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PHYSICAL REVIEW B 72, 165344 共2005兲 Quadratic response theory for spin-orbit coupling in semiconductor heterostructures Bradley A. Foreman* Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China 共Received 17 June 2005; revised manuscript received 31 August 2005; published 28 October 2005兲 This paper examines the properties of the self-energy operator in lattice-matched semiconductor heterostructures, focusing on nonanalytic behavior at small values of the crystal momentum, which gives rise to longrange Coulomb potentials. A nonlinear response theory is developed for nonlocal spin-dependent perturbing potentials. The ionic pseudopotential of the heterostructure is treated as a perturbation of a bulk reference crystal, and the self-energy is derived to second order in the perturbation. If spin-orbit coupling is neglected outside the atomic cores, the problem can be analyzed as if the perturbation were a local spin scalar, since the nonlocal spin-dependent part of the pseudopotential merely renormalizes the results obtained from a local perturbation. The spin-dependent terms in the self-energy therefore fall into two classes: short-range potentials that are analytic in momentum space, and long-range nonanalytic terms that arise from the screened Coulomb potential multiplied by a spin-dependent vertex function. For an insulator at zero temperature, it is shown that the electronic charge induced by a given perturbation is exactly linearly proportional to the charge of the perturbing potential. These results are used in a subsequent paper to develop a first-principles effective-mass theory with generalized Rashba spin-orbit coupling. DOI: 10.1103/PhysRevB.72.165344 PACS number共s兲: 73.21.⫺b, 73.61.Ey, 71.15.Ap I. INTRODUCTION Hamiltonian1 The Rashba is the prototype of a class of effective-mass Hamiltonians describing spin-orbit coupling in semiconductors.2 These models have been under intensive study in the past several years due to theoretical and experimental advances in spin-related phenomena such as the intrinsic spin Hall effect3–17 and the spin galvanic and circular photogalvanic effects.18–25 Such effects are generated by spin-orbit coupling terms in the conduction or valence bands of clean nonmagnetic semiconductors. Although a variety of different two- and three-dimensional semiconductor systems are under investigation, one of the most widely studied is a heterostructure between semiconductors with the zinc-blende structure, in which an external electric field can be used to tune the relative contributions from the Rashba and Dresselhaus spin-splitting terms.25–27 In a two-dimensional effective-mass model, the Rashba coupling has no coordinate dependence. But in a threedimensional theory, it is usually separated into 共1兲 a contribution proportional to the macroscopic electric field generated by gate voltages, dopants, and free carriers; and 共2兲 ␦ functions representing the contribution from the rapid change in potential at a heterojunction.2,28–39 However, the assumption that a heterojunction can be represented by a short-range ␦ potential has never been justified from first principles. In a self-consistent theory with electron-electron interactions, there are in general long-range Coulomb multipole potentials that are not well localized at the interface. These long-range potentials contribute spin-dependent terms to the Hamiltonian. Thus it is important to establish the conditions under which such terms will appear in the effective-mass Hamiltonian for a heterojunction. This paper examines the long-range terms in the selfenergy of a quasiparticle for the case of a lattice-matched semiconductor heterostructure. The potential energy of the ions is described using norm-conserving pseudo1098-0121/2005/72共16兲/165344共19兲/$23.00 potentials.40–42 The heterostructure pseudopotential is treated as a perturbation of a bulk reference crystal, with the selfenergy calculated to second order in the perturbation using quadratic response theory.43–48 This approach is well justified numerically, since the linear response alone has been shown to give excellent predictions for the valence band offset in a variety of material systems, including isovalent and heterovalent heterostructures.49–57 The approach used here follows closely earlier work by Sham58 on the theory of shallow impurity states in bulk semiconductors. Sham’s work is generalized to include nonlocal spin-dependent potentials and terms of higher order in the crystal momentum. Even for local spin-independent potentials, the present work includes terms neglected in Sham’s analysis, such as dipole potentials in the quadratic response. This paper is limited to a study of the electron self-energy in the limit of small crystal momentum. The derivation of an effective-mass Hamiltonian from these results is presented in the following paper.59 As a workable approximation scheme, the calculation of the linear response is carried out to terms two orders in q higher than the lowest nonvanishing term, where q is the crystal momentum transfer of the perturbing potential. The quadratic response is calculated to the same order in q as the lowest nonvanishing term in the linear response. 共See the following paper59 for further discussion of this approximation scheme.兲 Three classes of heterostructure perturbations are considered: 共I兲 Heterovalent perturbations with nonzero charge. In this case the perturbing potential includes a monopole of O共q−2兲, and the analysis is performed to an accuracy of O共q0兲 in the linear response and O共q−2兲 in the quadratic response. 共II兲 Isovalent perturbations for which the linear response has a nonzero dipole moment. In this case the leading term in the linear response is O共q−1兲, so the linear response is evaluated to O共q兲 and the quadratic response is evaluated to O共q−1兲. 165344-1 ©2005 The American Physical Society PHYSICAL REVIEW B 72, 165344 共2005兲 BRADLEY A. FOREMAN 共III兲 Isovalent perturbations for which the linear response has no dipole moment. In this case the leading term in the linear response is O共q0兲, so the linear response is evaluated to O共q2兲 and the quadratic response is evaluated to O共q0兲. The perturbations that make up a given heterostructure are generally a mixture of classes I, II, and III. The simplest situation is that of an isovalent heterostructure made up of semiconductors with the zinc-blende structure, such as GaAs/ AlAs or InAs/ GaSb. In this case, every ionic perturbation is an isovalent perturbation from class III. Most theoretical and experimental studies of the Rashba spin-splitting Hamiltonian have dealt with this type of heterostructure. This case is therefore studied in greatest detail here, by working out the explicit form of the self-energy from crystal symmetry. The results show that in this case the Rashba Hamiltonian contains only short-range terms 共to within the accuracy of the stated approximation scheme兲. However, there are long-range spin-dependent terms that are not of the Rashba form. In a heterovalent system such as Ge/ GaAs, the ionic perturbations are from class I. However, since macroscopic accumulations of charge are energetically unfavorable, real heterostructures tend to be macroscopically neutral.51,60 Such a nominally heterovalent class I problem can therefore often be reduced to an isovalent class II or III problem by grouping the ions together in neutral clusters and treating these clusters as the basic unit. This approach is discussed further in Appendix A. In wurtzite heterostructures such as GaN / AlN, the ionic perturbations are from class II, since the site symmetry of atoms in the wurtzite structure 共space group C46v兲 permits a dipole moment. These dipole terms produce spontaneous polarization along the hexagonal c axis in bulk wurtzite crystals, leading to macroscopic interface charge at heterojunctions.61–68 Such charge produces macroscopic electric fields that generate different piezoelectric strain fields in different materials. The present theory, which is restricted to lattice-matched heterostructures, is therefore not generally applicable to wurtzite systems 共except in the unrealistic special case61 where the interface polarization charge is exactly cancelled by an external interface charge兲. However, the results derived here provide a first step towards a more general theory dealing with lattice-mismatched heterostructures. The paper begins in Sec. II by establishing the basic definitions and notation for the Green function and self-energy that are used throughout the paper. The finite-temperature formalism is used 共both for generality and because it facilitates the derivation of Ward identities兲, although the main interest of this paper is the limit of an insulator at zero temperature. In Sec. III, the self-energy is expanded in powers of the perturbing potential using vertex functions. A