Design of semiconductor heterostructures with preset electron reflectance by inverse scattering techniques

Abstract

We present the application of the inverse scattering method to the design of semiconductor heterostructures having a preset dependence of the (conduction) electrons' reflectance on the energy. The electron dynamics are described by either the effective mass Schrödinger, or by the (variable mass) BenDaniel and Duke equations. The problem of phase (re)construction for the complex transmission and reflection coefficients is solved by a combination of Padé approximant techniques, obtaining reference solutions with simple analytic properties. Reflectance-preserving transformations allow bound state and reflection resonance management. The inverse scattering problem for the Schrödinger equation is solved using an algebraic approach due to Sabatier. This solution can be mapped unitarily onto a family of BenDaniel and Duke type equations. The boundary value problem for the nonlinear equation which determines the mapping is discussed in some detail. The chemical concentration profile of heterostructures whose self consistent potential yields the desired reflectance is solved completely in the case of Schrödinger dynamics and approximately for Ben-Daniel and Duke dynamics. The Appendix contains a brief digest of results from scattering and inverse scattering theory for the one-dimensional Schrödinger equation which are used in the paper.Contents *

Semiconductor intersubband laser/detector performance optimization using a simulated annealing algorithm

“…Little research effort has previously been expended in this area. Bessis et al [15] presented an inverse scattering technique which allows the design of intersubband devices with predetermined electron reflection/transmission coefficients. This method is restricted to optimization over a small set of parameters since only the reflection/transmission coefficient of the structure can be predefined.…”

“Smart Design” of Quantum Wells and Double-Quantum Wells Structures

“…Recently, step by step, the approach to the semiconductor heterostructures design comes into being, which perhaps could be termed as smart design and whose aim is to tailor the heterostructure shapes in order to obtain the predetermined characteristics (see [1][2][3]). In some of the cases, the "smart design" approach is based on the Inverse Scattering Problem Method (ISP) (see [2] and especially [4]). In others, when the space dependence of charge carriers effective mass is also taken into account, it is based on different techniques [5].…”

“…In others, when the space dependence of charge carriers effective mass is also taken into account, it is based on different techniques [5]. The paper [4], and then [6], in which the main ideas of [4] were implemented, were devoted mainly to the design of quantum filters with the predetermined reflection and transmission properties. On the other hand, the "palette" of quantum well (QW) potentials which are in use, is still limited to a few most popular ones, rectangular, parabolic or at most, semiparabolic or triangular [7,8] and this circumstance obviously restricts the possibility to choose and control the energy spectrum of QWs and the devices based on them.…”

“…The already cited papers [4,6] tackled the difficult problem of complex transmission and reflection coefficients reconstruction, if the potential does not support the bound states. As it was stated in [4], if bound states are present, knowledge of the accessible scattering data is not sufficient for unique recovery of the potential. One can notice, however, that, if from the very beginning, one restricts the searching for the shape of the potentials to some definite class of possible potentials, the uniqueness of potential reconstruction can be restored.…”

“…Surprisingly, it turns out that if one tries to solve the inverse problem with the aid of reflectionless Bargmann potentials which, above all, are symmetric, which is V(x) = V(−x), it suffices to know only the positions of energy levels, without knowing the normalizing constants of wave functions. There is also another problem with the QW potential reconstruction: this problem, as most of inverse problems, is ill-posed; this fact was emphasised also in [4] (However, there is certain well-posedness of the ISP in one dimension, in the sense that the mapping from the scattering data to potential is continuous in proper topology).…”