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Achieving self-consistent convergence for conventional effective-massapproaches at ultralow temperatures (below 4.2 K) is a challenging task, whichmostly lies in the discontinuities in material properties (e.g.,effective-mass, electron affinity, dielectric constant). In this article, wedevelop a novel self consistent approach based on cell centered Finite Volumediscretization of the Sturm-Liouville form of the effective mass Schrodingerequation and generalized Poisson's equation (FV-SP). We apply this approach tosimulate the one dimensional electron gas (1DEG) formed at the Si-SiO2interface via a top gate. We find excellent self-consistent convergence fromhigh to extreme low (as low as 50 mK) temperatures. We further examine thesolidity of FV-SP method by changing external variables such as theelectrochemical potential and the accumulative top gate voltage. Our approachallows for counting electron-electron interactions. Our results demonstratethat FV-SP approach is a powerful tool to solve effective mass Hamiltonians.

Finite-Volume Self-Consistent approach for electronic properties at  ultra-low temperatures: Theory and application to 1D electron gas at the  Si-SiO2 interface