set of Ward identities is derived for the vertex functions at finite and zero temperature. Section IV presents general expressions for the nonlinear density response, including Ward identities for the static polarization. The bare ionic perturbations are screened in Sec. V, where the proper vertex functions and proper polarization are introduced. A detailed analysis of the small wave vector properties of the linear screened potential is carried out in Sec. VI for the special case of a local spin-independent perturbation. The quadratic response for the same case is considered in Sec. VII, and the linear and quadratic contributions to the selfenergy are derived in Sec. VIII. In Sec. IX it is shown that in the norm-conserving pseudopotential formalism, the contributions from the nonlocal spin-dependent part of the perturbing potential merely renormalize the contributions from the local part of the perturbation. The main results of the paper are discussed and summarized in Sec. X. II. GREEN FUNCTION AND SELF-ENERGY This section establishes the notation, basic definitions, and symmetry properties of the Green function and selfenergy used in subsequent sections of the paper. The starting point is the definition of the one-particle thermal Green function69–71 † Gss⬘共x, ;x⬘, ⬘兲 = − 具T关ˆ s共x, 兲ˆ s⬘共x⬘, ⬘兲兴典, 共2.1兲 where s = ± 21 labels the z component of the spin, is the imaginary time, T is the time ordering operator, and ˆ ˆ ˆ ˆ ˆ s共x , 兲 = eKˆ s共x兲e−K and ˆ s†共x , 兲 = eKˆ s†共x兲e−K are field operators in the Heisenberg picture. The angular brackets denote a thermal average ˆ 具Ô典 = e⍀Tr共e−KÔ兲, 共2.2兲 where  = 1 / kBT is the inverse temperature, Tr denotes a trace over the many-particle Fock space, K̂ = Ĥ − N̂ is the grand Hamiltonian 共with the chemical potential and N̂ the ˆ number operator兲, and e−⍀ = Tr共e−K兲. The many-particle Hamiltonian is defined by Ĥ = 兺 s,s⬘ 冕冕 1 + 兺 2 s,s ⬘ ˆ s†共x兲hss⬘共x,x⬘兲ˆ s⬘共x⬘兲d3x d3x⬘ 冕冕 ˆ s†共x兲ˆ s†⬘共x⬘兲ˆ s⬘共x⬘兲ˆ s共x兲 兩x − x⬘兩 d 3x d 3x ⬘ , 共2.3兲 † where h = h is the Hamiltonian of a single noninteracting particle, and Hartree atomic units are used. Since K̂ is time independent, G has the form Gss⬘共x , ; x⬘ , ⬘兲 = Gss⬘共x , x⬘ , − ⬘兲, with Gss⬘共x , x⬘ , − 兲 = −Gss⬘共x , x⬘ , 兲 for 0 ⬍ ⬍ . This permits the Fourier series representation 共for − ⬍ ⬍ 兲 ⬁ 1 G共兲 = 兺 G共n兲e−in ,  n=−⬁ G共n兲 = 冕  G共兲eind , 共2.4a兲 共2.4b兲 0 where n = 共2n + 1兲 / , and G共兲 denotes a single-particle operator whose matrix elements in the 兩x , s典 basis are Gss⬘共x , x⬘ , 兲. A continuous Green function G共兲 may then 165344-2 QUADRATIC RESPONSE THEORY FOR SPIN-ORBIT … PHYSICAL REVIEW B 72, 165344 共2005兲 be defined by analytic continuation of G共n兲 from the discrete frequencies = + in. The Green function G共兲 satisfies Dyson’s equation 关 − h − ⌺共兲兴G共兲 = G共兲关 − h − ⌺共兲兴 = 1, 共2.5兲 which is an implicit definition for the self-energy operator ⌺共兲 = − h − G−1共兲. 共2.6兲 ext Vss⬘共x,x⬘, 兲 = vss⬘共x,x⬘兲 + ⌺ss⬘共x,x⬘, 兲. The external pseudopotential can be separated as vext = v共0兲 + v, where v共0兲 is the potential of some periodic reference crystal, and v is a nonperiodic perturbation associated with a heterostructure or an impurity. It is assumed that the total potential 共3.3兲 can be represented as a power series in the perturbation v: A formal solution to Eq. 共2.5兲 can be constructed by solving the nonhermitian eigenvalue equations 关h + ⌺共兲兴兩n共兲典 = En共兲兩n共兲典, 共2.7a兲 E*n共兲兩n共兲典, 共2.7b兲 关h + ⌺ 共兲兴兩n共兲典 = † which are also referred to as Dyson’s equations. It is usually assumed that the solutions to 共2.7兲 form a complete biorthonormal72 set with the properties 具n共兲兩n⬘共兲典 = ␦nn⬘ , 共2.8a兲 兺n 兩n共兲典具n共兲兩 = 1, 共2.8b兲 although this is difficult to prove in are valid, then G共兲 is given by72,74 G共兲 = 兺 n general.73 ⬁ Vs1s2共x1, 1 ;x2, 2兲 ⬅ V共12兲 = 兺 V共兲共12兲, =0 共2.9兲 which satisfies the Dyson equation 共2.5兲 by construction. Symmetries of the many-particle Hamiltonian Ĥ under time reversal and space group operations imply corresponding symmetries of the one-particle operators G and ⌺. Derivations of the most useful symmetry relations are presented in Appendix B. V共1兲共12兲 = ⌫共1兲共1243兲v共34兲, 1 V共2兲共12兲 = ⌫共2兲共124365兲v共34兲v共56兲, 2 v共34兲 = vs3s4共x3,x4兲␦共3 − 4兲. ⌫共1兲共1243兲 = ⌫共2兲共124365兲 = The single-particle Hamiltonian h is chosen here to have the form hss⬘共x,x⬘兲 = − 21 ⵜ2␦共x − x⬘兲␦ss⬘ + vss⬘共x,x⬘兲, ext ␦2V共12兲 ␦⌫共1兲共1243兲 = , ␦v共34兲␦v共56兲 ␦v共56兲 ⌫共1兲共1243兲 = ␦共13兲␦共24兲 + 共3.1兲 where the fixed external potential vext is a norm-conserving ionic pseudopotential,40–42 which accounts for both spinorbit coupling75–78 and scalar relativistic effects. The Dyson equation 共2.7a兲 is therefore = E共兲s共x, 兲, ␦V共12兲 , ␦v共34兲 共3.7兲 which may also be expressed as A. Definitions ⬘ 共3.6兲 The vertex functions are by definition functional derivatives of V with respect to v: In this section, a perturbative approach to the Dyson equation 共2.7a兲 is developed by using vertex functions to expand the self-energy in powers of the perturbing potential. 冕 共3.5兲 where ⌫共兲 is called the vertex function of order . Here a summation or integration of repeated coordinates is assumed, and the labels are ordered as the trace of a matrix product. The perturbation v is taken to be an instantaneous static potential of the form III. VERTEX FUNCTIONS 1 − ⵜ2s共x, 兲 + 兺 2 s 共3.4兲 where V共0兲 is the potential 共3.3兲 when vext = v共0兲, and the numerical arguments on the right-hand side are shorthand for the space, spin, and time coordinates 共1兲 = 共x1 , s1 , 1兲. Although the upper limit of the formal expansion 共3.4兲 is written as = ⬁, this may well be an asymptotic series, and in practice only a finite number of terms are retained. The linear and quadratic terms of 共3.4兲 are If Eqs. 共2.8兲 兩n共兲典具n共兲兩 , − E n共 兲 共3.3兲 Vss⬘共x,x⬘, 兲s⬘共x⬘, 兲d3x⬘ 共3.2兲 in which is a spinor wave function, and the total potential energy V is ⌫共2兲共124365兲 = ␦⌺共12兲 , ␦v共34兲 ␦2⌺共12兲 , ␦v共34兲␦v共56兲 共3.8兲 in which ␦共12兲 = ␦s1s2␦共x1 − x2兲␦共1 − 2兲. Note that upon application of the Fourier transforms defined in Appendix C, the above equations hold equally well in momentum and frequency space. It is convenient at this point to carry out the time integrals in Eq. 共3.5兲. This eliminates the variables 3 , 4 , … from ⌫共兲 and v, and reduces the time dependence of the ␦共13兲␦共24兲 term in Eq. 共3.8兲 to ␦共1 − 2兲. It is assumed below that this has been done. 165344-3 PHYSICAL REVIEW B 72, 165344 共2005兲 BRADLEY A. FOREMAN B. Ward identities ˆ s⬘s共x⬘,x兲 = ˆ s†共x兲ˆ s⬘共x⬘兲. 79–81 The vertex functions satisfy various Ward identities for certain limiting values of their arguments. One set of these can be derived by varying the chemical potential by a small amount ␦. Since K̂ = Ĥ − N̂, this is equivalent to varying v by ␦v共12兲 = −␦ ␦共12兲. For this special case, Eqs. 共3.7兲 and 共3.8兲 reduce to ⌫共1兲共1233兲 = ␦共12兲 − ␦⌺共12兲 , ␦ The mean nonlocal electron density in the perturbed system is defined as nss⬘共x,x⬘兲 = 具ˆ ss⬘共x,x⬘, 兲典, ⬁ nss⬘共x,x⬘兲 = 兺 nss⬘共x,x⬘兲, ␦⌫共1兲共1243兲 , ␦ 共3.9兲 共兲 ␦ → + . ␦ is the density of the reference crystal. 共The notation 具Ô典0 refers to a thermal average with respect to the reference crystal; see Appendix D.兲 The terms of order ⬎ 0 are given by n共兲共00⬘兲 = 2⌺共12兲 , 2 where ⌸共兲 is the th-order static polarization 共or density correlation function兲, which is defined in Eq. 共E1兲. Here and below, the numerical arguments of n共兲 , ⌸共兲, and v are timeindependent quantities of the form 共0兲 = 共x0 , s0兲. B. Ward identities n共1兲共12兲 = ⌸共1兲共1243兲v共34兲, 1 n共2兲共12兲 = ⌸共2兲共124365兲v共34兲v共56兲, 2 共3.11兲 ⌸共1兲共1243兲 = ⌸共2兲共124365兲 = IV. NONLINEAR DENSITY RESPONSE A. Definitions In this section, perturbation theory 共see Appendix D兲 is used to evaluate the electron density of the system with Hamiltonian Ĥ = Ĥ0 + Ĥ1, in which Ĥ0 is the Hamiltonian of the reference crystal and Ĥ1 is the perturbation due to v: s,s⬘ 冕冕 where ˆ is the density operator ␦n共12兲 , ␦v共34兲 ␦2n共12兲 ␦⌸共1兲共1243兲 = . ␦v共34兲␦v共56兲 ␦v共56兲 共4.8兲 For the special case ␦v共12兲 = −␦␦共12兲 representing a variation in chemical potential of ␦ 共see Sec. III B兲, these expressions give ⌸共1兲共1233兲 = − ˆ s⬘s共x⬘,x兲vss⬘共x,x⬘兲d3xd3x⬘ , ⌸共2兲共124355兲 = − 共4.1兲 共4.7兲 which shows that ⌸ may be defined as a functional derivative of n with respect to v: which generalize and extend the results derived for spinindependent local potentials in Refs. 58 and 80. Ĥ1 = tr共ˆ v兲 = 兺 1 共兲 ⌸ 共00⬘,1⬘1,…, ⬘兲v共11⬘兲 ¯ v共⬘兲, ! 共4.6兲 ⌫共1兲共1243兲 , ⌫共2兲共123344兲 = 共4.5兲 The linear and quadratic density response are given explicitly by 兺共12兲 ⌫ 共1233兲 = ␦共12兲 − , 共1兲 ⌫共2兲共124355兲 = − nss⬘共x,x⬘兲 = 具ˆ ss⬘共x,x⬘, 兲典0 共3.10兲 Now for the special case of an insulator at T = 0, the chemical potential can have any value in the range N−1 ⬍ ⬍ N, where N is the minimum energy needed to add one particle to the ground state of an N-particle system, and the energy gap is Eg = N − N−1. 关For small temperatures approaches the well-defined limit 21 共N + N−1兲,82 but at exactly T = 0 it becomes ill defined.