Simple theory for threshold-voltage modulation in short-channel m.o.s. transistors

The threshold-voltage modulation ¿Vr of m.o.s. transistors, due to substrate bias VR, is determined using a simple 2-dimensional approach. It is shown that, for a given substrate bias, the ¿VT of short-channel devices is less than that of long-channel devices. It is also shown that the intrinsic (zero substrate bias) threshold voltage of short-channel devices is less than that of long-channel devices. The functional dependence of ¿VT on VR is derived, and verified experimentally.