We present a set of closed form analytical solutions of the transport equation in the base of bipolar transistors. The presented 1D solutions hold for variable injection conditions, arbitrary doping profiles and arbitrary intrinsic carrier concentrations along the base, allowing to consider bandgap narrowing effects and a variable composition base (e.g. Si1-xGex graded base). The above solutions lay on the commonly used assumption of zero majority carrier current in the neutral zone, and are valid up to the onset of the Kirk effect. Two solutions are analysed; the zero-order solution, found as a correction of the low injection case, and the first order solution, derived as a correction of the zero-order case. The perturbative procedure can naturally be iterated leading to an exact solution, which is used to assess the accuracy of the proposed analytical solutions. The first-order solution shows the best trade-off between accuracy and complexity.

Peer Reviewed