Diffusion in a short base

Abstract

The usual diffusion-equation description of transport in the base of a junction transistor breaks down if the base length is comparable to the minority carrier mean free path. We present a rigorous analytic treatment of this problem, based on an exact solution of the Boltzmann transport equation (BTE). A key ingredient of our approach is formulation of the boundary conditions for the distribution function f( r, k, t) at the base-emitter interface. Numerical solution of the BTe shows that there are significant corrections (of order 100%) to the diffusion-equation estimates of both the static current gain and the frequency cut-off in a short-base bipolar junction transistor (BJT). In a model where the electron scattering mean free path and the recombination length are both assumed independent of energy, the steady-state BTE reduces to an integral equation for the electron concentration. We present an approximate analytical solution of this equation that gives an excellent agreement with the exact numerical solution. The analytic solution is asymptotically exact in the limit of ultra-short base lengths, where the minority-carrier transport can be regarded as thermionic, as well as in the long-base limit (where the diffusion equation is rigorously valid). On the basis of our analytic solution, we propose and test a new expression for the effective (concentration-dependent) diffusivity, that interpolates between the diffusive and thermionic limits.