Accurate mesh truncation for Schrödinger equations by a perfectly matched layer absorber: Application to the calculation of optical spectra

Abstract

Quasibound and continuum states are of particular importance for the numerical investigation of coherence properties and are sensitive with respect to the boundary condition chosen at the edge of the computational window. An open boundary condition will be derived which is particularly suitable for dynamical problems described by Schrödinger-type equations. With this approach, bound states as well as unbound states can be described adequately. The boundary condition is derived from a perfectly matched layer ͑PML͒ formalism commonly used in the field of electrodynamics. Consequently, the calculation domain is reduced leading to a calculation time reduction by orders of magnitude. From the physical point of view this formulation allows an adequate analysis of transport phenomena or absorption spectra, e.g., the results obtained by the PML formalism are compared with accurate numerical results calculated using a large mesh and show an excellent performance. For example, the Coulomb enhanced Franz-Keldysh effect is investigated, which cannot be analyzed adequately without using proper open boundary conditions. ͓S0163-1829͑99͒51332-5͔Numerical calculations are important for the design of semiconductor devices as well as in the investigation of novel effects like coherence properties. If one is not interested in bound states only, care must be taken at the boundary of the calculation window. If the boundary conditions are not appropriate, strong reflections occur, which can significantly affect the results. The usual solution is to choose a large calculation domain. Then, intrinsic losses lead to a vanishing wave amplitude at the mesh boundaries. However, this results in very long computation times, and, especially for multidimensional problems, the needed computer resources are not acceptable. Usually, all interactions take place within a limited area. Thus, computational efficiency can be enhanced by using open boundary conditions and more complex problems can be investigated.In this paper we apply the perfectly matched layer ͑PML͒ technique to the calculation of optical spectra of semiconductor heterostructures. Generally, the semiconductor Bloch equation is formulated in a six-dimensional hyperspace. Assuming rotational symmetry, quantum wells and superlattices can be calculated within a three-dimensional computation domain. If one is interested in bound-state absorption only, an expansion into a few single particle wave functions is a fast computational approach. 1 Using the full threedimensional semiconductor Bloch equation results in a wide range of possible applications, 2 but is numerically expensive without appropriate boundary conditions. For example, a mesh of up to 1 million discretization points has been used with about 10 points in the quantum well direction only. 2 This discretization is appropriate for demonstrating basic properties but is much too coarse for practical investigation purposes. We will show that using the PML boundary condition enhances the numerical efficiency, thus the range of applica...

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