兴 Since can vary arbitrarily within the gap for a system with finite Eg, the Ward identities for the insulator reduce to 共4.4兲 in which 共0兲 which involve a trace over the input variables of ⌫. If ⌺ is analytically continued as a function of = + in, the variation with respect to may be written as 共4.3兲 which is independent of . If n is evaluated using the perturbation theory formula 共D2兲, one obtains a power series in v: =0 ⌫共2兲共124355兲 = − 共4.2兲 n共12兲 , ⌸共1兲共1243兲 , 共4.9兲 which are the Ward identities for the static polarization. For an insulator at T = 0 , is indefinite, and these reduce to 165344-4 QUADRATIC RESPONSE THEORY FOR SPIN-ORBIT … ⌸共1兲共1233兲 = 0, ⌸共2兲共124355兲 = 0. PHYSICAL REVIEW B 72, 165344 共2005兲 共4.10兲 ⌸⬘共q;k,k + q + G兲 = V. SCREENING A. Potential In this section, the concept of screening is used to extract the long-range Coulomb interaction terms from the polarization and vertex functions. The first-order screened potential is defined by adding the Coulomb potential generated by n共1兲 to v: 共12兲 = v共12兲 + u共1243兲⌸共1兲共3465兲v共56兲. 共5.1兲 Here u represents the Coulomb interaction, which is spinindependent and local in coordinate space at both the input and output: 1 共1兲 ⌸␣␣,⬘共q + k2,k2 ;k,k + q + G兲. 兺 ⍀ k2 This simplified form of ⌸ is introduced because the Coulomb potential depends only on the local spin-independent density n共x兲 ⬅ n␣␣共x , x兲. B. Vertex functions Given Eqs. 共5.5兲 and 共5.9兲, one can rewrite the total potentials 共3.5兲 as V共1兲共12兲 = ˜⌫共1兲共1243兲共34兲, 1 V共2兲共12兲 = ˜⌫共2兲共124365兲共34兲共56兲 + ˜⌫共1兲共1243兲共2兲共34兲, 2 共5.12兲 u共1243兲 = ␦s1s2␦s3s4␦共x1 − x2兲␦共x3 − x4兲u共x1 − x3兲. 共5.2兲 In momentum space 共see Appendix C兲 this has the form u共1243兲 = ␦s1s2␦s3s4␦k1−k2,k3−k4u共k1 − k2兲, 共5.3兲 in which u共k兲 = vc共k兲 / ⍀, where ⍀ is the crystal volume and vc共k兲 = 再 4/k2 if k ⫽ 0, 0 if k = 0. 冎 in which the proper vertex function ˜⌫共兲 is defined in perturbation theory as the sum of all th-order vertex diagrams that cannot be separated into two disconnected parts by cutting one Coulomb interaction line or one electron propagator. 共An alternative definition would be as the th functional derivative of V with respect to .83,84兲 From the above results, ⌫ and ˜⌫ are related by 共5.4兲 ⌫共1兲共1243兲 = ˜⌫共1兲共1265兲⑀−1共5643兲, Another way of writing the screened potential is 共12兲 = ⑀−1共1243兲v共34兲, ⌫共2兲共124365兲 = ˜⌫共2兲共128709兲⑀−1共7843兲⑀−1共9065兲 共5.5兲 + ˜⌫共1兲共1287兲u共8709兲⌸共2兲共904365兲. 共5.13兲 in which the inverse static electronic dielectric matrix is ⑀−1共1243兲 = ␦共13兲␦共24兲 + u共1265兲⌸共1兲共5643兲. 共5.6兲 In an insulator, the Ward identities 共4.10兲 and 共5.8兲 yield ⌫共1兲共1233兲 = ˜⌫共1兲共1244兲, The dielectric matrix ⑀ satisfies ⑀共1243兲⑀−1共3465兲 = ⑀−1共1243兲⑀共3465兲 = ␦共15兲␦共26兲, 共5.7兲 and is given explicitly below in Eq. 共5.17兲. For an insulator, the Ward identity 共4.10兲 yields ⑀−1共1233兲 = ␦共12兲. 共5.8兲 The second-order potential 共2兲 is just the Coulomb potential generated by n共2兲: 共2兲共12兲 = u共1243兲n共2兲共34兲. 共5.9兲 ⌫共2兲共124355兲 = ˜⌫共2兲共128799兲⑀−1共7843兲, ⌫共2兲共123344兲 = ˜⌫共2兲共125566兲. By translation symmetry, ⌸ 共1243兲 = 0 unless k1 − k2 = k3 − k4 + G, where G is a reciprocal lattice vector of the reference crystal. Equation 共5.1兲 may therefore be written as ss⬘共k,k⬘兲 = vss⬘共k,k⬘兲 + ␦ss⬘vc共q兲 兺 兺 ⌸⬘共q;k⬙,k⬙ + q C. Polarization In a similar fashion, one can define the proper polarization ⌸̃ as the sum of all static polarization diagrams that cannot be split by cutting a Coulomb line. Thus ⌸共1兲共1243兲 = ⌸̃共1兲共1265兲⑀−1共5643兲, 共5.15兲 which has the form of a Dyson equation:81 ⌸共1兲共1243兲 = ⌸̃共1兲共1243兲 + ⌸̃共1兲共1265兲u共5687兲⌸共1兲共7843兲 k⬙ G + G兲v⬘共k⬙ + q + G,k⬙兲, 共5.14兲 Hence, for insulators, the Ward identities 共3.11兲 are valid for both ⌫ and ˜⌫. 共1兲 where q ⬅ k − k⬘ and 共5.11兲 = ⌸̃共1兲共1243兲 + ⌸共1兲共1265兲u共5687兲⌸̃共1兲共7843兲. 共5.10兲 共5.16兲 This can be used to verify that the dielectric matrix 165344-5 PHYSICAL REVIEW B 72, 165344 共2005兲 BRADLEY A. FOREMAN ⑀共1243兲 = ␦共13兲␦共24兲 − u共1265兲⌸̃共1兲共5643兲 共5.17兲 is indeed the inverse of Eq. 共5.6兲. Equations 共5.15兲 and 共5.16兲 can also be written as ⌸共1兲共1243兲 = ⑀−1共6512兲⌸̃共1兲共5643兲, 共5.18兲 in which the symmetry property 共E4兲 was used. The total and proper quadratic polarizations are likewise related by 共see Fig. 5 of Ref. 58兲 ⌸̃, it is apparent that the insulator Ward identities 共4.10兲 for ⌸ and ⌸̃ hold for P as well. VI. LINEAR RESPONSE TO A LOCAL PERTURBATION In this section the properties of the screened potential are examined in greater detail for the case of a local perturbing potential,58 which by definition has the form v共x,x⬘兲 = ␦共x − x⬘兲v共x兲, ⌸共2兲共124365兲 = ⑀−1共8712兲⌸̃共2兲共78092⬘1⬘兲 v共k,k⬘兲 = v共k − k⬘兲. ⫻⑀ 共9043兲⑀ 共1⬘2⬘65兲. −1 共5.19兲 −1 共1兲 In an insulator, Eqs. 共5.8兲 and 共5.15兲 give ⌸̃ 共1233兲 = ⌸共1兲共1244兲 = 0, which implies that ⑀共1233兲 = ␦共12兲. The inverses of Eqs. 共5.15兲 and 共5.19兲, i.e., ⌸̃共1兲共1243兲 = ⌸共1兲共1265兲⑀共5643兲, 共5.20兲 ⌸̃共2兲共124365兲 = ⑀共8712兲⌸共2兲共78092⬘1⬘兲⑀共9043兲⑀共1⬘2⬘65兲, Here the spin indices were omitted because a hermitian, time-reversal invariant, local potential must be a spin scalar 共see Sec. IX兲. Contributions from the nonlocal part of the perturbation are considered in Sec. IX. A. Screened potential With this simplification, all of the polarization matrices can be reduced to the scalar form 共5.23兲, and the screened potential 共5.10兲 simplifies to58 共5.21兲 then show that the insulator Ward identities 共4.10兲 are valid for both ⌸ and ⌸̃. Since the Coulomb interaction depends only on the reduced polarization matrix 共5.11兲, Eq. 共5.16兲 can be reduced to 共6.1兲 共q兲 = v共q兲 + vc共q兲 兺 ⌸共q,q + G兲v共q + G兲. 共6.2兲 G Likewise, the local version of Eq. 共5.24兲 is ⌸共q + G,q + G⬘兲 = P共q + G,q + G⬘兲 + P共q + G,q兲vc共q兲⌸共q,q + G⬘兲. ⌸ss⬘共q + G;k,k + q + G⬘兲 共6.3兲 = ⌸̃ss⬘共q + G;k,k + q + G⬘兲 + 兺 ⌸̃共q + G,q + G⬙兲 It is convenient at this point to define a macroscopic static electronic dielectric function88 G⬙ ⫻vc共q + G⬙兲⌸ss⬘共q + G⬙ ;k,k + q + G⬘兲, which may be used to express ⌸ as a function of the regular polarization P: in which a scalar version of ⌸ is defined by ⌸共q,q + G兲 = 兺 ⌸共q;k,k + q + G兲. ⑀共k兲 = 1 − vc共k兲P共k,k兲 = 1/关1 + vc共k兲⌸共k,k兲兴, 共6.4兲 共5.22兲 共5.23兲 ⌸共q,q + G兲 = ⑀−1共q兲P共q,q + G兲, 共6.5a兲 Now vc共q + G⬙兲 is nonsingular in the limit q → 0 when G⬙ ⫽ 0, so it is convenient to regroup the series expansion of Eq. 共5.22兲 so as to isolate the terms vc共q兲:85,86 ⌸共q + G,q兲 = ⑀−1共q兲P共q + G,q兲, 共6.5b兲 k ⌸共q + G,q + G⬘兲 = P共q + G,q + G⬘兲 ⌸ss⬘共q + G;k,k + q + G⬘兲 = Pss⬘共q + G;k,k + q + G⬘兲 + P共q + G,q兲⑀−1共q兲vc共q兲P共q,q + G⬘兲. 共6.5c兲 + P共q + G,q兲vc共q兲⌸ss⬘ ⫻共q;k,k + q + G⬘兲. 共5.24兲 Here P is the sum of all polarization diagrams that cannot be separated by cutting a Coulomb line labeled with q 共although they may be split by cutting lines labeled q + G⬙ with G⬙ ⫽ 0兲. This will be called the regular polarization; it is related to the proper polarization ⌸̃ by Eq. 共5.22兲 with ⌸ → P and G⬙ ⫽ 0. Both ⌸̃ and P are well behaved in the limit q → 0, but P is more convenient for analysis of the small-q behavior of ⌸ because, unlike the case for ⌸̃, it does not require the inversion of matrices 共see Ref. 87 for further discussion and an alternative approach兲. From the relationship between P and Here all of the nonanalytic behavior at small q is contained in the factors vc共q兲 and ⑀−1共q兲. Upon substituting 共6.5兲 into 共6.2兲, one obtains the screened potentials 共q兲 = v共q兲 vc共q兲 + P共q,q + G兲v共q + G兲, ⑀共q兲 ⑀共q兲 G⫽0 兺 共6.6兲 共q + G兲 = v共q + G兲 + vc共q + G兲P共q + G,q兲共q兲 + vc共q + G兲 兺 G⬘⫽0 P共q + G,q + G⬘兲v共q + G⬘兲, 共6.7兲 where both expressions are valid for arbitrary q and G, but 165344-6 QUADRATIC RESPONSE THEORY FOR SPIN-ORBIT … PHYSICAL REVIEW B 72, 165344 共2005兲 the latter is more useful for investigating the behavior of in the neighborhood of a nonzero reciprocal lattice vector. For small q, the first term in 共6.6兲 is the macroscopic screening that occurs even for slowly varying potentials 关with v共k兲 = 0 for k outside the first Brillouin zone兴, while the second term is a local-field correction89,90 arising from the microscopic inhomogeneity of the reference crystal. B. Power series expansions The next step is to establish the small-q properties of P. Since the only singular Coulomb terms in P共q + G , q + G⬘兲 are the factors vc共q + G⬙兲 with G⬙ ⫽ 0, the regular polarization P共q + G , q + G⬘兲 is analytic for q ⬍ Gmin, where Gmin is the magnitude of the smallest nonzero reciprocal lattice vector. In addition, the symmetry property 共E4兲 gives P共x , x⬘兲 = P共x⬘ , x兲 or P共k,k⬘兲 = P共− k⬘,− k兲, 共6.9兲 lim P共q,q + G兲 = 0. q→0 From these results, we see that the matrix PGG⬘共q兲 ⬅ P共q + G , q + G⬘兲 has the Taylor series expansion P00共q兲 = P0G共q兲 = 关1兴 P0G + + 关4兴 P00 关2兴 P0G 关0兴 + + 关6兴 P00 关3兴 P0G 关1兴 + + O共q 兲, 共6.10a兲 8 关4兴 P0G v共k兲 = van共k兲 + vc共k兲共k兲, + O共q 兲, 共6.10b兲 关2兴 5 PGG⬘共q兲 = PGG⬘ + PGG⬘ + PGG⬘ + O共q3兲, 共6.10c兲 关l兴 with PG0共q兲 = P0,−G共−q兲. Here PGG denotes a general poly⬘ nomial of order l in the Cartesian components of q; for example, 关2兴 关2兴 ␣ P00 = P00 共q兲 = q␣q P00 , 共k兲 = − van共k兲 = q→0 q ␣q  , q2 1 1 关4兴 关6兴 关4兴 2 = 关1 + wc共q兲共P00 + P00 兲 + w2c 共q兲共P00 兲 兴 + O共q6兲, ⑀共q兲 ⑀共q̂兲 共6.13兲 in which wc共q兲 = vc共q兲 / ⑀共q̂兲. 