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366
Se v e n
Cha p te r

exceeds that which would be expected from a noiseless multiplier on
the basis of Poissonian statistics (shot noise) alone.
The excess noise factor is a function of the carrier ionization ratio, k,
where (k) is usually defined as the ratio of the hole to electron
ionization probabilities. The excess noise factor may be calculated
using the model developed by McIntyre (3), which considers the sta-
tistical nature of
avalanche multiplication. The excess noise factor is given by:

F = k ⋅ M + (1 − k )(1 – 1/M) (4)
EFF EFF
Therefore, the lower the values of k and M, the lower the excess
noise factor.
The effective k factor (k ) for an APD can be measured experi-
EFF
mentally by fitting the McIntyre formula to the measured dependence
of the excess noise factor on gain. This is best done under illuminated
conditions.
It may also be theoretically calculated from the carrier ionization
coefficients and the electric field profile of the APD structure. The
ionization ratio k is a strong function of the electric field across the
APD structure, and takes its lowest value at low electric fields (only
in silicon). Since the electric field profile depends upon the doping
profile, the k factor is also a function of the doping profile. Depending
on the APD structure, the electric field profile traversed by a photo-
generated carrier and subsequent avalanche-ionized carriers may
therefore vary according to photon absorption depth. For indirect
band gap semiconductors such as silicon, the absorption coefficient
varies slowly at the longer wavelengths, and the “mean” absorption
depth is therefore a function of wavelength.
The value of k , and gain, M, for a silicon APD is thus a function
EFF
of wavelength for some doping profiles.
The McIntyre formula can be approximated for a k < 0.1 and
M > 20 without significant loss of accuracy as:

F = 2 + k ⋅ M (5)

Also often quoted by APD manufacturers is an empirical formula
used to calculate the excess noise factor, given as:

F = M (6)
x
where the value of X is derived as a log-normal linear fit of measured
F-values for given values of gain M.
This approximation is sufficiently appropriate for many applica-
tions, particularly when used with APDs with a high k factor, such as
InGaAs and germanium APDs. Table 7.3 provides typical values of k,
X, and F for silicon, germanium, and InGaAs APDs.