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Cooperation with: A. Arnold (Universität Münster), A. Jüngel (Universität Konstanz)
Supported by: DFG: Priority Program ``Analysis, Modellierung und Simulation von Mehrskalenproblemen'' (Analysis, modeling and simulation of multiscale problems)
Description: The role of semiconductor nanostructures will increase in the future due to the advent of quantum electronic devices which is assisted by the increasing sophistication of fabrication technology. To understand and to predict the physical properties of such devices, efficient and reliable computations of their electronic states is required.
The ab initio solution of the many-electron Schrödinger equations on the microscopic scale fails due to the immensity of the computational effort. To overcome this burden, it appears to be natural to utilize the presence of two scales inherent to the problem--the mesoscopic scale of the variation of the materials in the nanostructure and the microscopic scale of the atoms of each material.
The standard approach to this problem is the kp-method in combination
with the envelope function approximation. Within the
nanostructure, the wave function is represented in terms of
envelope functions, which describe the slowly-varying mesoscopic
part of the locally highly-oscillating microscopic wave functions. The
envelope functions are eigenfunctions of matrix-valued 
Schrödinger operators with discontinuous coefficients, whose
spectral properties and applications have been investigated by us in previous
projects. Usually kp-Schrödinger operators are constructed
by a physically intuitive but heuristic transfer of the kp-theory for
bulk materials to nanostructures. However, this approach to modeling has
two main drawbacks: the ambiguity of boundary conditions across
interfaces between two materials and the appearance of spurious modes for
some materials.
A viable approach to a kp-theory for heterostructures was first given by Burt ([1]), who points out a way to derive equations for the envelope functions in terms of the microscopic potential. Following this approach, we obtain pseudodifferential equations for the envelope functions in real space or integral equations for Fourier-transformed envelope functions (cf. [2]). A certain limit of these equations leads to kp-Schrödinger operators of the same formal structure as the ones in the standard approach, but with microscopically derived boundary conditions. This leads to new tools for the computation of electronic states in nanostructures (cf. [2]).
References:
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