Verification of the integral charge-control relation for high-speed bipolar transistors at high current densities

Gummel's integral charge-control relation (ICCR) IC= (const/Qp).exp (VBE/VT) is an important basis for developing self-consistent compact transistor models for the high-current region (including quasi-saturation). Such models are required for the simulation of future high-speed IC's with a high integration level. Unfortunately, the simplifying assumptions on which the ICCR is based seem to be doubtful especially for very fast transistors. Therefore, in this paper, the ICCR and its assumptions are checked via numerical simulation of such transistors (fT≈ 7-8 GHz). It is found that the one-dimensional ICCR is a fairly good approximation far into the high-current region. This satisfactory result is mainly due to the partial compensation of the influences of the spatially dependent doping concentration on both the electron mobility µnand the effective intrinsic density niewithin the product µnn²ie. Only in the emitter and in the emitter-base space-charge region there is a strong increase of this product which, in conjunction with the increasing contribution of the hole charge in these regions, was proved to be responsible for the errors observed at high current levels. The ICCR can also be applied to a two-dimensional transistor by additionally taking into account the excess hole charge stored outside the internal transistor for the determination of Qp. Thus the contribution of the minority charges can still be determined experimentally by measuring τf(IC).