Page 153 - Electrical Properties of Materials

A relationship between electron and hole densities 135

1 T =300 K
μ
e Ge

10 –1 μ h
Mobility (m 2 V –1 s –1 ) 10 –2

1

μ
e
10 –1 Si
μ
h Fig. 8.7
10 –2 Electron and hole mobilities in Ge
10 20 10 21 10 22 10 23 10 24 10 25 and Si as a function of impurity
–3
Impurity concentration (m ) concentration.

we can define the product N e N h exactly, whatever the Fermi energy and hence
whatever the impurity density. In particular, for an intrinsic material, where
N e = N h = N i , we get

2
N e N h = N . (8.47)
i
Let us think over the implications. We start with an intrinsic semiconductor;
so we have equal numbers of electrons and holes. Now add some donor atoms.
The number of electrons must then increase, but according to eqn (8.47) the
product must remain constant. At first this seems rather odd. One would think If the number of electrons incre-
that the number of electrons excited thermally from the valence band into the ases, the number of holes must
conduction band (and thus the number of holes left behind) would depend on decrease.
temperature only, and be unaffected by the presence of donor atoms. This is not
so. By increasing the concentration of donors, the total number of electrons in
the conduction band is increased, but the number of electrons excited across
the gap is decreased (not only in their relative proportion but in their absolute
number too). Why?
We can obtain a qualitative answer to this question by considering the
‘dynamic equilibrium’ mentioned briefly before. It means that electron–hole
pairs are constantly created and annihilated and there is equilibrium when the
rate of creation equals the rate of annihilation (the latter event is more usually
referred to as ‘recombination’).
Now it is not unreasonable to assume that electrons and holes can find each The rate of recombination must
other more easily if there are more of them present. For an intrinsic material be proportional to the densities of
we may write holes and electrons.

2
2
r intrinsic = aN , g intrinsic = aN , (8.48)
i
i
where a is a proportionality constant, and r and g are the rates of recombination
and creation, respectively.