The distribution of gains in uniformly multiplying avalanche photodiodes: Theory

Expressions are derived for the probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{n,m}</tex> that a pulse initiated by <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</tex> electrons (or holes) in a uniformly multiplying semiconductor diode will result in a total number of electrons (or holes) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</tex> , to give a gain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m/n</tex> , and for the probability <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q_{n,m}</tex> that the gain will be <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m/n</tex> or greater. It is shown that the distributions are far from Gaussian. The gain distribution <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">P_{1,m}</tex> for a single photoelectron, for example, is shown to have a maximum value for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m = 1</tex> for any value of the average gain <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M=m/n</tex> . The derivations are valid for any electric field distribution and assume only that the hole ionization coefficient <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta(E</tex> ) can be approximated by the relation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta(E) =k\alpha(E)</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\alpha(E)</tex> is the electron ionization coefficient and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> is a constant. A method of determining an effective value of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</tex> , for cases where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\beta=k\alpha</tex> 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.