Physics 489 Problem Set 6
Due Thursday, Oct. 29
(1) Kittel 9.3.
(2) 2-dimensional electron gas properties: Kittel 17.3.
(3) In the 10/22 class handout, there are some examples of FCC free-electron and actualmaterial electron energies.
(a) In the Ashcroft & Mermin figure for free-electrons, dots along the curves indicate the
degeneracy. The lowest curve along a W-L path is double-degenerate. Describe the (3D!)
relationship between k-vectors involved in this degeneracy, indicating which G vector joins
them.
(b) In the plot for aluminum on the next page, the lowest curves along X to W exhibit a splitting
into two nearly parallel energy curves. (i) Indicate what region of the FCC first zone is occupied
by these curves, and (ii) assuming a nearly-free electron model, indicate which Fourier
component of the crystal potential is involved in this splitting, and make a numerical estimate of
its value.
(c) Consider the case of copper, which is also FCC; it is considered to have one active valence
electron [it is at the end of the transition series on the periodic table, but the d-electrons are all
full, giving a band that is lower energy and separated from the valence bands, but optical
excitations from that band give copper its rosy color]. For this situation and based on the figures
in the handout, what would be the shape of the Fermi surface in the free-electron limit?
(4) 2D rectangular lattice in tight binding: Consider a set of identical localized states m, n on a
2D simple rectangular lattice, where the integers m, n index the positions of these states on the
2D lattice sites in x and y directions. Suppose that the Hamiltonian can be expressed as,
H = ∑ ⎡⎣ε o m, n m, n − t1 ( m, n m, n +1 + m, n m, n −1 ) –
m,n
t2 ( m, n m +1, n + m, n m −1, n )⎤⎦ ,
where εo , t1, t2 and are constants. This Hamiltonian couples nearest neighbors along y ( t1 term)
and x ( t2 term). Assume lattice constants a (along x) and b (along y).
(a) Write down a Bloch wave function composed of the localized states m, n . This should be
very similar to the form [1] in the handout, except there are two indices for the lattice positions
showing explicitly the 2-D positioning; also I streamlined by using bra-ket notation instead of the
function ϕ. In your answer express positions R explicitly in terms of a and b.
(b) Plug your wave-function into the Schrodinger equation, show that it is an eigenstate, and
solve for the energy vs. k for the resulting band. Find the lowest and highest energies for the
band, and identify the k-space coordinates of four saddle points in ε(k ) . Sketch the positions of
the extrema and saddle points in the first Brillouin zone for this lattice.