PHYSICAL REVIEW B 72, 165345 ͑2005͒
First-principles envelope-function theory for lattice-matched semiconductor heterostructures
Bradley A. Foreman* Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China ͑Received 17 June 2005; revised manuscript received 26 August 2005; published 28 October 2005͒
In this paper a multiband envelope-function Hamiltonian for lattice-matched semiconductor heterostructures is derived from first-principles self-consistent norm-conserving pseudopotentials. The theory is applicable to isovalent or heterovalent heterostructures with macroscopically neutral interfaces and no spontaneous bulk polarization. The key assumption—proved in earlier numerical studies—is that the heterostructure can be treated as a weak perturbation with respect to some periodic reference crystal, with the nonlinear response small in comparison to the linear response. Quadratic response theory is then used in conjunction with k·p perturbation theory to develop a multiband effective-mass Hamiltonian ͑for slowly varying envelope functions͒ in which all interface band-mixing effects are determined by the linear response. To within terms of the same order as the position dependence of the effective mass, the quadratic response contributes only a bulk band offset term and an interface dipole term, both of which are diagonal in the effective-mass Hamiltonian. The interface band mixing is therefore described by a set of bulklike parameters modulated by a structure factor that determines the distribution of atoms in the heterostructure. The same linear parameters determine the interface band-mixing Hamiltonian for slowly varying and ͑sufficiently large͒ abrupt heterostructures of arbitrary shape and orientation. Long-range multipole Coulomb fields arise in quantum wires or dots, but have no qualitative effect in two-dimensional systems beyond a dipole contribution to the band offsets. The method of invariants ⌫ ⌫ is used to determine the explicit form of the Hamiltonian for 6 and 8 states in semiconductors with the zinc-blende structure, and for intervalley mixing of ⌫ and X electrons in ͑001͒ GaAs/AlAs heterostructures.
DOI: 10.1103/PhysRevB.72.165345 PACS number͑s͒: 73.21.Ϫb, 73.61.Ey, 71.15.Ap
I. INTRODUCTION some of which ͑for reasons unrelated to symmetry͒ may hap- pen to be zero or negligibly small. To obtain this information A. Background and motivation ͑along with a deeper understanding of the physical origin of Envelope-function models continue to play a key role in the interface phenomena͒, one must turn to a more detailed the design and interpretation of experiments on semiconduc- microscopic model of the interface. Thus, numerous tor heterostructures. The canonical “envelope-function ap- envelope-function models have been derived directly from proximation,” which by definition makes use of only bulk the microscopic potential energy; these include interface effective-mass parameters and heterojunction band offsets, Hamiltonians47–71 and connection rules72–82 as well as nu- has been very successful in explaining a wide range of 1,2 merical approaches based on solving the k·p equations in experiments. However, recent studies have increasingly 83,84 emphasized interface-related effects lying outside the scope momentum space. of conventional envelope-function theory, such as optical3–5 However, all of the cited derivations are based on empiri- and electrical6–8 anisotropy, intervalley mixing,9–15 and spin cal pseudopotentials, in which the potential energy is deter- polarization phenomena.16–27 These effects are generated mined by the choice of some specific model for the interface. ͑wholly or in part͒ by interface band-mixing terms in the This leads to ambiguity in the results, as different choices heterostructure Hamiltonian, such as valence-band mixing of may yield conflicting predictions as to which interface pa- light and heavy holes,28–30 ⌫-X coupling,10,11 and the con- rameters are important,12–14,61,64,69 or whether interface band- duction or valence band Rashba coupling.31 mixing effects are directly related to band offsets.70,71,85,86 A great deal of progress in the modeling of these effects Such conflicts can only be resolved by deriving the can be made on the basis of symmetry information alone. envelope-function Hamiltonian from an ab initio self- The method of invariants, which was originally developed consistent potential, as suggested by Sham and Lu.87 The for bulk semiconductors,32–35 has proved a powerful tool in purpose of this paper is to present such a derivation and the study of heterostructures as well.6,15–31,36–38 The standard examine its implications for interface band-mixing effects. method of invariants uses symmetry information to construct Numerical applications of the theory are not considered here. an explicit interface Hamiltonian, but the same information A preliminary account of these results has been presented ͑supplemented by hermiticity or current-conservation re- elsewhere.88 quirements͒ may also be used to construct connection rules for the envelope functions on opposite sides of the interface.39–46 In either case, one obtains a phenomenological B. Basic assumptions and limitations model containing some interface parameters whose value is The current “standard model” for condensed-matter phys- determined by comparison with experiment. ics is based on density-functional theory in the local density Useful as this approach may be, it does not provide any approximation ͑LDA͒.89–91 However, it is well known that information about the magnitude of the interface parameters, this ground-state formalism does not accurately predict en-
1098-0121/2005/72͑16͒/165345͑23͒/$23.00165345-1 ©2005 The American Physical Society BRADLEY A. FOREMAN PHYSICAL REVIEW B 72, 165345 ͑2005͒ ergy gaps in semiconductors. Thus, the present state-of-the- of the present theory would be to isovalent or heterovalent art in band-structure theory involves calculations of the elec- heterostructures of semiconductors with the zinc-blende or tron self-energy in the shielded-interaction92,93 or GW diamond structure. However, the only examples treated ex- approximation.94–97 This field is not yet fully mature, as ap- plicitly here are from isovalent systems. parent early successes98–100 have been questioned in light of Lattice mismatch could not be included in this theory 101–105 recent developments. Nevertheless, any future refine- without fundamental changes to account for atomic relax- ments in approximation techniques for calculating the quasi- ation. Nevertheless, the present results provide a solid foun- particle band structure will continue to be based on the self- dation for subsequent extensions of the theory to include this energy in Dyson’s equation. For this reason, the present work effect. Other restrictions, such as macroscopic neutrality and relies only upon general properties of the self-energy opera- the lack of bulk dipole terms, could be lifted merely by ex- tor, not upon any specific approximation to this quantity. tending some power series expansions to higher order ͑al- For simplicity, however, this paper assumes that the ionic though, depending on the accuracy that is desired,126 this cores are described in terms of norm-conserving 106–110 may require the inclusion of certain nonanalytic terms ne- pseudopotentials. The projector augmented wave ͒ ͑PAW͒ method of Blöchl,103,105,111 which does not involve glected in Ref. 124 . The fact that ͑within the pseudopotential approximation͒ a this approximation, has become increasingly popular in re- 112–122 cent years. A generalization of the present theory to incorpo- typical heterostructure is a weak perturbation makes ͉͘ rate the PAW formalism should be possible, but to avoid possible the existence of energy eigenstates whose wave ͗ ͉͘ 127 ͉ ͘ undue complexity it is not considered here. In addition, this function nk in the Luttinger-Kohn representation nk paper considers only lattice-matched systems ͑with no bulk of the reference crystal is negligible outside a small region in or interface strain͒. k space near the high-symmetry points of the Brillouin The key assumption used in the present work is that the zone.128–131 These eigenstates are just the low-energy excita- heterostructure is a small perturbation112 with respect to tions that are of greatest interest experimentally. Their exis- ͑ some periodic virtual reference crystal such as Al0.5Ga0.5As tence, which has been verified by extensive numerical work for a GaAs/AlAs heterostructure͒. This assumption has on empirical pseudopotential models,128–131 is precisely the been verified within LDA for both isovalent and heterovalent condition needed for the validity of an effective-mass ap- systems, including GaAs/AlAs,113,114 Ge/GaAs,114,115 proximation for slowly varying envelope functions 114,116 ͑ ͒ 47–52,127 ͑ ͒ϵ͗ ͉͘ In0.53Ga0.47As/InP, GaAs/Si/AlAs and Fn x , since Fn k nk . The existence of slowly GaAs/Ge/AlAs,117,118 Si/Ge,119,120 and InAs/GaSb.121,122 varying envelopes50,51,128–131 provides the foundation for the Within the framework of pseudopotential theory, one would effective-mass theory derived here. expect it to be valid for other similar heterostructures also. The present approach, based on quadratic response This assumption makes it possible to describe the self- theory,124 fits well with the k·p perturbation formalism of consistent heterostructure perturbation in terms of nonlinear Leibler,47,48,65–69 in which the heterostructure perturbation is response theory, with the linear response predominant and treated as small in comparison to the energy separating the the quadratic response providing a weak correction.113–122 bands of interest from other remote bands. Leibler’s theory is Such an approach has been used by Sham123 to derive an used here to develop a multi-band effective-mass theory that effective-mass equation for shallow impurity states in semi- includes all terms of the same order as the position depen- conductors. Sham’s work was extended in the preceding dence of the effective mass. This includes cubic and quartic paper124 to obtain expressions for the self-energy at small dispersion terms in the Hamiltonian of the reference crystal, values of the crystal momentum in lattice-matched as well as the leading contributions from the quadratic re- heterostructures described by spin-dependent nonlocal sponse. As shown in Sec. II, higher-order terms cannot con- pseudopotentials. These expressions are used here to con- sistently be described in terms of local differential struct a multi-band effective-mass theory for lattice-matched equations.65–69 heterostructures. Although the basic formalism developed here is quite C. Summary of key results general, to keep the paper to a reasonable length it is neces- sary to impose some restrictions on the material systems that In this paper, it is shown that the dominant interface band- are treated in detail. ͑This allows the power series in Sec. V mixing terms are those arising from the linear response. In- to be terminated at a reasonably low order.͒ Since only deed, the linear response is the only contribution to band lattice-matched systems are considered here, it is assumed mixing that fits within the framework of the perturbation that the symmetry of the reference crystal does not support a theory defined above, and the only contribution that can con- spontaneous polarization, because that would generate mac- sistently be described in terms of local differential equations. roscopic electric fields and piezoelectric strain fields incon- To this level of accuracy, the quadratic response contributes sistent with the lattice-matching assumption. Therefore, the only a bulk band offset term and an interface dipole term, present theory is not directly applicable to wurtzite materials. neither of which produces any band mixing in the effective- In addition, it is assumed that the interfaces in a heterova- mass Hamiltonian. lent system such as Ge/GaAs are macroscopically This represents a major simplification, since it implies neutral,114,125 so that the atoms can be grouped together into that the interface band-mixing terms in the Hamiltonian are neutral clusters ͑of fractional atoms͒ that carry a dipole mo- just a superposition of parameters derived from the linear ment only at the interface.124,126 Thus, the main application response to individual ionic perturbations ͑or neutral cluster
165345-2 FIRST-PRINCIPLES ENVELOPE-FUNCTION THEORY… PHYSICAL REVIEW B 72, 165345 ͑2005͒ perturbations124,126 in the case of heterovalent substitutions͒. Volkov.65–69 The main results of the quadratic response These parameters are calculated once and for all for a given theory developed in the preceding paper124 are presented ͑in material system; they then appear as coefficients in front of modified form͒ in Sec. III. The basic envelope-function for- structure factors describing the distribution of atoms in the malism is developed in Sec. IV, while Sec. V describes the heterostructure. The Hamiltonian contains spatial derivatives power series expansions that are used to obtain approximate of ͑and differences between͒ these structure factors, which expressions valid for slowly varying envelopes. Perturbation generate ␦-like functions at an interface. The atomic distri- theory is used to eliminate the coupling to “remote” bands in bution functions change for heterostructures of different Sec. VI, yielding the basic expression for the multi-band shape ͑wells, wires, or dots͒ and orientation, but their coef- envelope-function Hamiltonian. Modifications to the Hamil- ficients do not. Thus, a single set of interface parameters tonian that are necessary if one wishes to describe the mate- governs the band mixing for any type of heterostructure132 in rial parameters of an abrupt heterojunction as piecewise con- a given material system. stant are discussed in Sec. VII. Symmetry properties of the If desired, one can express these linear parameters as a Hamiltonian are discussed in Sec. VIII, where explicit matrix difference between the properties of the various bulk materi- representations of the material properties are given for semi- als that make up the heterostructure. ͑A similar result was conductors with the zinc-blende structure. Finally, the sig- obtained for linear band offsets in Refs. 113–122.͒ However, nificance of the results and their relation to previous work in in a no-common-atom system such as InAs/GaSb, this must the literature are discussed in Sec. IX. include the ͑lattice-matched͒ “interface” materials GaAs and InSb. Also, these bulk-like linear response parameters ͑which II. ORDER OF TERMS INCLUDED in principle do not require a supercell calculation133͒ cannot be determined from experiments on bulk semiconductors. Takhtamirov and Volkov65–69 have recently demonstrated For some terms in the linear response ͑derived from the ͑using an instructive analogy to the leading relativistic cor- analytic part of the self-energy124͒, the interface Hamiltonian rections to the Schrödinger equation͒ that most derivations of itself has the form of a macroscopic average of the Dirac ␦ heterostructure effective-mass equations in the literature ͑in- function or its derivatives. For the remaining terms ͑derived cluding those of the present author͒ do not consistently in- from the nonanalytic part of the self-energy124͒, the effective clude all perturbative corrections of the same order. This charge density has this localized form, but the potential en- section presents a review and discussion of their results, with ergy is not as well localized, having the form of a long-range the objective of establishing which terms are to be retained multipole potential. However, in a quantum well or any other in subsequent perturbative approximations. heterostructure with two-dimensional translation symmetry, The first case to be considered is a wide-gap system sat- all terms are well localized except for the interface dipole isfying terms ͑which merely modify the band offsets͒. ⌬¯ Ӷ ¯ ͑ ͒ The band-mixing Hamiltonian has the same general form V Eg, 2.1 for slowly graded structures ͑within the virtual crystal ap- where ⌬¯V is a typical heterojunction band offset, and ¯E is a proximation͒ and abrupt heterojunctions, the only difference g typical energy gap ͑for the virtual bulk reference crystal͒ being the rate of change of the structure factor. That is, in separating the band in question from all other “remote” contrast to previous theories based on model potentials or bands. It is assumed that in the reference crystal this band is empirical pseudopotentials, no new band-mixing parameters describable by an effective-mass equation with effective appear at an abrupt junction. Indeed, the localized interface mass m*. This will be the case if the k·p interaction with terms in the Hamiltonian derived here are qualitatively iden- remote bands tical to the Hamiltonian derived by Leibler47,48 for slowly graded heterostructures. q¯k¯p 12–14,64,69 ϵ͑¯͒¯ ͑ ͒ These results shed light on recent suggestions k Eg 2.2 ⌫ m that the 1-X1z and X1x-X1y intervalley mixing potentials at ͑ ͒ an ideal 001 GaAs/AlAs heterojunction should be propor- can be treated as a small perturbation. Here m is the free- tional to ␦͑z͒. It is shown here that such mixing arises only electron mass, ¯kϳ2/L is a typical envelope function wave- from the quadratic response and is therefore negligible in number for a quantum well of width L,¯pϳ2q/a is a typi- comparison to ⌫ -X ,X -X , and X -X mixing. There is, 1 3z 1z 3z 3x 3y cal interband momentum matrix element ͑where a is the however, a linear contribution to the ⌫ -X and X -X mix- 1 1z 1x 1y lattice constant͒, and it is assumed that Lӷa. Equation ͑2.2͒ ing that is proportional to ␦Ј͑z͒, as well as a linear ␦͑z͒ ¯ ¯ ͑ mixing for nonideal interfaces or when spin-orbit coupling is defines a length parameter =qp/mEg, which if the free- * ͒ included. These contributions may help to explain the experi- electron contribution to m is negligible may also be written Ϸ ͑ *¯ ͒−1/2 mental observations in Refs. 12–14. as q 2m Eg . For GaAs, both expressions for give Ϸ6 Å, thus ϳa and ¯kӶ1 in a wide quantum well. For the remainder of this paper the parameters and a are used D. Outline of the paper interchangeably, although in general error estimates should The paper begins in Sec. II with a discussion of which be based on the larger of the two quantities. terms are to be retained in k·p perturbation theory, based In general one is interested in cases where the kinetic upon a review of important recent work by Takhtamirov and energy is comparable to the band offset:
