The linear stage of evolution of electron-hole avalanches in semiconductors

A complete analytical solution of the problem of the linear stage of evolution of electron-hole avalanches in the uniform time-independent electric field Eext is derived. The theory accounts for the drift, diffusion, and impact ionization of electrons and holes, thus providing a means for calculating the space-time distributions of fields and charges as well as all the basic parameters of the avalanches up to the onset of nonlinear effects at the time ta. Formulas for the group velocity of the avalanches and for the velocity of its leading fronts are derived. It is shown that the time ta must be determined from the condition that the impact ionization coefficient α in the center of the avalanche be reduced by a specified small quantity η. A transcendent equation is derived, which allows the calculation of the time ta as a function of the quantity η, the unperturbed coefficient α(Eext), and other parameters of the semiconductor. It is found that, when α(Eext) is increased by two orders of magnitude, the total number of electron-hole pairs generated up to the point ta decreases by nearly three orders of magnitude