Page 145 - Electrical Properties of Materials

Intrinsic semiconductors 127

leading to the final result
–(E g – E F )

N e = N c exp , (8.17)
k B T
where
∗ 3/2
2πm k B T
e
N c =2 . (8.18)
h 2
Thus, we have obtained the number of electrons in the conduction band as a
function of some fundamental constants, of temperature, of the effective mass
of the electron at the bottom of the band, and of the amount of energy by which
the bottom of the band is above the Fermi level.
We can deal with holes in an entirely analogous manner. The probability
of a hole being present (that is of an electron being absent) is given by the
function

1– F(E), (8.19)
which also declines exponentially along the negative E-axis. So we can choose
the lower limit of integration as –∞, leading to the result for the number of
holes in the valence band,
N h = N v exp(–E F /k B T), (8.20)

where
2 3/2
∗
N v =2(2πm k B T/h ) . (8.21)
h
For an intrinsic semiconductor each electron excited into the conduction
band leaves a hole behind in the valence band. Therefore, the number of
electrons should be equal to the number of holes (this would actually follow
from the condition of charge neutrality too); that is

N e = N h . (8.22)
Substituting now eqns (8.17) and (8.20) into eqn (8.22), we get

N c exp{–(E g – E F )/k B T}= N v exp(–E F /k B T), (8.23)
from which the Fermi level can be determined. With a little algebra we get

E g 3 m ∗ h Since k B T is small, and the ef-
E F = + k B T log e . (8.24)
2 4 m ∗ fective masses of electrons and
e
holes are not very much differ-
We may now ask how carrier concentration varies with temperature. Strictly
ent, we can say that the Fermi
speaking, the energy gap is also a function of temperature for the reason that
level is roughly halfway between
it depends on the lattice constant, which does vary with temperature. That is,
the valence and conduction bands.
however, a small effect on the normally used temperature range, so we are
nearly always entitled to disregard it. Substituting eqn (8.24) into eqns (8.17)
and (8.20), we find that both N e and N h are proportional to exp(–E g /2k B T), an
important relationship.
We know now everything we need to about intrinsic semiconductors. Let us
now look at the effect of impurities.