Analysis of the Kirk effect in silicon-based bipolar transistors with a nonuniform collector profile

In this paper, the Kirk effect has been analyzed for silicon-based bipolar transistors (BJTs) with a nonuniform collector profile. We show that, for any arbitrary collector doping profile, the Kirk effect starts when the electron concentration equals the average doping concentration in the depletion region. We present a basic guideline for determining the collector current density at the onset of Kirk effect (J_K) for any collector doping profile and simple expressions for J_K and the electrical field in the collector drift region for the case of a linearly graded collector drift region. These analytical expressions are verified with device simulations. The Kirk effect for this kind of transistor is substantially different from that presented previously for transistors having a uniform collector drift region. For example, the possibility of the onset of the Kirk effect in a partially depleted collector occurs, while in a uniform collector profile the effect can only occur in a fully depleted collector. Our expressions can be used to do approximate analytical calculations for optimizing future BJTs.