An efficient finite-difference scheme for computation of electron states in free-standing and coreshell quantum wires
The electron states in axially symmetric quantum wires are computed by means of the effective-mass Schrdinger equation, which is written in cylindrical coordinates ? ? , ? ? , and z z . We show that a direct discretization of the Schrdinger equation by central finite differences leads to a non-symmetric Hamiltonian matrix. Because diagonalization of such matrices is more complex it is advantageous to transform it in a symmetric form. This can be done by the Liouville-like transformation proposed by Rizea etal. (2008), which replaces the wave function ?(?) ? ( ? ) with the function F ( ? ) = ? ( ? ) ? and transforms the Hamiltonian accordingly. Even though a symmetric Hamiltonian matrix is produced by this procedure, the computed wave functions are found to be inaccurate near the origin, and the accuracy of the energy levels is not very high. In order to improve on this, we devised a finite-difference scheme which discretizes the Schrdinger equation in the first step, and then applies the Liouville-like transformation to the difference equation. Such a procedure gives a symmetric Hamiltonian matrix, resulting in an accuracy comparable to the one obtained with the finite element method. The superior efficiency of the new finite-difference scheme (FDM ) is demonstrated for a few ? ? -dependent one-dimensional potentials which are usually employed to model the electron states in free-standing and coreshell quantum wires. The new scheme is compared with the other FDM  schemes for solving the effective-mass Schrdinger equation, and is found to deliver energy levels with much smaller numerical error for all the analyzed potentials. It also gives more accurate results than the scheme of Rizea etal., except for the ground state of an infinite rectangular potential in freestanding quantum wires. Moreover, the PT P T symmetry is invoked to explain similarities and differences between the considered FDM schemes.
quantum wire effective-mass discretization finite difference finite element eigenvalue