Frequency response of PIN avalanching photodiodes

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 or
i
region and the photon density decreases with penetration depth according to
e^{-\alpha x}
. The frequency response for the case of photon absorption in the
p
or
n
region 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 the
p
side, shows the following: 1) The dc multiplication factor does not depend on v
n
and v
p
in 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 to
M_{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)],
where
M_{A}(0)
and
M_{B}(0)
are the dc multiplication factors for finite and infinite α, respectively, and
w
is 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 f
co
does not in any case depend on α but depends on α
n
and α
p
and on an "effective scattering-limited velocity" v
e
, defined as
v_{n}v_{p}/(v_{n}+v_{p})
. For
\alpha_{n} = \alpha_{p}
, the cutoff frequency is given by
f_{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 by
f_{co}\simeq\alpha_{n}\v_{e}/[2\piw \alpha_{p}M_{B}(0)]
.