First-principles envelope-function theory for lattice-matched semiconductor heterostructures

Abstract

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.

Theory of the electronic structure and carrier dynamics of strain-induced (Ga, In)As quantum dots

“…In order to write the BF Hamiltonian in terms of experimentally measurable bulk dispersion parameters, one is forced to approximate the exact form of the BF envelope function Hamiltonian (equation (6.4) of [73]) [78,77]. Doing so allows us to compare the conventional EFA model (see section 3.1) and the BF model [75] in terms of the individual elements of the eight-band Hamiltonians.…”

Reliable k⋅p band structure calculation for nanostructures using finite elements

“…(.). Other second order operators such as ∂ i ∂ j H and H∂ i ∂ j (2) do not appear in the standard k · p model but exist in the first-principles model of [1]. According to the dimensionality of the considered system, bulk second-order terms depending on the translationally invariant direction are effectively added to first and zero order terms of the differential operator:…”

“…Thus, the singularity of ∂ i H (1) i;R at the material interface is removed and no complicated tweaks in the numerical evaluation have to be applied. Let a(W, F) denote the left-hand side bilinear form and (W, F) 0 the right-hand side bilinear form of (19).…”

“…In contrast, the k·p envelope function method can be formulated as a partial differential equation system and therefore allows to determine the band structure efficiently using standard numerical techniques. Various simplifications during the perturbative treatment of remote bands obfuscate physical details, such as intervalley-and interface mixing [1,2]. This leads to an ongoing dispute about the precision and validity of the obtained results [3][4][5][6] where the raised arguments apply to the commonly used k·p standard model.…”

“…This leads to an ongoing dispute about the precision and validity of the obtained results [3][4][5][6] where the raised arguments apply to the commonly used k·p standard model. It was shown recently [1,2] that an exact derivation of the envelope equations from first principles does allow for a consistent incorporation of interface mixing effects and an exact description of the band structure.…”