Rev. 127, 126 (1962), Fig. 4 Fig. 25.10 J. C. Phillips, Solid State Physics, Vol. 18, Academic Press, New York (1966) Fig. 26.1 H. Kamerlingh Onnes, Comm. Phys. Lab. Univ. Leiden, No. 120b (1911) Fig. 26.2 M. K. Wu et al., Phys. Rev. Lett. 58, 908 (1987), Fig. 3 Fig. 26.3 L. H. Palmer and M. Tinkham, Phys. Rev. 165, 588 (1968), Fig. 3 Fig. 26.6 B. S. Deaver, Jr. and W. M. Fairbank, Phys. Rev. Lett. 7, 43 (1961), Fig. 2 (upper panel) Fig. 26.12 N. E. Phillips, Phys. Rev. 114, 676 (1959), Fig. 4 Fig. 26.14 I. Giaever and K. Megerle, Phys. Rev. 122, 1101 (1961), Figs. 7 and 8 Fig. 26.24 H. F. Hess, R. B. Robinson, R. C. Dynes, J. M. Valles, Jr., and J. V. Waszczak, Phys. Rev. Lett. 62, 214 (1989), Fig. 2 Fig. 26.27(b) R. C. Jaklevic, J. Lambe, J. E. Mercereau, and A. H. Silver Phys. Rev. 140, A1628 (1965), Fig. 1 Fig. 26.29 C. C. Grimes and S. Shapiro, Phys. Rev. 169, 397 (1968), Fig. 1 Fig. 26.33 D. E. Langenberg et al., Proc. IEEE 54, 560 (1966) Fig. 26.36 R. C. Jaklevic, J. Lambe, J. E. Mercereau, and A. H. Silver, Phys. Rev. 140, A1628 (1965), Fig. 7 Fig. 27.37(a) B. J. van Wees et al., Phys. Rev. Lett. 60, 848 (1988), Fig. 2 Fig. 27.37(b) B. J. van Wees et al., Phys. Rev. B 43, 12431 (1991), Fig. 6 Fig. 27.40 U. Meirav, M. A. Kastner, and S. J. Wind, Phys. Rev. Lett. 65, 771 (1990), Figs. 2 and 5 Fig. 27.41(a) J. H. F. Scott-Thomas, S. B. Field, M. A. Kastner, H. I. Smith, and D. A. Antoniadis, Phys. Rev. Lett. 62, 583 (1989), Fig. 2 (top panel) Fig. 27.41(b) U. Meirav, M. A. Kastner, M. Heiblum, and S. J. Wind, Phys. Rev. B 40, 5871 (1989), Fig. 1(a) Fig. 27.42 S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990) Name Index Page numbers in italics refer to Volume 1: Structure and Dynamics. Abrikosov, A. A., 4, 483, 493 Andersen, O. K., 177 Anderson, P. W., 6, 199, 466, 585 Azbel, M. I., 263 Baker, H. F., 441 Bardeen, J., 3, 4, 517 Becquerel, H., 52 Bednorz, J. G., 6, 466 Beer, A., 416 Berezinskii, V. L., 555 Bernal, J. D., 22 Bethe, H., 3, 564, 642 Binnig, G., 270 Blech, I., 3, 315 Bloch, F., 3, 65, 186, 509, 523, 528, 140, 333, 392 Bogoliubov, N. N., 542, 613 Bohr, N., 20 Boltzmann, L., 1, 4, 364 Born, M., 88, 185, 329 Bose, S. N., 396 Bouckaert, L. P., 647 Bouguer, P., 416 Bragg, W. H., 2, 241 Bragg, W. L., 2, 241, 242 Brattain, W. H., 4, 517 Bravais, A., 113 Bridgman, P. W., 64 Brillouin, L., 3, 123, 436 Brockhouse, B. N., 439 Buckminster Fuller, see Fuller, R. Buckminster Burgers, J. M., 285, 286 Cahn, J. W., 3, 315 Casimir, H. B. G., 474 Chambers, R. G., 373, 479 Chandrasekhar, S., 24 Clausius, R. J. E., 375 Cohen, M. H., 163 Coleman, S., 201 Compton, A. H., 188 Cooper, L. N., 4 Corbino, O. M., 62 Curie, M., 52 Curie, P., 52 Curl, R. F., Jr., 30 Cutler, M., 57 Darwin, C. G., 38, 259 Das, B., 576 Datta, S., 576 Debye, P., 267, 389, 442, 521 de Haas, W. J., 45, 314, 327 Dingle, R. B., 326 Dirac, P. A. M., 32, 37, 92, 464, 2 Dorda, G., 6, 405 Dresselhaus, G., 84 Drude, P., 1, 1 Dulong, P. L., 384 622 Name Index Dushman, S., 543 Dyson, F. J., 531 Dzyaloshinsky, I. E., 4 Eckart, C., 53 Einstein, A., 45, 387, 396, 190, 538 Esaki, L., 556 Escher, M. C., 115 Ettingshausen, A. v., 63 Evjen, H. M., 84 Ewald, P. P., 84, 120, 259, 264 Faigel, G., 252 Faraday, M., 513 Fedorov, J. S., 166 Fermi, E., 71, 2 Fert, A., 574 Feynman, R. P., 532, 502 Fibonacci, 317 Fick, A., 537 Fock, V., 291 Fourier, J. B. J., 9 Frank, F. C., 290, 297 Franz, R., 10 Frenkel, J., 283, 470 Fresnel, A. J., 418 Friedel, G., 24 Friedel, J., 71 Friedrich, W., 2, 241 Fröhlich, H., 355 Fuller, R. Buckminster, 29 Gantmakher, V. F., 270 Gell-Mann, M., 646 Gerber, Ch., 270 Ghosh, D. K., 577 Giaever, I., 460 Ginzburg, V. L., 4, 483 Goldschmidt, V. M., 236 Goldstone, J., 200, 588 Gorkov, L. P., 4, 485 Gorter, C. J., 474 Gossard, A. C., 6 Gratias, D., 3, 315 Griffith, R. B., 497 Grünberg, P., 574 Grüneisen, E., 419, 425, 392 Gutzwiller, M. C., 5 Hagen, E., 425 Haldane, F. D. M., 579 Hall, E. H., 11 Harrison, W. A., 115 Hausdorff, F., 441 Haüy, R.