165345-3 BRADLEY A. FOREMAN PHYSICAL REVIEW B 72, 165345 ͑2005͒
⌬ q2¯k2 the Brillouin zone boundary in the direction. Beyond such ϳ ⌬¯ ϳ͑¯͒2¯ ͑ ͒ * V k Eg. 2.3 an anticrossing the model is no longer valid and an error of 2m order ͑2.8͒ arises even in a k-space formalism. To eliminate This is the order of terms included in ordinary effective-mass such errors one must enlarge the Hamiltonian by treating theory,127 which may be compared with the nonrelativistic these remote bands explicitly, thus obtaining a full-zone k·p 74,83,137,138 ͑Schrödinger͒ approximation to the Dirac equation. The po- model. sition dependence of the effective mass can be calculated by Such a multiband envelope-function model is also needed ͑0͒ ͑0͒ ⌬¯ ¯ 47,48 for medium-gap ͑⌬¯VϳE ͒ and narrow-gap ͑E Ӷ⌬¯V͒ sys- treating V/Eg as a perturbation, yielding a correction of g g ͑0͒ order tems, where Eg is the energy gap of the reference crystal. The perturbative approach based on Eq. ͑2.1͒ can still be q2¯k2 ⌬¯V ͑⌬¯V͒2 ¯ ͩ ͪ ϳ ϳ͑¯k͒2⌬¯V, ͑2.4͒ used, provided that Eg refers to the energy gap between the 2m* ¯ ¯ bands of interest and those treated as remote perturbations. Eg Eg Takhtamirov and Volkov69 have considered the extreme 4 which is of the same order as the nonparabolic k terms in narrow-gap limit for the case in which the dispersion is the kinetic energy of the reference crystal. Hence, these dominated by linear-k terms for all energy ranges of interest, terms ͑analogous to the relativistic mass corrections to the and showed that in this case the error generated by the local ͒ Schrödinger equation must be included if one is to retain all approximation is of order ͑¯ka͒2⌬¯V. In such cases the terms terms of the same order as the position dependence of the ͑2.4͒ and ͑2.6͒ should be omitted for consistency. However, effective mass. in practical situations one is more likely to encounter cases As will be seen below, the interface band-mixing terms ⌬¯ ϳ ͑0͒ 2 proportional to a ␦ function are of order in which V Eg and the k dispersion terms are compa- rable to the linear-k terms. Thus, in this paper all terms of A͗␦͑z͒͘ϳ͑¯ka͒⌬¯V, ͑2.5͒ order ͑¯ka͒2⌬¯V are retained, while those at the level of the ͑ ͒ while those proportional to the derivative of a ␦ function are local approximation 2.8 are omitted. of order It should be noted that a fully self-consistent perturbation scheme is not always desirable. It would of course be impos- B͗␦Ј͑z͒͘ϳ͑¯ka͒2⌬¯V, ͑2.6͒ sible to achieve an accurate numerical prediction for the fine structure of the hydrogen atom without including the in which A and B are constants. Hence, the latter terms are relativistic-mass correction and the Darwin term in addition also comparable to the position dependence of the effective to the spin-orbit splitting. However, for qualitative consider- mass. ations one is often interested primarily in symmetry-breaking For a step-function discontinuity ⌬V in the potential en- effects, in which case the former two contributions may jus- ergy, the envelope function F has a discontinuity in its sec- tifiably be omitted. Likewise, although the present work re- ond derivative given by tains all terms of order ͑¯ka͒2⌬¯V, certain of these ͑such as the ⌬FЉ 2m* k4 bulk dispersion terms͒ may possibly be omitted for appli- = ⌬V, ͑2.7͒ F q2 cations in which the primary focus is on symmetry-breaking interface effects. ͓As an example, Takhtamirov and Volkov where FϳL−1/2. This gives rise to an asymptotic behavior in have proposed a model6 based on certain terms of order ͑ ͒ϳ͑⌬ Љ͒ −3 k space of F k F k . But in the exact Luttinger-Kohn ͑¯ka͒3⌬¯V, while neglecting larger terms of order ͑¯ka͒⌬¯V and envelope-function representation,127 the envelope functions ͑¯ ͒2⌬¯ ͔ are limited to wave vectors inside the first Brillouin zone. ka V. However, the neglect of these terms can only be Hence, the use of a local differential effective-mass equation, justified for specific individual applications and should not which gives rise to nonvanishing F͑k͒ outside the Brillouin be presumed to hold in general. zone, generates an error in the kinetic energy of order The discussion here has focused on wide quantum wells, but this should not be taken to imply that effective-mass q2¯k2 theory is inapplicable in other cases. For example, in a nar- ͑¯ka͒3ͩ ͪ ϳ͑¯ka͒3⌬¯V. ͑2.8͒ row quantum well,50,68 an effective-mass approximation can 2m* be developed along the same lines as for shallow Therefore, in the local approximation, the accuracy is limited impurities.123,127 In this case, however, the above estimates by Eq. ͑2.8͒, and contributions beyond the level of ͑2.4͒ and of the interface terms and local approximation are no longer ͑2.6͒ should for consistency be omitted. valid; see Ref. 68 for further details. Of course, this is not a fundamental limitation of the The remainder of this paper uses atomic units with envelope-function method, as one can always choose to work q=m=e=1. in k space ͑which is a common choice for numerical work134–136͒. However, there is another similar source of er- III. QUADRATIC RESPONSE THEORY ror that arises from anticrossings of the bands explicitly in- cluded in the envelope-function model with those treated as A. Basic definitions ⌫ remote perturbations. This occurs, for example, in the 6 The problem of interest is the Dyson eigenvalue equation conduction band of GaAs at about a third of the distance to ͑see Ref. 124 and references therein͒
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␣ ⌬v ͑x,xЈ͒ϵva ͑x,xЈ͒ − v¯aj ͑x,xЈ͒. ͑3.10͒ 2͑ ͒ ͵ ͑ Ј ͒͑ Ј ͒ 3 Ј ͑͒͑ ͒ ion ion ionٌ 1 − 2 x, + V x,x , x , d x = E x, , ␣ ͑ ͒ Here = a, j is a composite index, ¯aj is the label of some ͑3.1͒ given atom on site j, and the prime on the summation symbol ¯␣ ͑¯a j͒ E indicates that the values = j , are excluded. To simplify in which is a complex energy parameter, is the complex ͑ ͒ V the interpretation of 3.9 , the atom ¯aj is chosen to be the eigenvalue, and the self-consistent potential is the sum of a ͑ ͒ V same as the virtual atom on site j in the reference crystal. fixed norm-conserving ionic pseudopotential ion and the ͑ Ј͒ non-Hermitian self-energy operator ⌺: In this paper the self-consistent potential V x,x is treated using nonlinear response theory.124 The fundamental ͑ ͒ ͑ ͒ ⌺͑ ͒ ͑ ͒ V x,xЈ, = Vion x,xЈ + x,xЈ, . 3.2 assumption is that V can be expressed as a power series in ␣ For notational simplicity, the dependence will be sup- the variables R: ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ pressed in most of the equations that follow. In Eq. 3.1 , V͑x,xЈ͒ = V 0 ͑x,xЈ͒ + V 1 ͑x,xЈ͒ + V 2 ͑x,xЈ͒ + . and V are spinors, with V having the form ¯ ͑3.11͒ ͑ ͒ V = 1 Vsc + · V, 3.3 Here V͑0͒͑x,xЈ͒ is the self-consistent potential of the refer- ϫ in which 1 is the 2 2 spinor unit matrix, Vsc is the scalar ence crystal, which has the same periodicity as the ionic relativistic part of V, is the Pauli matrix, and V accounts potential ͑3.5͒: for spin-orbit coupling.139–142 Note that Eq. ͑3.1͒ incorpo- ͑ ͒ ͑ ͒ rates all relativistic corrections of order Z2␣2 ͑where Z is the V 0 ͑x,xЈ͒ = V 0 ͑x + R,xЈ + R͒. ͑3.12͒ atomic number and ␣ is the fine-structure constant͒, but ne- The linear response to the heterostructure perturbation has glects terms of order ␣2, such as spin-orbit coupling outside the form the atomic cores.107,108 In a heterostructure, it is convenient to partition the ionic Ј ͑1͒͑ Ј͒ ␣ ⌬ ␣ ͑ Ј͒ ͑ ͒ pseudopotential as V x,x = ͚ R vR x,x , 3.13 ␣ ͑ Ј͒ ͑0͒͑ Ј͒ ⌬ ͑ Ј͒ ͑ ͒ ,R Vion x,x = Vion x,x + Vion x,x , 3.4 ͑0͒ while the quadratic response is where Vion is the ionic pseudopotential of some periodic ref- erence crystal ͑which may be a virtual crystal͒. This is de- Ј Ј ͑2͒͑ Ј͒ ␣ ␣Ј⌬ ␣␣Ј ͑ Ј͒ ͑ ͒ fined as V x,x = ͚ ͚ R RЈ vRRЈ x,x . 3.14 ␣ ,R ␣Ј,RЈ ͑0͒͑ Ј͒ aj a ͑ Ј ͒ ͑ ͒ Vion x,x = ͚ f vion x − Rj,x − Rj , 3.5 a,j,R Here the expansion coefficients in the power series are de- a ͑ Ј͒ fined by in which vion x,x is the ionic pseudopotential for atomic aj 143 V͑x,xЈ͒ץ ␣ species a, f is the fractional weight associated with atom ⌬v ͑x,xЈ͒ = , ␣ץ a on site j at position j in the unit cell of the reference R R crystal, and Rj =R+ j, where R is a Bravais lattice vector of the reference crystal. faj must satisfy 2V͑x,xЈ͒ץ 1 ⌬ ␣␣Ј ͑ Ј͒ ͑ ͒ ഛ aj ഛ aj ͑ ͒ vRRЈ x,x = , 3.15 ␣Ј ץ ␣ץ f 1, ͚ f =1, 3.6 2 0 a R RЈ although the former constraint need not be strictly enforced. where the derivatives are evaluated with respect to the refer- ⌬ The term Vion is the perturbation due to the heterostruc- ence crystal. These derivatives can be evaluated numerically ͑ ͒ ␣ ture: by applying perturbations 3.9 with small values of R. The leading source of error in quadratic response theory ⌬ ͑ Ј͒ aj a ͑ Ј ͒ ͑ ͒ ͑3͒ Vion x,x = ͚ R vion x − Rj,x − Rj . 3.7 lies in the neglected cubic response term V . This error can a,j,R be reduced somewhat by a suitable choice of reference po- aj ͑ tential. For example, in a GaAs/AlAs heterostructure, choos- Here R is the change in fractional weight relative to the ͒ reference crystal of atom a at position Rj in the heterostruc- ing Al0.5Ga0.5As as a reference crystal instead of GaAs ture, which must satisfy would reduce the cubic error in AlAs by a factor of 8. How- ever, it would also create a comparable error in GaAs. Thus, ഛ aj aj ഛ aj ͑ ͒ 0 f + R 1, ͚ R =0. 3.8 the cubic error in the interface Hamiltonian for an a 1 Al0.5Ga0.5As reference potential would be about 4 that of a ͑ ͒ ⌬ GaAs or AlAs reference potential. This is certainly an im- The constraint 3.8 permits one to rewrite Vion as provement, but since the quadratic response is already quite Ј small ͑see Sec. III B͒, it is unlikely to make much practical ⌬ ͑ Ј͒ ␣ ⌬ ␣ ͑ Ј ͒ ͑ ͒ Vion x,x = ͚ R vion x − R␣,x − R␣ , 3.9 difference. ␣ ,R In the momentum representation, the Dyson equation in which ͑3.1͒ has the form
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1 This estimate is supported by the LDA calculations of k2͑k͒ + ͚ V͑k,kЈ͒͑kЈ͒ = E͑k͒. ͑3.16͒ 130 2 Wang and Zunger for GaAs/AlAs heterostructures, in kЈ which they found ͑see Table I of Ref. 130͒ that the interface Making use of the translation symmetry of the reference band-mixing terms arising from the linear response were on crystal, one can write the Fourier transforms of the potentials average about 1000 times larger than those arising from the ͑ ͑3.15͒ as quadratic response with the ratio between the smallest linear and largest quadratic terms being about 100, in agreement ⌬ ␣ ͑ Ј͒ϵ −i͑k−kЈ͒·R␣⌬ ␣͑ Ј͒ ͒ vR k,k e v k,k , with the estimate obtained above . However, the LDA calcu- lations for GaAs/AlAs and In0.53Ga0.47As/InP presented in ␣␣Ј ͑ Ј͒ ␣␣Ј Љ Refs. 114 and 116 indicate that the quadratic density re- ⌬v ͑k,kЈ͒ϵe−i k−k ·R␣⌬v R ͑k,kЈ͒, ͑3.17͒ RRЈ sponse is only about 10 times smaller than the linear density response ͑the linear and quadratic potentials were not given in which RЉ=RЈ−R, and the coordinate origin is R␣ for the ͒ modified functions on the right-hand side. The Fourier trans- in these papers . form of the linear response ͑3.13͒ can therefore be written as Nevertheless, this is still a sufficiently large ratio to estab- lish the validity of the quadratic approximation used here. Ј For the purposes of the perturbation scheme of Sec. II, the ͑1͒͑ Ј͒ ͚ ␣͑ Ј͒⌬ ␣͑ Ј͒ ͑ ͒ ⑀ ͑ ͒ V k,k = N k − k v k,k . 3.18 factor 4 F in the denominator of Eq. 3.22 will be treated ␣ ¯ ⌬ ͑2͒ formally as of order Eg, so that V is considered to be of ⍀ ⍀ ͑ ⍀ ͒ Here N= / 0 is the number of unit cells of volume 0 in the same order ͓͑⌬¯V͒2 /¯E ϳ͑¯ka͒2⌬¯V͔ as the smallest terms ͑ ⍀͒ ␣͑ ͒ g the reference crystal of volume , k is the Fourier retained in Sec. II. ␣ transform of R:
␣ ␣͑ ͒ ik·R␣ ͑ ͒ R = ͚ k e , 3.19a C. Functional form ⍀* k 0 As shown in Ref. 124, the Coulomb interaction gives rise to singularities in the linear and quadratic potentials 1 ⌬ ͑ Ј͒ Ј ␣͑ ͒ ␣ −ik·R␣ ͑ ͒ v k,k when k−k is equal to a reciprocal lattice vector. k = ͚ Re , 3.19b 124 N R⍀ The explicit form of the linear potential is ⍀* ͑ ͒3 ⍀ ⌬ ␣ ͑ Ј Ј͒ ␣ ͑ Ј Ј͒ and 0 = 2 / 0 is the volume of a unit cell in the recip- vR k + G,k + G = wR k,k ;G,G ␣ rocal lattice. Note from ͑3.19b͒ that ͑k͒ is quasiperiodic: ⌳ ͑ Ј͒␣ ͑ Ј͒ + GGЈ k,k R k − k , ␣ ␣ −iG· ͑k + G͒ = ͑k͒e ␣, ͑3.20͒ ͑3.23͒ where G is a reciprocal lattice vector of the reference crystal. where the potential w␣ ͑k,kЈ;G,GЈ͒ is an analytic function ͑ ͒ R In a similar fashion, inserting 3.17 into the Fourier of k and kЈ for wave vectors inside the first Brillouin zone of ͑ ͒ transform of 3.14 and relabeling RЈ gives the reference crystal. For small values of k and kЈ it can be Ј represented as a Taylor series, which is the basis for the ͑ ͒ ␣␣Ј Ј ␣␣Ј Ј V 2 ͑k,kЈ͒ = N ͚ R ͑k − kЈ͒⌬v R ͑k,kЈ͒, effective-mass theory developed in the following section. ⌳ ͑ Ј͒ ␣,␣Ј,RЈ The effective vertex function GGЈ k,k is also an analytic function of k and kЈ. ͑3.21͒ The singular contributions come from the screened poten- 124,146 in which ␣␣ЈRЈ͑k͒ is the Fourier transform of the pair dis- tial , which is a spin scalar of the form ␣Ј ␣␣ЈRЈ͑ ͒ϵ␣ ͑ ͒ ␣ ͑ ͒ tribution function R R R+RЈ. ␣ vc q n q ͑q͒ = R , ͑3.24͒ R ⑀͑q͒ B. Estimation of magnitude where ⑀͑q͒ is the static electronic dielectric function of the A crude estimate of the relative magnitudes of the linear reference crystal and and quadratic response can be obtained from a simple non- linear Thomas-Fermi model,89 with the result144 4/q2 if q 0, ͑ ͒ ͭ ͮ ͑ ͒ vc q = 3.25 ͑2͒ ⌬¯ 0 if q =0. ͑1͒ V V V ϳ ⌬¯V, ϳ , ͑3.22͒ ␣ ͑1͒ ⑀ ͑ ͒ V 4 F The function nR q is an effective electron density contain- ing partial contributions from the bare ionic pseudocharge ⌬¯ ⑀ where V is the typical band offset defined in Sec. II, and F and the screening charge. This is an analytic function of q is the Fermi energy. Now the difference in screened pseudo- ͑for small q͒ with the symmetry of site R␣ in the reference potentials between typical III-V semiconductors is roughly crystal. For isovalent substitutions in zinc-blende crystals, ⌬¯Vϳ0.02–0.05 Ry,145 which yields the estimate ͉V͑2͒ /V͑1͉͒ the leading nonanalytic terms in Eq. ͑3.24͒ are octopole and 2 ͑ 4 ϳ0.01. This suggests that the quadratic response is indeed hexadecapole potentials proportional to qxqyqz /q and qx 4 4͒ 2 124 very small. +qy +qz /q .
