The distribution of gains in uniformly multiplying avalanche photodiodes: Theory

Expressions are derived for the probability that a pulse initiated by electrons (or holes) in a uniformly multiplying semiconductor diode will result in a total number of electrons (or holes) , to give a gain , and for the probability that the gain will be or greater. It is shown that the distributions are far from Gaussian. The gain distribution for a single photoelectron, for example, is shown to have a maximum value for for any value of the average gain . The derivations are valid for any electric field distribution and assume only that the hole ionization coefficient ) can be approximated by the relation , where is the electron ionization coefficient and is a constant. A method of determining an effective value of , for cases where is not a good approximation, is presented. The results can be used to calculate the average gain and the mean square deviation from the average, giving results in agreement with previously published relations [1], [2]. The implications of this theory on the use of avalanche diodes for low-level photodetection are discussed. It is shown that in the near infrared, cooled avalanche photodiodes can compare favorably with the best available photomultiplier when used either in a photon-counting mode, or for the reliable detection of low-level laser pulses.