B-Splines and NURBS Based Finite Element Methods for Strained Electronic Structure Calculations

Abstract

This paper presents B-splines and nonuniform rational B-splines (NURBS)-based finite element method for self-consistent solution of the Schrödinger wave equation (SWE). The new equilibrium position of the atoms is determined as a function of evolving stretching of the underlying primitive lattice vectors and it gets reflected via the evolving effective potential that is employed in the SWE. The nonlinear SWE is solved in a self-consistent fashion (SCF) wherein a Poisson problem that models the Hartree and local potentials is solved as a function of the electron charge density. The complex-valued generalized eigenvalue problem arising from SWE yields evolving band gaps that result in changing electronic properties of the semiconductor materials. The method is applied to indium, silicon, and germanium that are commonly used semiconductor materials. It is then applied to the material system comprised of silicon layer on silicon–germanium buffer to show the range of application of the method.

An efficient pseudo-spectral method for the description of atomic electronic wave functions – Application to the hydrogen atom in a uniform magnetic field

“…Another application of FEMs are non-relativistic [127][128][129] and relativistic DFT calculations [130,131] of molecules. The role of FEMs for DFT-based simulations of solids have been discussed by Pask et al [132][133][134], Gavini et al [135] and Masud et al [4]. Grid-based methods for the solution of the TDSE and TISE allow for a direct integration of the resulting PE matrices in the Hamiltonian operators describing the nuclear motion, bypassing the need to develop an analytical form of the PE surfaces.…”

“…This is because grid-based methods have several attractive features for the numerical calculation of wave functions or probability densities compared to conventional basis set methods: (i) Whereas spectral methods are usually the most accurate option for solving partial differential equations due to their global nature, this is no longer the case if the functions to be approximated have sharp gradients or discontinuities (Gibbs phenomenon) [2,3]. In the context of electronic structure theory, the presence of time-dependent or time-independent strong internal or external fields may induce phenomena in the state solutions analogous to shock waves in fluid mechanics, thus by analogy suggesting the benefits of grid-based methods compared to spectral methods [4].…”

“…As mentioned under item (i) in Sec. I, the challenges global spectral methods have in describing solutions featuring sharp gradients or discontinuities can also affect quantumchemical calculations even in the absence of external fields, e.g., due to the "Coulomb singularity" or "Kato cusp condition"[3,4,20,57,58]. However, the negative effect ofthe Coulomb singularity on the convergence properties of global spectral methods is usually not relevant in practical applications since most applications do not focus on the computation of the absolute energies of electronic states.…”