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The quadratic potential has a similar form: ͒ ͵ ͑r,0͒͑ Ј ͒ ͑ Ј ͒ 3 Ј ͑ 2ٌ 1 − 2 Un x, + V x,x , Un x , d x ⌬ ␣␣Ј ͑ Ј Ј͒ ␣␣Ј ͑ Ј Ј͒ vRRЈ k + G,k + G = wRRЈ k,k ;G,G ͑͒ ͑ ͒͑͒ = En Un x, 4.1 ⌳ ͑ ͒␣␣Ј ͑ ͒ + GGЈ k,kЈ k − kЈ , RRЈ that satisfy the periodic boundary conditions ͑ ͒ 3.26 ͑ ͒ ͑ ͒ ͑ ͒ Un x, = Un x + R, , 4.2 ⌳ in which is the same vertex function as above. In this case, ͑r,0͒ ␣␣Ј in which V is the Hermitian part of the reference potential ͑ Ј Ј͒ ͑ ͒ however, the potential wRRЈ k,k ;G,G is not an analytic V 0 . For a general operator A, the “real” ͑Hermitian͒ and function of k and kЈ. Nevertheless, to within the accuracy “imaginary” ͑anti-Hermitian͒ parts are defined here as required here ͑namely, zeroth order in k and kЈ͒, the nonana- 1 i lytic part can be neglected for neutral perturbations. Note A͑r͒ = ͑A† + A͒, A͑i͒ = ͑A† − A͒. ͑4.3͒ that the analytic part of the quadratic potential is analytic 2 2 over a smaller region in k space than the linear potential. ␣ The solutions to Eq. ͑4.1͒ form a complete orthonormal Whereas w ͑k,kЈ;G,GЈ͒ was analytic for k and kЈ inside R set of periodic functions127 for any value of . In the Fourier the first Brillouin zone, the analytic part of series representation ␣␣Ј ͑ Ј Ј͒ wRRЈ k,k ;G,G is analytic only over the inner “half” of ͑ ͒ ͑͒ iG·x ͑ ͒ the Brillouin zone ͑i.e., over the Brillouin zone of a crystal Un x, = ͚ UnG e , 4.4 whose lattice constants are double those of the reference G ͒ crystal . the orthogonality and completeness relations are The quadratic screened potential is defined by ͚ U† ͑͒U ͑͒ = ␦ , ͑4.5a͒ ␣␣Ј nG nЈG nnЈ ͑ ͒ ͑ ͒ G ␣␣Ј vc q nRRЈ q ͑q͒ = , ͑3.27͒ RRЈ ⑀͑q͒ ͑͒ † ͑͒ ␦ ͑ ͒ ͚ UnG U = 1 GGЈ, 4.5b ␣␣Ј nGЈ ͑ ͒ n in which the effective density nRRЈ q is likewise not ana- lytic, but can be approximated as such. The leading terms † in which UnG denotes the hermitian conjugate of the spinor here are dipole and quadrupole potentials ͑since the mono- UnG. pole term vanishes for an insulator at zero temperature͒.In Note that the Bravais lattice chosen for the periodic zinc-blende crystals, only the interface dipole term is non- boundary conditions in Eq. ͑4.2͒ need not be the same as that negligible under the approximation scheme used in this pa- in ͑3.12͒; for certain applications it may be preferable to per. impose periodicity with respect to some ͑mathematically de- For use in Eqs. ͑3.18͒ and ͑3.21͒, one requires also ex- fined͒ supercell instead. For example, in treating intervalley pressions for the modified linear and quadratic potentials de- ⌫-X coupling in semiconductors with the zinc-blende struc- fined in Eq. ͑3.17͒. These are given by ture, it is convenient50,61 to choose a nonprimitive simple ␣ cubic unit cell of volume ⍀ =a3 ͑where a is the conven- ⌬v␣͑k + G,kЈ + GЈ͒ = w ͑k,kЈ͒ 0 GGЈ tional cubic lattice constant͒, which encompasses four primi- ͑ Ј͒ ␣ 147 X + ei G−G · ␣⌳ ͑k,kЈ͒ ͑k − kЈ͒, tive fcc cells. This folds the valleys onto the Brillouin GGЈ zone center of the supercell, thereby permitting intervalley ͑3.28a͒ ⌫-X mixing to be described in the same notation as that for ordinary ⌫ states ͑although the ⌫ and X states are of course ␣␣Ј Ј ␣␣ЈRЈ ͑ Ј͒ ͒ ⌬v R ͑k + G,kЈ + GЈ͒ = w ͑k,kЈ͒ + ei G−G · ␣ not coupled by the k·p interaction . GGЈ The wave function in the Luttinger-Kohn representation ϫ⌳ ͑ ͒␣␣ЈRЈ͑ ͒ ͑or envelope function͒ F ͑k͒ is defined as GGЈ k,kЈ k − kЈ , n ͑ ͒ ͑ ͒ † ͑ ͒ ͑ ͒ 3.28b Fn k = ͚ UnG k + G , 4.6a G in which the terms on the right-hand side are defined by ͑ ͒ expressions similar to 3.17 . ͑ ͒ ͑ ͒ ͑ ͒ k + G = ͚ Fn k UnG. 4.6b n IV. ENVELOPE-FUNCTION EQUATIONS This definition is valid for any value of k, but since only those values from one unit cell ⍀* are needed to determine In this section, the Dyson equation ͑3.16͒ for a hetero- 0 ͑x͒, it is convenient to set F ͑k͒ϵ0 when k⍀*.49,148 The structure is written in the Luttinger-Kohn representation,127 n 0 Fourier transform of ͑4.6b͒ is then the usual exact envelope- in which the basis functions are defined to be plane waves function expansion49,127 multiplied by the zone-center Bloch functions Un of the pe- riodic reference crystal.47,49 Here the set ͕U ͖ is chosen to be ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ n x = ͚ Fn x Un x . 4.7 a complete set of solutions to the equation ͓cf. Eq. ͑3.1͔͒ n
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The Dyson equation in the Luttinger-Kohn representation The physical interpretation of these results is considered be- is given by Eqs. ͑3.16͒, ͑4.5͒, and ͑4.6͒ as low. ͑ ͒ ͑ ͒ ͑ ͒ ⌬ ͑ ͒ ͑ ͒ EnFn k + ͚ LnnЈ k FnЈ k + ͚ ͚ VnnЈ k,kЈ FnЈ kЈ V. POWER SERIES EXPANSIONS Ј Ј Ј⍀* n n k 0 = EF ͑k͒. ͑4.8͒ In this section, power series expansions are used to obtain n approximate expressions for the Hamiltonian matrix ele- ͑ ͒ Here ⌬V=V−V 0 is the perturbation due to the heterostruc- ments in the envelope-function equations ͑4.8͒. This approxi- ture, the matrix elements of which are mation is justified by the existence50,51,128–131 of slowly vary- ing envelope functions F ͑x͒, for which F ͑k͒ is negligible ⌬ ͑ Ј͒ † ⌬ ͑ Ј Ј͒ n n VnnЈ k,k = ͚ UnG V k + G,k + G UnЈGЈ. unless k is small. This expansion provides a starting point for Ј G,G the development of an approximate effective-mass theory, ͑4.9͒ and also assists in the physical interpretation of the various ͑ ͒ terms in the Hamiltonian. The term LnnЈ k groups together all contributions from the bulk reference crystal Hamiltonian except En; i.e., ͑ ͒ ͑ ͒ A. Reference crystal Hamiltonian L ͑k͒ = V 0 ͑k͒ − V r,0 ͑0͒ + k · p + 1 k2␦ , nnЈ nnЈ nnЈ nnЈ 2 nnЈ ͑ ͒ The leading terms in the bulk Hamiltonian LnnЈ k of ͑4.10͒ Eq. ͑4.10͒ are derived from a Taylor series expansion of ͑0͒ ͑0͒ ͑0͒ ͑ ͒ ͑ ͒ ͑ ͒ VnnЈ k , and are given through terms of the fourth order in k in which VnnЈ k =VnnЈ k,k is the potential energy of the reference crystal and by ͑ ͒ ˜ ˜ ˜ † ͑ ͒ Lnn k = k + kkD + kkkC + kkkkQ . pnnЈ = ͚ GUnGUnЈG 4.11 Ј nnЈ nnЈ nnЈ nnЈ G ͑5.1͒ is the momentum matrix of the reference crystal. Here a summation with respect to the Cartesian indices Within quadratic response theory, the perturbation ⌬ ͑ ͒ ͑ ͒ ͑ ͒ , , , and is implicit, and the coefficients are VnnЈ k,kЈ is obtained by substituting Eqs. 3.18 , 3.21 , ͒ r,0͒͑͑ ץ and ͑3.28͒ into Eq. ͑4.9͒. The result is VnnЈ k = p + ͩ ͪ , ץ Ј͒ ͑ Ј͒ ⌳ ͑ Ј͒͑ Ј͒ nnЈ nnЈ ͑ ⌬ VnnЈ k,k = WnnЈ k,k + nnЈ k,k k − k , k k=0 ͑ ͒ 4.12 ͑ ͒ 2V r,0 ͑k͒ץ ˜ 1ͩ nnЈ ͪ where the vertex function D = ␦␦ + , ץ ץ nnЈ nnЈ 2 k k k=0 ⌳ ͑ Ј͒ † ⌳ ͑ Ј͒ ͑ ͒ nnЈ k,k = ͚ UnG GGЈ k,k UnЈGЈ 4.13 ͑ ͒ 3V r,0 ͑k͒ץ G,GЈ ˜ 1 ͩ nnЈ ͪ is again an analytic function of k and kЈ. The W term is CnnЈ = , k ץ k ץ kץ !3 defined by W=W͑1͒ +W͑2͒, where k=0 ͒ r,0͒͑͑ 4ץ Ј ͑ ͒ ␣ ␣ 1 VnnЈ k W 1 ͑k,kЈ͒ = ͚ ͑k − kЈ͒W ͑k,kЈ͒, ͑4.14͒ Q˜ = ͩ ͪ . ͑5.2͒ ץ ץ ץ ץ nnЈ nnЈ nnЈ ␣ 4! k k k k k=0 ͑r,0͒͑ ͒ ␣ ␣ The various derivatives of V k account for contributions W ͑k,kЈ͒ = N ͚ U† w ͑k,kЈ͒U ei͑GЈ−G͒·␣. nnЈ nnЈ nG GGЈ nЈGЈ from the nonlocal part of the potential energy to the disper- Ј G,G sion relation of the reference crystal. The quantity is the ͑ ͒ nnЈ 4.15 kinetic momentum matrix of the reference crystal, whereas The screened potential is defined by the other terms give partial contributions ͑see Sec. VI for the remaining contributions͒ to the effective-mass tensor and the ͑q͒n͑q͒ ͑ ͒ vc ͑ ͒ cubic and quartic dispersion terms of the reference crystal. q = , 4.16 ͑ ͒ ⑀͑q͒ In principle, LnnЈ k should include contributions from the ͑i,0͒ ͑i,0͒ ͑ ͒ ͑ ͒ antihermitian part of the self-energy ͚ ͑k,͒=V ͑k,͒. where n=n 1 +n 2 is an effective electron density for the nnЈ nnЈ heterostructure perturbation: However, in Appendix A it is shown that, for energies near the band gap of the reference crystal, the contributions from Ј ⌺͑i͒ are much smaller than the smallest terms retained in the ͑1͒͑ ͒ ␣͑ ͒ ␣͑ ͒ ͑ ͒ n q = N͚ q n q . 4.17 present approximation scheme. Such contributions were ␣ therefore neglected in the above expressions, and are like- The quadratic contributions W͑2͒ and n͑2͒ are given by obvi- wise neglected in subsequent analysis of the heterostructure ous generalizations ͓see Eq. ͑3.28͔͒ of the above expressions. perturbation. However, in any calculation where it is desired
165345-8 FIRST-PRINCIPLES ENVELOPE-FUNCTION THEORY… PHYSICAL REVIEW B 72, 165345 ͑2005͒ to include the effects of a finite quasiparticle lifetime, the ␣͑x͒ = ͚ B͑k͒␣͑k͒eik·x, ͑5.7͒ dominant terms may be restored to leading order by replac- k ing En with where the cutoff function B͑k͒ is defined in Appendix B. ͑͒ → ͑͒ ⌺͑i,0͒͑ ͒ ͑ ͒ En En + i nn k = 0, , 5.3 From Eq. ͑3.8͒, the constraint and then retaining the imaginary part only to first order in perturbation theory in all subsequent analysis. ͚ aj͑x͒ =0 ͑5.8͒ a
