Example: Simple Cubic
At first we have to figure out what the Fourier- series is for simple cubic. Just take the nearest
neighbors in k space. So choose gaussian atoms and put them on a simple cubic lattice and make
them pretty wide in real space so they are quite narrow in k-space. Then only the nearest neighbors
are relevant. For the potential only take contribution by the nearest neighbors (6 for sc). They will
correspond to these six terms:
U(r) =U 0 +U 1 (ei
2 πxa
+e−i
2 πxa
+ei2 πy
a +e−i
2 πy
a +ei^2 πza +e−i^2 πza )This is the reciprocal space expansion for the simple cubic. When we put this in the known algebraic
equations (48) and (49) we get a matrix and can solve it. There you get a 7 by 7 matrix. The term
in the middle of the matrix is always the central atom. This central atom is always connected equally
to the nearest neighbors. The terms in the diagonal represent the nearest neighbors (in this case 6)
in±x,±y and±z direction. The smallest G is^2 aπ, then one step further in this direction(kx+^4 aπ).
We can neglect the other interactions instead of the nearest neighbors because the gaussian function
falls exponentially.
So if you can construct this matrix you can calculate the band structure for an electron moving in a
periodic potential. If you increase the amplitude there will appear gaps in the band structure where
we thought they would appear in the empty lattice approximation, but you can’t tell what the band
structure would really be.
The next issue is band structure, where we try to calculate the dispersion relationshipE(k) for
electrons. Typically it starts out like a parabola and then bends over and continues on, so there opens
a gap like you see in fig. 29. In the density of states there are a lot of states in the range of the bands
and no states at the gaps.
Important differences for the DOS and the Fermi energy between the different types of materials:
A metal has partially filled bands with a high density of electron states at the Fermi energy. A
semimetal has partially filled bands with a low density of electron states at the Fermi energy. An
insulator has bands that are completely filled or completely empty with a bandgap greater than 3 eV.
The Fermi energy is about in the middle of the gap. A semiconductor has a Fermi energy in a gap
which is less than 3 eV wide. Some electrons are thermally excited across the gap. The difference
between a semimetal and a semiconductor is that a semiconductor looses all its conductivity during
cooling down. A semimetal does not loose all the conductivity. In fig. 29 you can see the dispersion
relation and the density of states over the energy. At the right you can see the Fermi energies of the
several possibilities.