Page 128 - Electrical Properties of Materials

110 The band theory of solids

We have now a large number of states so we should turn to eqn (5.24), which
looks formidable with j running through all the atoms of the one-dimensional
crystal. Luckily only nearest neighbours are coupled (or so we claim), so the
differential equation for atom j takes the rather simple form

dw j
i = E 1 w j – Aw j–1 – Aw j+1 , (7.24)
dt
where, as we mentioned before, E 1 is the energy level of the electron in the
absence of coupling, and the coupling coefficient is taken again as –A.We
could write down analogous differential equations for each atom, but fortu-
nately there is no need for it. We can obtain the general solution for the whole
array of atoms from eqn (7.24).
Let us assume the solution in the form

E is the energy to be found. w j = K j e –iEt/ . (7.25)

Substituting eqn (7.25) into eqn (7.24) we get

EK j = E 1 K j – A(K j–1 + K j+1 ). (7.26)
Note now that atom j is located at x j , and its neighbours at x j ± a,re-
spectively. We may therefore look upon the amplitudes K j , K j+1 , and K j–1 as
functions of the x-coordinate. Rewriting eqn (7.26) in this new form we get

EK(x j )= E 1 K(x j )– A{K(x j + a)+ K(x j – a)}. (7.27)

This is called a difference equation and may be solved by the same method
as a differential equation. We can assume the trial solution

K(x j )=e ikx j , (7.28)

which, substituted into eqn (7.27) gives
Ee ikx j = E 1 e ikx j – A{e ik(x j +a) +e ik(x j –a) }. (7.29)

Dividing by exp(ikx j ) eqn (7.29) reduces to the final form

E = E 1 –2A cos ka, (7.30)
which is plotted in Fig. 7.11. Thus, once more, we get the result that energies
within a band, between E 1 –2A and E 1 +2A, are allowed and outside that range
are forbidden.
It is a great merit of the Feynman model that we have obtained a very simple
mathematical relationship for the E – k curve within a given energy band. But
what about other energy bands that have automatically come out from the other
models? We could obtain the next energy band from the Feynman model by
planting our electron into the next higher energy level of the isolated atom,
E 2 , and following the same procedure as before. We could then obtain for the
The coupling coefficient between next band
nearest neighbours is now taken
as –B. E = E 2 –2B cos ka, (7.31)