B. Linear heterostructure potential is satisfied for any choice of B͑k͒. A similar Taylor series expansion technique is useful for The physical significance of the result ͑5.6͒ can be appre- ͑ ͒ ⌳ ͑ ͒ ͑ ͒ the terms WnnЈ k,kЈ and nnЈ k,kЈ in the heterostructure ciated by considering a specific example such as a 001 perturbation ͑4.12͒. This subsection begins by considering GaAs/AlAs heterojunction. In this case, as will be shown in ␣ the simple special case in which the screened atomic pseudo- Sec. VII, the function ͑x͒ depends only on the z coordinate ␣ ͑ Ј͒ ͑ ͒ ͑ ͒ and behaves like a smooth step function at the interface. The potentials wGGЈ k,k in Eqs. 3.28a and 4.15 have the form of a local potential; i.e., spatial derivatives in Eq. ͑5.6͒ therefore generate finite-width ␦-like terms at the interface, with the Z term proportional to ␣ ͑ Ј͒ ␣͑ Ј Ј͒ ͑ ͒ ␦͑ ͒ ␦ ͑ ͒ ͑ ͒ wGGЈ k,k = w k − k + G − G . 5.4 z and the Y term proportional to Ј z . Hence, Eq. 5.6 provides a first example of the interface band-mixing terms Such would be the case, for example, in an LDA calculation ͑ ͒ ͑ ͒ ͑1͒ alluded to previously in Eqs. 2.5 and 2.6 . based on local ionic pseudopotentials. In this case, WnnЈ The physical origin of these terms can be understood by ͑ ͒ ϫ͑k,kЈ͒ is also a local potential of the form W 1 ͑k,kЈ͒ going back one step further in the derivation. From Eq. nnЈ ␣ ͑1͒ ͑ ͒ ͑ ͒ ͑ Ј͒ 4.15 , it is clear that the Taylor series expansion of WnnЈ q =WnnЈ k−k . This simplification makes it easier to grasp in Eq. ͑5.5͒ is equivalent to a Taylor series expansion of the the physical significance of the power series expansion, and ͑ ͒ also facilitates a comparison between the present theory and screened atomic pseudopotential 5.4 with respect to q= k−kЈ. Hence, the physical origin of the linear and quadratic earlier envelope-function models based on local empirical ͑ ͒ ͑ ͒ pseudopotentials. in q terms in Eq. 5.5 is simply the finite slope and curva- ture of the screened atomic pseudopotentials in momentum 1. Local analytic terms space. This demonstrates that the Hamiltonian of a heterostruc- ͑1͒ ͑ Ј͒ The approximation technique used here for WnnЈ k−k is ture depends not just on the values of the atomic pseudopo- ␣ ͑ Ј͒ w␣͑k͒ G to expand the term WnnЈ k−k on the right-hand side of Eq. tentials at the reciprocal lattice vectors , but also at a ͑4.14͒ in a Taylor series in qϵk−kЈ: range of k values in a finite neighborhood of each G ͑the size of the neighborhood depending on how rapidly varying Ј the envelope function is͒. The necessity for an accurate fit- ͑1͒ ͑ ͒ ͚ ␣͑ ͓͒ ␣ ͑ ͒␣ ͑ ͒␣͔ WnnЈ q = q WnnЈ + iq ZnnЈ − qq YnnЈ . k ␣ ting of empirical pseudopotentials over a range of values has been emphasized particularly in the work of Mäder and ͑5.5͒ Zunger.149 The truncated expansion ͑5.5͒ shows that the ␣ Here the terms W ,͑Z ͒␣, and ͑Y ͒␣ are q-independent present perturbation scheme relies for its accuracy upon the nnЈ nnЈ nnЈ validity of a quadratic extrapolation of w␣͑k͒ in the neigh- expansion coefficients. The series ͑5.5͒ has been truncated at borhood of each G. Inspection of the form of typical atomic the second order in q because such terms are of order pseudopotentials145,150 shows this to be a good approxima- ͑¯ ͒2⌬¯ ka V, which are the smallest corrections permitted in the tion. present perturbation scheme. The definition ͑5.5͒ of the interface band-mixing param- The result ͑5.5͒ may be used directly in a k-space eters Z and Y highlights a significant difference between the envelope-function calculation based on Eq. ͑4.8͒, in which present approach and previous derivations of envelope- ⍀* the k values are expressly limited to the unit cell 0. How- function Hamiltonians. In previous derivations, the hetero- ever, to obtain a local differential equation, one must allow k structure potential was chosen to have the form to range over all possible values. Since ␣͑k͒ is quasiperi- ͓ ͑ ͔͒ odic see Eq. 3.20 , this local approximation will generate V͑x͒ = ͚ Vl͑x͒l͑x͒, ͑5.9͒ large-k terms in the envelope functions unless a k-space cut- l off is introduced. If this is done, the Fourier transform of ͑5.5͒ is the local potential in which Vl͑x͒ is the periodic potential for the bulk material l͑ ͒ Ј l, and x is a form factor determining the composition of ␣͑ ͒ the heterostructure. Within this model, V͑x͒ depends only͔ ץ ץ␣͒ ͑ ץ␣͒ ͑ ␣ ͓ ͒ ͑ 1͒͑ WnnЈ x = ͚ WnnЈ + ZnnЈ + YnnЈ x , ␣ upon the atomic pseudopotentials at the reciprocal lattice vectors G, with the magnitude of the interface ␦-like terms ͑5.6͒ determined by l͑x͒. In the limit of slowly varying l͑x͒, the .x and ͓cf. Eq. ͑3.19a͔͒ interface terms vanishץ/ץ= ץ in which
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␣ ␣ ␣ ␣ ␣ The present results show that this behavior is an unphysi- n ͑q͒ = qn + qqn + qqqn + qqqqn, cal artifact of the model ͑5.9͒. In the present theory, the ͑5.14͒ strength of the interface terms is the same ͑in the virtual crystal approximation͒ for slowly graded and abrupt hetero- which is valid for neutral perturbations. In a heterovalent ␣ structures, with the rate of change of ͑x͒ affecting only the zinc-blende heterostructure described by neutral width of the interface terms. perturbations,124,126 the dipole and traceless quadrupole ␣ ␣ 1 ␣ terms n and n − n␦ are nonvanishing only at inter- 3 ␣ 2. Nonlocal analytic terms faces. For isovalent substitutions in zinc-blende, n =0 and ␣ 1 ␣ n = n ␦ everywhere, so the contribution from the latter The more general case of a nonlocal potential involves 3 term is analytic and can be absorbed into the definition of the only a straightforward extension of these results. The matrix ␣ ␣ ␣ analytic potential ͑5.10͒. The remaining terms n and n element W ͑k,kЈ͒ of Eq. ͑4.15͒ is expanded to second or- nnЈ describe octopole and hexadecapole moments.124 der in k and kЈ as follows: The inverse dielectric function ⑀−1͑q͒ in Eq. ͑4.16͒ can
␣ ␣ ␣ ␣ * ␣ also be expanded in a power series. For isovalent perturba- W ͑k,kЈ͒ = W + k͑˜J ͒ + kЈ͓͑˜J ͒ ͔ + kk͑M˜ ͒ nnЈ nnЈ nnЈ nЈn nnЈ tions in zinc-blende this has no qualitative significance, since ˜ ␣ * ˜ ␣ the leading correction merely renormalizes the hexadecapole + kЈkЈ͓͑M ͒ ͔ + kkЈ͑R ͒ . ͑5.10͒ ␣ 124 nЈn nnЈ term n. However, the wave vector dependence of ⑀−1͑ ͒ Here the anti-Hermitian part of the self-energy was ne- q does generate a qualitatively new contribution from ͑ ͒ ͑ ͒ glected, as discussed above Eq. ͑5.3͒ and in Appendix A. the interface dipole term in Eq. 5.14 ; see Eq. 6.23 of Ref. This approximation simplifies the expansion by providing a 124 for the explicit form of this term. ͑ ͒ ͑ ͒ relationship between ͑for example͒ the coefficients of k and With the expansion 5.14 , the effective density 4.17 has kЈ. The physical interpretation of the various expansion co- the same form as that derived above for the local potential in ͑ ͒ efficients is discussed below ͑in Sec. VI͒, after perturbation Eq. 5.5 . In coordinate space, it involves a series of deriva- ␣͑ ͒ ͑ ͒ theory has been used to eliminate the coupling to remote tives of x , similar to the result shown in Eq. 5.6 . This is bands. the same as the usual multipole expansion of the macro- 151 When this expansion is substituted into Eq. ͑4.14͒ for scopic charge density in classical electromagnetism. In ͑ ͒ W 1 ͑k,kЈ͒, the result can be written as particular, note that for a perturbation consisting of a single nnЈ impurity atom, the function ␣͑x͒ is just the macroscopic ͑1͒ ͑1͒ * average of a Dirac ␦ function, in complete agreement with W ͑k,kЈ͒ = W ͑q͒ + k˜J ͑q͒ + kЈ͓˜J ͑− q͔͒ nnЈ nnЈ nnЈ nЈn the expressions given in Ref. 151. * Since the Taylor series expansion for ⌳ ͑k,kЈ͒ is iden- + kkM˜ ͑q͒ + kЈkЈ͓M˜ ͑− q͔͒ nnЈ nnЈ nЈn ␣ ͑ Ј͒ tical in form to that for WnnЈ k,k , the contribution from Ј˜ ͑ ͒ ͑ ͒ this term will not be written out explicitly here. All subse- + kkRnnЈ q , 5.11 quent perturbation theory analysis for the two terms is for- in which q=k−kЈ as before, and the various q-dependent mally identical, except that the vertex function is multiplied functions are defined by expressions of the form by an extra factor of ͑q͒.
Ј C. Quadratic heterostructure potential ˜ ͑ ͒ ␣͑ ͒͑˜ ͒␣ ͑ ͒ RnnЈ q = ͚ q RnnЈ . 5.12 ␣ 1. Analytic terms These functions all have a step-function-like behavior in x Because the quadratic response is already of order ͑ ͒ ␣ ͑⌬¯ ͒2 ¯ ͑ ͒ space at a heterojunction see Sec. VIII . Since R is real, V /Eg, the quadratic version of Eq. 4.14 can be replaced they also have the hermiticity and symmetry properties by the zeroth-order approximation ͑ ͒ ͑ ͒ ͓W 1 ͑q͔͒* = W 1 ͑− q͒, Ј nnЈ nЈn ͑2͒ ͑ Ј͒ ␣␣ЈRЈ͑ ͒ ␣␣ЈRЈ ϵ ˜ ͑2͒ ͑ ͒ WnnЈ k,k = ͚ q WnnЈ WnnЈ q , ␣,␣Ј,RЈ ͓˜ ͑ ͔͒* ˜ ͑ ͒ RnnЈ q = RnЈn − q , ͑5.15͒ in which ˜ ͑ ͒ ˜ ͑ ͒ ͑ ͒ MnnЈ q = MnnЈ q . 5.13 ␣␣ЈRЈ ␣␣ЈRЈ͑ Ј͒ ͑ ͒ WnnЈ = lim WnnЈ k,k , 5.16 k,kЈ→0 3. Nonanalytic terms where the limit is well defined for neutral perturbations.124 ⌳ ͑ Ј͒͑1͒͑ ͒ This contributes a local potential-energy term similar to that The term nnЈ k,k q describing the nonanalytic ͑ ͒ ͑ ͒ 1 ͑ ͒ ͑ ͒ contributions to Eq. 4.12 is handled in much the same way. given by WnnЈ q in Eq. 5.11 . At a heterojunction, the func- ⌳ ͑ ͒ The vertex function nnЈ k,kЈ is expanded in a Taylor se- tional dependence in x space is similar to that of a smooth ries of the form ͑5.10͒, while the linear electron density in ͑macroscopically averaged͒ step function, with possible de- Eq. ͑4.17͒ is expanded as viations in the vicinity of the junction.