共k兲 = 0 + 关1兴 + 关2兴 + ¯ , 共6.16兲 共6.17兲 although the linear response can always be treated as a superposition of individual ions. For G ⫽ 0 , v共q + G兲 is analytic for q ⬍ G, with the Taylor series 关1兴 关2兴 关3兴 4 v共q + G兲 = v关0兴 G + vG + vG + vG + O共q 兲, 共6.18兲 关0兴 in which vG = v共G兲. A similar expansion is valid for vc共q + G兲. D. Effective macroscopic density It is now convenient to rewrite Eq. 共6.6兲 in a form modeled after the familiar expressions for screening in a homogeneous system:92 共q兲 = van共q兲 + C. Pseudopotential To proceed further it is necessary to make some assumptions about the perturbing pseudopotential v共k兲. This is taken 共6.15兲 Therefore, the only term in vc共k兲共k兲 that does not vanish in the limit k → 0 is the divergent term −4Zv / ⍀k2. This term has been eliminated91 at k = 0 关by the definition 共5.4兲 of vc共k兲兴 because the ion is assumed to be accompanied by Zv electrons, so that the crystal remains neutral after the perturbation. If the perturbation contains more than one ion 共e.g., the quasiatoms defined in Appendix A兲, 共k兲 is just a general Taylor series 共6.12兲 in which q̂ = q / q. This behavior contributes nonanalytic terms in the small-q expansion of ⑀−1共q兲: 1 2 关2Zvr20 + g共k2r20兲兴e−共kr0兲 /2 , ⍀ 共k兲 = 0 + 4k4 + 6k6 + ¯ . ␣ P00 ␣ lim ⑀共q兲 ⬅ ⑀共q̂兲 = 1 − 4 P00 Zv 2 1 + 21 共kr0兲2兴e−共kr0兲 /2 , 关 ⍀ where Zv is the charge of the ion, r0 is a core radius parameter, and g共x兲 is a cubic polynomial given in Refs. 41 and 42. Here 共k兲 has been defined in such a way that its Taylor series contains no term proportional to k2: 共6.11兲 is a constant, and a sum over the Cartesian in which components ␣ and  is implicit. In the limit of small q, the dielectric function of an insulator therefore tends toward a finite but direction-dependent limit: 共6.14兲 in which van共k兲 and 共k兲 are entire analytic functions of k, and 共k兲 represents a portion of the pseudocharge density of the perturbation 关the other portion being given by k2van共k兲 / 4兴. For a single ion, these functions have the form of a Gaussian times a polynomial in k2:41,42 共6.8兲 while the Ward identity 共4.10兲 for an insulator implies that 关2兴 P00 to be a superposition of spherically symmetric ionic perturbations, each of which has the Gaussian form used in Refs. 40–42. The pseudopotential can therefore be written as in which 165344-7 vc共q兲n̄共q兲 , ⑀共q兲 共6.19兲 PHYSICAL REVIEW B 72, 165344 共2005兲 BRADLEY A. FOREMAN n̄共q兲 = 共q兲 + P00共q兲van共q兲 + 兺 P0G共q兲v共q + G兲 G⫽0 关4兴 关1兴 共q兲 = van共q兲 + wc共q兲共n̄关1兴 + n̄关2兴 + n̄关3兴兲 + w2c 共q兲P00 n̄ 共6.20兲 is an effective macroscopic electron density, which is analytic for q ⬍ Gmin. This includes the 共partial兲 bare charge 共q兲, the macroscopic charge induced by van共q兲, and the local-field corrections. Note that since 共q兲 = v共q兲 + vc共q兲n共1兲共q兲, one can also write n̄共q兲 = 关共q兲 + n共1兲共q兲兴⑀共q兲. Also note that n̄0 = 0, because the second term in 共6.20兲 is O共q2兲 and the last term is O共q兲. The leading contributions to 共q兲 for small q are therefore 关4兴 关6兴 关4兴 2 共q兲 = 0关wc共q兲 + w2c 共q兲共P00 + P00 兲 + w3c 共q兲共P00 兲 兴 + wc共q兲 关4兴 关1兴 ⫻共n̄关1兴 + n̄关2兴 + n̄关3兴 + n̄关4兴兲 + w2c 共q兲P00 共n̄ + n̄关2兴兲 + van共q兲 + O共q3兲, 共6.21兲 in which van共q兲 is to be replaced by its Taylor series expansion. The first set of terms in 共6.21兲 is proportional to 0 = −Zv / ⍀. These terms are just the power series expansion for 0vc共q兲 / ⑀共q兲. Such terms are present in general, but they vanish for isovalent perturbations 共e.g., Al substituting for Ga in GaAs兲. The remaining nonanalytic terms depend on n̄关l兴, where l 艌 1. The symmetry of n̄关l兴 may differ from that of 关l兴. For example, the term wc共q兲n关1兴共q兲 would contribute a dipole field if it were present, but 关1兴 vanishes for a spherically symmetric atom. However, since the symmetry of P is the same as that of the reference crystal, the symmetry of n̄ is just the site symmetry at the position of the ionic perturbation 共i.e., the maximal common subgroup of the reference crystal space group and the full rotation group at the given atomic site兲. Therefore, for crystals of sufficiently low symmetry 共e.g., wurtzite兲, n̄关1兴 may contribute a dipole field even though 关1兴 does not. + O共q2兲, 共6.23兲 which includes additional octopole terms. Finally, for isovalent perturbations in crystals with site symmetry that does not support a dipole moment 共class III兲, one would use 关4兴 关2兴 共q兲 = van共q兲 + wc共q兲共n̄关2兴 + n̄关3兴 + n̄关4兴兲 + w2c 共q兲P00 n̄ + O共q3兲. 共6.24兲 As an explicit example, consider the case of isovalent substitutions in a crystal with the zinc-blende or diamond structure 共space group T2d or O7h兲, both of which have site symmetry Td at the atomic sites. In this case 0 = 0, and the only invariants of order q4 or lower are 1, q2 , qxqyqz , q4, and ␣ q4x + q4y + qz4. Hence, the quadratic terms are isotropic 共P00 关2兴 2 = P2␦␣ , n̄ = n̄2q 兲 and the long-wavelength dielectric function reduces to a constant 关⑀共q̂兲 = ⑀ = 1 − 4 P2兴. The leading contributions to 共q兲 may be written as 共q兲 = 4n̄2 q xq y q z 共1 − ␦q0兲 + C1 + C2q2 + C3 2 ⑀ q + C4 q4x + q4y + qz4 + O共q3兲, q2 where Ci is a constant. The terms C3 and C4 represent octopole and hexadecapole moments, respectively. F. Nonzero reciprocal lattice vectors Turning now to 共q + G兲, Eq. 共6.7兲 can be written in the condensed notation 共q + G兲 = RG0共q兲共q兲 + G共q兲, 关4兴 共q兲 = van共0兲 + wc共q兲共0 + n̄关1兴 + n̄关2兴兲 + 0w2c 共q兲P00 + O共q兲, 共6.22兲 which contains monopole, dipole, and quadrupole terms, plus a correction to the monopole term describing the wave vector dependence of the dielectric function. For isovalent perturbations in crystals with atomic site symmetry that supports a dipole moment 共class II兲, a suitable approximation would be 共6.26兲 in which RG0共q兲 and G共q兲 are analytic for q ⬍ Gmin: RG0共q兲 = 再 1 if G = 0, vc共q + G兲PG0共q兲 if G ⫽ 0 冋 冎 共6.27兲 G共q兲 = 共1 − ␦G0兲 v共q + G兲 + vc共q + G兲 E. Special cases The general expression 共6.21兲 is quite cumbersome and is unlikely to be used in its entirety for any particular material system. In many cases one would only be interested in retaining terms that are two orders in q higher than the lowest nonvanishing term. Thus, for heterovalent perturbations with 0 ⫽ 0 共i.e., class I of the Introduction兲, Eq. 共6.21兲 could be simplified to 共6.25兲 ⫻ 兺 G⬘⫽0 册 PGG⬘共q兲v共q + G⬘兲 . 共6.28兲 For G ⫽ 0, Eqs. 共6.10b兲 and 共6.18兲 show that RG0共q兲 is of order q or higher. The explicit form of the Taylor series for RG0共q兲 is determined by finding the invariants of the group of the wave vector G in the reference crystal, where different G vectors are treated as inequivalent. Thus, for general G, the linear term is nonvanishing. However, the leading term in the Taylor series for G共q兲 共with G ⫽ 0兲 is a constant. Hence, for G ⫽ 0, the nonanalytic terms in 共q + G兲 are at least one order in q higher than those in 共q兲. In the zincblende example discussed above, one has ␣ 共q + G兲 = C3RG0 q ␣q xq y q z + G共q兲 + O共q3兲, 共6.29兲 q2 ␣ is the linear coefficient in which G共q兲 is analytic, and RG0 in the Taylor series for RG0共q兲. The nonanalytic term in 165344-8 QUADRATIC RESPONSE THEORY FOR SPIN-ORBIT … PHYSICAL REVIEW B 72, 165344 共2005兲 共6.29兲 is a hexadecapole moment that is invariant with respect to the group of G. VII. QUADRATIC RESPONSE TO A LOCAL PERTURBATION To calculate the quadratic density n共2兲共k兲, it is helpful to begin by considering the following partial density obtained from the local version of Eqs. 共4.7兲, 共5.5兲, and 共5.19兲: ñ共2兲共k兲 = 1 兺 ⬘ 兺 ⌸̃共2兲共k,k1 + G1,k − k1 + G2兲 2 k1 G1G2 ⫻共k1 + G1兲共k − k1 + G2兲, 共7.1兲 where the summation on k1 is limited to the first Brillouin zone of the reference crystal. Here the proper polarization ⌸̃共2兲共k , k1 , k2兲 vanishes unless k = k1 + k2 + G, where G is any reciprocal lattice vector. It satisfies the symmetry relations 共E4兲, the reduced form of which is R0G共q兲 = n̄A共2兲共q兲 = n̄B共2兲共q兲 = 共2兲 ␥ 共k,k1,k2兲 = k1k2␥⌸̃G00 + O共k3兲, ⌸̃G00 共2兲 ⌸̃GG 1G2 1G2 + O共k2兲, 共k,k1,k2兲 = ⌸̃GG1G2 + O共k兲, 共7.4兲 where the order of the leading term is equal to the number of G vectors that are zero. Here O共kn兲 denotes a term of order k pkq1kr2, where n = p + q + r. The partial quadratic density 共7.1兲 generates a Coulomb potential v共2兲共q兲 = vc共q兲ñ共2兲共q兲, which is then screened to produce 共2兲共q兲 of Eq. 共5.9兲. This potential is calculated by replacing v共q兲 with v共2兲共q兲 in Eqs. 共6.6兲 and 共6.7兲. The result may be written as 共2兲共q兲 = vc共q兲n̄共2兲共q兲 , ⑀共q兲 共7.5兲 where n̄共2兲共q兲 = n共2兲共q兲⑀共q兲 is an effective “external” density n̄共2兲共q兲 = 兺 R0G共q兲ñ共2兲共q + G兲, G in which 1 2 1 2 兺k ⬘A共q,k,q − k兲共k兲共q − k兲, 兺k ⬘关B共q,k,q − k兲共q − k兲 + B共q,q − k,k兲共k兲兴, 1 2 兺k ⬘C共q,k,q − k兲. 共7.8兲 A共k,k1,k2兲 = k␣k1k2␥A␣␥ + O共k4兲, 共7.3兲 共2兲 ␣␥ 共k,k1,k2兲 = k␣k1k2␥⌸̃000 + O共k4兲, ⌸̃000 ␣ 共k,k1,k2兲 = k␣⌸̃0G 共7.7兲 B共k,k1,k2兲 = k␣k2␥B␣␥ + O共k3兲, With these constraints, the Taylor series expansion of the 共2兲 共k , k1 , k2兲 = ⌸̃共2兲共k + G , k1 polarization matrix ⌸̃GG 1G2 + G1 , k2 + G2兲 has a form similar to that given for P in Eq. 