Download Quadratic response theory for spin-orbit coupling in semiconductor

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.
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©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 = eK␶␺ˆ s共x兲e−K␶ and ␺ˆ s†共x , ␶兲 = eK␶␺ˆ 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−i␨n␶ ,
␤ n=−⬁
G共␨n兲 =
冕
␤
G共␶兲ei␨n␶d␶ ,
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
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PHYSICAL REVIEW B 72, 165344 共2005兲
be defined by analytic continuation of G共␨n兲 from the discrete frequencies ␻ = ␮ + i␨n.
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
− ⵜ2␺s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.
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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 ␻ = ␮ + i␨n, 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
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⌸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 −4␲Zv / ⍀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2␲Zvr20 + 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 =
4␲n̄2
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
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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 = k1␤k2␥⌸̃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␣k1␤k2␥A␣␤␥ + O共k4兲,
共7.3兲
共2兲
␣␤␥
共k,k1,k2兲 = k␣k1␤k2␥⌸̃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␣q␤n̄␣␤
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:
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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␣q␤n̄␣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␣q␤Wss⬘ 共G,G⬘ ; ␻兲 + Q␣Q␤Wss⬘ 共G,G⬘ ; ␻兲
共␣兩␤兲
+ q␣Q␤Wss⬘ 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
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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4␲n̄2q2/⑀ + 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␣q␤wc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
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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
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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.
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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−1/2␺ˆ −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−1/2␺ˆ −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
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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
␤
ei␨n␶V共␶, ␶⬘兲e−i␨n⬘␶⬘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0␤Ĥ1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
d␶1 ¯
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.
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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
d␶1 ¯
冕
␤
d␶␯D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
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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. In these expressions, some special values of the coefficients
in which RG0
for the case G⬙ = 0 are given by the Ward identity 共3.11兲:
共0兲
˜⌫共1兲 共G,G⬘ ;0; ␻兲 = ␦ ␦
ss⬘ GG⬘ −
ss⬘
冏
⳵ ⌺ss⬘共G,G⬘ ; ␻兲
⳵␻
共0兲
,
⳵ ⌺ss⬘共k + G,k + G⬘ ; ␻兲
˜⌫共·兩␣兲共G,G⬘ ;0; ␻兲 = − ⳵
ss⬘
⳵ k␣
⳵␻
冏
共0兲
冏
,
k=0
2
⳵ ⌺ss⬘共k + G,k + G⬘ ; ␻兲
˜⌫共·兩␣␤兲共G,G⬘ ;0; ␻兲 = − 1 ⳵
ss⬘
2 ⳵ k␣ ⳵ k␤
⳵␻
.
共H3兲
k=0
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