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␣␣ЈRЈ ␣ ␣ Now since W depends only on the part of ͑ ͒ ͑ ͚͒ ␦ −iGʈ· ␣ ͑ ͒ nnЈ k = k3 kʈGʈe , 5.18 ␣␣ЈRЈ͑ Ј͒ Ј Gʈ WnnЈ k,k that is analytic in k and k , it is a short-range quantity that is significant only when RЈ+␣Ј−␣ is compa- where kʈ =k1b1 +k2b2. For small kʈ, the kʈ dependence is rable to the lattice constant a. Therefore, at a heterojunction, simply ␦ . The same conclusion holds for the pair distribu- ͑2͒ kʈ0 replacing W˜ ͑q͒ with an ideal step function would generate ␣␣ЈRЈ nnЈ tion function ͑k͒. an error of order ¯ka in a term of order ͑¯ka͒2⌬¯V. Hence, the Therefore, the nonanalytic potential ͑4.16͒ has the form ͑ ͒ ͑ ͒ ͑ ͒␦ error is of the same order as the local approximation 2.8 q = q3 qʈ0, which is independent of the direction of q. and can be neglected. As a result, all terms in ͑q͒ except the monopole and dipole Thus, within the present perturbation scheme there is no terms reduce to analytic functions of q3, which can be ab- ˜ ͑2͒ ͑ ͒ sorbed into the definition of the analytic potential W.Inre- interface contribution from WnnЈ q . This is an important re- sult, as it simplifies the analysis of interface effects in later gard to the monopole and dipole terms, this paper considers sections of this paper. only neutral perturbations in crystals with no bulk dipole Within the context of an empirical pseudopotential moment. Thus the only nonanalytic contributions are the ͑ ͒ 1/q terms generated by the interface dipoles in Eqs. ͑5.14͒ model,149 the nonlinear bulk term W˜ 2 ͑q͒ derived here can 3 nnЈ and ͑5.17͒. be represented as an environment dependence of the These have the same q dependence as the Fourier trans- ͑ ͒ 3 screened empirical pseudopotential, in which for example form of a step function, and merely add extra terms to the the pseudopotential for an As atom in GaAs is different from band offsets at a heterojunction. Hence, the interface dipole that for an As atom in AlAs. The importance of accounting ͑1͒ ͑ ͒ contributions can be absorbed into the definition of WnnЈ q for such effects has been emphasized by Mäder and ͑2͒ 149 ˜ ͑ ͒ Zunger. and WnnЈ q . Note that in a no-common-atom system such as InAs/GaSb, the contribution from the quadratic interface di- 2. Nonanalytic terms pole to the band offset has a different value for GaAs-like and InSb-like interfaces.152 The leading terms in the quadratic density are the dipole In summary, for the material systems considered in this and quadrupole terms paper, the nonanalytic potential ͑q͒ does not contribute anything qualitatively new in a heterostructure with two- dimensional translation symmetry. Only the interface dipole ␣␣ЈRЈ͑ ͒ ␣␣ЈRЈ ␣␣ЈRЈ ͑ ͒ n q = qn + qqn , 5.17 term is truly nonanalytic, and that can be absorbed into the definition of the band offsets ͑although this contribution does which again is valid for neutral perturbations. ͑Here the con- depend on the microscopic structure of the interface152͒. stant term vanishes even for charged perturbations as long as Long-range potentials arising from the direction dependence the system is insulating,124 but in this case there is an addi- of ͑q͒ appear only in structures with lower translation sym- tional nonanalytic term of order q2.124͒ In zinc-blende mate- metry, such as quantum wires and dots. ͑2͒ ␣␣ЈRЈ rials, the net contributions to n ͑q͒ from n and the ␣␣ЈRЈ traceless part of n both vanish in the bulk regions of a VI. ELIMINATION OF INTERBAND COUPLING ␣␣ЈRЈ heterostructure. The interface part of n is negligible un- 31,34,127 der the current approximation scheme, while the bulk part In this section, perturbation theory is used to derive can be absorbed into the definition of the analytic potential a multi-band effective-mass Hamiltonian from the infinite- ͑ ͒ dimensional matrix equations ͑4.8͒. The method is outlined W˜ 2 ͑q͒. Therefore, only the interface dipole term remains nnЈ briefly here; for further details, see Refs. 34 and 31. under the current approximation scheme. The zone-center states n of the reference crystal are di- vided into a class A containing the states of interest, and a class B containing all other states. The total Hamiltonian is D. Two-dimensional systems written as H=H0 +HЈ, where H0 has matrix elements ͑ ͒ ␦ ͑ ͒ In a heterostructure ͑such as a quantum well͒ with two- H0 mmЈ=Em mmЈ, and m= n,k is a composite index. The dimensional translation symmetry, the nonanalytic terms unperturbed Hamiltonian H0 is assumed to be Hermitian, but arising from ͑q͒ have a particularly simple form. Let the the perturbation HЈ need not be. A similarity transformation ¯ −S S dimensionless coordinates xi and kj be defined by x=xiai and H=e He is used to eliminate the coupling between sets A ͑ k=kjbj, where ai and bj are basis vectors for the direct and and B to any desired order in the perturbation HЈ. The trans- ␦ reciprocal lattices of the reference crystal, with ai ·bj =2 ij. formation is unitary if HЈ is Hermitian, as is approximately ͒ In these coordinates, the lattice sites are defined by xi =Ri and the case here. This yields a finite-dimensional effective kj =Gj, where Ri and Gj are integers. In such a two- Hamiltonian for states m,mЈ A, which is given explicitly ͑ ͒ ͑ ͒ dimensional system, one can choose a1 and a2 to lie parallel to third order in HЈ in Eq. C1 of Appendix C. to the junction plane, so that the atomic distribution function For the case considered here, the matrix elements of H0 ␣ ␣͑ ͒ Ј ͑ ͒ ͗ ͉ ͉ Ј Ј͘ ␦ ␦ R is independent of R1 and R2. The Fourier transform k and H are given by Eq. 4.8 as nk H0 n k =En nnЈ kkЈ then has the form and
165345-11 BRADLEY A. FOREMAN PHYSICAL REVIEW B 72, 165345 ͑2005͒
͗ ͉ ͉ ͘ ͑ ͒␦ ͑1͒ ͑ ͒ ˜ ͑2͒ ͑ ͒ only on the function immediately to its right. nk HЈ nЈkЈ = LnnЈ k kkЈ + W k,kЈ + W q nnЈ nnЈ All of the x-dependent functions are defined in terms of ⌳ ͑ ͒͑ ͒ ͑ ͒ ͑ ͒ + nnЈ k,kЈ q , 6.1 the k-space cutoff 5.7 . The quantity ͑ ͒ ͑ ͒ ͑1͒ ͑ Ј͒ ͑x͒ = + J ͑x͒ + ͓J ͑x͔͒* ͑6.4͒ in which LnnЈ k is defined in Eq. 5.1 , WnnЈ k,k in Eq. nnЈ nnЈ nnЈ nЈn ͑ ͒ ˜ ͑2͒ ͑ ͒ ͑ ͒ 5.11 , and WnnЈ q in Eq. 5.15 . Upon inserting these ma- is the material-dependent kinetic momentum matrix for the trix elements into Eq. ͑C1͒, one obtains the effective-mass heterostructure, while Hamiltonian ͑for n,nЈA͒ ͑ ͒ ͑ ͒ ͓ ͑ ͔͒* ͑ ͒͑ ͒ DnnЈ x = DnnЈ + MnnЈ x + MnЈn x + RnnЈ x 6.5 ͗nk͉H¯ ͉nЈkЈ͘ = ͑E ␦ + k + kkD + kkkC n nnЈ nnЈ nnЈ nnЈ is half the inverse effective mass tensor for the heterostruc- ͒␦ ͑1͒ ͑ ͒ ͑2͒ ͑ ͒ ture. This has symmetric and antisymmetric parts: + kkkkQnnЈ kkЈ + WnnЈ q + WnnЈ q ͕͖ ͑ ͒ 1 ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ Ј͓ ͑ ͔͒* ͑ ͒ DnnЈ x = 2 DnnЈ x + DnnЈ x , + kJnnЈ q + k JnЈn − q + kkMnnЈ q Ј Ј͓ ͑ ͔͒* Ј ͑ ͒ ͓⌳ ͓͔ 1 + kk M − q + kkR q + nnЈ ͑ ͒ ͓ ͑ ͒ ͑ ͔͒ ͑ ͒ nЈn nnЈ DnnЈ x = 2 DnnЈ x − DnnЈ x , 6.6 ˆ Ј͑ˆ ͒* ˆ Ј Ј͑ ˆ ͒* + kJnnЈ + k JnЈn + kkMnnЈ + kk MnЈn although the antisymmetric part has a nonvanishing contri- bution only in the presence of a magnetic field. Ј ˆ ͔͑ ͒ ͑ ͒ ⌫ ⌽ ͑ ͒ + kkRnnЈ q . 6.2 The functions Z,Y , , and in Eq. 6.3 are all interface terms. The first two terms In this expression, is the kinetic momentum matrix nnЈ ͑ ͒ Z ͑x͒ =−i 1 J ͑x͒ − ͓J ͑x͔͒* , ͑6.7͒ 5.2 of the reference crystal, 2DnnЈ is the inverse effective nnЈ 2 „ nnЈ nЈn … mass tensor of the reference crystal ͓see Eq. ͑C2͔͒, and C nnЈ ͕͖ Y ͑x͒ = 1 R ͑x͒, ͑6.8͒ and QnnЈ are the coefficients of the cubic and quartic dis- nnЈ 2 nnЈ persion terms in the reference crystal ͓see Eqs. ͑C3͒ and are renormalized versions of the ␦ and ␦Ј mixing potentials ͑C4͔͒. These are just renormalized versions of the quantities considered previously in Eqs. ͑5.5͒ and ͑5.6͒. However, the ˜ ˜ ˜ ͑ ͒ DnnЈ,CnnЈ , and QnnЈ defined previously in Eq. 5.2 . other two terms ͑2͒ ͑ ͒ ͑ ͒ ͑ ͒ Likewise, the functions WnnЈ q ,JnnЈ q ,MnnЈ q , and ͕͖ ͕͖ * ⌫ ͑x͒ =−i M ͑x͒ − ͓M ͑x͔͒ , ͑6.9͒ ͑ ͒ nnЈ „ nnЈ nЈn … RnnЈ q are all renormalized versions of quantities defined previously. The renormalized functions are given explicitly ⌽ ͑ ͒ ͓͔͑ ͒ ͑ ͒ in Appendix C. nnЈ x = iRnnЈ x , 6.10 The terms multiplying the screened nonanalytic potential were not present in Eqs. ͑5.5͒ and ͑5.6͒. The term ⌽ ͑x͒, ͑q͒ are derived from the Taylor series expansion and k· nnЈ ⌳ ͑ Ј͒ which is antisymmetric in and , is just a generalized renormalization of the vertex function nnЈ k,k . The vari- 31,153–156 Rashba coefficient for multiband Hamiltonians. ous constant coefficients ͑e.g., Rˆ ͒ are defined in the same nnЈ However, the term ⌫ ͑x͒, which is symmetric in and ͓ ͑ ͔͒ nnЈ way as the analogous q-dependent functions e.g., RnnЈ q ,157 has received little attention in the literature. Its physical ͑1͒ ͑ ͒ given in Appendix C, but with WnnЈ q replaced by the con- interpretation will be discussed below in Sec. VIII. ⌳ ⌳ ͑ ͒ ⌫ ⌽ stant nnЈ= nnЈ 0,0 . Since the functions Z,Y , , and behave to lowest order Equation ͑6.2͒ can now be rearranged47,48 and Fourier as step functions at an abrupt junction, the Hamiltonian ͑6.3͒ transformed to obtain the effective-mass Hamiltonian shows explicitly that these functions produce interface terms proportional to ␦ or ␦Ј. ¯ ͑ ͒ ␦ ͕ ͑ ͖͒ ͕ ͕͖͑ ͖͒ HnnЈ x,p = En nnЈ + p, nnЈ x + pp,DnnЈ x The remaining terms ͑ˆ ,Zˆ , etc.͒ in Eq. ͑6.3͒ that appear in ͑1͒ front of ͑x͒ are defined by the obvious generalizations of + pppC + ppppQ + W ͑x͒ nnЈ nnЈ nnЈ Eqs. ͑6.4͒–͑6.10͒. The interpretation of these terms parallels ͑2͒ that of the terms already discussed, except that the contribu- Y ͑x͒ץץ + Z ͑x͒ץ + W ͑x͒ + nnЈ nnЈ nnЈ tions from are not as well localized at the interface. ⌽ ͑ ͔͒ ⌳ ͑ ͒ Here it is worth noting that in the GW approximation,94,95 ץ͓ ͖͒ ͑ ⌫ ץ ͕ + p, nnЈ x + nnЈ x p + nnЈ x ⌳ ␦ ˆ ͑ ͒ one has nnЈ= nnЈ and ˆ =D =0, so the potential x ͕ ͑ ͖͒ ˆ ͕͖͕ ͑ ͖͒ nnЈ nnЈ + ˆ nnЈ p, x + DnnЈ pp, x does not contribute to the momentum matrix or the effective- mass tensor. In LDA, these simplifications are also valid, and ͑ ͖͒ ץ ͕ ˆ⌫ ͒ ͑ ץ ץ ˆ ͒ ͑ ץ ˆ + ZnnЈ x + YnnЈ x + nnЈ p, x ˆ one has in addition ZnnЈ=0, since the exchange-correlation potential is short-ranged. ͑x͔͒p, ͑6.3͒ץ͓ ˆ⌽ + nnЈ For isovalent zinc-blende systems, the leading term in ͑1͒ ͕ ͖ ͕ ͖ 1 ͑ ͒ is an octopole potential, so the contributions from the in which A,B = AB = 2 AB+BA is the symmetrized prod- x acts second-rank tensors Dˆ ,Yˆ ,⌫ˆ , and ⌽ˆ are negligible. Theseץ/ץ= ץ is the momentum operator, and uct, p=−i١
165345-12 FIRST-PRINCIPLES ENVELOPE-FUNCTION THEORY… PHYSICAL REVIEW B 72, 165345 ͑2005͒ terms are non-negligible only for the linear interface dipole not depend on the choice of reference crystal.͔ The transfor- term in a heterovalent zinc-blende system ͑or for a slowly mation from ͑7.1a͒ and ͑7.1b͒ is then simply varying external potential, which is not considered explicitly ͑1͒ Al Al Al AlAs ͒ ͑2͒ ⌳ W ͑x͒ = ͑x͒W = ͑x͒W . ͑7.3͒ here . Likewise, is negligible in all terms except nnЈ. nnЈ nnЈ nnЈ ͑ ͒ Equation 6.3 is the main result obtained in this paper. Hence The qualitative form of this Hamiltonian is very similar to the Leibler Hamiltonian47,48 for slowly graded heterostruc- AlAs = Al, GaAs = Ga. ͑7.4͒ 65,66,68 tures, as amended by Takhtamirov and Volkov. The For a no-common-atom system such as InAs/GaSb, four differences are primarily due to the use of atomic pseudopo- different bulk potentials can be defined: tentials ͑rather than a model based on periodic bulk poten- ͒ InAs In As tials , the inclusion of long-range Coulomb potentials, and WnnЈ = WnnЈ + WnnЈ, the use of linear response theory to simplify the interface Hamiltonian, as discussed in Sec. IX. WInSb = WIn + WSb , The explicit form of the various material parameters in nnЈ nnЈ nnЈ ͑ ͒ Eq. 6.3 for semiconductors with the zinc-blende structure is GaAs Ga As given below in Sec. VIII. First, however, the possibility of WnnЈ = WnnЈ + WnnЈ, representing the material parameters as piecewise constant is GaSb Ga Sb ͑ ͒ considered. WnnЈ = WnnЈ + WnnЈ. 7.5 In If the reference crystal is chosen to be InAs, then WnnЈ VII. SIMPLIFIED MATHEMATICAL REPRESENTATION As OF HETEROSTRUCTURE MATERIAL PROPERTIES =WnnЈ=0. In the atomic description, the linear response po- tential is This section discusses several ways in which the math- ͑1͒ ͑ ͒ Ga͑ ͒ Ga Sb͑ ͒ Sb ͑ ͒ ematical description of material properties can be simplified. WnnЈ x = x WnnЈ + x WnnЈ, 7.6 The first is to label the materials in terms of bulk compounds which can be rewritten ͑using In+Ga=0 and As+Sb=0͒ in ͑e.g., GaAs͒ rather than atoms; the second is to approximate several different ways, two of which are the properties of an abrupt junction using piecewise constant ␦ ͑1͒ Ga GaSb ͑In Sb͒ InSb ͑ ͒ material parameters with functions and their derivatives at WnnЈ = WnnЈ + + WnnЈ 7.7a interfaces. For simplicity, only two-dimensional isovalent systems are considered here ͑see Sec. V D͒. and ͑1͒ Sb GaSb ͑Ga As͒ GaAs ͑ ͒ WnnЈ = WnnЈ + + WnnЈ . 