共6.10兲: 1G2 冎 Here the functions A , B, and C, which are defined in Appendix F, have the Taylor series expansions It also satisfies the Ward identity 共4.10兲: 共2兲 ⌸̃0G if G = 0, P0G共q兲vc共q + G兲 if G ⫽ 0 n̄C共2兲共q兲 = 共7.2兲 lim ⌸̃共2兲共k,k1,k − k1 + G2兲 = 0. 1 In Eq. 共7.6兲, ñ共2兲共q + G兲 is given by 共7.1兲, where 共k + G兲 can be expressed in terms of 共k兲 using Eq. 共6.26兲. The resulting expression for 共7.6兲 can be written as n̄共2兲共q兲 = n̄A共2兲共q兲 + n̄B共2兲共q兲 + n̄C共2兲共q兲, in which ⌸̃共2兲共k,k1,k2兲 = ⌸̃共2兲共k,k2,k1兲 = ⌸̃共2兲共− k1,− k,k2兲. k→0 再 共7.6兲 C共k,k1,k2兲 = k␣C␣ + O共k2兲. 共7.9兲 For q values inside the first Brillouin zone, the functions A , B, and C in 共7.8兲 are analytic for all k values included in the summation, but 共k兲 is nonanalytic at k = 0. Therefore it is possible that the terms in 共7.8兲 may produce nonanalytic behavior in n̄共2兲共q兲. n̄C共2兲共q兲 is obviously analytic in q, as is the second term in n̄B共2兲共q兲. The first term in n̄B共2兲共q兲 is as well, since a small variation ␦q can be eliminated from 共q − k兲 with an equal variation ␦k = ␦q. This slightly shifts the zone boundary in the summation, but 共q − k兲 is analytic at the zone boundary, so n̄B共2兲共q兲 is analytic for small q. However, for n̄A共2兲共q兲 this argument is no longer valid. The singularities in 共k兲 and 共q − k兲 merge when q = 0, producing nonanalytic behavior in n̄A共2兲共q兲 at this point. The contribution from the nonanalytic part of n̄A共2兲共q兲 is examined in Appendix G, where it is shown to be negligible under all three approximation schemes defined in the Introduction. Therefore, only the analytic part of n̄共2兲共q兲 is retained here. The leading contributions to the quadratic screened potential are therefore 共2兲 共2兲共q兲 = wc共q兲共q␣n̄␣共2兲 + q␣qn̄␣ 兲 + O共q兲, 共7.10兲 共2兲 where n̄␣共2兲 and n̄␣ are Taylor series coefficients for the ana共2兲 lytic part of n̄ 共q兲. The absence of a constant term in the power series for n̄共2兲共q兲 is a consequence of the Ward identity 共7.3兲. Equation 共7.10兲 is used in its full form only for isova共2兲 term is neglent class III perturbations. For class II, the n̄␣ ligible, while for class I, the entire contribution from 共2兲共q兲 is negligible.93 In the vicinity of a nonzero reciprocal lattice vector, 共2兲共q + G兲 can be written in a form similar to 共6.26兲: 165344-9 PHYSICAL REVIEW B 72, 165344 共2005兲 BRADLEY A. FOREMAN 共2兲共q + G兲 = RG0共q兲共2兲共q兲 + 共2兲 G 共q兲, 共7.11兲 in which 共2兲 G 共q兲 is given by Eq. 共6.28兲 with v共k兲 replaced by v共2兲共k兲 = vc共k兲ñ共2兲共k兲. Using the same type of analysis as before, one finds that the nonanalytic part of 共2兲 G 共q兲 is O共q兲 for class I, O共q3兲 for class II, and O共q5兲 for class III. Therefore, 共2兲 共0兲 is well defined, and the leading terms in the limit G 共2兲 共q + G兲 are given by 共1兲 Wss⬘共k,k⬘ ;G,G⬘ ; 兲 = G⬙⫽0 Since the leading terms here are O共q0兲, this contribution is negligible for classes I and II. Note that the quadratic response for a heterostructure cannot be written as a superposition of spherically symmetric atomic perturbations; one must also include diatomic perturbations with axial symmetry C⬁v 共for a heteronuclear diatomic molecule兲 or D⬁h 共for a homonuclear diatomic molecule兲.94 The symmetry of n̄共2兲共q兲 is determined by the maximal common subgroup of the reference crystal space group and the molecular point group. For example, for a perturbation at neighboring atomic sites in zinc-blende, the symmetry of n̄共2兲共q兲 is C3v, which supports a nonvanishing dipole moment n̄␣共2兲. In general, n̄␣共2兲 is nonvanishing for any heteronuclear perturbation, because C⬁v itself permits the existence of a dipole. Such dipoles therefore always appear in heterostructures involving more than one type of atomic perturbation 共e.g., InAs/ GaSb兲. 共This property of the nonlinear response was deduced from numerical calculations of band offsets in 共2兲 1 共2兲 − 3 n̄ ␦␣ is Ref. 95.兲 Furthermore, the quadrupole term n̄␣ nonvanishing for any diatomic perturbation, since isotropy requires cubic symmetry. VIII. SELF-ENERGY FOR A LOCAL PERTURBATION A. Linear terms The above results may now be used to calculate the selfenergy ⌺ and the total potential V defined in Eqs. 共3.3兲 and 共5.12兲. The total linear potential 共5.12兲 is given for the case of a local perturbation by 共1兲 Vss⬘共k + G,k⬘ + G⬘ ; 兲 共8.1兲 G⬙ where q = k − k⬘. Here 共q + G⬙兲 can be expressed in terms of 共q兲 using Eq. 共6.26兲; this yields 共1兲 Vss⬘共k + G,k⬘ + G⬘ ; 兲 = ⌳ss⬘共k,k⬘ ;G,G⬘ ; 兲共q兲 共1兲 in which 共8.3兲 is an analytic function of k and k⬘ 共and therefore also of q兲. The nonanalytic terms are all contained in the screened potential 共q兲 = van共q兲 + vc共q兲n̄共q兲 / ⑀共q兲, which is multiplied by the effective macroscopic vertex function 共1兲 = 兺 ˜⌫ss⬘共k + G,k⬘ + 共G⬘ ;q + G⬙ ; 兲RG⬙0共q兲. 共7.12兲 + Wss⬘共k,k⬘ ;G,G⬘ ; 兲, ˜⌫共1兲 共k + G,k⬘ + G⬘ ;q + G⬙ ; 兲 共q兲 G⬙ ss⬘ ⌳ss⬘共k,k⬘ ;G,G⬘ ; 兲 共2兲  共2兲共q + G兲 = wc共q兲共q␣qn̄␣共2兲RG0 兲 + G 共0兲 + O共q兲. 共1兲 = 兺 ˜⌫ss⬘共k + G,k⬘ + G⬘ ;q + G⬙ ; 兲共q + G⬙兲, 兺 共8.2兲 共8.4兲 G⬙ Since ⌳ss⬘ is an analytic function of k and k⬘, it can be expanded in a Taylor series 关treating q = k − k⬘ and Q = 21 共k + k⬘兲 as the independent variables兴, with the result 共0兲 ⌳ss⬘共k,k⬘ ;G,G⬘ ; 兲 = 关␦ss⬘␦GG⬘ − ⌺ss⬘共G,G⬘ ; 兲/兴 共␣兩·兲 共·兩␣兲 + q␣⌳ss⬘ 共G,G⬘ ; 兲 + Q␣⌳ss⬘ 共G,G⬘ ; 兲 共␣兩·兲 共·兩␣兲 + q␣q⌳ss⬘ 共G,G⬘ ; 兲 + Q␣Q⌳ss⬘ 共G,G⬘ ; 兲 共␣兩兲 + q␣Q⌳ss⬘ 共G,G⬘ ; 兲 + O共q3兲. 共8.5兲 Here the lowest-order term is determined by the Ward iden共␣兩兲 tity 共3.11兲, and the Taylor series coefficients such as ⌳ss ⬘ are given in Appendix H. The analytic potential 共8.3兲 can be expanded in the same way: 共1兲 共1兲 Wss⬘共k,k⬘ ;G,G⬘ ; 兲 = Wss⬘共0,0;G,G⬘ ; 兲 共␣兩·兲 共·兩␣兲 + q␣Wss⬘ 共G,G⬘ ; 兲 + Q␣Wss⬘ 共G,G⬘ ; 兲 共␣兩·兲 共·兩␣兲 + q␣qWss⬘ 共G,G⬘ ; 兲 + Q␣QWss⬘ 共G,G⬘ ; 兲 共␣兩兲 + q␣QWss⬘ 共G,G⬘ ; 兲 + O共q3兲. 共8.6兲 In the expansion 共8.5兲, a term such as q␣ appears in the total potential 共8.2兲 as a multiplicative factor in front of the screened potential 共q兲. In coordinate space, this term therefore takes the gradient of 共x兲, generating the ␣ component of the macroscopic electric field produced by the perturbation v共x兲. Likewise, the term Q␣ has the form of a 共symmetrized兲 crystal momentum operator that acts upon the envelope functions in an effective-mass theory. The Taylor series 共8.6兲 for the analytic potential 共8.3兲 is interpreted in the same way, except that these terms produce only short-range localized potentials because they are analytic functions of q. The various terms in Eq. 共8.5兲 therefore give rise to various long-range spin-dependent potentials whose particular 共␣兩·兲 form is determined by the symmetry of the coefficients ⌳ss , ⬘ etc. The specific term that generates the long-range Rashba 共␣兩兲 effect is ⌳ss , since this term is linear in the electric field ⬘ q␣共q兲 and linear in the crystal momentum Q. The usual short-range part of the Rashba coupling is generated by the 共␣兩兲 analogous term Wss in Eq. 共8.6兲. ⬘ The complete expression for V共1兲 is obtained by inserting the expansion 共8.5兲 for ⌳ and one of the three expansions 165344-10 QUADRATIC RESPONSE THEORY FOR SPIN-ORBIT … PHYSICAL REVIEW B 72, 165344 共2005兲 共6.22兲, 共6.23兲, and 共6.24兲 for 共q兲 into Eq. 共8.2兲. For the specific example of isovalent perturbations in zinc-blende materials treated in Eq. 共6.25兲, one finds 共1兲 Vss⬘共k = 共2b兲 Vss⬘ 共k + G,k⬘ + G⬘ ; 兲 = 兺 ˜⌫ss⬘共k + G,k⬘ + G⬘ ;q + G⬙ ; 兲共2兲共q + G⬙兲, 共8.10兲 共1兲 G⬙ + G,k⬘ + G⬘ ; 兲 where 共2兲共q + G⬙兲 was given previously in Eq. 共7.11兲. Inserting this expression into Eq. 共8.10兲, one obtains 1 − ␦q0 共0兲 兵关␦ss⬘␦GG⬘ − ⌺ss⬘共G,G⬘ ; 兲/兴 q2 共2b兲 Vss⬘ 共k + G,k⬘ + G⬘ ; 兲 = ⌳ss⬘共k,k⬘ ;G,G⬘ ; 兲共2兲共q兲 ⫻ 关4n̄2q2/⑀ + C3qxqyqz + C4共q4x + q4y + qz4兲兴 共␣兩·兲 共2b兲 + Wss⬘ 共k,k⬘ ;G,G⬘ ; 兲, 共·兩␣兲 + C3qxqyqz关q␣⌳ss⬘ 共G,G⬘ ; 兲 + Q␣⌳ss⬘ 共G,G⬘ ; 兲兴其 + analytic terms + O共q3兲, 共8.11兲 in which ⌳ was defined in Eq. 共8.4兲, and 共8.7兲 where the analytic terms include W共1兲 and contributions from the analytic part of 共q兲. From this result it can be seen that the Rashba effect in isovalent zinc-blende materials does not include any long-range terms 共to within the accuracy of the 共␣兩兲 present approximation scheme兲, since ⌳ss contributes only ⬘ to O共q3兲. However, there are other long-range spin-splitting terms of O共q2兲 or lower, and the Rashba effect does contribute nonnegligible long-range terms for perturbations in classes I and II. 共2b兲 Wss⬘ 共k,k⬘ ;G,G⬘ ; 兲 = 兺 G⬙⫽0 ˜⌫共1兲 共k + G,k⬘ + G⬘ ;q + G⬙ ; 兲共2兲 共q兲. ss⬘ G⬙ 共8.12兲 Unlike the case for Eq. 共8.3兲, this is not an analytic function of q. However, as discussed below Eq. 共7.11兲, the nonanalytic portion is O共q兲 and therefore vanishes at q = 0. An explicit expression for V共2b兲 can now be obtained by inserting the expansion 共7.10兲 for 共2兲共q兲 into Eq. 共8.11兲: 共2b兲 Vss⬘ 共k + G,k⬘ + G⬘ ; 兲 共0兲 = q␣wc共q兲兵n̄␣共2兲关␦ss⬘␦GG⬘ − ⌺ss⬘共G,G⬘ ; 兲/ B. Quadratic terms 共兩·兲 共2a兲 Vss⬘ 共k 共2兲 + G,k⬘ + G⬘ ; 兲 = 兺 ⬘ 兺 兺 ˜⌫ss⬘共k + G,k⬘ + G⬘ ;k1 1 2 k1 G1 G2 + G1,q − k1 + G2 ; 兲共k1 + G1兲 ⫻共q − k1 + G2兲. 共8.8兲 Upon inserting Eq. 