Page 125 - Electrical Properties of Materials

The Ziman model 107

may argue that the propagation of electrons is strongly disturbed whenever
eqn (7.13) is satisfied. In one dimension the condition reduces to

nλ =2a. (7.14)

Using the relationship between wavelength and wavenumber, the above
equation may be rewritten as

nπ
k = . (7.15)
a
Thus, we may conclude that our free-electron model is not valid when
eqn (7.15) applies. The wave is reflected, so the wave function should also
contain a term representing a wave in the opposite direction,

ψ –k =e –ikx . (7.16)
Since waves of that particular wavelength are reflected to and fro, we may
expect the forward- and backward-travelling waves to be present in the same
proportion; that is, we shall assume wave functions in the form

1 ikx –ikx √ cos kx
ψ ± = √ (e ± e )= 2 , (7.17)
2 isin kx
where the constant is chosen for correct normalization.
Let us now calculate the potential energies of the electrons in both cases.
Be careful; we are not here considering potential energy in general but the
potential energy of the electrons that happen to have the wave functions ψ ± .
These electrons have definite probabilities of turning up at various places, so
∗
their potential energy may be obtained by averaging the actual potential V(x) ∗ You may also look upon eqn (7.18)
2
weighted by the probability function |ψ ± | . Hence, as an application of the general formula
given by eqn (3.43).
1 2
V ± = |ψ ± | V(x)dx
L
2
1 2 cos kx
= 2 V(x)dx. (7.18)
L 2sin kx
Since k =nπ/a, the function V(x) contains an integral number of periods of L is the length of the one-
2
|ψ ± | ; it is therefore sufficient to average over one period. Hence, dimensional ‘crystal’ and V(x)is
the same function that we met be-
2
1 a 2 cos kx fore in the Kronig–Penney model
V ± = 2 V(x)dx
a 0 2sin kx but now, for simplicity, we take
1 a 1 + cos 2kx 2w = a.
= V(x)dx
a 0 1 – cos 2kx
1 a
= ± cos 2kx V(x)dx, (7.19)
a 0

since V(x) integrates to zero. Therefore,
V ± = ±V n . (7.20)