Gate current modeling through high-k materials and compact modeling of gate capacitance
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High dielectric constant materials are expected to replace SiO2 when the direct tunneling currents become intolerable for circuit design considerations. These dielectrics exhibit a trend of decreasing barrier height with increasing dielectric constant and hence the performance trade offs of choosing various dielectrics was performed. This work involves the numerical solution of the Schrodinger's and Poisson's equation to obtain the direct tunneling current through high dielectric constant materials allowing for wave function penetration into the gate electrode. This approach has been validated for oxides as thin as 5Å. A Franz-type complex energy band structure model with energy dependent effective mass was used to calculate tunneling across the dielectric. In addition, with scaling of the oxide to ultra-thin dimensions, the effects of wave function penetration into the gate electrode on the gate capacitance become significant. It was observed that allowing the wave function to penetrate into the gate electrode shifts the centroid of the inversion charge closer to the interface resulting in higher gate capacitance. This phenomenon has been comprehensively studied in the context of gate electrodes, gate dielectrics and scaling. The characterization of ultra-thin oxides is becoming non-trivial with transmission line effects and high direct tunneling currents. The physical thickness characterization must now be performed in conjunction with capacitance and tunneling current measurements. However, most compact direct tunneling current models have a large number of parameters that are sometimes represent incorrect physics and thus, cannot be used as a predictive tool. Hence, there is a strong need for a first principles compact gate capacitance and gate current model. In this work a gate capacitance model based on the characterization of the quantized subbands is presented. This model is fast and accurate and can be extended to evaluate tunneling currents from each subband. The advantages and disadvantages of using simpler, unphysical models to estimate the effective oxide thickness of capacitors are also presented.