BSIM3v3.2.2 MOSFET Model - The University of Texas at Dallas

Non-Uniform Doping and Small Channel Effects on Threshold Voltagegaussian distribution, as shown in Figure 2-1. This non-uniformity willmake γ in Eq. (2.1.2) a function of the substrate bias. If the depletionwidth is less than X t as shown in Figure 2-1, N a in Eq. (2.1.2) is equal toN ch ; otherwise it is equal to N sub .In order to take into account such non-uniform substrate doping profile, thefollowing V th model is proposed:( Φs−Vbs− Φs) K VbsVth= VTideal+ K1 −2(2.1.4)For a zero substrate bias, Eqs. (2.1.1) and (2.1.4) give the same result. K 1and K 2 can be determined by the criteria that V th and its derivative versusV bs should be the same at V bm , where V bm is the maximum substrate biasvoltage. Therefore, using equations (2.1.1) and (2.1.4), K 1 and K 2 [3] willbe given by the following:K= γ2− K212Φ−sV bm(2.1.5)K2( γ1− γ )( Φs− Vbx− Φs)Φs( Φs− Vbm− Φs) + Vbm=22(2.1.6)where γ 1 and γ 2 are body bias coefficients when the substrate dopingconcentration are equal to N ch and N sub , respectively:γ1=2qεNCsioxch(2.1.7)2-4 BSIM3v3.2.2 Manual Copyright © 1999 UC Berkeley