2.4 Fermi Surface and Brillouin Zone
The Fermi surface (or Fermi sphere) is defined as the surface of constant energy E F
in k-space inside which all the energy states are filled by the valence electrons,
while all the energy states lying outside it are empty. The shape of the Fermi surface
(FS) is determined by the geometry of the contours in a Brillouin Zone (B-Z). Well
within the B-Z, the FS remains spherical in shape while its shape changes near the
B-Z boundaries as shown in Fig. 2.28.
Total number of free/valence electrons, n within the sphere of radius k F is
equivalent to the number of possible filled energy states of electrons and can be
determined by the equation [where each particle in k-space occupies a volume (2p/a)]:
n ¼
Volume of FS
Volume of BZ
 2
¼
4p=3
ð
Þk
3
F
2p=a
ð
Þ
3
 2
This gives us
k F ¼
3p
2 n
a 3
1=3
where the factor 2 is included for two spin states in each energy level and a is a
direct unit cell parameter. For monovalent metals (e.g., alkali metals), n is the
number of atoms per unit cell.
Fig. 2.28 The evolution of
FS with the increase in
concentration of valence
electrons
2.4 Fermi Surface and Brillouin Zone
81