Physics and Engineering of Radiation Detection

5.1. Semiconductor Detectors 263

Eg

Eimp

Hole

Electron

Conduction Band

Valence Band

E

Photons

Figure 5.1.6: Trap-recombination with sub-

sequent emission of two photons.

number of charge pairs present at that time. Mathematically, we can write

dN

dt

∝−N

= −krN, (5.1.19)

wherekris the proportionality constant, generally known as therecombination rate
constant. The integration of this equation yields

N=N 0 e−krt, (5.1.20)

where we have used the initial conditionN=N 0 andt=0. Thefactorkrin the
above equation can be used to define carrier lifetimeτthrough the relation

τ=

1

kr

. (5.1.21)

To understand the meaning ofτwe substitutet=1/kris equation 5.1.20 and get

N

N 0

=e−^1 ≈ 0. 37. (5.1.22)

This implies that the carrier lifetimeτrepresents the time it takes the carrier pop-
ulation to decrease by approximately 63%.
The value ofkrandτ for a particular charge carrier (electron or hole) depend
on the type of material, the donor and acceptor impurities and the temperature.
for example at 300Kin a p-type silicon having acceptor density of 10^19 cm^3 ,the
lifetime of an electron is approximately 0. 1 μsbut if the acceptor density is reduced
by two orders of magnitude the lifetime increases to about 10μs. This trend is
typical of all semiconductors.

Example:

Compute the percentage of holes lost within 4μsof their generation in an

n-type silicon kept at 300K. The number density of the dopant impurity