PHY 140A Midterm: PHYS 140B Midterm Exam Spring 2012
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Consider a simple cubic lattice with lattice constant a. That is, the real space lattice vectors are ~a1 = a x, ~a2 = a y, and ~a3 = a z. Draw the free electron energy bands e(~k + ~g) = h2|~k + ~g|2/2m folded back into the rst brillouin zone. For simplicity, set (kx, ky, kz) = (k, 0, 0) so that you are just plotting e as a function of a single variable k. do at least the rst seven smallest. Consider electrons hopping in one dimension with a dispersion relation e(k) = 2t cosk. Connect any features you notice in e(k) to the plot of e(k) versus k. Sketch the density of states for the dispersion relation e(kx, ky) = 2t [coskx + cosky]. Why might it be a reasonable choice (that is, what characteristic of this class of materials makes one want to use this dispersion relation?)
