# On a finite volume scheme for the computation of quantum confined states in semiconductor nanostructures

## Thomas Koprucki

We consider the computation of the lowest eigenmodes for a Schrödinger operator with variable coefficients in BenDaniel-Duke form. This type of Schrödinger operators are the basic model for the description of the electronic states in semiconductor nanostructures. Semiconductor nanostructures are heterostructures consisting of different semiconductor materials. Characteristic for these heterostructures is that the material properties experience a step-like change across the interface between two materials. This leads to a Schrödinger operator with piecewise constant coefficients. The coefficient function of the second order part is defined by the effective mass of the semiconductor material and the potential is defined by the variation of the band-edge energy.

An approximation of the lowest eigenmodes by a finite volume scheme is presented. We prove the convergence of the finite volume eigenvalues and eigenmodes. The finite volume scheme results in large sparse symmetric matrix eigenvalue problems. We present numerical results for 1d, 2d, and 3d test problems obtained with the iterative eigenvalue solver ARPACK in combination the the sparse direct solver PARDISO and PDELIB2.