Threshold voltage of m.o.s. transistors doped nonuniformly near the surface

Due to impurity redistribution and ion implantation, m.o.s. transistors are usually doped nonuniformly beneath the gate. From an analytical solution of Poisson's equation a modified expression for the threshold voltage has been derived. The change in the impurity concentration is taken into account, on the one side, in the usual manner by its contribution to the surface charge and, on the other side, on the basis of its contribution to the surface potential instead of the rather arbitrary approximation by an ‘effective’ impurity concentration. The modified formula gives a better description of the measured threshold voltages at least for an extended region, if not for the entire range of the applied substrate bias voltage. As theoretically predicted, the substrate bias coefficient corresponds to the bulk impurity concentration. A comparison with experimental results shows exceptionally close agreement if the depletion layer fully, or largely, covers the region of altered impurity concentration. If this region extends substantially beyond the depletion layer, deviations are observed. This result has been confirmed from impurity profiles determined for the purpose of verification.