Depth and State Analyses of Deuterium and Helium in D₂-Annealed He⁺-Implanted Yttrium Iron Garnet

Abstract

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:where D is constant, y an n-dimensional vector, and g andfscalar functions ofy. The special case f(Y) = c WIZ, I = 1 with 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 to the calculation of the Auger rate in indirect materials is adapted to the calculation of impact ionization thresholds for any parabolic band structure. Two classes of threshold are identified. In class ct thresholds arise away from the extrema due to energy and momentum constraints. In class p thresholds occur at extrema of the band contamng the hot carrier. Various special cases are considered and the links with previously published results reviewed.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' (in the general sense of thresholds, anti-thresholds and saddle-point thresholds), which depends upon the effective masses. This classification is completed by the identification of two further subgroups of transitions, the criterion being whether or not there is an impact ionization 'window' in terms ofenergies. Simple examples of anti-thresholds are constructed.