Transverse instability of a plane front of fast impact ionization waves

The transverse instability of a plane front of fast impact ionization waves in p ⁺-n-n ⁺ semiconductor structures with a finite concentration of donors N in the n layer has been theoretically analyzed. It is assumed that the high velocity u of impact ionization waves is ensured owing to the avalanche multiplication of the uniform background of electrons and holes whose concentration ??_b ahead of the front is high enough for the continuum approximation to be applicable. The problem of the calculation of the growth rate s of a small harmonic perturbation with wavenumber k is reduced to the eigenvalue problem for a specific homogeneous Volterra equation of the second kind containing the sum of double and triple integrals of an unknown eigenfunction. This problem has been solved by the method of successive approximations. It has been shown that the function s(k) for small k values increases monotonically in agreement with the analytical theory reported in Thermal Engineering 58 (13), 1119 (2011), reaches a maximum s _M at k = k _M, then decreases, and becomes negative at k > k ₀₁. This behavior of the function s(k) for short-wavelength perturbations is due to a decrease in the distortion of the field owing to a finite thickness of the space charge region of the front and ??smearing?? of perturbation of concentrations owing to the transverse transport of charge carriers. The similarity laws for perturbations with k ? k _M have been established: at fixed ??_b values and the maximum field strength on the front E _0M, the growth rate s depends only on the ratio k/N and the boundary wavenumber k ₀₁ ?? N. The parameters s _M, k _M, and k ₀₁, which determine the perturbation growth dynamics and the upper boundary of the instability region for impact ionization waves, have been presented as functions of E _0M. These dependences indicate that the model of a plane impact ionization wave is insufficient for describing the operation of avalanche voltage sharpers and that fronts of fast streamers in the continuum approximation should be stable with respect to transverse perturbations in agreement with the previously reported numerical simulation results. The results have been confirmed by the numerical simulation of the evolution of small harmonic perturbations of the steady-state plane impact ionization wave.