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Physics-Based Analytical Model of Quantum-Mechanical Electron Wave Function Penetration into Thin Dielectric Films and Capacitance Evaluation

We propose an analytical expression of the electron distribution function inside silicon and SiO2 films under the surface-accumulation condition, to simulate the capacitance of a metal–oxide–semiconductor system. We assume a three-dimensional electron gas system and discuss the penetration of the electron wave function into SiO2 or high-k material films. It is demonstrated that the penetration of the wave function into the insulator film slightly influences the capacitance-voltage characteristics. It is also demonstrated that the penetration of the wave function into high-k materials is deeper than that into SiO2 film for identical equivalent oxide thickness (EOT).