Page 136 - Electrical Properties of Materials

118 The band theory of solids

Using our definition of effective mass [eqn (7.42)] we may rewrite eqn
(7.48) in the following manner:

dI 2 1
= e E , (7.61)
dt m ∗
i i
where the summation is over the occupied states.
If there is only one electron in the band, then

2
dI e e E
= . (7.62)
dt m ∗
If the band is full, then according to eqn (7.52) the effective number of
electrons is zero; that is,

dI 2 1
= e E = 0. (7.63)
dt m ∗
i i
Assume now that somewhere towards the top of the band an electron, de-
noted as j, is missing. Then, the summation in eqn (7.61) must omit the state j,
which we may write as

dI h 2 1
= e E . (7.64)
dt m ∗
i i
i =j
But from eqn (7.63)
⎛ ⎞
⎜ 1 1 ⎟
e E + ⎟ = 0. (7.65)
2 ⎜
m j i m i
⎝ ∗ ∗ ⎠
i =j
Equation (7.64) therefore reduces to
dI h 2 1
=–e E . (7.66)
dt m ∗ j
In the upper part of the band, however, the effective mass is negative;
therefore

Hence, an electron missing from e E
2
dI h
the top of the band leads to exactly = ∗ . (7.67)
dt |m |
j
the same formula as an electron
present at the bottom of the band. Now there is no reason why we should not always refer to this phenomenon
as a current due to a missing electron that has a negative mass. But it is a
lot shorter, and a lot more convenient, to say that the current is caused by a
positive particle, called a hole. We can also explain the reason why the signs of
eqn (7.62) and of eqn (7.67) are the same. In response to an electric field, holes
move in an opposite direction carrying an opposite charge; their contribution
to electric current is therefore the same as that of electrons.