Problem
An electron of mass m moves in a square lattice of lattice spacing a. The nearly free-electron approximation applies.
(a) With one electron per site in the crystal, draw the Fermi surface on
the 
plane. Is this a metal or an insulator?
(b) With two electrons per site, draw the Fermi surface. Is this a metal or an insulator?
Solution
(a) With one electron per site, the band becomes half-filled and will
therefore lead to a metal. The Fermi surface is sketched by taking a constant
energy cut such that the area within the Fermi line is half of the area of the
Brillouin zone (see left panel of Figure 1.1). Since the Fermi line does not get
too close to the Brillouin zone boundary, in good approximation the cut is a
circle. Note the difference between this Fermi surface and the Fermi surface
in the tight-binding approximation, Figures 
and .
The Fermi wavevector is
Figure 1.1: Left panel: Contour plot of the energy surface for the lower-energy band.
Darker shades correspond to higher energies. The
dashed line is the Fermi surface for one electron per site;
the solid line is that for two electrons per site. Right panel:
Contour plot for the upper band.
The solid line is the Fermi surface
for two electrons per site.
(b) With two electrons per site, assuming that the constant energy lines are circles, we obtain
This value is larger than 
, therefore
we have to consider the two bands, as illustrated in Figure
1.2. There will be two Fermi surfaces, one in the upper band and
another one in the lower band. The Fermi line in the upper
band is shown in the right panel of Figure 1.1.
Each of the bands is partially filled; with two electrons per site the
NFE approximation leads to a metal. In contrast, the tight-binding model
(Problem ) results in an insulator.
Figure 1.2: One quadrant of the lowest two energy bands for a square lattice.