共6.26兲 for 共k + G兲 into the right-hand side, one obtains an expression for V共2a兲 very similar to that found in Eqs. 共7.8兲 and 共F1兲 for the effective quadratic density n̄共2兲共q兲. Just as before, there are both analytic and nonanalytic contributions. The nonanalytic contributions can be evaluated using the method outlined in Appendix G; the results show that the nonanalytic terms in V共2a兲 are O共q−1兲 for class I, O共q兲 for class II, and O共q3兲 for class III. 共An explicit expression for the leading nonanalytic term in class I was given previously by Sham.58,96兲 Therefore, the nonanalytic contributions are negligible in all three cases, and the leading term in V共2a兲 is just a constant: 共2a兲 共·兩兲 + q⌳ss⬘ 共G,G⬘ ; 兲 + Q⌳ss⬘ 共G,G⬘ ; 兲兴其 Turning now to the quadratic response, the two contributions to V共2兲 in Eq. 共5.12兲 will be denoted V共2a兲 and V共2b兲, respectively. The first of these is given by 共0兲 共2兲 + q␣qwc共q兲兵n̄␣ 关␦ss⬘␦GG⬘ − ⌺ss⬘共G,G⬘ ; 兲/兴其 共2b兲 + Wss⬘ 共0,0;G,G⬘ ; 兲 + O共q兲. 共8.13兲 The first and second terms are dipole and quadrupole potentials, respectively, while the last term is just a constant. In class II, only the leading O共q−1兲 dipole term is retained; in class I, the entire expression 共8.13兲 is neglected. IX. NONLOCAL PERTURBATIONS An arbitrary nonlocal potential can be separated into local and nonlocal parts 共although this separation is not unique兲: loc nl vss⬘共x,x⬘兲 = vss⬘共x,x⬘兲 + vss⬘共x,x⬘兲. 共9.1兲 loc 共x , x⬘兲 has the form of Eq. 共6.1兲 and is Here the local part vss ⬘ treated according to the methods developed above. This section considers the changes in the preceding expressions that may be necessary in the case of nonlocal perturbations, particularly those involving spin-orbit coupling. A. Analytic form 共2a兲 Vss⬘ 共k + G,k⬘ + G⬘ ; 兲 = Vss⬘ 共G,G⬘ ; 兲 + O共q兲. A general nonlocal potential can be written as 共8.9兲 This term is negligible under the approximation schemes for classes I and II. Finally, the second contribution to V共2兲 in Eq. 共5.12兲 is given by vss⬘共x,x⬘兲 = ␦ss⬘v0共x,x⬘兲 + ss⬘ · v共x,x⬘兲, 共9.2兲 where v0 is a scalar relativistic potential, is the Pauli matrix, and v is a pseudovector 共similar to orbital angular momentum兲 that accounts for spin-orbit coupling. If v is Hermitian and time-reversal invariant, then v0 is real and 165344-11 PHYSICAL REVIEW B 72, 165344 共2005兲 BRADLEY A. FOREMAN symmetric, while v共x , x⬘兲 = −v共x⬘ , x兲 is imaginary and antisymmetric. Thus v can have no local component, and a local time-reversal-invariant potential must be a spin scalar: loc vss⬘共x,x⬘兲 = ␦ss⬘␦共x − x⬘兲vloc共x兲. 共9.3兲 In the norm-conserving pseudopotential formalism, the nl 共x , x⬘兲 is connonlocal part of the ionic pseudopotential vss ⬘ fined to a small region near the nucleus, typically either having the form of a polynomial times a Gaussian40–42 or vanishing absolutely outside a core region of radius rc.97,98 As a nl result, vss 共k , k⬘兲 is an entire analytic function of k and k⬘. ⬘ It is important to note that this analytic form relies upon a physical approximation. In an all-electron calculation where the pseudopotential approximation is not used, the spin-orbit coupling does in general include a contribution from the long-range Coulomb part of the ionic potential. The choice nl of an analytic pseudopotential vss 共k , k⬘兲 is therefore an ap⬘ proximation, in which the spin-orbit coupling is assumed to be dominated by the contribution from the ionic core. Conventional norm-conserving pseudopotentials incorporate all relativistic corrections of order Z2␣2 共where Z is the atomic number and ␣ is the fine-structure constant兲, but neglect various terms of order ␣2,99,100 including the spin-orbit coupling from the long-range 共but slowly varying兲 Coulomb potential outside the core region. This approximation is used in all that follows. It greatly simplifies the analysis of spin-dependent perturbations, as shown below. B. Screening Consider now the description of screening for a spindependent perturbation. The relationship between the total polarization ⌸ and the regular polarization P was given above in Eq. 共5.24兲 for a general nonlocal potential. Setting G = 0 in this equation gives ⌸ss⬘共q;k,k + q + G⬘兲 = ⑀−1共q兲Pss⬘共q;k,k + q + G⬘兲, 共9.4兲 where ⑀共q兲 is the same scalar dielectric function defined above in Eq. 共6.4兲. Substituting this result into Eq. 共5.24兲 then yields nnl共q兲 = 兺 兺 P⬘共q;k⬙,k⬙ + q + G兲v⬘共k⬙ + q + G,k⬙兲 nl k⬙ 共9.7兲 is a correction to n̄共q兲 from the nonlocal part of the perturbation. Since nnl共q兲 = O共q兲 is an analytic function with the same site symmetry as n̄共q兲, this term does not produce any qualitative changes in ; it merely renormalizes n̄共q兲. The only qualitatively new contribution to is the spinnl dependent term vss 共k , k⬘兲 itself, which is analytic. ⬘ For wave vectors in the vicinity of a nonzero reciprocal lattice vector, it is convenient to write Eq. 共5.10兲 in the following alternative form: G ss⬘共k + G,k⬘兲 = RG0共q兲ss⬘共k,k⬘兲 + ss 共k,k⬘兲. 共9.8兲 ⬘ Here RG0共q兲 was defined in Eq. 共6.27兲, and G ss 共k,k⬘兲 = ␦ss⬘关G共q兲 + 共1 − ␦G0兲vc共q + G兲nnl共q + G兲兴 ⬘ nl an 共9.9兲 is a generalization of the function G共q兲 defined in Eq. 共6.28兲. This is an analytic function of k and k⬘ for q ⬍ Gmin. If Eqs. 共9.6兲 and 共9.8兲 are inserted into the nonlocal version of Eq. 共8.1兲 关i.e., Eq. 共5.12兲兴, it is apparent that the nonlocal part of the perturbation produces no qualitative change in the total linear potential V共1兲. The only change is a simple renormalization of the analytic and nonanalytic terms in V共1兲. The same conclusion also holds for the quadratic potential V共2兲. Thus, the correct qualitative form of V共1兲 and V共2兲 can be derived by ignoring the nonlocal 共and spin-dependent兲 part of the perturbing potential, and including spin only in the vertex function ⌳ and the analytic parts of V共1兲 and V共2兲. This is precisely the approach used in Secs. VI–VIII. The key to obtaining this simple result is the fact that nl vss⬘共k , k⬘兲 is analytic. As shown above, this relies upon the approximation of neglecting spin-orbit coupling outside the atomic cores. Such an approximation would also be possible 共and even desirable for its simplicity兲 in an all-electron calculation where the core electrons are treated explicitly. X. SUMMARY AND CONCLUSIONS 共9.5兲 Equations 共9.4兲 and 共9.5兲 replace the scalar equations 共6.5兲 derived previously. If the perturbation is now separated into local and nonlocal parts, the linear screened potential 共5.10兲 can be written in a form similar to 共6.19兲: ss⬘共k,k⬘兲 = vss⬘共k,k⬘兲 + ␦ss⬘ nl + vss⬘共k + G,k⬘兲 − RG0共q兲vss⬘共k,k⬘兲 ⌸ss⬘共q + G;k,k + q + G⬘兲 = Pss⬘共q + G;k,k + q + G⬘兲 vc共q兲 P 共q;k,k + q + G⬘兲. + P共q + G,q兲 ⑀共q兲 ss⬘ G vc共q兲关n̄共q兲 + nnl共q兲兴 , ⑀共q兲 共9.6兲 nl an 共k , k⬘兲 = vss 共k , k⬘兲 + ␦ss⬘van共q兲 , q = k − k⬘ , n̄共q兲 is in which vss ⬘ ⬘ the effective density 共6.20兲 for the local potential, and This paper has presented an analysis of the self-energy of an electron in a lattice-matched semiconductor heterostructure for small values of the crystal momentum. A general theory of nonlinear response for nonlocal spin-dependent perturbations was developed in terms of vertex functions and the static polarization, and applied to the case of quadratic response in a periodic insulator at zero temperature. A set of Ward identities was established for nonlocal spin-dependent potentials. The heterostructure perturbation was separated into a local spin-independent part and a nonlocal spindependent part, and the contributions from these were analyzed separately. Due to the neglect of spin-orbit coupling outside the atomic cores, the nonlocal part of the potential is analytic in momentum space. As a result, the nonlocal part of 165344-12 QUADRATIC RESPONSE THEORY FOR SPIN-ORBIT … PHYSICAL REVIEW B 72, 165344 共2005兲 the perturbation merely renormalizes the contributions from the local part. The main results of the paper are presented in Eqs. 共8.2兲, 共8.9兲, and 共8.11兲. The total linear potential 共8.2兲 has the form V共1兲 = ⌳ + W共1兲, in which all of the nonanalytic contributions come from the screened scalar potential . This has a form 共q兲 = van共q兲 + vc共q兲n̄共q兲 / ⑀共q兲 similar to that for screening in a homogeneous system, except that the effective density n̄共q兲 has the site symmetry of the perturbation and the macroscopic dielectric function ⑀共q兲 has the symmetry of the reference crystal. Spin-dependent contributions come from the analytic part W共1兲 and the vertex function ⌳. The vertex function can be expanded in a Taylor series 共8.5兲, in which q␣ takes the gradient 共in coordinate space兲 of 共q兲, while Q␣ is a crystal momentum operator in effective-mass theory. The generalized Rashba effect comes from the term linear in q␣ and Q, but there are other spin-splitting contributions from the lower-order terms as well. A more detailed analysis of the various terms is given in the following paper on effectivemass theory.59 The total quadratic potential of Eqs. 