7.7b A. Transformation from atomic to bulk-crystal description Equation ͑7.7a͒ is useful for describing a ͑001͒ heterojunc- All of the linear-response terms can be transformed im- tion with an InSb-like interface: mediately to a bulk-crystal representation similar to that de- ͑ ͒ scribed above in Eq. ͑5.9͒: ¯-As-In-As-In-Sb-Ga-Sb-Ga- ¯ , 7.8a Ј while Eq. ͑7.7b͒ is useful for describing a heterojunction ͑1͒ ͑ ͒ ␣͑ ͒ ␣ ͑ ͒ with a GaAs-like interface: WnnЈ x = ͚ x WnnЈ 7.1a ␣ ͑ ͒ ¯-In-As-In-As-Ga-Sb-Ga-Sb- ¯ . 7.8b Ј In the first case one can identify the bulk functions =͚ l͑x͒Wl , ͑7.1b͒ nnЈ GaSb = Ga, InAs = As, l in which l labels the different bulk materials, and ͓cf. Eq. InSb = In + Sb, GaAs =0, ͑7.9a͒ ͑5.8͔͒ while in the second case l͑ ͒ ͑ ͒ ͚ x =0. 7.1c GaSb = Sb, InAs = In, l For example, in a GaAs/AlAs heterostructure, the linear po- GaAs = Ga + As, InSb =0. ͑7.9b͒ tentials for the two bulk media are defined by This type of transformation can be used for any term in the GaAs Ga As linear response. For the quadratic response, such a descrip- WnnЈ = WnnЈ + WnnЈ, tion is not appropriate, but all contributions ͑including the interface dipole potential͒ can be approximated as abrupt AlAs Al As ͑ ͒ WnnЈ = WnnЈ + WnnЈ. 7.2 step functions, as discussed in Secs. V C 1 and V D. Ga If the reference crystal is chosen to be GaAs, then WnnЈ B. Piecewise constant material parameters As ͓ =WnnЈ=0. This choice is made for clarity of exposition; the The next step in simplifying the description of the mate- final results given in Eqs. ͑7.4͒, ͑7.9͒, ͑7.14͒ and ͑7.16͒ do rial properties is to approximate an ideal heterostructure as
165345-13 BRADLEY A. FOREMAN PHYSICAL REVIEW B 72, 165345 ͑2005͒ piecewise constant. As a specific example, the case of a ͑001͒ ͑¯ka͒3⌬¯V or higher, which can be neglected. Thus for an ideal heterojunction between semiconductors with the zinc-blende GaAs/AlAs junction, all of the material parameters in ͑6.3͒ ͑ ͒ structure is considered here. except W 1 ͑x͒ can be replaced by abrupt step functions. For this case, a convenient slab-adapted158,159 unit cell is nnЈ defined by the basis vectors For the case of a no-common-atom InAs/GaSb junction, the situation is more complicated. The example considered a a a here is the GaAs-like junction in Eq. ͑7.8b͒, where the origin ͑ ͒ ͑ ͒ ͑ ͒ a1 = 1,− 1,0 , a2 = 1,1,0 , a3 = 1,0,1 . 1 ͑ ͒ 2 2 2 is the midpoint of an As-Ga bond and c =− a = 8 a 1,1,1 . In this case, the small-k behavior of the atomic distribution ͑7.10͒ functions is Periodic boundary conditions are applied over the crystal volume ⍀=L ·͑L ϫL ͒, where L =N a and N is an integer a a2 1 2 3 i i i i Ga͑ ͒Ӎ⌰͑ ͒ ⌰ ͑ ͒ ⌰ ͑ ͒ ͑ ⍀ ⍀ ͒ k k + z k − zz k , thus =N 0, where N=N1N2N3 . 8 384 The bulk properties of the heterojunction are to be repre- sented in terms of the periodic step function a a2 Ͻ Ͻ 1 Sb 1, 0 z Lz, ͑k͒Ӎ⌰͑k͒ − ⌰ ͑k͒ − ⌰ ͑k͒. ͑7.15͒ ⌰͑x͒ = ⌰͑z͒ = ͭ 2 ͮ ͑7.11͒ 8 z 384 zz 1 Ͻ Ͻ 0, − 2 Lz z 0, 1 Upon inserting these results into Eq. ͑7.7b͒, one finds that for in which Lz = 2 N3a is the period in the z direction, and ⌰͑ ͒ ⌰͑ ͒ ͉z͉Ͻ 1 L , the linear band offset may be approximated by x = x+Li . The interface properties are to be repre- 2 z ͒ ͑ ⌰ ץ ⌰ץ ͒ ͑ ⌰ sented by the derivatives z x = / z and zz x z2 ␦ ␦Ј ͑1͒ GaSb InAs a GaAs InAsץ ⌰2ץ = / , which are periodic arrays of and functions. W ͑x͒ = ͑W − W ͓͒⌰͑z͒ − x͔ + ͑2W − W The first example of an actual ͑001͒ heterojunction to be nnЈ nnЈ nnЈ 8 nnЈ nnЈ considered is a common-atom GaAs/AlAs junction. The co- a2 ordinate origin is chosen to be an interface As atom, with − WGaSb͒␦͑z͒ − ͑WGaSb − WInAs͒␦Ј͑z͒, 1 ͑ ͒ nnЈ 384 nnЈ nnЈ unit-cell basis vectors a =0 and c = 4 a 1,1,1 for anions ␣ ␣ ͑ ͒ and cations, respectively. The discrete function R for 7.16a =Al is therefore 1 which is written in a form suitable for the virtual reference 1, 0 ഛ R ഛ N −1, Al 3 2 3 ͑ ͒ ͑ ͒ = ͭ ͮ ͑7.12͒ crystal InAs 1−x GaSb x. Note that this has the form R 1 ഛ ഛ 0, − 2 N3 R3 −1, ͑ ͑ ͒ ͑ ͒ ͑ ͒ a ͑ ͒ ͑ ͒ where the dimensionless coordinate R3 is an integer see Sec. W 1 ͑x͒ = ͑W + − W − ͓͒⌰͑z͒ − x͔ + ͑2W i − W + VD͒. From the Fourier transform of this function, one finds nnЈ nnЈ nnЈ 8 nnЈ nnЈ that for small k, 2 ͑ ͒ a ͑ ͒ ͑ ͒ − W − ͒␦͑z͒ − ͑W + − W − ͒␦Ј͑z͒, ͑7.16b͒ a2 nnЈ nnЈ nnЈ Al͑k͒Ӎ⌰͑k͒ − ⌰ ͑k͒. ͑7.13͒ 384 96 zz in which ͑+͒ and ͑−͒ label the bulk materials to the right and Upon inserting this result into Eq. ͑7.3͒, one finds that for 1 left of the junction, ͑i͒ labels the “interface” material, and the ͉z͉Ͻ L , the linear band offset at a GaAs/AlAs heterojunc- 2 z reference crystal is ͑+͒ ͑−͒ . This result is valid for any tion can be represented as 1−x x no-common-atom ͑001͒ junction, regardless of the conven- ͑ ͒ W 1 ͑x͒ = ͑WAlAs − WGaAs͓͒⌰͑z͒ − x͔ tion chosen for the vectors. Thus for a no-common-atom nnЈ nnЈ nnЈ ͑ ͒ junction, W 1 ͑x͒ contains terms that renormalize both the Z a2 nnЈ ͑ AlAs GaAs͒␦Ј͑ ͒ ͑ ͒ and Y matrices. − WnnЈ − WnnЈ z , 7.14 96 The ␦͑z͒ term in ͑7.16͒ has a very simple physical inter- ͑ ͒ which has been written in a general form suitable for a vir- pretation in which half a monolayer a/4 at the interface is ͑ ͒ ͑ ͒ occupied by GaAs instead of either InAs or GaSb. This con- tual reference crystal GaAs 1−x AlAs x. Thus, if the band offset is treated as piecewise constant, there is an additional cept has been used previously in the construction of 160 161 correction proportional to ␦Ј͑z͒ that has the effect of renor- envelope-function models for electrons and phonons. ͑ ͒ malizing the matrix Y in the Hamiltonian ͑6.3͒. This merely Note that if one replaces Sb with As in Eq. 7.16a , the reflects the fact that Al͑z͒ is a smooth ͑macroscopically av- result does not reduce directly to an expression of the form ͑ ͒ eraged͒ step function, and the difference between a smooth 7.14 . However, the difference is merely due to the different step function and an abrupt step function is, to leading order, choice of coordinate origin in the two cases. To leading or- ⌰͑ 1 ͒Ӎ⌰͑ ͒ 1 ␦͑ ͒ 1 2␦Ј͑ ͒ ␦͑ 1 ͒Ӎ␦͑ ͒ proportional to the macroscopic average of ␦Ј͑z͒. der, z− 8 a z − 8 a z + 128a z , z− 8 a z 1 ␦Ј͑ ͒ ␦Ј͑ 1 ͒Ӎ␦Ј͑ ͒ The remaining material-dependent parameters in Eq. ͑6.3͒ − 8 a z , and z− 8 a z . Thus, replacing Sb with As 1 ͑ ͒ can be treated using the same approach, but for these param- and shifting the origin by 8 a 1,1,1 does indeed reduce ⌰ ͑ ͒ ͑ ͒ ͑ ͒ eters the term proportional to zz yields a correction of order 7.14 , 7.15 , and 7.16a .
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The result ͑7.16b͒ can be applied to all other linear- TABLE I. Basis functions for constructing invariants of Td. Here response terms in the Hamiltonian ͑6.3͒, but since the ⌬͑x͒ P is a vector operator whose components need not commute, while I J ͑ ͒ and Z͑x͒ terms are already of order ͑¯ka͒⌬¯V, the ␦Ј͑z͒ term in , , and are angular momentum pseu- dovector operators cor- responding to angular momentum 1 ,1,and 3 , respectively. Cyclic ¯ 3 ¯ 2 2 ͑7.16b͒ yields a negligible correction of order ͑ka͒ ⌬V. Thus permutations of x,y, and z also yield acceptable basis functions. ͑ ͒ ͑ ͒ Z ͑x͒ = ͑Z + − Z − ͓͒⌰͑z͒ − x͔ nnЈ nnЈ nnЈ Rep. Basis functions
a ͑i͒ ͑+͒ ͑−͒ ⌫ 2 ͑ ͒␦͑ ͒ ͑ ͒ 1 1, P + 2ZnnЈ − ZnnЈ − ZnnЈ z , 7.17 8 ⌫ 2 JxJyJz +JzJyJx ⌫ 2 1 2 2 1 2 2 1 2 and we see that Z generates a correction to Y, while gen- 12 Px − 3 P ,Ix − 3 I ,Jx − 3 J ⌫ ⌫ ͕ ͖ ͕ ͖ ͕ ͖ ϵ͕ ͑ 2 2͖͒ erates a correction to . However, the position dependence 15 Px , PxPy , IxIy , JxJy ,Vx Jx Jy −Jz ͑ ͒͑ ⌫ ͓ ͔ 3 of all remaining terms in 6.3 including the quadratic re- 25 i Px , Py , x ,Ix ,Jx ,Jx sponse͒ can be represented as a simple step function: ͑ ͒ ͑ ͑+͒ ͑−͓͒͒⌰͑ ͒ ͔ ͑ ͒ YnnЈ x = YnnЈ − YnnЈ z − x , 7.18 combination of l͑x͒ functions. The corresponding results for ⌫ and ⌫ states are obtained by replacing J →I ͑spin 1͒ ͑¯ ͒2⌬¯ 15 6 since these terms are already of order ka V. → ͑ 1 ͒ and J spin 2 , respectively. This yields zero in both ͑ ͒ ⌫ cases, hence x occurs only for 8 and is a relativistic VIII. THEORY OF INVARIANTS effect. In this section the method of invariants32–36 is used to For the effective-mass tensor D ͑x͒ one obtains the well- construct the explicit form of the Hamiltonian ͑6.3͒ for ⌫ known result32 electrons in semiconductors with the zinc-blende 1 2 1 2 − D ͑x͒ = ␥ ͑x͒␦1−␥ ͑x͒␦͑J − J ͒ structure.162,163 The results are then compared with the inter- 2 1 2 3 ␥ ͑ ͒͑ ␦ ͕͒ ͖ face Hamiltonian given by the method of invariants for a − 3 x 1− JJ ͑ ͒ ͑ common-atom 001 junction, including all linear and non- 3 + i⑀͓͑x͒J + q͑x͒J ͔, ͑8.2͒ linear͒ interface terms of order ͑¯ka͒2⌬¯V that are permitted by ␥ ͑ ͒ ␥ ͑ ͒ ␥ ͑ ͒ ͑ ͒ symmetry. Finally, the interface terms of order ͑¯ka͒⌬¯V are in which the Luttinger parameters 1 x , 2 x , 3 x , x , ͑ ͒ l͑ ͒ considered for the case of ⌫-X coupling in GaAs/AlAs. The and q x are all linear combinations of x functions, and ⑀ ⌫ results in this section make use of the time reversal and crys- is the antisymmetric unit tensor. For 15 the parameter q ͑ 3 ͒ ⌫ ␥ tal symmetry properties of the self-energy that were derived is not independent since I =I , whereas for 6 only 1 and in Ref. 124. are independent. The Z matrix has the same symmetry as the coordinate ͑ ͒ A. Generalized Leibler Hamiltonian operator i.e., a vector that is even under time reversal ,soit has the form30 As demonstrated above, to within terms of order ͑¯ka͒2⌬¯V, ͑ ͒ ͉⑀ ͉͕ ͖͑ ͒ ͑ ͒ the position dependence of the interface parameters can be Z x = JJ x , 8.3 calculated entirely in terms of bulk-like matrix elements where ͑x͒ is a linear combination of l͑x͒ functions. This modulated by the atomic form factors ␣͑x͒. Thus, the ⌫ ⌫ result has the same form for 15, but vanishes for 6. The Hamiltonian for a zinc-blende heterostructure of arbitrary coupling ͑8.3͒ generates a zone-center mixing of heavy and composition can be determined by constructing invariants light holes, and was proposed independently in Refs. 3, 28, ⌫ transforming as 1 under the symmetry operations of the Td 59, and 60. ␣ group. This amounts to treating ͑x͒ as a “slowly varying” The Y interface matrix has the same symmetry as the 48 function that is an invariant of Td. symmetric part of D, so it can be written The relevant basis functions for the representations of Td ͑ ͒ 1 ͑ ͒␦ ͑ ͒␦ ͑ 2 1 2͒ are given in Table I. The specific example to be considered Y x = 2 1 x 1− 2 x J − 3 J ⌫ ⌫ ⌫ here is that of the 8 valence band, with 15 and 6 derivable ͑x͒͑ ␦ ͕͒J J ͖ ͑ ͒ ⌫ − 3 1− . 8.4 as special cases of the 8 results. The extension of these ͑ ⌫ ⌫ ⌫ ͒ ⌫ results to multi-band Hamiltonians e.g., 6 7 8 can be This term has not been studied previously for 8 states, al- ⌫ ͑ handled using the methods of Refs. 33–35, but is not consid- though the corresponding term for 6 electrons a direct ana- ered explicitly here. log of the Darwin term from the Dirac equation͒ is well The momentum matrix ͑x͒ must transform as a vector known.47,48,53 ͑⌫ ͒ ⌫ 15 that is odd under time reversal, which leaves only one The matrix has the form possibility: ⌫ ͑x͒ = ͉⑀͉V͑x͒, ͑8.5͒ ͑ ͒ ͑ ͒ ͑ ͒ x = V x , 8.1 in which ͑x͒ is a linear combination of l͑x͒ functions. The ϵ͕ ͑ 2 2͖͒ ˆ in which Vx Jx Jy −Jz is a product of angular momentum closely related coupling ⌫ arising from an external electric 3 ͑ ͒ matrices J for a particle with spin 2 , and x is some linear field was proposed recently in Ref. 18 as a possible mecha-
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͑ ͒ TABLE II. Basis function for D2d. The components of ,I, and TABLE IV. Terms in the 001 GaAs/AlAs interface Hamil- ͑ ϫ ͒ B=ic P P transform as those of J. tonian constructed from invariants of C2v.
