Impact ionization thresholds as generalizations of Van Hove singularities

An analogy is drawn between impact ionization probability integrals and joint density of states integrals in radiative processes by consideration of the class of integrals: I n g (D) identical to integral g(y) delta (D-f(y))dy (- infinity to + infinity ) where D is constant, y an n-dimensional vector, and g and f scalar functions of y. The special case f(y)= Sigma sigma r y r 2 (r=1 to n), with sigma r =+or-1, is considered, and the critical point concept in radiative transitions is extended to six dimensions, some details of these integrations being given, and explicit expressions for the impact ionization probability per unit time discussed. The approach of Hill and Landsberg (1976) to the calculation of the Auger rate in indirect materials is adapted to the calculation of impact ionization thresholds for any parabolic band structure. The analogy between radiative and Auger processes previously drawn by Landsberg is used to extend the usual classification of van Hove singularities to a classification of impact ionization 'thresholds', which depends upon the effective masses.