共8.9兲 and 共8.11兲 has a similar form V共2兲 = ⌳共2兲 + W共2兲, in which W共2兲 is analytic to within the accuracy of the approximation scheme defined in the Introduction. The quadratic screened potential is 共2兲共q兲 = vc共q兲n̄共2兲共q兲 / ⑀共q兲, where the effective external density n̄共2兲共q兲 has the site symmetry of a diatomic perturbation in the reference crystal. Due to the Ward identities for an insulator, the leading term in the power series expansion of n̄共2兲共q兲 is a dipole term. The Rashba term in the quadratic potential is always negligible under the approximation scheme used here. The results derived in this paper are used to develop a first-principles effective-mass theory in the following paper.59 The present results are of crucial importance in establishing clearly defined limitations on the validity of this theory. Most previous formulations of effective-mass theory have been based on non-self-consistent empirical pseudopotentials, for which the possibility of long-range Coulomb interactions is not even considered. However, as shown here, long-range potentials arising from nonanalytic terms in the screening potential—and even the charge density itself— must be considered in general. The omission of such terms is partially justified 共to a certain order of approximation兲 in isovalent zinc-blende systems, where high crystal symmetry eliminates the contributions from dipole and quadrupole terms in the linear response.51 However, it is not fully justified even in zincblende, since the leading octopole terms are of a lower order than the position dependence of the effective mass, which is often included in heterostructure effective-mass calculations. The following paper59 accounts for all terms of the same order as the position dependence of the effective mass, including the octopole and hexadecapole potentials derived here in Eqs. 共6.25兲 and 共8.7兲. A pioneering paper by Sham on effective-mass theory for shallow impurity states58 dealt with many of the same issues 共for local spin-independent potentials兲, but at a lower order of approximation. In particular, Sham considered only the lowest-order terms in cubic crystals. At this level of approximation, the total polarization can be treated as analytic 关see Eq. 共4.8兲 of Ref. 58兴, whereas the present Eq. 共6.5兲 shows that this is no longer true for terms of higher order 共such as those needed for the analysis in Ref. 59兲 or crystals of lower symmetry. The present work provides a systematic framework for extending Sham’s analysis to crystals of general symmetry and terms of arbitrary order. The value of establishing such a framework is demonstrated by the result 共7.10兲 derived here for the leading dipole term in the quadratic density response. Sham has stated that the quadratic density response contains no dipole terms,101 but the justification for this statement is not clear because no details of his calculation were given. However, numerical evidence to the contrary was subsequently provided by Dandrea, Duke, and Zunger in a first-principles study of band offsets in InAs/ GaSb superlattices.95 They deduced that the calculated difference between the macroscopic interface dipoles for InSb and GaAs interfaces must be a nonlinear effect 共because such differences do not exist in linear response theory51 in cubic crystals兲, but did not inquire further into its origin. To the author’s knowledge, the present derivation provides the first direct explanation for their result, and the first demonstration that dipole terms are a general feature of the quadratic density response. The magnitude of such dipoles is small—contributing 50 to 100 meV to the band offset of typical no-common-atom heterojunctions95,102—but they play an important role in explaining the experimentally observed asymmetry of band offsets in such systems.102 As a final note, it is worth drawing attention to a fundamental property of the nonlinear response of insulators that apparently is not widely known. For example, in Refs. 51 and 52, Baroni et al. have pointed out that “within linear response theory, the electronic charge induced by a given perturbation is proportional to the charge of the perturbing potential,”52 which implies that “within linear response theory, isovalent substitutional impurities carry no net charge.”51 Although the restriction to linear response theory is necessary in general, the results derived here 共in Sec. VII兲 demonstrate that the quadratic density response of an insulator to a charged perturbation also carries no net charge 共i.e., it vanishes in the limit of small wave vectors兲. Indeed, upon replacing Eqs. 共4.10兲 and 共5.19兲 with their higher-order generalizations, one finds that this statement remains true for the nonlinear density response 共4.6兲 of arbitrary order. This result stems from the Ward identities 共4.10兲 for the total static polarization and proper polarization 共see Sec. V C兲 of an insulator at zero temperature. As a consequence of these identities, one can therefore state that in an insulator, the total electronic charge induced by a given perturbation is exactly linearly proportional to the charge of the perturbing potential. Of course, this statement assumes that the system remains insulating over the full range of the perturbation 共from zero to full strength兲; otherwise, the perturbation theory used here is no longer valid. ACKNOWLEDGMENTS This work was supported by Hong Kong UGC Grant HIA03/04.SC02. 165344-13 PHYSICAL REVIEW B 72, 165344 共2005兲 BRADLEY A. FOREMAN APPENDIX A: REDUCING HETEROVALENT PERTURBATIONS TO ISOVALENT PERTURBATIONS In a heterovalent system such as Ge/ GaAs,60 the ionic perturbations are from class I. However, it is often possible to reduce such problems to an equivalent class II or III problem, because accumulations of macroscopic charge are energetically unfavorable; therefore, the interfaces grown in real heterojunctions tend to be macroscopically neutral. For Ge/ GaAs, an ideal 共110兲 heterojunction is already neutral, but an ideal 共001兲 or 共111兲 interface would have a large macroscopic interface charge, leading to a large compensating interface free-carrier density.60 Since this is not observed experimentally, the atoms in a real interface are believed to be arranged in one or more layers of mixed composition, such that the net macroscopic interface charge is zero.51,60 共This is similar to the concept of surface reconstruction, but the interface layers differ from the bulk only in chemical composition, not structure.兲 In such a system it is possible to replace the heterovalent ionic perturbations with a set of equivalent isovalent perturbations, simply by grouping the atoms together in clusters. The first step is to define quasiatomic building blocks using a modified version of Evjen’s technique.103 Let ⍀0共r兲 be a Wigner-Seitz cell 共of the reference crystal兲 that is centered on position r, and let N be the number of atoms in any primitive cell of the reference crystal. For a given atom a at position ra in the heterostructure, the quasiatomic potential v̄a is defined in terms of the ionic potentials va⬘ for all atoms a⬘ via v̄a = 兺 wa⬘共a兲va⬘ . neutral兲, and for some choices of compositional mixing, it may also have no dipole moment.60 Thus, if one treats these slab-adapted unit cells as the fundamental perturbations, this type of heterovalent class I perturbation can be replaced by an equivalent neutral perturbation from class II or class III. In general the interface cells do have a dipole moment, so the interface perturbations are class II, while the bulk perturbations are class III. However, an interface dipole term of O共q−1兲 is physically equivalent to a bulk quadrupole term of O共q0兲. Therefore, the approximation scheme defined in the Introduction yields results of the same overall accuracy for both the class II interface and class III bulk perturbations in Ge/ GaAs. APPENDIX B: SYMMETRY PROPERTIES This appendix considers some symmetry properties of G and ⌺. Time-reversal symmetry is developed from the properties of zero- and one-particle states. The vacuum state 兩0典 is defined to be time-reversal invariant: ⍜̂兩0典 = 兩0典, where ⍜̂ is the antiunitary time-reversal operator. The phase of ⍜̂ may be partially defined by letting the single-particle basis states 兩x , s典 = ˆ s†共x兲兩0典 satisfy ⍜̂兩x,s典 = 共− 1兲s−1/2兩x,− s典. 