Analytical modeling of metal gate granularity induced Vt variability in NWFETs

Metal gate granularity (MGG) induced threshold voltage (Vt) variability is a critical process random variations for sub-20nm transistors [1]. It has been studied using either by stochastic TCAD simulations[2] or by analytical modeling of probability distribution[3]. This analytical modeling is based on the approach of finding an effective WF. This is a weighted average on the area of WFs of all the grains. The probability distribution of this effective WF is expected to be correlated with distribution of Vt. The problem with this approach is that positional randomness is ignored and the results depend only on the fraction of the area covered by a particular grain. Hence a physics based analytical model is attractive to address this issue. The electrostatics of NWFETs has been well explained and studied to model Vt and subthreshold slope (SS) are found [4] and [5]. The objective of this work is to develop an analytical model to estimate the metal gate granularity (MGG) induced Vt variability in Silicon nanowire FETs (NWFETs) using analytical solution of equilibrium electrostatics.