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First-Principles Envelope-Function Theory for Lattice-Matched Semiconductor Heterostructures

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First-Principles Envelope-Function Theory for Lattice-Matched Semiconductor Heterostructures 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.
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