# HW-9 - Theory of Solids HW-9 Solution By Qifeng Shan...

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1 Theory of Solids HW-9 Solution By Qifeng Shan Ashcroft and Mermin’s book 12-1 For free electrons ε ( k ) = ħ 2 k 2 / 2 m . Calculate ǝ A ( ε , k z ) / ǝ ε and show that the general expression (12.42) for the period in a magnetic field reduces to the free electron result (12.43). Solution: A is the k-space area enclosed by the orbit in its plane. For free electrons, the constant energy surfaces are spheres, whose intersections with planes are circles. Thus, A = π k 2 (1) Using Eq. (12.42), the period of the motion is:  z z k A eH c k T , , 2 (2) Insert the dispersion relation for free electron: m k k 2 2 2 (3) We obtain Eq. (12.43): eH mc k T z 2 , (4) Q.E.D 12-2 For electrons near a band minimum (or maximum) ε ( k ) has the form , 2 constant 0 1 0 2 k k M k k k (12.64) where the matrix M is independent of k . (Electrons in semiconductors are almost always treated in this approximation.) (a) Calculate the cyclotron effective mass from (12.44) and show that it is independent of ε and k z , and given by ) (cyclotron , 1/2 * zz m M M (12.65) where | M | is the determinant of the matrix M .

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2 (b) Calculate the electronic specific heat (2.80) resulting from the band structure (12.64), and by comparing it with the corresponding free electron results, show that the band structure contribution to the specific heat effective mass (page 48) is given by   heat) (specific , 1/3 * M m (12.66) Solution: (a) The constant energy surface is an ellipsoid with anisotropic masses. The cyclotron effective mass m * is defined by Eq. (12.44):  z k A m , 2 * 2 (5) If we choose the magnetic field direction as z direction, the orbit in k-space is perpendicular to the z-axis. To calculate the cyclotron effective mass
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## This note was uploaded on 10/04/2011 for the course PHYS 4720 taught by Professor Shan during the Fall '09 term at Rensselaer Polytechnic Institute.

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HW-9 - Theory of Solids HW-9 Solution By Qifeng Shan...

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