408 Chapter 6. Scintillation Detectors and Photodetectors
approximations to this probability density function have been proposed, including
the one that assumes the shape of a Gaussian distribution, though with longer tails.
We will see later that even though the gain fluctuates around a mean value,
still with proper attention to the noise sources one can build a detector based on
APD that works at the limit of quantum fluctuations of the multiplication process.
The mean value of the gain is an important parameter for design optimization, noise
consideration, and calibrations. For the output signal due to electrons the expression
for the mean gain can be written as (22)
〈Ge〉=1 −u
e−αd(1−u)−u, (6.5.57)
whereu= 1 is the ratio of the hole ionization rate to the electron ionization rate,
αis the electron ionization rate, anddis the thickness of the depletion region. The
ionization ratioucan have any value between 0 and 1 but is generally very small,
on the order of 10−^2 or 10−^3. Let us see what happens when we substituteu=0in
the above expression. This gives
〈Ge〉≈eαd, (6.5.58)which, the reader might recall, looks remarkably similar to the expression that was
obtained for gas multiplication in the chapter on gas filled detectors. From this
expression one might conclude that higher gain can be obtained by simply increasing
the depletion width. Although this, in principle, is true but is far away from the
actual practice as it has the downside of increasing the fluctuations in the gain. We
will return to this discussion in the next section. One thing to note is that this
simplified expression for mean gain should not be used in actual computations as
the effect ofuon gain is not negligible (see example below).
Example:
An APD has a mean gain of 50 at hole-to-electron ionization rate ratiou
of 0.1. Compute the relative change in its mean gain ifuchanges to 0.01.
Assume that all other parameters remain the same.Solution:
Using the given values we first compute the value ofαdin equation 6.5.57.〈Ge〉 =1 −u
e−αd(1−u)−u⇒αd = −ln[
1 −u
〈G〉+u]
1 −u= −ln