We have studied the magnitude and length scale of potential fluctuations in the channel of metal–oxide–semiconductor field-effect transistors due to the random positions of ionized impurities in the depletion layer. These fluctuations effect the threshold voltage of deep submicron devices, impede their integration, and reduce yield and reliability. Our simple, analytic results complement numerical, atomistic simulations. The calculations are based on a model introduced by Brews to study fluctuations due to charges in the oxide. We find a typical standard deviation of 70 mV in the potential below threshold, where the channel is empty, falling to 40 mV above threshold due to screening by carriers in the channel. These figures can be reduced by a lightly dopedepitaxial layer of a few nm thickness. The correlation function decays exponentially in an empty channel with a length scale of 9 nm, which screening by carriers reduces to about 5 nm. These calculations of the random potential provide a guide to fluctuations of the threshold voltage between devices because the length of the critical region in a well-scaled transistor near threshold is comparable to the correlation length of the fluctuations. The results agree reasonably well with atomistic simulations but detailed comparison is difficult because half of the total standard deviation comes from impurities within 1 nm of the silicon–oxide interface, which is a single layer of the grid used in the simulations.
Slavcheva, G., Davies, J. H., Brown, A. R., & Asenov, A. (2002). Potential fluctuations in metal-oxide-semiconductor field-effect transistors generated by randomly distributed impurities in the depletion layer. Journal of Applied Physics, 91(7), 4326-4334 . https://doi.org/10.1063/1.1450031