Physics 617 Problem Set 3
Due Weds, Feb. 17
(1) 2-dimensional crystal reciprocal lattice and free-electrons: Consider a triangular net, with
60° angles and equal near-neighbor distances a (e.g. connecting yields equilateral triangles).
(i) Find the reciprocal lattice, with orientation and cell dimensions.
(ii) Carefully construct a sufficient number of perpendicular bisector planes, and indicate the
first 5 Brillouin Zones. (Carefully means at least with a ruler, or a computer if you like.)
(iii) Calculate the area in k-space per zone.
(iv) If there are 4 electrons per primitive cell, find kF for the case of free electrons.
Comparing to the calculated dimensions of your zones, which zone boundaries does the
Fermi surface (Fermi circle) cross? Plot the Fermi surface on your diagram.
(v) Sketch reduced-zone parts of the Fermi surface as connected closed curves for each zone.
(2) Ashcroft-Mermin #9.3
(3) Free-electron bands in a 3D simple cubic lattice
(a) Write down a general free-electron formula for ε(k ) along the (100) and the (111) directions,
in terms of k , and using h, k, and l to label the indices of the K vectors used to fold the band
segments into the first zone, in a reduced-zone scheme.
(b) What are the energies of the lowest two band crossings at the
Γ and X points, in terms of the lattice constant a and fundamental
constants? The figure shows a slice at kz = 0; by convention Γ is
the origin, and X is the center of the square 1st BZ face.
(c) In the nearly-free electron model, which Fourier components
of the crystal potential would be involved in determining band
splittings at the lowest crossing at Γ?
(d) Sketch the three lowest energy bands in the Γ to X direction in the NFE model; what are the
degeneracies of these bands? (Equivalent to the dots in figure 9.5 in the text for the fcc case.)
(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
2D lattice sites in x and y directions. Suppose that the Hamiltonian can be expressed as,
H = ∑ εo 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 (aong x) and b (along y).
(a) Write down a single-particle Bloch wave function composed of the localized states m, n .
(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.
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