⌫ ͑ ͒ Rep. Basis functions Term D2d 1 Td Origin ␦͑ ͒ X1 1, 1 z X1 no 2 2 2 2 2 2 ͕ ͖ 2 1 2 Px + Py , Pz ,Jx +Jy ,Jz , Pz JxJy , PzVz , ͑J − J ͒␦͑z͒ X no 3 3 z 3 1 P J P J P J P J P V P V x x − y y , x x − y y , x x + y y , ͕J J ͖␦͑z͒ X yes Z P ͕J J ͖+ P ͕J J ͖ x y 3 x y z y z x ␦ ͑ ͒ 2 2 2 2 3 1 Ј z X3 yes Y X2 Px − Py ,Jx −Jy , PzJz , PzJz , PxJx + PyJy , 2 1 2 3 3 ͑J − J ͒␦Ј͑z͒ X yes Y P J + P J , P V − P V , P ͕J J ͖− P ͕J J ͖ z 3 3 x x y y x x y y x y z y z x ͕ ͖␦ ͑ ͒ ͕ ͖ ͕ ͖ 2 JxJy Ј z X1 no X3 Pz , PxPy , JxJy ,Vz , PzJz , PxJy − PyJx , 3 3 ͕ ͖ ͕ ͖ V ͕P ␦͑z͖͒ X no PxJy − PyJx , PxVy + PyVx , Px JxJz + Py JyJz z z 1 3 ͑ 2 2͒ ͑P V + P V ͒␦͑z͒ X no X4 Jz ,Jz , Pz Jx −Jy , PxJy + PyJx , x x y y 1 3 3 ͕ ͖ ͕ ͖ ͑ ͒␦͑ ͒ ⌫ PxJy + PyJx , PxVy − PyVx , Px JxJz − Py JyJz PxVy + PyVx z X3 yes ͑ ͒ ͕͑ ͖ ͕ ͖͒ ͑ ͒ ͑ 3 3͒ ͑ ͒␦͑ ͒ X5 Px , Py , PyPz , PzPx , Jx ,−Jy , Jx ,−Jy , PxJx − PyJy z X1 no ͑V ,V ͒,͕͑J J ͖,͕J J ͖͒,͑P J ,−P J ͒, ͑ ͒␦͑ ͒ ⌽ x y y z z x z y z x PxJy − PyJx z X3 yes ͑ 3 3͒ ͑ ͕ ͖ ͕ ͖͒ ͑ ͒ PzJy ,−PzJx , Pz JzJx , Pz JzJy , PzVy , PzVx , ͑P J3 − P J3͒␦͑z͒ X no ͑ ͒ ͑ 2 2͒ ͑ 3 3͒ x x y y 1 PyJz ,−PxJz , PxJz , PyJz , PyJz ,−PxJz , ͑ 3 3͒␦͑ ͒ ⌽ PxJy − PyJx z X3 yes ͑PyVz , PxVz͒,͑Py͕JxJy͖, Px͕JxJy͖͒
allowed by symmetry for a GaAs/AlAs ͑001͒ heterojunction, nism for the linear and circular photogalvanic effects. This which has the point group C2v. To better understand this ⌫ ⌫ term is relativistic in origin and does not occur for 15 or 6. comparison, it is helpful to begin by studying the symmetry The matrix ⌽ has the same symmetry as the antisymmet- properties of the linear and quadratic response in this system. ric part of D, except that it is odd rather than even under time The starting point is the observation that although the ͑ reversal. It can therefore be written as point group of the heterojunction is C2v with the coordinate origin at an interface As atom͒, if the reference crystal is 3 ⌽ ͑x͒ = ⑀͓Ј͑x͒J + qЈ͑x͒J͔. ͑8.6͒ chosen to be the virtual crystal Al0.5Ga0.5As, then the pertur- ⌫ ͑ Ј ͒ bation due to the ionic pseudopotentials has a higher sym- For 6 electrons where q is not independent this is the metry: it transforms according to the representation X of the analog of the Rashba spin-splitting effect for low-symmetry 3 153 D2d group. In other words, it transforms according to the bulk semiconductors, which occurs also in heterostruc- ⌬ ͑ 48,154,164,165 identity representation 1 of the C2v group none of whose tures of cubic semiconductors due to the reduced operations change the z coordinate͒, but in addition it has symmetry at a surface or interface. The corresponding ⌫ odd parity with respect to those elements of D2d that change Hamiltonian for 8 states was proposed in Ref. 58, and has the sign of z. This occurs because the atomic mole fraction received renewed attention as the valence-band Rashba ␣ ͑ ͒ R itself transforms as X3 D2d , whereas the atomic pertur- coupling27,31,155,156 in recent years. ␣ bation ⌬v ͑x,xЈ͒ has the site symmetry ⌫ ͑T ͒ or X ͑D ͒. The coefficients of the various terms involving in Eq. ion 1 d 1 2d Therefore, ⌬V ͑x,xЈ͒ transforms as X X =X under the ͑6.3͒ have the same form as those already given. The sym- ion 1 3 3 ͑ ͒ D metry restrictions on the Taylor series expansions for n 1 and operations of 2d. n͑2͒ were given in Ref. 124. To determine the behavior of the screened potential ⌬V͑x,xЈ͒, note that since the vertex functions124 ⌫͑1͒ and ⌫͑2͒ ⌫ ͑ ͒ B. GaAs/AlAs (001) heterojunction of the reference crystal transform as 1 Td , the linear re- ͑1͒͑ ͒ sponse V x,xЈ transforms as X1 X3 =X3, while the qua- The significance of these results is now investigated by ͑2͒͑ ͒ ͑ dratic response V x,xЈ transforms as X1 X3 X3 =X1. In ͑¯ ͒2⌬¯ comparing them with all interface terms of order ka V general, all odd-order terms in the response transform as X3, ͒ while all even-order terms transform as X1. Thus the total ⌬ ͑ ͒ ͑1͒͑ ͒ ͑2͒͑ ͒ TABLE III. Basis functions for C2v. The components of ,I, response V x,xЈ =V x,xЈ +V x,xЈ transforms as ͑ ϫ ͒ ⌬ ͑ ͒ and B=ic P P transform as those of J. neither X3 nor X1, but as 1 C2v . If the quadratic response is of the same order as the linear Rep. Basis functions response, then one must include all possible invariants of C2v ¯ 2 ¯ ⌬ ͕ ͖ 2 2 2 ͕ ͖ 2 2 2 3 that are of order ͑ka͒ ⌬V when constructing the interface 1 1, Pz , PxPy , Px + Py , Pz , JxJy ,Jx +Jy ,Jz ,Jz , Hamiltonian. Basis functions for D and C are given in Vz 2d 2v ⌬ P2 − P2 ,J ,J2 −J2 Tables II and III, respectively; note that both X1 and X3 are 2 x y z x y ⌬ ⌬ ͕ ͑ ͖͒ ͕ ͑ ͖͒ compatible with 1. All interface invariants of C2v that are 3 Px + Py , Pz Px + Py ,Jx −Jy , Jz Jx +Jy , 3 3 ͑1͒ Hermitian, ͑2͒ time-reversal invariant, and ͑3͒ of order J −J ,Vx +Vy x y ¯ 2 ¯ ⌬ ͕ ͑ ͖͒ ͕ ͑ ͖͒ ͑ka͒ ⌬V or less are listed in Table IV. 4 Px − Py , Pz Px − Py ,Jx +Jy , Jz Jx −Jy , 3 3 The second column in this table lists the symmetry of Jx +Jy ,Vx −Vy each term under the operations of D2d. The third column
165345-16 FIRST-PRINCIPLES ENVELOPE-FUNCTION THEORY… PHYSICAL REVIEW B 72, 165345 ͑2005͒ indicates whether the term can be constructed as an invariant a discussion of the differences between the present theory ⌰͑ ͒ ͑ of Td, treating z as an invariant of Td. As discussed and previous envelope-function theories that have appeared above, the latter approach can be used for all linear-response in the literature. The principal differences are that ͑1͒ the ͒ interface terms. Note that in the Td symmetry analysis, present Hamiltonian is constructed from atomic pseudopo- ␦͑ ͒ ⌰ ͑⌫ ͒ ␦ ͑ ͒ 2 z =d /dz transforms as z 15 and Ј z transforms as z tentials rather than the periodic potential of a bulk crystal; ͑⌫ ⌫ ͒ ␦͑ ͒ ͑ ͒ 1 or 12 , whereas in the D2d symmetry analysis, z 2 the present theory is self-consistent, accounting fully for 2 ͑ ͒ ␦ ͑ ͒ ͑ ͒ ͑ ͒ transforms as z X1 and Ј z transforms as z X3 . electron-electron Coulomb interactions; and 3 the present In agreement with the general symmetry properties of V͑1͒ approach uses linear and quadratic response theory to sim- and V͑2͒ derived earlier, Table IV shows that all linear inter- plify the functional form of the heterostructure Hamiltonian. ͓ ⌫ ͑ ͔͒ ͑ ͒ Starting with point ͑1͒, previous envelope-function face terms derivable from 1 Td transform as X3 D2d . The 47–82 remaining terms that transform as X all originate in the qua- theories were based on model potentials of the form 1 ͑ ͒ dratic or higher-order response. Thus, if one accounts for the 5.9 , which is a linear combination of periodic bulk poten- l͑ ͒ l͑ ͒ smallness of the quadratic response ͑Sec. III B͒, seven out of tials multiplied by steplike functions x . The form of x the thirteen possible interface invariants for ͑001͒ near an interface was either specified as part of the model GaAs/AlAs heterojunctions can be omitted because they are ͑e.g., abrupt step function͒ or treated as unknown—and in actually of order ͑¯ka͒3⌬¯V or higher. This represents a major principle unknowable, at least within the model potential ap- proach. simplification over the general case in which all response Although the present theory led to a similar expression terms are of the same order. ͑7.1b͒, this was a derived result based on linear response For a no-common-atom heterojunction, the results of Sec. l͑ ͒ ͑ theory, in which the functions x have a known form de- VIII A remain valid as they depend only on the bulk sym- ␣ ͒ termined by the distribution R of atoms in the heterostruc- metry , but the linear response now has only the symmetry ͑ ͒ ⌬ ͑C ͒. Thus the terms labeled X in Table IV now arise ͑in ture. The potential 7.1b includes contributions from the “in- 1 2v 1 terface” materials ͑e.g., GaAs and InSb in an InAs/GaSb general͒ even in linear-response theory, due to the renormal- heterostructure͒, which are often omitted in the model- ization effects described above in Eqs. ͑7.16͒ and ͑7.17͒. For potential approach. Furthermore, for the model potential, the example, the ␦͑z͒ term in Eq. ͑7.16͒ generates the term in the coefficients of the ␦ terms in the Hamiltonian depend on ͑i͒ first row of Table IV. However, the term in the second row the values of the bulk potentials at the reciprocal lattice vec- of Table IV is still zero ͑within linear response͒, because l ͑ ͒ tors G; and ͑ii͒ whether the functions ͑x͒ are smooth ͑zero W 1 ͑x͒ in Eq. ͑7.16͒ couples only states of the same nnЈ outside the Brillouin zone͒ or sharp, with some ␦ terms van- ͑ ͒ ͑ ͒ ͑ ͒ ͑ ͒ 1 ͑ ͒ ␦ 1 ͑ ͒ 1 ͑ ͒ 1 ͑ ͒ 65–69 symmetry, with WnnЈ x = nnЈWnn x and Wnn x =WnЈnЈ x ishing in the smooth case. In the present theory these l͑ ͒ whenever En =EnЈ. coefficients are determined not by x , but instead by the properties of the linear atomic pseudopotentials in a finite C. ⌫-X coupling neighborhood of each G. For point ͑2͒, the present theory includes long-range mul- If one extends the above analysis to the case of intervalley tipole Coulomb potentials ͑because the self-energy is ⌫ ͑ ͒ -X coupling at a 001 GaAs/AlAs junction, it is immedi- nonanalytic at q=G͒, which were not considered in previous ͑ ately apparent because the Z matrix has the symmetry of a studies. These terms have no qualitative effect in two- ͒ ␦͑ ͒ coordinate matrix that the linear response produces z dimensional systems ͑other than to renormalize the band off- coupling between the bands sets͒, but they are present in quantum wires and dots, even ⌫ − X , X − X , X − X , ͑8.7͒ for isovalent heterostructures. 1 3z 1z 3z 3x 3y Finally, for point ͑3͒, if the functions l͑x͒ are treated as but no ␦͑z͒ coupling between the bands unknown in the model potential approach, one has no way of knowing how large the various interface terms are, and all ⌫ − X , X − X . ͑8.8͒ 1 1z 1x 1y terms permitted by symmetry should in principle be Coupling of the latter type only occurs in the quadratic re- included.65–69 However, since the linear response is dominant sponse, or in the linear response from terms proportional to ͑for typical heterostructures͒ in a first-principles theory, one ␦ ͑ ͒ ͕ ␦͑ ͖͒ Ј z or i Pz z . These conclusions agree with those of Ref. can eliminate all nonlinear interface band-mixing terms, 61, which were derived from a model potential constructed thereby simplifying the Hamiltonian considerably. from a linear superposition of atomic-like pseudopotentials. Aside from these differences, the qualitative form of the Note, however, that these results hold only for an ideal Hamiltonian derived here is very similar to those derived by heterojunction. If interdiffusion of Ga and Al atoms breaks Leibler47,48 for slowly graded heterostructures and Ta- 65–69 the X3 symmetry of the linear response, then there will be khtamirov and Volkov for abrupt heterostructures. The ␦͑z͒ coupling between the bands ͑8.8͒ due to terms similar to basic structure of these Hamiltonians is similar because the those derived in Eq. ͑7.16͒. same type of perturbation theory31,34,127 was used in their derivation. IX. DISCUSSION Other differences arise in theories using different approxi- mation techniques. The well-known theory of Burt50 is based Since a summary of the main results of this paper has on the Luttinger-Kohn representation47,127 with an energy- already been given in Sec. I C, this section will be limited to dependent approximation similar to Löwdin perturbation
