The cell-centered Finite-Volume self-consistent approach for heterostructures: 1D electron gas at the Si-SiO2 interface

Achieving self-consistent convergence with the conventional effective-mass approach at ultra-low temperatures (below $4.2~K$) is a challenging task, which mostly lies in the discontinuities in material properties (e.g., effective-mass, electron affinity, dielectric constant). In this article, we develop a novel self-consistent approach based on cell-centered Finite-Volume discretization of the Sturm-Liouville form of the effective-mass Schr{\"o}dinger equation and generalized Poisson's equation (FV-SP). We apply this approach to simulate the one-dimensional electron gas (1DEG) formed at the Si-SiO$_2$ interface via a top gate. We find excellent self-consistent convergence from high to extremely low (as low as $50~mK$) temperatures. We further examine the solidity of FV-SP method by changing external variables such as the electrochemical potential and the accumulative top gate voltage. Our approach allows for counting electron-electron interactions. Our results demonstrate that FV-SP approach is a powerful tool to solve effective-mass Hamiltonians.