Computing-Inexpensive Matrix Model for Estimating the Threshold Voltage Variation by Workfunction Variation in High-κ/Metal-gate MOSFETs

In high-${\kappa}$/metal-gate (HK/MG) metal-oxide-semiconductor field-effect transistors (MOSFETs) at 45-nm and below, the metal-gate material consists of a number of grains with different grain orientations. Thus, Monte Carlo (MC) simulation of the threshold voltage ($V_{TH}$) variation caused by the workfunction variation (WFV) using a limited number of samples (i.e., approximately a few hundreds of samples) would be misleading. It is ideal to run the MC simulation using a statistically significant number of samples (>~$10^6$); however, it is expensive in terms of the computing requirement for reasonably estimating the WFV-induced $V_{TH}$ variation in the HK/MG MOSFETs. In this work, a simple matrix model is suggested to implement a computing-inexpensive approach to estimate the WFV-induced $V_{TH}$ variation. The suggested model has been verified by experimental data, and the amount of WFV-induced $V_{TH}$ variation, as well as the $V_{TH}$ lowering is revealed.