-J., 109 Heine, V., 163 Heisenberg, W., 3, 92, 464, 470 Heitler, W., 90 Hermann, C., 125 Herring, C., 158 Hertz, H., 190 Higgs, P. W., 201 Hofstadter, D. R., 304 Holstein, T., 530, 349 Hooke, R., 365 Hopfield, J. J., 432 Hubbard, J., 5 Hund, F., 42, 96 Hückel, E., 521 Ising, E., 472 Jahn, H., 353 Jordan, P., 533 Josephson, B. D., 497, 501 Joule, J. P., 15 Kadanoff, L. P., 498 Kamerlingh Onnes, H., 2, 449 Kaner, E. A., 263, 268 Kármán, T. von, 185 Kasuya, T., 466 Kelvin, Lord, 59 Kerr, J., 513 Kittel, C., 465, 466 Kleinman, L., 160 Klitzing, K. von, 6, 405 Knight, W. D., 72 Knipping, P., 2, 241 Kohn, W., 168, 351 Kondo, J., 5, 394 Korringa, J., 71, 168 Kosevich, A. M., 323 Kosterlitz, J. M., 555 Kramers, H. A., 64, 182, 466 Kronig, R., 64 Kroto, Sir H. W., 30 Subject Index Page numbers in italics refer to Volume 1: Structure and Dynamics. A1 structure, 204, 217 A2 structure, 204, 213 A3 structure, 204, 226 A3′ structure, 204, 226 A4 structure, 204, 221 A8 structure, 204, 231, 232 A9 structure, 204, 234 A15 structure, 204, 209, 464 abrupt junction, 526 absorption in ionic crystal, 430 of light, 413 two-phonon, 445 absorption coefficient, 416 absorption edge, 269, 442 absorption index, 416 absorption region, 424 AC conductivity, 14, 378 acceptor, 220 acceptor level, 223 thermal population of, 227 accidental degeneracy lifting of, 126 acoustic branch, 343 acoustic vibrations, 340, 360–361 actinoids, 179, 218 adiabatic approximation, 332 adiabatic decoupling, 330 AFM, see atomic force microscope Ah structure, 204, 207 AlFe3 structure, 220 alkali halides, 76, 83, 219, 282, 200, 349 alkali metals, 213, 2, 3, 22, 41, 75, 91, 178, 423 alkaline-earth metals, 213, 218, 3, 91, 182 almost periodic functions, 311 aluminum band structure of, 84 Fermi surface of, 184 amorphous materials, 21, 303 amorphous semiconductors, 200 Anderson insulators, 92 Anderson model, 611 angle-resolved photoemission spectroscopy, 192 angular frequency, 339 angular momentum, 665–672 angular wave number, 25 anharmonicity, 421 anisotropy ∼ constant, 507 magnetic, 471–473, 504–513 annihilation operator, 589 of antiferromagnetic magnon, 542 of Bloch electron, 92 of electron state, 31 of magnon, 525 of phonon, 394–395 of Wannier state, 103 anomalous dimension, 499 antibonding state, 98 antiferromagnetic ground state, see ground state, of antiferromagnet 626 Subject Index antiferromagnetic materials, 453–459 antiferromagnetic ordering Néel type absence of in one-dimensional systems, 551 presence of in two-dimensional systems, 551 antiferromagnetism, 453–459 in the Heisenberg model, 464 antiferromagnon, see magnons, antiferromagnetic antiperiodic boundary conditions, see boundary conditions, antiperiodic antiphase boundary, 300, 301 antiphase domains, 301 antisite defects, 283 anti-Stokes component, 434, 446 antiunitary operator, 182 APW method, 165 ARPES, 192 atomic force microscope, 270 atomic form factor, 249 atomic packing factor, see packing fraction atomic-sphere approximation, 168 augmented-plane-wave method, 164, 165 avalanche breakdown, 554 avalanche effect, 556 Azbel–Kaner resonance, 262 B1 structure, 204, 219 B2 structure, 204, 207, 208 B3 structure, 204, 221 B4 structure, 204, 228 B8 structure, 204, 227, 228 Baker–Hausdorff formula, 441 band gap, 89 band index, 81 band structure, 78 calculation of, 151 experimental study of, 187 in empty lattice, 110 of aluminum, 84 of calcium, 183 of compound semiconductors, 208 of copper, 181 of diamond, 89 of germanium, 206 of lead, 185 of magnesium, 183 of rare-earth metals, 186 of semiconductors, 201 of silicon, 204 of sodium, 178 of transition metals, 185 barrier layer, see depletion layer basis, 114 BCS theory, 4 BEDT-TTF, 309 β-(BEDT-TTF)2I3, 327 (BEDT-TTF)2I3, 386 benzene-hexa-n-alkanoate, 24 Berezinskii–Kosterlitz–Thouless phase, 556 Berezinskii–Kosterlitz–Thouless transition, 555 critical exponent in, see critical exponent, in Berezinskii– Kosterlitz–Thouless transition Bernal model, 22, 23, 304 Bernoulli numbers, 618 Bessel functions, 620 Bethe ansatz, 564–566 biased junction, 541 biaxial nematic phase, 26 BiF3 structure, 220 bipartite lattice, 520 bipolar transistor, see transistor, bipolar BIS, 192 BKT phase, see Berezinskii–Kosterlitz– Thouless phase black-and-white group, 167 black-and-white lattice, 167 Bloch electrons, 92 annihilation operator of, 92 creation operator of, 92 density of states for, 96 diamagnetic susceptibility of, 305 effective mass of, 93 Green function of, 105 in magnetic field, 248 in strong magnetic field, 297 specific heat of, 98 susceptibility of, 98 velocity of, 240 Bloch equations, 65 Subject Index 627 Bloch function, 79 Bloch states, 78 energy spectrum of, 80 Bloch T 3/2 law, 528 corrections to, 529 Bloch wall, 509 Bloch–Grüneisen relation, 392 Bloch’s theorem, 186, 79 Bloch–Wilson insulator, 92 body-centered cubic structures, 210–214 Bogoliubov transformation application of, 546 for antiferromagnets, 542 Bohr magneton, 44, 588 Bohr radius, 50, 588 Bohr–van Leeuwen theorem, 20 Boltzmann distribution, see Maxwell–Boltzmann distribution Boltzmann equation, 361, 