Theory of Low-Temperature Impact Ionization in High-Purity Germanium

An analysis of low-temperature electrical breakdown of semiconductors shows that the main theoretical problem is to find the distribution function $f(\epsilon , E)$ of the charge carriers, depending on the applied electric field $E$. In the case of high-purity $n$-type germanium ($N_D+N_A<4\mathrm{}\times {10}^{13}$ ${\mathrm{cm}}^{-3\mathrm{}}$), it is shown that the only important interactions are the electron-acoustical phonon ones. The distribution function has been calculated under equivalent assumptions by Stratton. It is used here in order to find the rates of creation and recombination of the charge carriers. Direct radiative recombination is found to be negligible; and the prebreakdown characteristics are mainly governed by the decrease of the thermal recombination rate. The impact ionization rate is calculated under the simplifying assumption of a cross section independent of the energy, thus introducing the only adjustable parameter. The breakdown field dependence on both temperature and compensation of the material is obtained. The results are compared with the data of Koenig and Brown. The agreement is found to be satisfactory. Conclusions to be drawn from this agreement are discussed, together with the limitations introduced by the chosen models.