Elimination of spurious solutions from eight-bandk⋅ptheory

Abstract

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...

Design of low bias voltage Ge/SiGe multiple quantum wells electro-absorption modulator at 1550 nm

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Operator ordering, ellipticity and spurious solutions in k · p calculations of III-nitride nanostructures

“…He concluded that, the operator ordering is hermitian, but not necessarily symmetric. Foreman (1997) then derived the operator ordering for zinc-blende models, which is referred to as the Burt-Foreman (BF) ordering.…”