Kirk effect in bipolar transistors with a nonuniform dopant profile in the collector

We have calculated the threshold current density of the Kirk effect in bipolar transistors with a nonuniform doping concentration in the collector. The threshold current is enhanced by more than 50% compared to the uniform doping case if the dopant profile is weighed toward the base and if velocity overshoot is small. Significant velocity overshoot restores the threshold value obtained with uniform doping.