Realistic semiconductor heterostructures design using inverse scattering

A semiconductor heterostructure can be modeled by a system of equations describing (with a certain degree of completeness and precision) the state of the system. The equations depend on a set of structural and compositional data: the spatial dependence of the chemical composition (including dopant profiles), the applied external fields, etc. The system behavior (response) is described by functional data, such as the electric or thermal conductance, the energy dependence of the electron transmittance, the wavelength dependence of the optical absorption coefficient, etc. The functional data can be computed using the solution of the equations, and are thus functionals of the structural and compositional data. To design a heterostructure for a certain application is to find a set of structural and compositional data, which is physically (and technologically) achievable, such that the values of a chosen subset of functional data will be within a desirable range. The designer, in principle, solves an inverse problem: inverting the dependence of the functional data on the structural and compositional data. In the usual approach, the starting point is a proposed configuration of the structural and compositional data which is then varied to optimize the component’s performance; at each iteration the spectral-scattering data describing the performance have to be re-calculated. In this method, much depends on the initial choice of parameters as in general several non equivalent local extrema of the response are present in the parameter space. Inverse methods, when available, are not subject to this drawback. Contrary to the previous situation, the designer starts from data describing the system’s performance, such as current spectral response curves and determines the optimal structural and compositional data. He automatically gets an optimized set of parameters. Even if the model used to make the optimization calculation feasible lacks all the details of the full physical situation, one has still the possibility to use the traditional approach to fine tune the results from such a new starting point. The inverse method we use requires three different steps [1,2]: 1. The reconstruction of the phase of the transmittivity of a quantum device using Finite Dispersion Relations under the very convenient form of the Padé approximations. 2. The Standard Inverse Scattering method, making use of the results of Kay and Sabatier [3,4] for the inverse problem with rational coefficients (Padé approximations). This allows to build an efficient algorithm for solving the inverse problem and obtaining the potential. 3. A unitary transformation that maps the place dependent mass problem of a BenDaniel equation onto the usual constant mass Schroedinger problem keeping the potential in the equation local [2]. By applying the previous procedure, it becomes possible to design new realistic quantum wells [5,6] submitted to optimization constraints. In particular, one can use inverse methods to design and fabricate novel color sensitive QWIPs in the mediumand long-wave IR. Moreover, one of the most unexpected and astonishing results of the inverse method is that inclusion of the self-consistent potential of the conduction electrons is far much simpler than in the usual direct approach [2]. As a first application, we will use the versatility of the inverse scattering methods for the Schroedinger/BenDaniel-Poisson model [1,2], for designing and building prototypes of improved electronic filters. An aspect that has not yet been dealt with in depth is how to discretize the potential profiles obtained by inverse scattering methods so as to make their production viable while mantaining at the same time their properties, their electron wave reflectance in particular. An added complication is the necessary presence in the devices of substantial densities of conduction electrons: the electric potentials they generate do not, if suitably compensated by doping the devices [2], significantly alter the reflectance of the ”smooth” potentials obtained by inverse scattering; but what about the discretized potentials? The aim of the present paper is to deal with the problems stemming from this necessity to both discretize the potentials and have substantial densities of conduction electrons. To avoid unnecessary complica-