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MIT OpenCourseWarehttp://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 8.512 Theory of Solids II Problem Set 5 Due March 17, 2008 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 field →→→ E (r, t) =E0 e i( →→ q·r −iωt) (1) The Boltzmann equation in the relaxation time approximation is ∂f ∂f + · → +v ∂f f − f0→ (−e)E= − (2) →→∂t τ ∂r ∂k where f0 is the equilibrium distribution 1 f0(ǫ) = (3) eβǫ + 1 Write →→→ f(r,k, t) = f0(ǫk) + Φ( k)e i → → q·r −iωt (4) →→ and work to first order in Φ( k) and E. Show that the conductivity is given by → τ(ˆe· v)2 ∂f0 2 →e → σ(q, ω) = d 4π3 − (5) k →→∂ǫk1 − iτ(ω− q · v) → where ˆe is the unit vector in the direction of E0. (b) A simple way to derive the K ubo formula is to compare the energy dissipation rate σE02 with the rate of photon absorption. At finite temperature, we need to include both absorption and emission processes. Show that for free electrons (including spin) 22e 1 (f0(Eα) − f0(Eβ)) � →→→ > |2σ ′ (q, ω) = iq·r| < β|e eˆ· p |α δ (�ω − (Eβ − Eα)) 2 V (Eβ − Eα)/�m α,β (6) Using the Kramers-Kronig relation, show that the complex conductivity is →→→ 2e 2 1 | < β|e > |2 (−i) (f0(Eα) − f0(Eβ)) iq·r eˆ· p |α → σ(q, ω) (7) = m2 V (Eβ − Eα)� (Eβ − Eα − �ω − iη)α,β Problem Set 4 Due March 9, 2009� 2 (c) For |q| ≪ kF, show that Eq.(7) reduces to Eq.(5) under the assumptions that |α >, |β > are plane waves and η is identified with τ 1 . → 2. Equation (5) in Problem 1 is va lid for any relation between q and ˆe. In an isotropic → material t he response can be separated into the longitudinal ( q � eˆ) and transverse → parts ( q⊥ eˆ). The latter is appropriate for the propagation of electromagnetic waves. (a) For T ≪ ǫF, show that the transverse conductivity can be written as an integra-tion over the Fermi surface. 3 1 1 −2σ0→x σ⊥(q, ω) = dx (8) 1 − iωτ 4 −1 1 + sx where iqvFτ s = (9) 1 − iωτ In Eq.(8) σ0 = ne2τ/m is the DC Boltzmann conductivity and the integration variable x stands for cos θ in an integration over the Fermi surface. (b) The integra l in Eq.(8) can be done analytically. For our purposes, find 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 t he electron mean free path ℓ is much greater tha n the wavelength of light. Note t hat it is reduced from σ0 by the factor 1/(qℓ). 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 qℓ ≪ 1 , all the electrons can absorb energy fr om the electric field. However, for qℓ ≫ 1, only a fraction travelling almost parallel to ˆe can do so. The a r gument was first given by


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