Boundary Conditions For Envelope Functions Research Articles

Interface states in two-dimensional electron systems with spin–orbital interaction

Interface states at a boundary between regions with different spin–orbit interactions (SOIs) in two-dimensional (2D) electron systems are investigated within the one-band effective mass method with generalized boundary conditions for envelope functions. We have found that the interface states unexpectedly exist even if the effective interface potential equals zero. Depending on the system parameters, the energy of these states can lie in either or both forbidden and conduction bands of bulk states. The interface states have chiral spin texture similar to that of the edge states in 2D topological insulators. However, their energy spectrum is more sensitive to the interfacial potential, the largest effect being produced by the spin-dependent component of the interfacial potential. We have also studied the size quantization of the interface states in a strip of 2D electron gas with SOI and found an unusual (non-monotonic) dependence of the quantization energy on the strip width.

Interface electronic states and boundary conditions for envelope functions

The envelope-function method with generalized boundary conditions is applied to the description of localized and resonant interface states. A complete set of phenomenological conditions which restrict the form of connection rules for envelope functions is derived using the Hermiticity and symmetry requirements. Empirical coefficients in the connection rules play role of material parameters which characterize an internal structure of every particular heterointerface. As an illustration we present the derivation of the most general connection rules for the one-band effective mass and 4-band Kane models. The conditions for the existence of Tamm-like localized interface states are established. It is shown that a nontrivial form of the connection rules can also result in the formation of resonant states. The most transparent manifestation of such states is the resonant tunneling through a single-barrier heterostructure.

General boundary conditions for the envelope function in the multibandk⋅pmodel

We have derived general boundary conditions (BC) for the multiband envelope functions (which do not contain spurious solutions) in semiconductor heterostructures with abrupt heterointerfaces. These BC require the conservation of the probability flux density normal to the interface and guarantee that the multiband Hamiltonian be self--adjoint. The BC are energy independent and are characteristic properties of the interface. Calculations have been performed of the effect of the general BC on the electron energy levels in a potential well with infinite potential barriers using a coupled two band model. The connection with other approaches to determining BC for the envelope function and to the spurious solution problem in the multiband k.p model are discussed.

InAs/GaSb interfaces; the problem of boundary conditions

We present here a derivation of phenomenological boundary conditions for envelope functions at the interface between an InAs layer and a GaSb layer. The overlap of the valence band of GaSb with the conduction band of InAs leads to a coupling between electrons with total angular momentum 1/2 and holes with total angular momentum 3/2. The method that we use to match these wave-functions is that of the minimizing of the total energy of the system including the surface energy. We consider the different spinor characters of the states explicitly, and derive the required boundary conditions, which are not dependent on any specific microscopic model. The proposed approach is general, and may be used to obtain boundary conditions for other complicated cases of interfaces.

Boundary conditions for envelope functions at interfaces between dissimilar materials.

In the effective-mass approximation, the validity of the boundary conditions for an envelope function F acoss an interface between two different materials is predicated on the similarity of the nearest band-edge Bloch functions. Such approximations break down when the two materials are very dissimilar, e.g., a metal and a semiconductor. By studying one-dimensional model potentials we derive more accurate functional relations between the total wave function and its envelope. From these the usual boundary conditions are restored (continuity of F and F'/${\mathit{m}}^{\mathrm{*}}$) when similiar band edges are aligned at abrupt interfaces in semiconductor heterostructures. More importantly, we also derive appropriate boundary conditions for the case when band edges with qualitatively different Bloch functions are aligned. In particular, we find unique boundary conditions for interfaces between regions with and without periodic potentials. These boundary conditions are shown to apply to semiconductor heterostructures described within the nested effective-mass approximation and are arguably reasonable approximations for metal-semiconductor interfaces.

Boundary conditions for envelope functions in heterostructures.

In this paper a detailed derivation of the boundary conditions for envelope functions is carried out ming use of the k.p method. The resulting condition are different from those of Harrison by an extra term originating from a surface energy. This term removes the difference between expression for the transmission coefficient obtained with the help of microscopical calculations and the envelope-function method