Analytical description of the injection ratio of self-biased bipolar transistors under the very high injection conditions of ESD events

This paper proposes a 1D-analytical description of the injection ratio of a self-biased bipolar transistor under very high current injection conditions. Starting from an expression of the current gain based on the stored charge into the emitter and base regions, we derive a new analytical expression of the current injection ratio. This analytical description demonstrates the presence of an asymptotic limit for the injection ratio at very high current densities, as the ratio of electron/hole mobilities in the case of an NPN transistor and to the ratio of hole/electron saturation velocities for a PNP. Moreover, for the first time, a base narrowing effect is demonstrated and explained in the case of a self-biased PNP, in contrast with the base widening effect (Kirk effect [Kirk CT, A theory of transistor cutoff frequency (fT) falloff at high current densities, IRE Trans Electr Dev 1961: p. 164–73]) reported for lower current density. These results are validated by numerical simulation and show a good agreement with experimental characterizations of transistors especially designed to operate under extreme condition such as electrostatic discharge (ESD) events.