Evolution of electron-hole avalanches and streamers in indirect gap semiconductors

Abstract

A numerical simulation of the origination and evolution of streamers in semiconductors has been performed using the diffusion-drift approximation including the impact and tunnel ionization. It is assumed that an external electric field E 0 is static and uniform, an avalanche and a streamer are axisymmetric, background electrons and holes are absent, and all their kinetic coefficients are identical. The linear evolution of an electron-hole avalanche, an avalanche-to-streamer transition, and two successive stages of the evolution of the streamer—intermediate “diffusion” and main exponentially self-similar—have been examined in detail. It has been shown that a streamer is similar to a dumbbell with conical weights. The bases of these cones, streamer fronts, are thin shells, which contain almost the entire streamer charge and are close in shape to the halves of ellipsoids of revolution. A front propagates so that its shape and the shape of the weight of the dumbbell, the maximum field on the front, and the electron-hole plasma density in weights remain unchanged. The field strength behind the front is much smaller than E 0, but increases with approaching the bar of the dumbbell whose diameter increases with the time t owing to the transverse diffusion. The electron and hole densities in the bar increase due to the impact ionization in an almost uniform field, which is only slightly lower than E 0. At the diffusion stage, the length of the streamer and the curvature radius of its front increase with constant rates, which are determined not only by the impact ionization and drift, but also by diffusion. In relatively low fields (E 0 ≲ 0.4 MV/cm for silicon) this stage ends due to the appearance of the instability of the front. In higher fields, the tunnel ionization is manifested before the appearance of instability and gives rise to the appearance of a new-type streamer. Its main feature is the stable exponential increase in all spatial scales with the same response time t R , so that the charge-carrier density and field strength at large times t depend only on one vector variable $$ \hat R $$ = Rexp(−t/t R ). This means that the solution of a Cauchy problem describing the evolution of the streamer in the uniform field is asymptotically exponentially self-similar.