364 for electrons, 363 for phonons, 366 bonding state, 98 Born–Mayer approximation, 88 Born–Oppenheimer approximation, 329 Born–von Kármán boundary condition, see boundary conditions, Born–von Kármán boron group, 91 Bose–Einstein statistics, 396 bound state, 107 around impurities, 73 boundary conditions antiperiodic, 186 Born–von Kármán, 185, 186, 338, 26 for Ginzburg–Landau equations, 484 twisted, 186 Bragg condition, 243 Bragg peaks, 243 shape of, 252 temperature dependence of, 443–444 Bragg plane, 245, 85, 124 Bravais cell, 117, 118, 144 Bravais group, 140 Bravais lattices, 113 types in three dimensions, 141, 146–154 types in two dimensions, 140, 142–145 breakdown avalanche, 556 electric, 274 magnetic, 274 of the diode, 554 Zener, 555 bremsstrahlung isochromat spectroscopy, 192 Bridgman relations, 64 Brillouin function, 57 Brillouin scattering, 436–437, 443, 445 Brillouin zone, 123, 190 higher, 85 of bcc lattice, 212 of fcc lattice, 216 size of, in antiferromagnets, 519 Brillouin–Wigner perturbation theory, 582 buckyball, 30 built-in potential, 529 bulk defects, see volume defects bulk modulus, 365, 369 bulk susceptibility, see susceptibility, volume Burgers circuit, 286, 287 Burgers dislocation, see screw dislocation Burgers vector, 286, 287 C1 structure, 204, 219 C1b structure, 223 C2 structure, 204 C3 structure, 204, 209, 210 C4 structure, 204 C14 structure, 204 C15 structure, 204, 223 calamitic nematic phase, 25 calcium band structure of, 183 canonical structure constant, 177 canonical transformation, 355, 603 carbon group, 196 carrier lifetime, 536 630 Subject Index Debye function, 416, 611 Debye length, 521 Debye model, 389–390 Debye–Hückel theory, 404, 521 Debye–Scherrer method, 267 Debye temperature, 412, 596 Debye–Waller factor, 442 deep level, 224 defects, see also line defects; planar defects; point defects; volume defects dislocations, 283–292 grain boundaries, 298 deformation potential, 339 degeneracy accidental, 173, 126 lifting of, 173, 126 degeneracy temperature, 37 degenerate semiconductor, 215 de Haas–van Alphen effect, 314 delta function, 615 dense random packing model, 304 density of states in strong magnetic field, 283 in two-dimensional electron gas, 283 integrated, 399 of Bloch electrons, 96 of electron gas, 32 of phonons, 398–409 density-of-states mass, 99, 213 depletion layer, 519, 522, 530 depletion region, see depletion layer destruction operator, see annihilation operator detailed balance, see principle of detailed balance dhcp structure, see hexagonal crystal structures, double close-packed diamagnetic resonance, see cyclotron resonance diamagnetic susceptibility of Bloch electrons, 305 of electron gas, 47 diamagnetism perfect, 453 diamond band structure of, 89 diamond structure, 204, 221–224, 197 electron states of, 201 dielectric constant, 18, 435 of bound electrons, 435 of germanium, 221 of ideal gas of electrons, 18 of silicon, 221 transverse, 423 dielectric function, 18 diffraction dynamical theory of, 258–260 theory of, 242–260 diffusion, 537 diffusion coefficient, 358, 538, 539 diffusion length, 539 diffusion potential, 529 diffusion region, 548 digamma function, 619 dihedral group, 130 dimerized chain, 337, 345–348 Dingle factor, 326 Dingle temperature, 326 dipole approximation, 435, 436 dipole–dipole interaction, 70, 80, 463, 661 Dirac delta function, 615 direct exchange, see exchange, Heisenberg direct gap, 211 direct lattice, 122 direct-gap semiconductors, 211 director, 25 Dirichlet’s construction, 117, 85 in reciprocal lattice, 123 disclinations, 290 screw, 291 wedge, 291 discotic columnar phase, 28 discotic nematic phase, 25, 26 dislocation line, 284 dislocations, 283–292 edge, 284, 285 mixed, 285, 286 partial Frank, 297 Shockley, 295 screw, 285 Subject Index 631 dispersion in ionic crystal, 430 dispersion relation for antiferromagnetic magnons, 520, 543 for ferromagnetic magnons, 518, 524 for phonons, 395 of spinons in an isotropic antiferromagnet, 571 of XY model, 573 distribution function Bose–Einstein, see Bose–Einstein statistics Fermi–Dirac, see Fermi–Dirac statistics in relaxation-time approximation, 373 Maxwell–Boltzmann, see Maxwell– Boltzmann distribution nonequilibrium, 362 stationary, 49 divacancy, 280 divalent metals, 182 domain wall, 504 width of, 511 domains, 508–513 donor, 219 donor level, 222 thermal population of, 227 doped semiconductors, 219 double exchange, 468–469 double group, 179, 641–642 double hexagonal close-packed structure see hexagonal crystal structures, double close-packed Dresselhaus