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A method is developed for eliminating spurious solutions from eight-band k•p theory. It reduces the bulk dispersion to a cubic equation ͑quadratic along ͗001͘ and ͗111͘), yet gives results virtually indistinguishable from ordinary k•p theory. A unique operator ordering is established for heterostructures, and example calculations on superlattices show good results. ͓S0163-1829͑97͒51244-6͔Multiband k•p models are frequently used to describe the electronic band structure of semiconductors in situations where nonparabolicity is important.1,2 It was noted early in the development of envelope structure theory 3 that such models can produce spurious solutions with very large wave vectors. If these solutions are evanescent 3 they are little more than a numerical nuisance, wreaking havoc in computer calculations 4,5 but having no physical significance. Spurious oscillatory modes 6,7 are more troublesome, however, since the resulting model contradicts the most basic of experimental facts: that semiconductors possess a band gap.The problem is an old one, but no fully satisfactory solution exists, despite a recent resurgence of activity in this area.2-13 Three general approaches have been suggested. The first is to modify the Hamiltonian by discarding those terms responsible for the spurious solutions.2-4 This costs some accuracy in the band structure, since it is no longer possible to fit all experimental effective masses. A second possibility is to keep the original Hamiltonian ͑which is accurate near kϭ0), but reject the large-k solutions as unphysical. [8][9][10][11][12] This is problematic in heterostructures, since the discarded solutions are needed to satisfy the boundary conditions associated with the model, and it is unclear which boundary conditions should be eliminated for mathematical consistency. The third approach therefore advocates retaining all solutions on the grounds that spurious bands have negligible influence on the properties of bound-state eigenfunctions. 5,13 This is better justified 13 than most people think, 8,12 but it still runs into trouble with oscillatory modes ͑as shown below͒.This paper presents an eight-band k•p Hamiltonian ͑for the ⌫ point of zinc-blende crystals͒ that has no spurious solutions, yet permits an accurate fit to all experimental effective masses.14 The basic idea is very simple: to set the coefficient of the k 2 term in the conduction-band ͑CB͒ matrix element to zero, and fit the CB mass using a spatially varying momentum matrix element. This yields a complex band structure virtually identical with that of ordinary k•p theory. However, the dispersion is simpler, being given by a cubic equation for general k and a quadratic along ͗001͘ and ͗111͘.For heterostructures, one must also determine the proper arrangement of differential operators with respect to material parameters. The main new feature of this theory is its position-dependent momentum matrix. A unique operator ordering is derived for this case, giving a Hamiltonian that is Hermitian but not symmetric. This model is...

An analytical theory of intervalley mixing at semiconductor heterojunctions is presented. Burt's envelope-function representation is used to analyze a pseudopotential Hamiltonian, yielding a simple d-function mixing between G and X electrons and light and heavy holes. This coupling exists even for media differing only by a constant band offset (i.e., with no difference in Bloch functions).[S0031-9007(98)06534-X] PACS numbers: 73.20. Dx, 71.15.Th, 73.61.Ey It is well known in semiconductor physics that bulk effective-mass theory [1] is not valid at an abrupt heterojunction, since the rapid change in potential at the interface causes a mixing of wave functions in different energy bands, and the neglect of such mixing is a key approximation in the development of this theory. It is consequently almost universally believed that a realistic description of the interface can only be achieved numerically, by performing a microscopic supercell calculation. The purpose of this paper is to demonstrate that a careful application of modern envelope-function theory yields a fully analytical description of interface band mixing.The most widely used form of envelope-function theory is Bastard's "envelope-function approximation" (EFA) [2], which openly ignores any interband mixing not found in bulk k ? p theory [3]. Less well known is Burt's theory of the envelope-function representation [4,5], which is an exact representation of the Schrödinger equation, fully capable of describing any effect found in pseudopotential theory. Thus far, the main applications of this theory have been a one-dimensional proof [5] that in long-period superlattices, interface-induced mixing is a small perturbation on the EFA, and the resolution of an ambiguity in the EFA ordering of differential operators [6,7].Unfortunately, the former work [5] is often misconstrued as implying that Burt's theory is no different from the EFA [8][9][10][11]. This interpretation is not warranted, because even small perturbations can have a dramatic impact when they introduce couplings of a qualitatively different nature. In this paper the envelope-function representation is used to analyze an empirical pseudopotential model [12] of the GaAs͞AlAs (001) heterojunction. The result is a simple analytical theory of the interface-induced mixing between G and X electrons [13][14][15][16] and light and heavy holes [17][18][19], in which the coupling takes the form of a finite-width d function whose strength is given directly in terms of pseudopotential form factors. The most striking outcome is that there is no limit in which the EFA is valid for abrupt heterojunctions, since the coupling exists even for identical Bloch functions.I begin by presenting a paraphrased (nonrigorous) version of Burt's theory. Let the microscopic Hamiltonian be H p 2 ͞2m 1 V ͑r͒, and choose as basis functions the complete orthonormal Luttinger-Kohn functions [1] x n ͑r͒ U n ͑r͒e ik?r , where U n ͑r͒ is a periodic Bloch function from some bulk reference crystal (e.g., the virtual crystal Al 0.5 Ga ...