On the avalanche initiation probability of avalanche diodes above the breakdown voltage

In the calculation of the turn-on probabilities per unit time of avalanche diode microplasmas, or of the single-photon detection probabilities of avalanche photodiodes used in the photon-counting mode, it is desirable to know how the avalanche initiation probability varies with voltage above the breakdown voltage. It is shown that the two coupled differential equations derived by Oldham et al. for the probabilities that a self-sustaining avalanche will be initiated in an avalanche diode biased above the breakdown voltage by an injected electron or by an injected hole (avalanche initiation probabilities) can be combined to provide a single integral equation for each of the electron, hole, and electron-hole pair initiation probabilities. These equations can be integrated for the special case in which the electron and hole ionization rates α
e
and α
h
are related by
\alpha_{h} \approx k\alpha_{e}
where
k
is a constant. A method of computing an effective value of
k
for other cases in which this approximation is not a good one is presented. The resulting expressions are shown to be consistent with previously published calculations by McIntyre of the breakdown probabilities both for the case
k = 1
and for the more general case
k \neq 1
.