splitting, 84, 209 drift velocity, 5 Drude–Lorentz model, 2, 1, 23 Drude model, 1–23 failures of, 22 Drude peak, 15 Drude weight, 16 Drude–Zener model, 424 Dulong–Petit law, 384 dynamical interactions, 534 dynamical matrix, 356 dynamical structure factor, 440, 659 Dyson–Maleev transformation, 531 and the Hamiltonian of interacting magnons, 534 E2 structure, 204 E21 structure, 209 easy axis of magnetization, 472 easy plane of magnetization, 472 edge dislocation, 284, 285 edge states, 294, 409 effective Hamiltonian, 603 effective interaction, 344 effective magnetic moment, 58 effective magneton number, 58 effective mass, 41 in compound semiconductors, 210 of Bloch electrons, 93 of electrons in germanium, 208 of electrons in silicon, 205 of holes, 96 of holes in germanium, 208 of holes in silicon, 206 tensor, see effective-mass tensor effective-mass tensor, 95, 246 inverse, see inverse effective-mass tensor Einstein model, 387–389 Einstein relation, 538 elastic constants Lamé, see Lamé constants of crystals, 367–371 Voigt, see Voigt elastic constants elastic waves, 363–367 electric breakdown, 274 electric current, 53 electrical conductivity, 5 electrochemical potential, 53, 360 electrodynamics of superconductors, 474 electron conduction, 2 core, 2, 157 valence, 2 electron gas at finite temperatures, 34 classical, 2 compressibility of, 43 density of states for, 32 entropy of, 43 632 Subject Index equation of state for, 42 ground state of, 28 magnetic properties of, 20 specific heat of, 40 susceptibility of, 44 electron–electron interaction effective, 344 electron–hole excitations, 35 electron–hole pair, 446, 534 electron–ion interaction, 334 electron paramagnetic resonance, 61 electron–phonon interaction, 193 consequences of, 343 Hamiltonian of, 333 electron–photon interaction, 440 electron spectroscopy for chemical analysis, 191 electron spin resonance, 61 electron states finite lifetime of, 344 localized, 73 elementary excitations in magnetic systems, 515 phonons, see phonons Eliashberg equations, 452 empty-lattice approximation, 109 energy current, 248 energy gap, see gap energy spectrum of Bloch states, 80 entropy of electron gas, 43 of superconductors, 460 of vortices in XY model, 554 EPR, 61 equation of state for crystal, 418–420 for electron gas, 42 equipartition theorem, 384, 4 equivalent wave vectors, 190, 82 Esaki diode, 556 ESCA, 191 ESR, 61 Ettingshausen effect, 63 Euler–Lagrange equation, 359 Euler–Maclaurin formula, 295 Euler–Mascheroni constant, 620 Euler’s constant, see Euler–Mascheroni constant Euler’s equation, 510 Euler’s gamma function, 619 Evjen’s method, 84 Ewald construction, 264–265 for Laue method, 266 for powder method, 268 for rotating-crystal method, 267 Ewald’s method, 84 Madelung energy of NaCl crystal, 86 Ewald sphere, 264 EXAFS, 269, 309 exchange, 463 direct, see exchange, Heisenberg double, see double exchange Heisenberg, 463–464 RKKY, see RKKY interaction super-, see superexchange exchange energy, 464 exchange integral, 92 exchange interaction, 42 excited states, 31 exciton, 440 extended-zone scheme, 85 extinction coefficient, 416 extinction length, 263 extrinsic range, 230 face-centered cubic structures, 214–221 factorial function, 619 faithful representation of point groups, 130 Faraday effect, 513 F -center, 282 Fermi contact term, 71, 661 Fermi energy, 28 for Bloch states, 88 Fermi–Dirac distribution function, 34, 49 Fermi–Dirac statistics, 2, 34, 47, 92 Fermi integral, 612, 36 Fermi momentum, 28 Fermi pseudopotential, 247, 439 Fermi sea, 28 Fermi sphere, 28 Fermi surface, 89 for nearly free electrons, 136 in empty lattice, 115 Subject Index 635 quadratic and quartic parts, 534–535 Heitler–London approximation, 90 Helmholtz free energy, 278, 470 of phonon gas, 414 Hermann–Mauguin symbols, 125 Hermite polynomials, 623 heterojunction, 532 heteropolar bond, 83 Heusler alloy, see Heusler phase Heusler phase, 204, 220 hexa-n-alkoxy triphenylene, 24 hexagonal crystal structures, 224–229 close-packed, 204, 224, 226 double close-packed, 224, 226 simple, 224 hexatic phase, 28 Higgs bosons, 201 high-Tc superconductors, 466 high-temperature expansion, 503–504 HMTTF-TCNQ, 232 Hofstadter butterfly, 304 Hofstadter spectrum, 304 hole states, 31 holes, 95 creation operator of, 31 in Fermi sphere, 31 in semiconductors, 212 motion of, 247 Holstein model, 349 Holstein–Primakoff transformation, 530 and the Hamiltonian of interacting magnons, 534 application of, 534, 546 homeopolar bond, 89 homopolar bond, 89 honeycomb lattice, 113, 114 Hooke’s