1. 1 NFE Bands in 2D and 3D Consider a square lattice with side a. The 1 st BZ is also a square lattice, but with side 2  /a. (I) Situation: One electron per unit cell. The fraction of occupied states is exactly ½, and hence the Fermi circle has radius k F  0.798  /a. All the filled states are within the 1 st BZ. There are empty states located close to the Fermi surface, low-energy excitations are possible, and the system is a metal.  A very weak periodic potential, i.e., looks like the free electron model: kxkx kyky /a/a /a/a -/a-/a -/a-/a 1 st BZ  For an intermediate periodic potential: - Gaps open up at the zone boundaries. States close to the zone boundary get moved down in energy – and the closer they are to the boundary, the more they are moved down. - As a result, states close to the boundary get filled up preferentially at the expense of states further from the boundary: There are still empty states located close to the Fermi surface, low-energy excitations are possible, and the system is a metal.

2. 2  For a very strong periodic potential: - The Fermi surface may even touch the Brillouin zone boundary. Nevertheless, there are empty states in the vicinity of parts of the Fermi surface. Low-energy excitations are possible, and the system is a metal. Comparison with Fermi surfaces of real monovalent metals (The wire frames mark off the 1 st BZ, which are half filled.) One electron per unit cell (con’t):

3. 3 (II) Situation: Two electrons per unit cell. No Periodic Potential: Strong periodic potential - Without a periodic potential, the fraction of occupied states is exactly 1. The number of electrons is precisely enough to fill a single zone. - With a periodic potential, a gap opens at the zone boundary. This gap opening pushes down the energy of all states within the first zone and pushes up the energy of all states in the second zone. No Periodic Potential Strong Periodic Potential - If the periodic potential is sufficiently strong, then the states in the 1 st BZ are all lower in energy than states in the 2 nd BZ. The entire lower band is filled, and the upper band is empty. Since there is a gap at the zone boundary, there are no low-energy excitations possible, and this system is an insulator. - Without a periodic potential, the Fermi circle has radius k F  1.13  /a. The Fermi surface crosses into the second Brillouin zone.

4. -Potential produces a gap: lowers energy of states inside 1 st BZ and raises those outside, particularly near the BZ boundary. - Standing waves at the BZ boundary implies that (dE/dk)  = 0, i.e. constant energy contours are perpendicular to the BZ boundary. 4 - The part of the Fermi circle in 2 nd BZ shrinks, and that in the 1 st BZ grows. Intermediate Periodic Potential - Some states will remain occupied in the 2 nd BZ and some states will remain empty within the 1 st BZ. -Electric field E can displace part of the electron distribution, indicated by the dashed lines in the figure below, since there exist nearby empty states. (- Those parts of the electron distribution having an interface at the zone boundaries are characterized by having an energy gap and so cannot “move” in the E-field.) Crystal is a metal although it is divalent. Two electrons per unit cell (con’t):

5. 5 1 st BZ2 nd BZ Fermi surface of the (fcc) divalent metal Ca. Left: The first band is almost completely filled with electrons. The solid region is where the Fermi surface is inside the Brillouin zone boundary. Right: A few electrons fill small pockets in the second band. Comparison with Fermi surface of a real divalent metal: Reduced zone scheme