Modeling Impact Ionization in a Bipolar Transistor by means of a Direct Solution of the BTE

In this paper we study impact ionization in a BJT, by means of a newly develope'd technique based on a deterministic one-dimensional solution of the BTE. We evaluate the collector current multiplication factor and compare it with measurements in a thinbase transistor. Impact ionization is likely to become the major physical limiting factor toward the miniaturization of silicon bipolar transistors. In fact the collector-base breakdown voltage poses a serious limit to the maximum supply voltage and to the maximum doping level inside the collector. For this reason the collector profile itself must be carefully designed, and the ionization mechanism properly predicted by means of an efficient and accurate mlodeling tool. The limitations of the drift-diffusion approach in short devices have long been recognized. Traditional alternative methods are either based on the energy transport or hydrodynamic model or on the Monte Carlo particle simulation. While the former is reasonably fast but inadequate for describing the high-energy ionizing electrons, the latter suffers a major drawback in the huge required CPU time. In recent papers a deterministic approach for finding approximate solutions of the Boltzmann Transport Equation (BTE) has been proposed, based on a Spherical Harmonics Expansion of the distribution function [l, 21. This method gave quite good results in both the homogeneous and the one-dimensional problems, providing a sufficiently accurate description of the electron energy distribution inside the device with reasonable CPU times. The purpose of this work is to extend the procedure described in [2] to account for impact ionization. We test and validate the model by simulating a one-dimensional section of a BJT with realistic impurity profiles taken from the literature [3]. The ionization model contains only one adjustable parameter, which is calibrated by comparying the ionization coefficient with Monte Carlo homogeneous simulations and with measurements obtained in long diodes. We conclude showing how the model can approximately predict the collector current multiplication factor in the B JT under investiga.tion.