law, 365, 368 Hubbard model, 5 Hund’s rules, 42 hybrid states, 103–105, 233 hydrogen bond, 106 hyperfine structure, 69 icosahedral group, 131 ideal crystal, 14, 109 improper rotation, 126 impurities acceptor, 220 bound states around, 73, 107 donor, 219 electron states around, 104 in semiconductors, 219 magnetic, 394 scattering by, 64 incoherent scattering, 660 incommensurate structures, 312 magnetic, see spiral structures index of refraction, see refractive index indirect exchange, see RKKY interaction indirect gap, 211 indirect-gap semiconductors, 211 inelastic neutron scattering, see neutron scattering, inelastic, 547 infrared absorption, 431–433, 443 infrared active mode, 432, 447 insulators, 90 integer quantum Hall effect, 407 integrated density of states, 399 interaction electron–ion, 334 electron–phonon, 193, 333 electron–photon, 440 phonon–phonon, 423–424 phonon–photon, 443 RKKY, 466 s–d, 465, 606, 611 spin–orbit, 38, 83, 204 van der Waals, 78 with radiation field, 412 interface, 518 metal–semiconductor, 518 interfacial defects, 274 international notation, see Hermann– Mauguin symbols interstitials, 278–280 split, 279 intrinsic carrier density, 217 intrinsic range, 232 intrinsic semiconductors, 201, 212 inverse AC Josephson effect, 507 inverse effective-mass tensor, 94, 246 inverse photoemission spectroscopy, 192 inversion, 126 inversion layer, 406, 524 ionic bond, 83 ionic–covalent bond, 94 ionic crystals, 83–89, 430 optical vibrations in, 373–377 ionic radius, 236 636 Subject Index IPES, see inverse photoemission spectroscopy IQHE, see integer quantum Hall effect irreducible representations, 637 Ising model, 472 and Mermin–Wagner theorem, 551 two-dimensional, 551 isomer shift, 73 isotope effect, 452 Jahn–Teller distortion, 353 Jahn–Teller theorem, 354 JFET, 560 jj coupling, 40 Jones symbol, 130 Jordan–Wigner transformation, 533 application of, 572 Josephson constant, 587, 506 Josephson effect, 501 AC, 506 DC, 504 in magnetic field, 509 inverse AC, 507 Josephson inequality, 497, 499 Joule heat, 15, 60, 61, 359 junction abrupt, 526 step, 526 junction transistor, 4, 560 kagome lattice, 113, 114 Kelvin relations, see Thomson relations Kerr effect, 513 kinematical interaction, 536 kinetic coefficient, 380 kinetic theory of gases, 1, 23, 43 KKR method, 168 Knight shift, 72 Kohn anomaly, 350 Kondo effect, 5, 394 Korringa law, 71 Korringa relaxation, 71 Kosterlitz–Thouless transition, see Berezinskii–Kosterlitz– Thouless transition Kramers–Kronig relation, 64, 16, 19, 416, 434 Kramers’ theorem, 182 L1 structure, 204, 207 L2 structure, 204 L′3 structure, 204 La2−xBaxCuO4, 466 ladder operator, 393, 589 Lagrange’s equation, 34, 359 Laguerre polynomials, 624 Lambert–Beer law, 416 Lamé constants, 365 Landau diamagnetism, 295 Landau gauge, 278 Landau levels, 277 degree of degeneracy of, 281 Landau–Peierls instability, 28, 201 Landau theory of phase transitions, 199, 489–492 Landauer formula, 568 Landé g-factor, 54 Langevin diamagnetism, see Larmor diamagnetism Langevin function, 57 Langevin susceptibility, 53 Langmuir frequency, 20 Langmuir oscillation, 20 lanthanoids, 179, 218 Larmor diamagnetism, 49 Larmor frequency, 50 Larmor’s theorem, 21 laser, 562 lattice parameters, 229 lattice vibrations classical description of, 331–385 Einstein model, 387–389 quantum description of, 387–427 lattices that are not Bravais lattices, 113, 114 Laue condition, 194, 244, 245 Laue method, 265 Laves phase, 204, 223 law of mass action, 218 layered structures, 229–233 LCAO method, 97, 154 lead band structure of, 185 Fermi surface of, 185 LED, 534 LEED, 262 Legendre polynomials, 625 Lennard-Jones potential, 81 Subject Index 637 level splitting in crystals, 173–182 Lie groups, 643 Lieb–Schultz–Mattis theorem, 578, 582 lifetime carrier, 536 of electron states, 344 of electrons, 69 of magnons, 536, 548 of phonons, 424 Lifshitz–Kosevich formula, 323 light interaction with bound electrons, 427 reflection of, 417 refraction of, 417 scattering by free electrons, 421 light holes, 206 Lindemann criterion, 413 line defects, 274, 283–292 linear chain diatomic, 341–345 dimerized, 345–348 monatomic, 337–341 liquid crystals, 24–29 liquid phase, 22–23 little group, 126 LMTO method, 175 localized excitations lattice vibrations, 377–383 localized state, 73, 107 of electron, 104 London equations, 475 London penetration depth, 477 long-range order, 14 longitudinal magnetoresistance, 61 longitudinal mass, 208 longitudinal vibrations, 341 Lorentz formula, 374 Lorentz–Lorenz equation, 375 Lorentz model, 