Systematic analytical solutions for minority-carrier transport in semiconductors with position-dependent composition, with application to heavily doped silicon

For quasi-neutral regions of semiconductor devices with position-dependent composition, we have derived expressions for the position dependence of the excess minority-carrier density and for relevant recombination currents. To make the development concrete, we study nonuniformly and heavily doped emitter regions of silicon p-n junction devices. The expressions developed differ from those previously advanced in that they are in the form of a multiple integral series, yielding, by truncation, many different orders of approximation. Correspondences exist between some of the different orders of approximation and various solutions previously obtained. All of the mechanisms relating to hole and electron transport in position-dependent heavily doped semiconductors are accounted for in the new expressions. These mechanisms include bandgap narrowing, majority-carrier degeneracy, Auger recombination lifetime, etc. To assess the accuracy of the various orders of approximation, we compare their predictions with a numerical solution. We determine that the simplest approximation, which contains only one term of the integral series, is accurate to within about 5 percent of the numerical solution for thin emitters (∼0.2 µm) provided the surface recombination velocity is less than 10
⁵
cm/s. It thus applies directly to bipolar transistors with polysilicon contacts, and to surface-passivated solar cells. This new solution, which we call the zeroth-order or quasi-neutral quasi-equilibrium approximation, is simpler than solutions previously put forward. If the emitter junction is deep or the contact is ohmic, the higher-order approximations provide whatever accuracy is needed. We separate the emitter recombination current into charge and a characteristic time, enabling the calculation of the contribution of emitter to capacitance and to delay in the time domain or phase shift in the frequency domain.