Eigensolutions of the Wigner-Eisenbud problem for a cylindrical nanowire within finite volume method

We present a node-centered finite volume method for computing a representative range of eigenvalues and eigenvectors of the Schrödinger operator on a three-dimensional cylindrically symmetric bounded domain with mixed boundary conditions. More specifically, we deal with a semiconductor nanowire which consists of a dominant host material and contains heterostructure features such as double-barriers or quantum dots. The three-dimensional Schrödinger operator is reduced to a family of two-dimensional Schrödinger operators distinguished by a centrifugal potential. Ultimately, we numerically treat them by means of a finite volume method. We consider a uniform, boundary conforming Delaunay mesh, which additionally conforms to the material interfaces. The 1/r singularity is eliminated by approximating r at the vertexes of the Voronoi boxes. We study how the anisotropy of the effective mass tensor acts on the uniform approximation of the first K eigenvalues and eigenvectors and their sequential arrangement. There exists an optimal uniform Delaunay discretization with matching anisotropy with respect to the effective masses of the host material. This anisotropic discretization yields best accuracy also in the presence of a mildly varying scattering potential, shown exemplarily for a nanowire resonant tunneling diode. For a centrifugal potential one retrieves the theoretically established first-order convergence, while the second-order convergence is recovered only on uniform grids with an anisotropy correction.