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theory166 used to eliminate the coupling to remote bands. was argued that the valence-band mixing generated by Estimation of the error involved in this approximation shows Bloch-function differences is negligible in comparison to that Burt’s general theory50 includes all terms of order that generated by a discontinuous model potential. Although ͑¯ka͒⌬¯V, but retains some terms of order ͑¯ka͒2⌬¯V while omit- the latter does not appear in the present first-principles theory, its analog is the Z term shown in Eq. ͑5.6͒. The ting others. In particular, his results give an essentially cor- former contribution is given by the term proportional to rect ͑within the model potential approach͒ description of the ͑ ͒ W 1 ͑q͒ in Eqs. ͑6.7͒ and ͑C6͒. Inspection of these terms ␦-function term Z and the Rashba term ⌽, but omit the con- inЈ tributions from remote-band coupling to the terms Y and ⌫. shows that both are of order ͑¯ka͒⌬¯V. Therefore, neither the In Ref. 58, I used Burt’s theory, but introduced an addi- model of Ref. 60 ͓which omits the interface contributions tional approximation ͑also adopted in Refs. 52 and 62͒ from Eq. ͑5.5͔͒ nor the model of Refs. 66–68 ͑which omits whereby the energy eigenvalue in the denominator of the the contribution from Bloch-function differences͒ gives a energy-dependent effective mass was replaced with the correct value for the valence-band mixing coefficient. position-dependent energy of the bulk valence band maxi- In Refs. 12–14, 64, and 69 it was proposed that interval- ⌫ mum. This has the effect of replacing the Rashba coefficient ley 1-X1z and X1x-X1y coupling should be proportional to ͓͔ ␦͑ ͒ ⌫ ͑ ͒ ͑ ͒ z and of the same order of magnitude as 1-X3z ,X1z-X3z, in Eq. 6.10 with iDnnЈ x . In other words, the Rashba pa- Ј Ј ͑ ͒ and X3x-X3y coupling. However, as shown above in Sec. rameters and q of Eq. 8.6 were replaced by the Lut- VIII C, the former types of coupling should generally be ͑ ͒ tinger parameters and q of Eq. 8.2 . This simplifies appli- substantially weaker than the latter for ideal interfaces. The cations of the theory because and q are known from bulk same conclusion was reached in Ref. 130, where it was 167,168 magnetoabsorption measurements, and can also be shown that despite the very small magnitude of the direct estimated from ␥ ,␥ , and ␥ .58,167 However, since the true ⌫ ⌫ ͑ 1 2 3 1-X1z coupling, there is a net effective 1-X1z coupling as Rashba parameters Ј and qЈ cannot be determined from revealed by the anticrossing of superlattice subbands in Fig. ͒ ⌫ bulk measurements, this simplification will produce values 1 of Ref. 130 generated by 1-X3z ,X1z-X3z, and k·p cou- that have the correct order of magnitude but are quantita- plings that is only a factor of two or three smaller than the ⌫ tively incorrect. net 1 −X3z coupling. A similar approximation was used in Refs. 3, 70, and 71 However, in Refs. 12–14 it was found that Xx-Xy mixing to estimate the ␦-function mixing parameter for light and experiments could not adequately be explained without in- ␦ heavy holes ͓see Table IV and Eq. ͑8.3͔͒ from the valence- cluding -function X1x-X1y coupling of comparable magni- band offset. The original proposal was based on the HBF tude to X3x-X3y coupling. Nevertheless, since Refs. 12–14 did 3 ⌫ not test the effect of ␦Ј͑z͒X -X coupling, it is not yet clear model, in which the zone-center 15 Hamiltonian is written 1x 1y 1 1 ␦͑ ͒ as a sum of operators B= +͕I I ͖ and F= −͕I I ͖ multi- that their experiments require such a large z coupling. If 2 x y 2 x y ␦ ͑ ͒ ͕ ␦͑ ͖͒ Ј z or i Pz z coupling is unable to explain their results, plied by position-dependent bulk valence-band energies Ev. ͑ ͒ ⌫ then their data may indicate the presence of nonideal inter- For a 001 GaAs/AlAs heterojunction this yields the 15 interface Hamiltonian faces, as discussed in Sec. VIII C. Another possible explana- tion would be spin-orbit coupling, which generates a direct ␦͑ ͒ a z coupling between X6x and X6y even for ideal interfaces. ͕I I ͖͑EGaAs − EAlAs͒␦͑z͒, ͑9.1͒ 4 x y v v Two important physical effects not considered in this pa- per are the contributions from alloying ͑beyond the virtual whereas for an InAs/GaSb junction with a GaAs-like inter- crystal approximation͒ and from strain ͑in the bulk or at an face one obtains interface͒. The former effect is in principle already encom- passed by the present formalism ͑for lattice-matched systems a in which linear response is dominant͒, although for practical ͑1−2͕I I ͖͓͒EGaAs − 1 ͑EInAs + EGaSb͔͒␦͑z͒. ͑9.2͒ 4 x y v 2 v v applications one would need to perform a detailed analysis of the significance of different intervalley mixing effects.130 The ⌫ The corresponding results for 8 are given by the substitu- contribution from strain would require a nontrivial extension ͕ ͖→ 1 ͕ ͖ 32 ͓ tion IxIy 3 JxJy . Now the diagonal term proportional of this theory, but since an ab initio linear-response approach to 1␦͑z͔͒ in Eq. ͑9.2͒ is identical ͑within linear response͒ to to strain has already been successfully applied to 114,116 119,120 122 the ␦͑z͒ term derived above in Eq. ͑7.16͒. However, the In0.53Ga0.47As/InP, Si/Ge, and InAs/GaSb valence-band mixing term derived in Eqs. ͑6.7͒ and ͑8.3͒ is within LDA, the general perturbation scheme used in this not related to the valence-band offset. Therefore, although paper should work for lattice-mismatched systems also. the model of Refs. 3, 70, and 71 yields a coupling of the ACKNOWLEDGMENTS correct symmetry and order of magnitude, its numerical value is not reliable.85 The same conclusion was reached in I am grateful to Eduard Takhtamirov and Roland Winkler Ref. 86. for valuable comments on an earlier version of the manu- In Ref. 60, I proposed a smooth model potential169 for script. This work was supported by Hong Kong RGC Grant abrupt heterostructures that yields a valence-band mixing de- HKUST6139/00P and UGC Grant HIA03/04.SC02. termined chiefly by the change in Bloch functions at an in- terface. This result is of course not valid for general model APPENDIX A: IMAGINARY PART OF THE SELF-ENERGY potentials60,61 ͑despite claims to the contrary56,57͒, and it is Luttinger170 has shown for homogeneous systems that ͚͑i͒ not supported by the present work. However, in Ref. 68 it vanishes at the Fermi surface to all orders in perturbation
165345-18 FIRST-PRINCIPLES ENVELOPE-FUNCTION THEORY… PHYSICAL REVIEW B 72, 165345 ͑2005͒ theory. The generalization of his result to the case of inho- most obvious choice ͓from Eq. ͑3.19a͔͒ would be ͚͑i͒͑ +͒ ϯ ͑͒ mogeneous systems is ±i0 = J , where is real * 171,172 1, k ⍀ , and B͑k͒ = ͭ 0 ͮ ͑B1͒ ⍀* ͑ ͒2 ജ 0, k 0, C+ − + , +, J͑͒ = 0, ഛ ഛ , ͑A1͒ which was used previously in Refs. 49 and 50. However, the Ά − + · ⍀* sharp cutoff at the boundary of has the undesirable effect ͑ ͒2 ഛ 0 C− − − , −, of producing Gibbs oscillations in x space. Other possibili- ϵ 1 ͑ ties that eliminate this problem are B͑k͒=͓1+͑ka͒2͔ for near ± ± 2 Eg. Here is the chemical potential in ͒ ϫexp͓−͑ka͒2͔ and B͑k͒=exp͓−͑ka͒4͔, in which  is some the limit of zero temperature , Eg is the energy gap, and C± ജ ͑ ͒ 0 is independent of . The magnitude of C± may be esti- number of order 1. These choices of B k are smooth, spheri- mated from the calculations of Quinn and Ferrell173 for a cally symmetric, and introduce a negligible error of order degenerate homogeneous electron gas. Their results may be ͑¯ka͒4⌬¯V. ͗ ͘ 173,174 expressed as C± =1/Ei, where For some applications one need not introduce any cutoff 2 function at all, as an alternative power-series approximation 256 ⑀ ␣ E = F , ͑A2͒ for ͑k͒ is often more convenient. This approximation is i ͱ 2 3 p discussed in Sec. VII B. ⑀ in which F is the Fermi energy and p is the plasma fre- quency. If one treats semiconductors with the diamond or APPENDIX C: RENORMALIZED EFFECTIVE-MASS zinc-blende structure as a homogeneous gas with a density of PARAMETERS eight valence electrons per primitive unit cell, one obtains E Ӎ E Ӎ This appendix presents some details of the perturbation values of i 126 eV for GaAs and i 140 eV for Si. This Ј is very large in comparison to the energy gap and suggests theory used in Sec. VI. For states m,m A, the effective that ͚͑i͒ will have a negligible influence on the band struc- Hamiltonian H¯ is given by31,34 ture in typical semiconductor heterostructures ͑although it B 1 1 1 has an important qualitative effect in producing a finite qua- H¯ = H + ͚ HЈ HЈ ͩ + ͪ ͒ mmЈ mmЈ mi imЈ siparticle lifetime . 2 i Emi EmЈi Since a real semiconductor is not a degenerate homoge- B B neous electron gas, one may question whether this estimate 1 1 1 + ͚ ͚ HЈ HЈHЈ ͩ + ͪ is reliable. However, the calculations of Fleszar and mi ij jmЈ 2 i j EmiEmj EmЈiEmЈj 175 տ ͑ Hanke for Si yield Ei 100 eV estimated from Fig. 3 of Ref. 175͒, with some asymmetry between electrons and B A HЈ HЈ HЈ HЈ HЈ HЈ 1 ͩ mmЉ mЉi imЈ mi imЉ mЉmЈͪ holes. Thus it is reasonable in heterostructures to assume that − ͚ ͚ + 2 EmiEmЉi EmЈiEmЉi ӷ¯ ͑ ͒2 ͑⌬¯ ͒2 ͚͑i͒ i mЉ Ei Eg. Since − ± is of order V , this means that ͑ ͒4 ͑ ͒ is negligible under the present perturbation scheme. Thus it + O„ HЈ …, C1 will, for the most part, not be considered explicitly in the envelope-function equations derived here. However, the in which Emi=Em −Ei. This expression can be used to derive leading contribution from ͚͑i͒ is noted in Eq. ͑5.3͒ so that it most of the terms in the effective-mass Hamiltonian ͑6.2͒. may be included if desired. The coefficients of the quadratic, cubic, and quartic disper- sion terms of the reference crystal are given by APPENDIX B: CUTOFF FUNCTION B 1 1 1 This appendix considers possible definitions of the Bril- ˜ ͚ ͩ ͪ ͑ ͒ DnnЈ = DnnЈ + ni inЈ + , C2 louin zone cutoff function B͑k͒ introduced in Eq. ͑5.7͒. The 2 i Eni EnЈi
B B B 1 1 1 1 1 1 ˜ ͚ ͑ ˜ ˜ ͒ͩ ͪ ͚ ͚ ͩ ͪ CnnЈ = CnnЈ + niDinЈ + Dni inЈ + + ni ij jnЈ + 2 i Eni EnЈi 2 i j EniEnj EnЈiEnЈj B A 1 nnЉ nЉi inЈ ni inЉ nЉnЈ − ͚ ͚ ͩ + ͪ, ͑C3͒ 2 E E E E i nЉ ni nЉi nЈi nЉi
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B 1 1 1 ˜ ͚ ͑˜ ˜ ˜ ˜ ͒ͩ ͪ QnnЈ = QnnЈ + Dni DinЈ + niCinЈ + Cni inЈ + 2 i Eni EnЈi B B 1 1 1 ͚ ͚ ͑˜ ˜ ˜ ͒ͩ ͪ + Dni ij jnЈ + niDij jnЈ + ni ijDjnЈ + 2 i j EniEnj EnЈiEnЈj B A ˜ ˜ ˜ ˜ ˜ ˜ 1 DnnЉ nЉi inЈ + nnЉDnЉi inЈ + nnЉ nЉiDinЈ Dni inЉ nЉnЈ + niDinЉ nЉnЈ + ni inЉDnЉnЈ ͩ ͪ 4 − ͚ ͚ + + O͑ ͒. 2 E E E E i nЉ ni nЉi nЈi nЉi ͑C4͒
In the last expression, the symbol O͑4͒ denotes terms of the B 1 ͑ ͒ 1 1 fourth order in the kinetic momentum . These cannot be M ͑q͒ = M˜ ͑q͒ + ͚ D˜ W 1 ͑q͒ͩ + ͪ nnЈ nnЈ 2 ni inЈ E E obtained from Eq. ͑C1͒, but they can be derived easily from i ni nЈi Eq. ͑B15e͒ on p. 205 of Ref. 31. B ͑ ͒ 1 1 1 2 ͑ ͒ + ͚ ˜J ͑q͒ͩ + ͪ The term WnnЈ q is a renormalized second-order contri- ni inЈ 2 Eni EnЈi bution to the band offsets, given by i B B 1 1 1 ͚ ͚ ͑1͒ ͑ ͒ͩ ͪ + ni ijWjnЈ q + B 2 i j EniEnj EnЈiEnЈj ͑2͒ ͑2͒ 1 1 1 ͑1͒ ͑ ͒ ͑ ͒ ˜ ͑ ͒ ͚ ͩ ͓ͪ ͑ ͒ ⌳ 1 ͑ ͔͒ B A ͑1͒ ͑1͒ WnnЈ x = WnnЈ x + + Wni x + ni x ͑ ͒ ͑ ͒ 2 E E 1 nnЉ nЉiWinЈ q ni inЉWnЉnЈ q i ni nЈi − ͚ ͚ ͩ + ͪ, ͑ ͒ 2 E E E E ϫ͓ 1 ͑ ͒ ⌳ ͑1͒͑ ͔͒ ͑ ͒ i nЉ ni nЉi nЈi nЉi WinЈ x + inЈ x , C5 ͑C7͒
B ͑1͒ ͑2͒ where W ͑q͒ and W˜ ͑q͒ were defined in Eqs. ͑5.11͒ and 1 * nnЈ nnЈ R ͑q͒ = ˜R ͑q͒ + ͚ ͓˜J ͑− q͔͒ + ˜J ͑q͒ ͑1͒ nnЈ nnЈ ni nЈi ni inЈ ͑5.15͒. The only contribution from to be included here is 2 i „ … the linear interface dipole term ͑5.14͒ for heterovalent sys- 1 1 tems; but note that this contribution is zero in the GW ϫͩ + ͪ approximation94,95 and in density-functional theory89,90 Eni EnЈi ͑ ⌳ ␦ ͒ since nnЈ= nnЈ in these cases . B B 1 ͑ ͒ 1 1 The remaining terms in ͑6.2͒ are renormalized versions of + ͚ ͚ W 1 ͑q͒ ͩ + ͪ ni ij jnЈ ˜ ͑ ͒ ˜ ͑ ͒ ˜ ͑ ͒ 2 i j EniEnj EnЈiEnЈj the functions JnnЈ q , MnnЈ q , and RnnЈ q defined in Eqs. ͑5.11͒ and ͑5.12͒: B A ͑1͒͑ ͒ ͑1͒͑ ͒ 1 nnЉWnЉi q inЈ niWinЉ q nЉnЈ − ͚ ͚ ͩ + ͪ. 2 E E E E i nЉ ni nЉi nЈi nЉi B 1 1 1 ͑C8͒ ͑ ͒ ˜ ͑ ͒ ͚ ͑1͒͑ ͒ͩ ͪ ͑ ͒ JnnЈ q = JnnЈ q + niWinЈ q + , C6 2 i Eni EnЈi ͑ ͒ ͑ ͒ ͑ ͒ Note that MnnЈ q MnnЈ q , in contrast to Eq. 5.13 .
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