Iteration acceleration for the Kohn-Sham system of DFT for semiconductor devices

Abstract

In Density Functional Theory the main object of relevance is the systems particle density, which, thanks to Hohenberg and Kohn [42], is known to uniquely determine the systems ground-state and all other properties. It was due to Kohn and Sham [57] to benefit from this information, by introducing a single-particle formulation for describing the multi-particle problem. For incorporating interaction effects, they included an exchange and correlation term in the effective potential. The resulting Kohn-Sham system is a nonlinearly coupled system of partial differential equations, that has to be solved self-consistently. In this work we are mainly interested in the quantum mechanical description of semiconductor heterostructures. Thus, we will look at the involved Schrodinger operator in effective mass approximation and have to deal with discontinuous coefficients and potentials. For numerically treating the Kohn-Sham system, we will use the fixed point formulation on basis of the particle density. A commonly used scheme for solving this problem is the well-known linear mixing scheme, that corresponds to a damped Picard (or Banach) iteration. However, this method is known to suffer from slow convergence and thus the use of acceleration methods is advised. Using well-established acceleration schemes based on the Newton-method, is possible, but the numerical costs for computing the needed information about the Jacobian are quite big. The aim of this work is the introduction of a fast and efficient acceleration method that generalises the linear mixing scheme to higher dimensions. The basis of our approach will be the direct inversion in the iterative subspace (DIIS) method from quantum chemistry. In Hartree-Fock and Coupled Cluster calculations DIIS is used to accelerate the calculation of electron orbitals. However, a straight forward transfer to our problem is dangerous. This is due to the extrapolation ability of the DIIS scheme, leading to negative mixing coefficients. Applied to our density approach, this may result in a negative density, meaning an iterate lying outside of the solution space. Thus, when applying the DIIS scheme to our problem, we have to ensure positivity of the produced density. We do this by introducing further constraints on the coefficients, that ensure positivity of the computed iterates. The resulting convex DIIS (CDIIS) scheme is then tested on exciton calculation in a three-dimensional quantum dot example. The results show, that the CDIIS method considerably accelerates the linear mixing approach, while in every step only a single function evaluation is performed. Thus, the CDIIS scheme accelerates the calculation while keeping the computational costs low and ensuring the quality of the iterates.