427 Lorentzian function Fourier transform of, 609 Lorenz number, 10, 56 low-angle grain boundary, 298 low-dimensional magnetic systems, 548f lower critical field, 455, 497 LS coupling, 40 Luttinger’s theorem, 139 Lyddane–Sachs–Teller relation, 377, 431 M -center, 282 Madelung constant, 87 Madelung energy, 84 magnesium band structure of, 183 magnetic breakdown, 274 magnetic force microscope, 513 magnetic form factor, 249 magnetic group, see color group, black-and-white magnetic lattice, see black-and-white lattice magnetic length, 250 magnetic space groups, see space groups, black and white magnetic structures, 453–462 magnetic-field dependence of resistivity, 11 magnetism antiferro-, 453–459 atomic, 51 ferri-, 461, 462 ferro-, 450–453, 470 magnetization, see also sublattice magnetization definition of, 48 temperature dependence of, see Bloch T 3/2 law magnetoacoustic oscillations, 265 magnetomechanical ratio, see gyromag- netic ratio magnetoresistance, 61, 384 colossal, 573 giant, 573 longitudinal, 61 transverse, 61 magnon energy temperature-dependent corrections to, 535 magnons antiferromagnetic, 540–547 approximate bosonic character of, 521, 525 as magnetic counterparts of phonons, 521 bound states of, 536–540, 565–566 cutoff for, 527, 545 ferrimagnetic, 546–547 ferromagnetic, 521 640 Subject Index Pauli exclusion principle, 36, 464 Pauli matrices, 646, 674 Pauli susceptibility, 45 of Bloch electrons, 99 Pauling ionic radius, 236 Pearson symbol, 205 Peierls instability, 350 Peierls insulators, 92 Peierls substitution, 297 Peltier coefficient, 59 in superconductors, 458 Peltier effect, 59 penetration depth in Ginzburg–Landau theory, 486 London, 477 temperature dependence of, 489 Penrose tiling, 323–326 perfect diamagnetism, 453 periodic boundary condition, see boundary conditions, Born–von Kármán periodic potential, 77 periodic table, 593 permittivity, 17, 415 relative, see dielectric constant perovskite structure, 204, 209 perpendicular susceptibility, 484 persistent current, 451 perturbation theory, 579 Brillouin–Wigner, 582 degenerate, 583 nondegenerate, 579 Rayleigh–Schrödinger, 581 time-dependent, 584 time-independent, 579 PES, see photoelectron spectroscopy phase diagram of high-Tc superconductors, 498 of type I superconductors, 456 of type II superconductors, 457 phase shift, 67, 173 phonon drag, 58 phonon softening, 352 phonon–phonon interaction, 423–424 phonon–photon interaction, 443 phonons, 395 acoustic, 397 density of states of, 398–409 experimental study of, 429–447 interaction among, see phonon–phonon interaction lifetime of, 446 specific heat of, 413–418 photoelectric effect, 190 photoelectron spectroscopy, 190 photoemission spectroscopy, 190 physical constants, 587–588 planar defects, 274, 293–301 planar model, see XY model planar regime, 566 plane groups, 162 plane-wave method, 156 plasma frequency, 20 plasma oscillations, 20 plastic crystals, 29–30 p–n junction, 526 biased, 545 breakdown in, 554 point defects, 274–283 point group of the crystal, 161 point groups, 127–135 point-contact transistor, 4 Poisson equation, 520 Poisson summation formula, 319 Poisson’s ratio, 369 polar covalent bond, 94 polariton, 432 polarization factor, 339 polarization vector, 357 polarons, 346 polycrystals, 21, 293 polymers, 30 positron annihilation, 187 powder method, 267 p-polarization, 417 primitive cell, 115 primitive vectors, 110 choice of, 111–113 principle of detailed balance, 369 proximity effect, 485 p-type semiconductors, 224 pseudo-wavefunction, 162 pseudopotential, 162 pseudopotential method, 160 pyrite structure, 204 quantum critical point, 500 quantum dot, 564, 567 Subject Index 641 quantum Hall effect, 6, 195, 405 quantum oscillations, 306 quantum phase transitions, 500 quantum point contact, 568 quantum well, 564, 565 quantum wire, 564, 567 quantum-well laser, 567 quasicrystals, 3, 21, 315–330 quasimomentum, 191 quasiparticles, 92 quasiperiodic functions, 311 quasiperiodic structures, 309–330 quasiperiodic tiling, see Penrose tiling QW laser, see quantum-well laser radial distribution function, 17, 305 in amorphous silicon, 306 in quasicrystals, 316 radiofrequency size effect, 270 Raman active mode, 435, 447 Raman scattering, 433–436, 443, 445 two-phonon, 446 rapidity, 563 rare-earth garnet, 547 rare-earth metals, 452, 23, 42 band structure of, 186 Rashba term, 38, 576 Rayleigh–Schrödinger perturbation theory, 581 R-center, 282 reciprocal lattice, 120–124 definition, 120 of bcc lattice, 212 of fcc lattice, 216 of hexagonal lattice, 225 primitive