Fig. 33 shows the dispersion relationship of a simple cubic. It starts fromΓto X because in the
x-direction there is a nearest neighbor. It starts like a parabola, close to the brillouin zone boundary
it turns over and hits with about 90 degrees. M is in the 110 direction and R is in the 111 direction.
So the energy is higher at the corner and at the edge of the cube. That is what we expected to happen
from the empty lattice approximation. At the bottom we can calculate the effective massm∗= ~
2
2 ta^2
which is the second derivation of the dispersion relation. So if the overlap between the atomic wave-
functions is small (that means a smallt) the effective mass is high. This makes it difficult to move
the electron through the crystal. This also means that we get very flat bands because of eqn. (59).
In fig. 33 you see just one band. If you want to have the next band you have to take the next atomic
state (2s instead of 1s) and redo the calculation.
If you knowkx,kyandkzyou can calculate the energy and you can plot the fermi sphere, which
Figure 33: Dispersion relationship for a simple cubic.looks like a ball. If you go to higher energies there are no more possible states left, so you get a hole
at the brillouin zone boundaries. If you increase the energy the hole becomes bigger and bigger and
at one point there will be no more solutions in this band (from the atomic wavefunction you chose), if
you took a very high energy. For a metal the fermi surface then separates into the occupied and the
unoccupied states in a band.