共B2兲 This relation is consistent with the operator equation106 共A1兲 † ⍜̂ˆ s†共x兲⍜̂† = 共− 1兲s−1/2ˆ −s 共x兲. a⬘ Here wa⬘共a兲 is a weight factor, defined as wa⬘共a兲 = 1 / N if atom a⬘ lies inside ⍀0共ra兲 , wa⬘共a兲 = 0 if a⬘ lies outside ⍀0共ra兲, and wa⬘共a兲 = 1 / mN if a⬘ lies on the surface of ⍀0共ra兲 关where m is the number of cells ⍀0共ra + R兲 that share atom a⬘, with R any Bravais lattice vector of the reference crystal兴. In a bulk crystal, these quasiatoms are neutral objects with the site symmetry of atom a in the reference crystal. Therefore, in a heterostructure, the quasiatoms carry a charge only near the heterojunctions. For Ge/ GaAs, each quasiatomic building block is constructed from 21 of the potential for a given ion plus 81 of the potential for each of its four nearest neighbors. In a bulk zinc-blende crystal, these quasiatoms have Td symmetry and possess no charge, no dipole moment, and no quadrupole moment. Therefore, in a heterostructure, the quasiatoms have monopole, dipole, and quadrupole moments only near the heterojunctions. It is assumed here that the ions in the mixed-composition interface layers form a periodic array, so that a twodimensional superlattice translation symmetry exists in the directions parallel to the junction plane. In this case, one can define a three-dimensional “slab-adapted”104,105 unit cell of quasiatoms that is large enough to contain 100% of the ions in the mixed-composition layers. This unit cell has no net charge 共since the interface is assumed to be macroscopically 共B1兲 共B3兲 One may therefore define ⍜̂ over the entire many-particle Fock space by Eqs. 共B1兲 and 共B3兲. The next step is to use the identity107 具兩Â兩␣典 = 具˜兩⍜̂Â⍜̂†兩˜␣典* = 具˜␣兩⍜̂†⍜̂†兩˜典, 共B4兲 in which  is a linear operator and 兩˜␣典 = ⍜̂兩␣典. If the manyparticle Hamiltonian Ĥ is time-reversal invariant 共i.e., 关⍜̂ , Ĥ兴 = 0兲, one has † 共x,− 兲, ⍜̂关ˆ s共x, 兲兴†⍜̂† = 共− 1兲s−1/2ˆ −s 共B5兲 which holds for complex . From this and Eq. 共B4兲 one immediately obtains Gss⬘共x, ;x⬘, ⬘兲 = 共− 1兲s−s⬘G−s⬘,−s共x⬘,− ⬘ ;x,− 兲, 共B6兲 which is the generalization of an ordinary Green-function “reciprocity” relation108 to the interacting-particle case. 共A similar expression was given in Ref. 79, but with the sign term omitted.兲 Since the change of time variables in 共B6兲 does not alter − ⬘, the Fourier transform of 共B6兲 is just Gss⬘共x,x⬘, 兲 = 共− 1兲s−s⬘G−s⬘,−s共x⬘,x, 兲. 共B7兲 Now Ĥ is time-reversal invariant if and only if h is, which from 共B2兲 and 共B4兲 implies that 165344-14 QUADRATIC RESPONSE THEORY FOR SPIN-ORBIT … hss⬘共x,x⬘兲 = 共− 1兲s−s⬘h−s⬘,−s共x⬘,x兲. PHYSICAL REVIEW B 72, 165344 共2005兲 Equation 共2.6兲 then shows that ⌺ has the same time-reversal properties as G: ⌺ss⬘共x,x⬘, 兲 = 共− 1兲s−s⬘⌺−s⬘,−s共x⬘,x, 兲. 共B9兲 G and ⌺ may also satisfy other conditions derived from linear symmetries of Ĥ. Consider the linear many-particle operator Q̂ defined by Q̂ = 兺 s,s⬘ 冕冕 V共n, n⬘兲 = V共n兲␦nn⬘ . 共B8兲 For many-variable quantities such as the vertex function ⌫共兲, the Fourier integrals for ⌫共兲共k , k⬘ ; q , q⬘ ; …兲 have the same form as 共C1兲 for each pair of 共k , k⬘兲 variables. m For a function of the form f共x兲 = f共r兲Y m l 共x̂兲, where Y l is a spherical harmonic, the Fourier transform is of the form f共k兲 = f共k兲Y m l 共k̂兲, where for k ⬎ 0, ˆ s†共x兲qss⬘共x,x⬘兲ˆ s⬘共x⬘兲d3x d3x⬘ , 共B10兲 in which q is a linear single-particle operator. Q̂ obeys the commutation relations 冕 关ˆ s共x兲,Q̂兴 = 兺 s⬘ qss⬘共x,x⬘兲ˆ s⬘共x⬘兲d3x⬘ , 冕 关ˆ s†共x兲,Q̂兴 = − 兺 s⬘ q3 ⬅ 关q1,q2兴. 关G,q兴 = 0, ⌺ss⬘共x,x⬘, 兲 = ␦ss⬘⌺共x,x⬘, 兲. 共B14兲 However, if h includes spin-orbit coupling 共which is the case studied here兲, ⌺ is nondiagonal. V共n, n⬘兲 = 冕冕 1  ⍀ ⍀ 冕冕  0  einV共, ⬘兲e−in⬘⬘d d⬘ . 具T关ÂI共兲Û兴典0 具Û典0 共D1兲 , the thermal average 共2.2兲 with respect to K̂ , 具Ô典0 is a thermal average with respect to K̂0 = Ĥ0 − N̂, and Û = T兵exp关−兰0Ĥ1共兲d兴其. If Û is expanded in a power series, terms of equal order in the numerator and denominator can be grouped together as ⬁ =1 共− 1兲 ! 冕  0 d1 ¯ 冕  d 0 ⫻具T关⌬Ĥ1共1兲 ¯ ⌬Ĥ1共兲⌬ÂI共兲兴典0 , 共D2兲 where ⌬ÂI,H共兲 = ÂI,H共兲 − 具ÂI共兲典0. A more compact expression for 共D2兲 is 共C2兲 具⌬ÂH共兲典 = 具T关⌬ÂI共兲Ŵ兴典0 , 0 Since V共 , ⬘兲 = V共 − ⬘兲, the latter integral is always diagonal in n: 共C5兲 where ÂH共兲 is a Heisenberg picture operator, ÂI共兲 is the same operator in the interaction picture, 具Ô典 denotes 具ÂH共兲典 = 具ÂI共兲典0 + 兺 e−ik·xV共x,x⬘兲eik⬘·x⬘d3x d3x⬘ , 共C1兲 共C4兲 0 冑⌫关 21 共l − n + 3兲兴 n−3 4 k , 共− i兲l n−1 1 ⍀ 2 ⌫关 2 共l + n兲兴 具ÂH共兲典 = The Fourier transforms of the potential with respect to momentum and frequency are defined by 1 ⍀ r2 f共r兲jl共kr兲dr, The starting point for the perturbation theory used in Sec. VI is the standard formula69,70 APPENDIX C: FOURIER TRANSFORMS V共k,k⬘兲 = ⬁ APPENDIX D: PERTURBATION THEORY 共B13兲 which further implies that 关⌺ , q兴 = 0. This result is applied to lattice translations throughout this paper, and to other space group operations in the following paper.59 Also of interest in the present paper is the spin operator sss⬘共x , x⬘兲 = 21 ss⬘␦共x − x⬘兲, where is the Pauli spin matrix vector. If h is independent of spin 共i.e., 关h , s兴 = 0兲, then 关G , s兴 = 0, and G and ⌺ have the scalar form 冕 in which ⌫共z兲 = 兰⬁0 e−ttz−1dt. Equation 共C5兲 is valid for k ⬎ 0 , n ⬎ 1, and l ⬎ n − 3.109 共B12兲 Now suppose that the Hamiltonian has the symmetry 关K̂ , Q̂兴 = 关Ĥ , Q̂兴 = 0. From 共B12兲, this is possible only if 关h , q兴 = 0. One can then use Eqs. 共2.1兲 and 共B11兲 and the cyclic property of the trace to show that 4 共− i兲l ⍀ in which jl共kr兲 is a spherical Bessel function. For the special case f共r兲 = r−n, Eq. 共6.561.14兲 of Ref. 109 gives f共k兲 = The commutator of any two such operators is another operator with the same form: 关Q̂1,Q̂2兴 = Q̂3, f共k兲 = 共B11a兲 ˆ s†⬘共x⬘兲qs⬘s共x⬘,x兲d3x⬘ . 共B11b兲 共C3兲 where Ŵ = T兵exp关−兰0⌬Ĥ1共兲d兴其. 165344-15 共D3兲 PHYSICAL REVIEW B 72, 165344 共2005兲 BRADLEY A. FOREMAN B共k,k1,k2兲 = 兺 兺 兺 R0G共k兲⌸̃共2兲 APPENDIX E: POLARIZATION The static polarization 共4.6兲 is defined by G G1 G2 ⫻共k + G,k1 + G1,k2 + G2兲G1共k1兲RG20共k2兲, ⌸共兲共00⬘,11⬘,…, ⬘兲 = 冕  0 d1 ¯ 冕  dD共兲共00⬘,11⬘,…, ⬘兲, 共E1兲 0 C共k,k1,k2兲 = 兺 兺 兺 R0G共k兲⌸̃共2兲 G G1 G2 ⫻共k + G,k1 + G1,k2 + G2兲G1共k1兲G2共k2兲. where D is the dynamic polarization 共F1兲 D共兲共00⬘,11⬘,…, ⬘兲 = 共− 1兲具T关⌬ˆ 共00⬘兲⌬ˆ 共11⬘兲 ¯ ⌬ˆ 共⬘兲兴典0 . 共E2兲 „2… APPENDIX G: NONANALYTIC TERMS IN n̄A „q… Here a superfluous second time variable has been added 共for notational convenience兲 to the interaction picture operators according to the definition ˆ ss⬘共x, ;x⬘, ⬘兲 ⬅ ˆ ss⬘共x,x⬘, − ⬘兲, 1 n̄A共2兲共q兲 = A␣␥q␣ 兺 ⬘k共q␥ − k␥兲共k兲共q − k兲 + O共q3兲, 2 k 共E3兲 where ⬘ ⬅ 0. Since D共兲 is periodic 共with period 兲 in all time variables , but depends on time only via the intervals − 0 共for = 1,2 , … , 兲, it follows that ⌸共兲 is independent of time. By definition, D is symmetric with respect to interchange of any pair of operators ⌬ˆ ; thus D共兲共…,ii⬘,…, jj⬘,…兲 = D共兲共…, jj⬘,…,ii⬘,…兲. To leading order, the term n̄A共2兲共q兲 in Eq. 共7.8兲 is 共E4兲 Another constraint on D can be derived from time-reversal symmetry using the methods of Appendix B: D共兲共00⬘,11⬘,…, ⬘兲 = 共− 1兲sD共兲共0̄⬘0̄,1̄⬘1̄,…,¯⬘¯兲. 共G1兲 in which A␣␥ is the Taylor series coefficient 共7.9兲. The only contribution to Eq. 共G1兲 that is of order q2 comes from the monopole terms in 共q兲. For any perturbation comprising a finite number of atoms, the expansion 共6.21兲 begins as 共q兲 = −Zvwc共q兲 / ⍀ + O共q−1兲, where Zv is the net valence charge of the ionic perturbation. If this lowest-order term is considered, the value of Eq. 共G1兲 can be estimated by extending the summation to all values of k. For a cubic crystal with scalar ⑀, the convolution can be performed using the Fourier transforms in Appendix C; the result is 共E5兲 n̄A共2兲共q兲 = ¯ 兲 = 共x , −s , 兲 and Here 共 s = 兺 共s − s⬘ 兲. 共E6兲 =0 These symmetries are valid for the static polarization ⌸ as well 共with the time variables omitted兲. APPENDIX F: FUNCTIONS A , B, AND C The functions A , B, and C introduced in Eq. 共7.8兲 are defined by A共k,k1,k2兲 = 兺 兺 兺 R0G共k兲⌸̃共2兲 G G1 G2 ⫻共k + G,k1 + G1,k2 + G2兲RG10共k1兲RG20共k2兲, 2Z2v q ␣q q ␥ + O共q3兲. A␣␥ q 8⍀⑀2 共G2兲 This shows explicitly that the leading nonanalytic term in n̄A共2兲共q兲 is O共q2兲. However, this term exists only for the heterovalent perturbations of class I in the Introduction. For the isovalent perturbations 共Zv = 0兲 of classes II and III, there is no O共q2兲 term in n̄A共2兲共q兲. A similar analysis shows that for class II, n̄A共2兲共q兲 = O共q4兲, while for class III, n̄A共2兲共q兲 = O共q6兲. Therefore, according to the approximation scheme adopted in this paper, n̄A共2兲共q兲 is negligible in all three cases. APPENDIX H: VERTEX FUNCTION TAYLOR SERIES Since the proper vertex function ˜⌫共1兲 is an analytic function of k and k⬘, it can be expanded in a Taylor series in the variables q = k − k⬘ and Q = 21 共k + k⬘兲. This yields an expression similar to Eq. 共8.5兲: ˜⌫共1兲 共k + G,k⬘ + G⬘ ;q + G⬙ ; 兲 = ˜⌫共1兲 共G,G⬘ ;G⬙ ; 兲 + q ˜⌫共␣兩·兲共G,G⬘ ;G⬙ ; 兲 + Q ˜⌫共·兩␣兲共G,G⬘ ;G⬙ ; 兲 + q q ˜⌫共␣兩·兲 ␣  ss⬘ ␣ ss⬘ ␣ ss⬘ ss⬘ ss⬘ 共·兩␣兲 共␣兩兲 ⫻共G,G⬘ ;G⬙ ; 兲 + Q␣Q˜⌫ss⬘ 共G,G⬘ ;G⬙ ; 兲 + q␣Q˜⌫ss⬘ 共G,G⬘ ;G⬙ ; 兲 + O共q3兲. 共H1兲 The Taylor series coefficients in Eq. 共8.5兲 for the effective vertex function ⌳ are therefore given by 165344-16 QUADRATIC RESPONSE THEORY FOR SPIN-ORBIT … PHYSICAL REVIEW B 72, 165344 共2005兲 共␣兩·兲 共␣兩·兲 ⌳ss⬘ 共G,G⬘ ; 兲 = ˜⌫ss⬘ 共G,G⬘ ;0; 兲 + 兺 G⬙⫽0 ˜⌫共1兲 共G,G⬘ ;G⬙ ; 兲R␣ , G⬙0 ss⬘ 共·兩␣兲 共·兩␣兲 ⌳ss⬘ 共G,G⬘ ; 兲 = ˜⌫ss⬘ 共G,G⬘ ;0; 兲, 共␣兩·兲 共␣兩·兲 ⌳ss⬘ 共G,G⬘ ; 兲 = ˜⌫ss⬘ 共G,G⬘ ;0; 兲 + 兺 G⬙⫽0 ˜ 共␣兩·兲共G,G⬘ ;G⬙ ; 兲R + ˜⌫共1兲 共G,G⬘ ;G⬙ ; 兲R␣ 兴, 关⌫ G⬙0 G⬙0 ss⬘ ss⬘ 共·兩␣兲 共·兩␣兲 ⌳ss⬘ 共G,G⬘ ; 兲 = ˜⌫ss⬘ 共G,G⬘ ;0; 兲, 共␣兩兲 共␣兩兲 ⌳ss⬘ 共G,G⬘ ; 兲 = ˜⌫ss⬘ 共G,G⬘ ;0; 兲 + 兺 G⬙⫽0 ˜⌫共·兩兲共G,G⬘ ;G⬙ ; 兲R␣ , G⬙0 ss⬘ 共H2兲 ␣ ␣ and RG0 are the Taylor series coefficients for RG0共q兲. 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