Physics and Engineering of Radiation Detection

410 Chapter 6. Scintillation Detectors and Photodetectors

voltage can be written as
dG
dV

=kvG %V−^1 , (6.5.59)

whereV is the applied voltage andkvis a constant that depends on the APD
material and construction. Generally its value lies between 3 and 4.
Another parameter on which the gain depends heavily is the temperature. The
gain increases with decrease in temperature according to the simple relation

dG

dT

=−ktG %^0 C−^1 , (6.5.60)

whereTis the temperature in Celcius andktis a constant that depends on the type
of APD. It normally assumes a value between 2 and 3. The negative sign simply
signifies the inverse relationship between the gain and the temperature.
The above equations imply that operating an APD at a fixed temperature and
voltage would yield a fixed gain. What these equationsdo notimply is that there
is no uncertainty associated with the gain. As a matter of fact, the biggest prob-
lem with APD gain is actually its inherent uncertainty and not its variation with
temperature and voltage. This uncertainty stems from the noisy nature of the car-
rier impact ionization process that multiplies the charges. Any measurement from
an APD is therefore subject to these fluctuations as well, which of course can be-
come undesirable for high precision measurements such as single photon counting.
The fluctuations in the APD gain has thus become an active area of research and
development.
Although some researchers prefer to work directly with the gain but most prefer
the gain fluctuations that are characterized by the so calledexcess noise factor
defined as

F=

〈G^2 〉

〈G〉^2

, (6.5.61)

whereGis the APD gain and〈〉represents the mean value. A number of mathemat-
ical models exist that transform the above equation into a numerically computable
form (see, for example (23)). We will not go into that discussion but it is worth men-
tioning here that the excess noise factor and hence fluctuations in the gain depend
on

the mean APD gain,

the ionization coefficients of the charge carriers (electrons and holes), and

the multiplication region.

The variation of the excess noise factor with the mean gain is the most profound
one among the three. Though some APDs behave a bit differently, but generally
there exists linear relationships between these two parameters, that is

F≈AG+B, (6.5.62)

whereAandBare constants that depend on the APD material and construction.
TypicallyAhas a value on the order of 10−^3 , whileBassumes a value of around 2.