Quantum-mechanically corrected variational principle for metal-oxide-semiconductor devices, leading to a deep sub-0.1 micron capacitor model

We treat an integral expression for electrostatic energy as a variational principle, with electric potential as the sole argument. We modify this principle to incorporate the quantum-mechanical (QM) effect of metal-oxide-semiconductor (MOS) interface charge confinement. The result is a QM-corrected variational principle for application to MOS devices. We apply this principle to develop a model of the sub-0.1 mum MOS capacitor. This variational-quantum-mechanical (VQM) model gives closed-form expressions for the behavior of threshold voltage, V-th, oxide capacitance, C-ox, and depletion capacitance, C-j, as functions of the perimeter and area of the gate, thickness of the oxide, doping level, and temperature. Using this model, we further obtain closed-form expressions for QM-corrected dopant fluctuation-induced statistical deviation of V-th and C-total, as functions of gate dimensions, oxide thickness, doping level, and temperature. Excellent agreement is obtained when comparison with published detailed three-dimensional Poisson-Schroedinger numerical simulations are made for the behavior of V-th and deviation of V-th, sigma (V-th), as functions of gate dimensions and doping levels. Our model further shows that the oxide and depletion capacitance both consist of one term which is proportional to the gate area plus another term proportional to the perimeter. For very large devices, only the area term is significant, while, for very small devices, the perimeter term dominates. The perimeter term is clearly to be associated with the contribution of the fringe field at the edge of the capacitor. The decomposition of capacitance into four parts clearly indicates that the standard equivalent circuit model for MOS capacitance, i.e., two capacitors in series, should be modified to account for the contributions of the fringe field. Interestingly, our model reveals that there exists a gate perimeter-to-area ratio for which the capacitance is wholly insensitive to oxide thickness, and another for which the standard deviation of the depletion capacitance due to dopant fluctuations, sigma(C-j), has a minimum, when the doping level is high. (C) 2004 American Institute of Physics.