Homework08_solution

# Homework08_solution - Homework 8 Solution Problem#1 A...

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Homework # 8 - Solution Problem #1 A two-dimensional metal has one atom of valence one in a simple rectangular primitive cell of a 1 = 2Å and a 2 = 4Å. (a) Draw the first and the second Brillouin zones. (b) Calculate the radius of the free electron Fermi sphere and draw this sphere to scale on the drawing of the Brillouin zones. (c) Draw the Fermi surface in reduced zone scheme and show schematically the effect of a weak crystal potential. (a) 2 π / a 1 2BZ 1BZ 2 π / a 2 (b) In two dimensions the “volume” of the Fermi “sphere” is 2 F k π . The “volume” of k-space per k-point is 12 22 aa × . Therefore, taking into account spin degeneracy we have 2 2 FF ka a k N ππ == ⎛⎞ ⎜⎟ ⎝⎠ , which results in 1/2 -1 -1 2 Å0 . 8 9 Å 2 F N k . This value is bigger than the shortest distance from the origin to the boundary of the first Brillouin zone (0.79Å), so that the Fermi sphere appears in the second zone.

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(c) The Fermi surface in the reduced zone scheme is shown by the red line, and the effect of a weak crystal potential is shown by the blue line.
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## This note was uploaded on 09/21/2011 for the course PHYSICS 101 taught by Professor Wormer during the Spring '08 term at Aarhus Universitet.

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Homework08_solution - Homework 8 Solution Problem#1 A...

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