Numerical Computation of Cubic Eigenvalue Problems for a Semiconductor Quantum Dot Model with Non-parabolic Effective Mass Approximation

We consider the three-dimensional Schrödinger equation simulating nanoscale semiconductor quantum dots with non-parabolic effective mass approximation. To discretize the equation, we use non-uniform meshes with half-shifted grid points in the radial direction. The discretization yields a very large eigenproblem that only several eigenpairs embedded in the spectrum are interested. The eigenvalues and eigenvectors correspond to the energy states and wave functions of the quantum dots, respectively. Effective and efficient numerical algorithms for computing these values are essential for exploring their physical phenomena and related practical applications. We provide insights into the resulting matrix structures that reduce the 3D problem to a set of independent 2D eigenproblems. The reduction results in cubic λ-matrix polynomial eigenproblems. Several numerical algorithms, such as the nonlinear Jacobi-Davidson method and the fixed point method based on the linear Jacobi-Davidson method, are then proposed for the solutions of these eigenproblems. For computing the successive eigenvalues, we suggest and analyze a novel explicit non-equivalence deflation technique with low-rank updates. Furthermore, we offer various acceleration schemes including Newton’s method to improve computational speed. All of the proposed algorithms have been implemented and successfully tested for solving the eigenproblems with sizes up to 76 millions. Numerical results are given to demonstrate the usefulness and efficiency of these algorithms.