The modeling of the electrical properties of ultra-thin (<2 nm thick) oxide metal-oxide-semiconductor (MOS) structures requires the self-consistent solution of the Schrodinger and the Poisson equations. To calculate the change density profile required by the Poisson equation, the occupancy of the quantum electronic states solution of the Schrodinger equation is a key issue. The most widely used approximation consists in assuming that the states that impinge from cathode and anode are occupied according to the Fermi-Dirac distribution with a quasi-Fermi (imref) level equal to that of the corresponding reservoir. The cathode and anode quasi-Fermi levels differ in the applied bias. In this work, we study the failure of this quasi-equilibrium approximation in the case of a MOS structure biased in accumulation. Our approach consists in considering the balance between inelastic scattering of electrons in the accumulation layer and the tunneling through the oxide. Using this procedure, we estimate that this quasi-equilibrium approximation fails for oxide thickness between 1 and 2 nm. Finally, we argued that kinetic treatments of transport are required for thinner oxides.
Suñé, J., Oriols, X., & Autran, J. L. (2001). Non-equilibrium gate tunneling current in ultra-thin (<2 nm) oxide MOS devices. Journal of Non-Crystalline Solids, 280(1–3), 127–131. https://doi.org/10.1016/S0022-3093(00)00364-1