7.2.1 Empty Lattice Approximation
The empty lattice approximation is the easiest one for most materials. We know from previous
experiments that gaps will appear in the DOS for solids in a periodic potential. We also know that
if we have a metal, the electrons are almost free - the dispersion-relationship is a parabola. But at
the Brillouin zone boundary there will be some diffraction that will cause a gap. That’s the same for
the next boundary. For most materials that gap is approximately 1 eV big (The energy where the
gap occurs can be calculated withk=πa andE= ~
(^2) k 2
2 m). After the^1
stBrillouin zone the dispersion
curve would move on to higherk-vectors, but we map it back into the 1 stBrillouin zone (see fig.
24). This helps saving paper, but the more important reason is that the crystal lattice conserves the
momentum (i.e. if a photon excites an electron) by absorbing or giving momentum to the electrons
(the whole crystal starts to move with the opposite momentum of the electron). The third reason is
the symmetry (the HamiltonianHcommutes with the symmetry operatorS). You can take any atom
and pick it up and put it down on another atom. So the empty lattice approximation just tells you
that you can draw the parabola and then draw another one just one reciprocal lattice vector shifted.
So you can draw the dispersion relationship in the first Brillouin zone. The dispersion relationships
are always symmetric inkxandk−x, so they are in y- and z-direction.
Figure 24: Dispersion relationship for a simple cubic lattice in 3D for the empty lattice approximation
If you have a simple cubic metal the curve plotted in fig. 25 shows the dispersion relation for the
empty lattice approximation. It starts atΓ, wherekis zero, to different directions for example to M
at the 110 direction. This complicated looking thing is just the free electron model, where k starts at
zero and grows withk^2. If there is a potential, gaps will be formed.
You can do the same for fcc (most metals). Fig. 26 is an equivalent kind of drawing for fcc. Again it
starts atΓwhere k is zero and goes to X (to the y direction, in this crystal x, y and z direction have the
same symmetry (6 symmetry related directions)) or L (is closer in k space toΓbecause the reciprocal
lattice to fcc is bcc and in bcc there are 8 nearest neighbors which are in the 111 direction).