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homework12 - (10 points Consider a two-dimensional crystal...

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Physics 481: Condensed Matter Physics - Test prep homework Problem 1: Tightly bound electrons in 1D (10 points) Consider a one-dimensional electron system with lattice constant a in tight binding approximation. The energy-momentum relation reads ( k ) = - 2 t cos( ka ) . a) Calculate the electronic density of states D ( ). b) Does it have van-Hove singularities? If so, discuss their character! c) Calculate the Fermi energy for 0.5, 1, and 2 electrons per unit cell. d) For one electron per unit cell, calculate the low-temperature specific heat (per cell)! Problem 3: Two-dimensional band structure
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Unformatted text preview: (10 points) Consider a two-dimensional crystal with a rectangular unit cell of length a = 5 ˚ A and width b = 3 ˚ A. a) Determine the 1st Brillouin zone. b) For free electrons, calculate the Fermi wavevector for 1, 2, 3 electrons per unit cell. Sketch the projected Fermi surfaces in the extended and reduced zone schemes. c) Now assume a weak periodic potential. Where will gaps open in k-space? Sketch the Fermi surfaces for nearly free electrons for 1, 2, 3 electrons per unit cell....
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