5.1. Semiconductor Detectors 289
This expression can be used to determine the individual depletion depths provided
we make use of another equation containingxpandxn. For this we can use the
charge conservation relation
NDxn=NAxp, (5.1.65)
which simply implies that the total charge remains constant no matter how much
of it gets transferred between the two regions. The above two relations give the
following depletion depths on p and n sides.
xp =[
2 V 0
eNA(1 +NA/ND)] 1 / 2
xn =[
2 V 0
eND(1 +ND/NA)] 1 / 2
(5.1.66)
The total depletion depthdis then just the sum of these two depths, that is
d=xp+xn. (5.1.67)
Usually in semiconductor detectors, the disparity in the dopant levels on p and
n sides is so large that the depth on one side can be safely ignored. This greatly
simplifies the expression for total depletion depth. To see this let us assume that the
acceptor impurity level is much higher than the donor impurity level (NA>> ND).
In such a case the n side depletion depth will be much greater than the p side.
This can also be seen from the charge conservation relation stated above, which for
NA>> NDimplies thatxn>> xp. In such a case the depletion depth on p side
will be so small that it can simply be ignored. The total depletion depth as deduced
from the relation 5.1.66 becomes
d[
2 V 0
eND] 1 / 2
. (5.1.68)
Similarly for the caseND>> NAwe get
d[
2 V 0
eNA] 1 / 2
. (5.1.69)
These relations show that once the junction has been physically established with
fixed dopant levels, the applied voltage is the only parameter that can be varied
to change the depletion depth. We will see later in the chapter that the depletion
width in detectors exposed to high radiation environments decreases with time due
to the damage caused by the radiation. We can see from the above relation that
at fixed applied voltage, this would mean that the minority charge concentration
has changed and the only way to compensate for this increase would be to increase
the reverse bias. In fact this is what is routinely done. Since the depletion width
is inversely proportional to the dopant level, therefore close monitoring of leakage
current is done to see if the charge concentration has changed and based on the
result the reverse bias is increased to increase the depletion width.
Sometimes it is more convenient to know the depletion depth in terms of resis-
tivity and mobility since these parameters are generally known. The resistivity of a
doped semiconductor is given by
ρ1
eN μ