Probability distribution functions of threshold voltage fluctuations due to random impurities in deca–nano MOSFETs

Random dopant effects on threshold voltage in ultra-small metal-oxide-semiconductor field-effect transistors (MOSFETs) are theoretically studied. The probability distribution of threshold voltage fluctuation δVth for arbitrary vertical dopant profiles is systematically derived by including the quantum correction associated with gate capacitance variations for the first time. It is shown that the distribution of δVth gradually changes its shape from the Gaussian to a Poisson-like distribution as MOSFETs are miniaturized. Also, it is found that a lower limit corresponding to the extreme case that there is no dopant in the depletion region appears in the δVth distribution for ultra-small n-MOSFETs.