Three-dimensional derivations of the bipolar reciprocity theorem and Gummel´s equation

A proof of the bipolar reciprocity theorem valid for three-dimensional transistors is presented. The derivation is quite general in that mobility, carrier lifetime, bandgap narrowing, and doping are permitted to have an arbitrary spatial dependence. It has still been necessary to retain the usual low-injection assumption. A derivation of a Gummel-like formula expressing the relationship in 3-D transistors with negligible base current is also obtained. An example application of the formula is provided, and several interpretations of the 3-D Gummel number are suggested.