MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 1 Problem Set 4Set Problem 8 512 Theory of Solids II Due March 9 2009 Due March 17 2008 5 1 This problem reviews the Boltzmann equation and compares the result with the Kubo formula For a derivation of the Boltzmann equation read p 319 of Ashcroft and Mermin a Consider an electron gas subject to an electron eld i E r t E 0 e q r i t 1 The Boltzmann equation in the relaxation time approximation is f f f f f0 v e E t r k where f0 is the equilibrium distribution f0 1 e 1 2 3 Write f r k t f0 k k ei q r i t 4 and work to rst order in k and E Show that the conductivity is given by e v 2 f0 e2 q 3 d k 5 4 k 1 i q v where e is the unit vector in the direction of E 0 b A simple way to derive the Kubo formula is to compare the energy dissipation rate E02 with the rate of photon absorption At nite temperature we need to include both absorption and emission processes Show that for free electrons including spin q f E f0 E 2e2 1 iq r p 2 0 e e E E 2 m V E E 6 Using the Kramers Kronig relation show that the complex conductivity is 2e2 1 ei q r e p 2 i f0 E f0 E q 2 m V E E E E i 7 2 c For q kF show that Eq 7 reduces to Eq 5 under the assumptions that are plane waves and is identi ed with 1 2 Equation 5 in Problem 1 is valid for any relation between q and e In an isotropic material the response can be separated into the longitudinal q e and transverse parts q e The latter is appropriate for the propagation of electromagnetic waves a For T F show that the transverse conductivity can be written as an integra tion over the Fermi surface 0 3 q 1 i 4 1 dx 1 1 x2 1 sx 8 where s iqvF 1 i 9 In Eq 8 0 ne2 m is the DC Boltzmann conductivity and the integration variable x stands for cos in an integration over the Fermi surface b The integral in Eq 8 can be done analytically For our purposes nd the small s and large s limits The small s limit is the Drude conductivity while the large s limit is called the extreme anomalous region It describes the situation when the electron mean free path is much greater than the wavelength of light Note that it is reduced from 0 by the factor 1 q Produce a simple argument to show that this reduction factor can be understood on the basis of kinetic theory of classical particles Hint Consider a low frequency transverse electromagnetic wave For q 1 all the electrons can absorb energy from the electric eld However for q 1 only a fraction travelling almost parallel to e can do so The argument was rst given by Pippard
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