vectors of, 122 recombination current, 554 recombination lifetime, 536 recombination of carriers, 534 rectification by p–n junction, 553 by Schottky diode, 544 rectifying contact, 544 reduced-zone scheme, 85 reduction, 638 reflectance, 419 reflection coefficient, 417, 419 reflectivity, 419 of semiconductors, 426 refractive index, 416 relative permittivity, see dielectric constant relativistic effects, 36–38, 83 relaxation function, 63 relaxation region, 425 relaxation time, 4, 8, 370 spin–lattice, 64 spin–spin, 65 transport, 69 relaxation-time approximation, 370 renormalization, 501 renormalization-group transformation, 500–502 repeated-zone scheme, 85 residual resistivity, 69, 389 resistivity, 7 contribution of electron–phonon interaction, 389 of metals, 8 residual, see residual resistivity resonance absorption, 61, 72 resonance fluorescence, 72 resonance integral, 98 resonant absorption, 61 resonating valence bond spin liquid, 585 Reuter–Sondheimer theory, 479 reverse bias, 542 Richardson–Dushman equation, 543 Riemann zeta function, 617 Righi–Leduc effect, 63 rigid-ion approximation, 337 RKKY interaction, 466, 574, 605 rock-salt structure, see sodium chloride, structure Rodrigues’ formula for generalized Laguerre polynomials, 625 for Hermite polynomials, 623 for Laguerre polynomials, 624 rotating-crystal method, 265 rotation axis, 125 rotation group, 642, 665 rotation–inversion, see symmetry operations, rotation–inversion rotation–reflection, see symmetry operations, rotation–reflection rotational symmetry, 125 642 Subject Index rotoinversion, see symmetry operations, rotation–inversion rotoreflection, see symmetry operations, rotation–reflection Ruderman–Kittel oscillation, 465 Rushbrook inequality, 497 Russell–Saunders coupling, 40 Rutgers formula, 473 rutile structure, 204 RVB, see spin liquid, resonating valence bond Rydberg energy, 588 satellite peaks, 314, 459 saturated bond, 99 saturation current, 554 saturation range, 230 scaling laws, 496–500 scanning tunneling microscope, 270 scattering by impurities, 64, 387 by lattice defects, 365 by magnetic impurities, 394 electron–phonon, 341 scattering amplitude, 65 scattering cross section of impurity scattering, 68 scattering length, 247, 440 scattering of light by bound electrons, 427 by free electrons, 421 scattering theory methods based on, 164 scattering vector, 548 Schoenflies symbols, 125 Schottky barrier, 523 Schottky defect, 280–282 Schottky diode, 541 biased, 541 current–voltage characteristics of, 545 Schrieffer–Wolff transformation, 612 Schwinger boson, 532 screening length, 521 screw axis, 158 screw dislocation, 285 screw rotation, 158 s–d interaction, 465, 606, 611 second quantization, 589 Seebeck coefficient, 56 Seebeck effect, 58 Seitz symbol, 157 selection rules, 184, 650 γ-selenium structure, 204 semiclassical dynamics, 239 limitations of, 271 semiclassical equation of motion, 243 semiclassical quantization, 300 semiconductor laser, 562 II–VI semiconductors, 198 III–V semiconductors, 198 semiconductor quantum devices, 564 semiconductors, 76 band structure of, 201 conduction band in, 196 degenerate, 215 electronic structure of, 195 elemental, 196 gap in, 198, 199 multivalley, 207 n-type, 223 p-type, 224 valence band in, 196 semimetals, 91 separation energy, 77 Shapiro steps, 508 shear modulus, 365, 369 shift operator, see ladder operator Shockley partial dislocation, 295 Shockley’s law, 553 short-range order, 14, 305–309 Shubnikov groups, 167 Shubnikov phase, 456 Shubnikov–de Haas effect, 327 silicon, 91, 197 band structure of, 204 dielectric constant of, 221 effective mass of electrons in, 205 effective mass of holes in, 206 simple cubic crystals, 205–210 single crystal, 20 single-electron transistor, 569, 571 size effects, 268 skin depth, 262 skutterudite structure, 204 Slater determinant, 101, 24, 78 small polaron, 349 Subject Index 645 thermomagnetic effects, 61 thermopower, 56 Thomson effect, 60 Thomson heat, 60 Thomson relations first, 59 second, 61 tight-binding approximation, 139 p-band, 145 s-band, 143 in magnetic field, 302 tilt grain boundary, 298 time reversal, 196 TM polarization, 417 top-hat function Fourier transform of, 608 topological quantum number, 554 torsional waves, 366 trajectory of electrons in real space, 250 transistor bipolar, 517, 558 field-effect, 560 junction, 4, 560 point-contact, 4, 517 single-electron, 571 spin-field-effect, 576 transition metals, 213, 218, 23, 42, 393, band structure of, 185 translational symmetry, 110 transmission coefficient, 419 