WIAS Preprint No. 1709, (2012)

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


  • Racec, Paul N.
  • Schade, Stanley
  • Kaiser, Hans-Christoph

2010 Mathematics Subject Classification

  • 65N30 65Z05 35P99

2008 Physics and Astronomy Classification Scheme

  • 62.23.Hj 73.63.-b 71.15.-m


  • Finite element method, Schrödinger operator, cylindrical coordinates, R-matrix formalism, Wigner--Eisenbud problem, nanowire


We present a 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. 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 potentials with 1/r singularity one retrieves the theoretically established first order convergence, while the second order convergence is recovered only on uniform grids with an anisotropy correction.

Appeared in

  • J. Comput. Phys., 252 (2013) pp. 52--64.

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