A charge-sheet model of the MOSFET

Abstract

Intuition, device evolution, and even efficient computation require simple MOSFET (metal-oxide-semiconductor field-effect transistor) models. Among these simple models are charge-sheet models which compress the inversion layer into a conducting plane of zero thickness. It is the purpose of this paper to test one such charge sheet model to see whether this approximation is too severe. This particular model includes diffusion which is expected to be important in the subthreshold and saturation regions. As a test the charge sheet model is applied to long-channel devices. Long-channel MOSFET behavior has been thoroughly studied, and is very well explained by the Pao-Sah double-integral formula for the current. Hence, a clear-cut test is a comparison of the charge sheet model with the Pao-Sah model. We find the charge sheet model has two advantages over the Pao-Sah model. (1) It leads to a very simple algebraic formula for the current of long-channel devices. The same formula applies in all regimes from subthreshold to saturation. Neither splicing nor parameter changes are needed. No discontinuities occur in either the current or the small-signal parameters, or in the derivatives of the small-signal parameters. (2) It is simpler to extend the charge sheet model to two or three dimensions than the Pao-Sah model. This simplification is a result of dropping the details of the inversion layer charge distribution. An important aspect of the gradual channel approximation is brought out by the analysis. Suppose the boundary condition relating the quasi-fermi level at the drain, φ fL , to that at the source, φ fo , namely φ ƒL=φ ƒ0+V D where V D is the drain voltage, is applied in all bias regimes. Then it is shown that this means the potential at the drain end of the channel, φ sL is not related to the potential at the source end of the channel, φ so , by φ sL=φ s0+V D Instead, φ sL is computed, not imposed as a boundary condition. It is suggested that this failure of the potential to satisfy the boundary condition at the drain is justifiable. That is, φ sL should be reinterpreted as the potential at the point in the channel where the gradual channel approximation fails. Hence, (2) may be relaxed. However, the “channel length” in the gradual-channel approximation now becomes a fitting parameter and is not the metallurgical source-to-drain separation. In addition several aspects of the long-channel MOSFET are brought out: (1) Pinch-off is achieved only asymptotically as the drain voltage tends to infinity. This is in marked contrast to the often-stated, textbook view that pinch-off is achieved for some finite drain voltage, the saturation voltage. (2) The channel or drain conductance approaches zero only asymptotically. (3) The transconductance saturates only asymptotically. Figures comparing the simple charge-sheet model formulas with the usual textbook formulas are included for direct-current vs drain voltage, channel conductance vs drain voltage, and transconductance vs drain voltage. The charge-sheet model agrees with the original Pao-Sah double-integral formula for the current at all gate and drain voltages, and possesses the correct subthreshold behavior. The textbook formulas do not.