transmittance, 419 transport in magnetic field, 383 transport coefficients, 54, 379 in semiconductors, 402 transport phenomena, 357 transport relaxation time, 69 transverse magnetoresistance, 61 transverse mass, 208 transverse vibrations, 341 triple-axis spectrometer, 439, 445 trivalent metals, 183 TTF-TCNQ, 76 tungsten Fermi surface of, 186 tunnel diode, 556 tunneling, 542, 555, 556 in SIN junction, 460 in SIS junction, 460 of normal electrons, 501 single-particle, 505 twin crystals, 299, 301 twist grain boundary, 298 twisted boundary conditions, see boundary conditions, antiperiodic two-fluid model, 474 two-phonon absorption, 445 two-phonon Raman scattering, 436, 446 type I superconductors, 455 type II superconductors, 456, 473 ultraviolet photoelectron spectroscopy, 190 ultraviolet photoemission spectroscopy, 190 umklapp process, 193, 424, 427, 337, 392, 400 uniaxial anisotropy, 472 unsaturated bond, 99 upper critical field, 455, 497 UPS, 190 vacancies, 275–278 formation energies of, 276 vacancy pair, 280 valence band, 158, 191 in semiconductors, 196 valence electron, 2 valence-bond method, 90 valence-bond-solid state, 580 van der Waals bond, 79–81 van der Waals interaction, 78 Van Hove formula, 440, 656 Van Hove singularities, 405–409, 97 Van Vleck paramagnetism, 60 Van Vleck susceptibility, 60 variational methods, 164 vector operator definition of, 53 velocity of Bloch electrons, 240 virtual bound state, 68, 106 Voigt elastic constants, 368 Volterra construction, 284 646 Subject Index volume defects, 274, 302 volume susceptibility, see susceptibility, volume von Klitzing constant, 587, 407 Voronoi polyhedron, 117 vortex lattice, 497 vortices, 553–559, 457, 493 W structure, 213 wallpaper groups, see plane groups Wannier functions, 100 Wannier’s theorem, 297 Weiss field, 474 Weiss indices, 119 Wiedemann–Franz law, 10, 23, 56, 74, 399, 457 Wigner–Eckart theorem, 53 Wigner–Seitz cell, 117 Wigner–Seitz radius, 3 Wigner–Seitz sphere, 116, 3 Wigner’s theorem, 173 Wilson ratio, 46, 100 work function, 518 wurtzite structure, 204, 228 X-ray diffraction experimental methods of, 261–268 theory of, 242–260 X-ray photoelectron spectroscopy, 191 X-ray photoemission spectroscopy, 191 XPS, 191 XY model, 472, 551, 572 entropy of vortices in, see entropy, of vortices in XY model free energy of vortices in, see free energy, of vortices in XY model Hamiltonian of, 551 phase transition in, see Berezinskii–Kosterlitz– Thouless transition YBa2Cu3O7−δ, 450, 466 YBCO, 450, 466 YIG, see yttrium–iron garnet Young’s modulus, 369 yttrium–iron garnet, 547 Yukawa function, 404 Fourier transform of, 609 Zener breakdown, 554 Zener diode, 556 Zener effect, see Zener breakdown Zener tunneling, 275 zero-point energy, 392, 543 zero-point spin contraction in antiferromagnetic ground state, 544 in two-dimensional antiferromagnets, 551 zero-point vibrations, 89, 392 zeta function, 617 zinc group, 182 zincblende structure, see sphalerite structure zone folding, 113 Fundamental physical constants Name Symbol Value Bohr magneton μB = e/2me 9.274 009 × 10−24 JT−1 Bohr radius a0 = 4πε0 2/mee 2 0.529 177 × 10−10 m Boltzmann constant kB 1.380 650 × 10−23 JK−1 Conductance quantum G0 = 2e2/h 7.748 092 × 10−5 S Electron g-factor ge = 2μe/μB −2.002 319 Electron gyromagnetic ratio γe = 2|μe|/ 1.760 860 × 1011 s−1 T−1 γe/2π 28 024.9540 MHz T−1 Electron magnetic moment μe −9.284 764 × 10−24 J T−1 −1.001 160 μB Electron mass me 9.109 382 × 10−31 kg Electric constant ε0 = 1/μ0c 2 8.854 188 × 10−12 F m−1 Elementary charge e 1.602 176 × 10−19 C Hartree energy Eh = e2/4πε0a0 4.359 744 × 10−18 J in eV 27.211 383 eV Josephson constant KJ = 2e/h 483 597.9 × 109 Hz V−1 Magnetic constant μ0 4π × 10−7 N A−2 Magnetic flux quantum Φ0 = h/2e 2.067 834 × 10−15 Wb Nuclear magneton μN = e/2mp 5.050 783 × 10−27 JT−1 Neutron mass mn 1.674 927 × 10−27 kg Neutron magnetic moment μn −0.966 236 × 10−26 J T−1 −1.913 043 μN Neutron g-factor gn = 2μn/μN −3.826 085 Planck constant h 6.626 069 × 10−34 J s in eV h/{e} 4.135 667 × 10−15 eV s Proton g-factor gp = 2μp/μN 5.585 695 Proton gyromagnetic ratio γp = 2μp/ 2.675 222 × 108 s−1 T−1 γp/2π 42.577 482 MHz T−1 Proton magnetic moment μp 1.410 607 × 10−26 JT−1 2.792 847 μN Proton mass mp 1.672 622 × 10−27 kg Reduced Planck constant  = h/2π 1.054 572 × 10−34 J s in eV /{e} 6.582 119 × 10−16 eV s Rydberg constant R∞ = α2mec/2h 10 973 731.569 m−1 Rydberg energy Ry = R∞hc 2.179 872 × 10−18 J in eV 13.605 692 eV Speed of light c 299 792 458 m s−1 Von Klitzing constant RK = h/e2 25 812.807 572 Ω