Frequency response of PIN avalanching photodiodes

Abstract
The frequency response of a PIN avalanching photodiode has been analyzed without the usual assumption that the electrons and the holes have equal ionization rates (\alpha_{n} = \alpha_{p}) and equal scattering-limited velocities (v_{n} = v_{p}). A general formula for the frequency response is first derived for the case in which the photons are absorbed in the space charge region oriregion and the photon density decreases with penetration depth according toe^{-\alpha x}. The frequency response for the case of photon absorption in thepornregion is obtained simply by letting α approach infinity in the general formula and that for the case of uniform absorption of photons in the space charge region by allowing α to approach zero. The analysis, based on a model in which the photons come from thepside, shows the following: 1) The dc multiplication factor does not depend on vnand vpin any manner. It does not depend on α if\alpha_{n} = \alpha_{p}. When\alpha_{n} > \alpha_{p}, the dc multiplication factor increases with increasing α according toM_{A}(0)/M_{B}(0) = [1 - \exp (-\alpha w - \alpha_{n}w +\alpha_{p}w)]/[1 + (\alpha_{n} - \alpha_{p})/ \alpha] \cdot [1 - \exp (-\alpha w)],whereM_{A}(0)andM_{B}(0)are the dc multiplication factors for finite and infinite α, respectively, andwis the space charge region width.M_{B}(0)is given by(1- \alpha_{n}w)^{-1}when\alpha_{n} = \alpha_{p}and by[\exp (-\alpha_{n}w)-\alpha_{p}/ \alpha-{n}]^{-1}when\alpha_{n} \gg \alpha_{p}. 2) The cutoff frequency fcodoes not in any case depend on α but depends on αnand αpand on an "effective scattering-limited velocity" ve, defined asv_{n}v_{p}/(v_{n}+v_{p}). For\alpha_{n} = \alpha_{p}, the cutoff frequency is given byf_{co}\simeq3v_{e}/[\piwM_{B}(0)]\simeq3\alpha_{n}\v_{e}/[\pi M_{B}(0)]. For\alpha_{n} \gg \alpha_{p}, f_{co}is greatly improved and is given byf_{co}\simeq\alpha_{n}\v_{e}/[2\piw \alpha_{p}M_{B}(0)].

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