Ashcroft Mermin eng · Problem 4

Surface Effects

Physics — Statistical Mechanics · impurity-scattering-boltzmann-equation

Problem

Consider the Boltzmann equation (16.13) for a metal in a static uniform electric field, with a collision term (16.18) appropriate to elastic impurity scattering. (a) Assuming a nonequilibrium distribution function of the form g(k)=f(k)+δg(k),(16.36)g(\mathbf{k}) = f(\mathbf{k}) + \delta g(\mathbf{k}), \tag{16.36} where ff is the equilibrium Fermi function and δg(k)\delta g(\mathbf{k}) is of order EE, derive to linear order in EE an integral equation obeyed by δg\delta g, and show that the conductivity can be written in the form σ=e2∫dk4π3(−∂f∂E)v(k)u(k),(16.37)\boldsymbol{\sigma} = e^2 \int \frac{d\mathbf{k}}{4\pi^3} \left(-\frac{\partial f}{\partial \mathcal{E}}\right) \mathbf{v}(\mathbf{k})\mathbf{u}(\mathbf{k}), \tag{16.37} where u(k)\mathbf{u}(\mathbf{k}) is a solution to the integral equation v(k)=∫dk′(2π)3 Wk,k′[u(k)−u(k′)].(16.38)\mathbf{v}(\mathbf{k}) = \int \frac{d\mathbf{k}'}{(2\pi)^3}\, W_{\mathbf{k},\mathbf{k}'}[\mathbf{u}(\mathbf{k}) - \mathbf{u}(\mathbf{k}')]. \tag{16.38} (b) Let α(k)\alpha(\mathbf{k}) and γ(k)\gamma(\mathbf{k}) be any two functions of k\mathbf{k}. Define (α,γ)=e2∫dk4π3(−∂f∂E)α(k)γ(k)(16.39)(\alpha, \gamma) = e^2 \int \frac{d\mathbf{k}}{4\pi^3} \left(-\frac{\partial f}{\partial \mathcal{E}}\right) \alpha(\mathbf{k})\gamma(\mathbf{k}) \tag{16.39} so that (16.37) can be written compactly as σμν=(vμ, uν).(16.40)\sigma_{\mu\nu} = (v_\mu,\, u_\nu). \tag{16.40} Define {α,γ}=e2∫dk4π3(−∂f∂E)α(k)∫dk′(2π)3 Wk,k′[γ(k)−γ(k′)].(16.41)\{\alpha, \gamma\} = e^2 \int \frac{d\mathbf{k}}{4\pi^3} \left(-\frac{\partial f}{\partial \mathcal{E}}\right) \alpha(\mathbf{k}) \int \frac{d\mathbf{k}'}{(2\pi)^3}\, W_{\mathbf{k},\mathbf{k}'}[\gamma(\mathbf{k}) - \gamma(\mathbf{k}')]. \tag{16.41} Show that {α,γ}={γ,α}\{\alpha, \gamma\} = \{\gamma, \alpha\}, and that Eq. (16.38) implies {uμ,γ}=(vμ, γ),(16.42)\{u_\mu, \gamma\} = (v_\mu,\, \gamma), \tag{16.42} so that the conductivity can also be written in the form σμν={uμ, uν}.(16.43)\sigma_{\mu\nu} = \{u_\mu,\, u_\nu\}. \tag{16.43} (c) Prove for arbitrary α\alpha and γ\gamma that {α,α}≥{α,γ}2{γ,γ}.(16.44)\{\alpha, \alpha\} \geq \frac{\{\alpha, \gamma\}^2}{\{\gamma, \gamma\}}. \tag{16.44} (Hint: Prove that {α+λγ,α+λγ}≥0\{\alpha + \lambda\gamma, \alpha + \lambda\gamma\} \geq 0 for arbitrary λ\lambda, and choose a λ\lambda that minimizes the left-hand side of this inequality.) (d) With the choice α=ux\alpha = u_x, deduce that σxx\sigma_{xx} satisfies the inequality σxx≥e2 ⁣[∫dk4π3(−∂f∂E)vx(k)γ(k)]2∫dk4π3(−∂f∂E)γ(k)∫dk′(2π)3 Wk,k′[γ(k)−γ(k′)](16.45)\sigma_{xx} \geq \frac{e^2\!\left[\displaystyle\int \frac{d\mathbf{k}}{4\pi^3} \left(-\frac{\partial f}{\partial \mathcal{E}}\right) v_x(\mathbf{k})\gamma(\mathbf{k})\right]^2}{\displaystyle\int \frac{d\mathbf{k}}{4\pi^3} \left(-\frac{\partial f}{\partial \mathcal{E}}\right) \gamma(\mathbf{k}) \int \frac{d\mathbf{k}'}{(2\pi)^3}\, W_{\mathbf{k},\mathbf{k}'}[\gamma(\mathbf{k}) - \gamma(\mathbf{k}')]} \tag{16.45} for arbitrary functions γ\gamma. (e) Suppose that W=W(1)+W(2)W = W^{(1)} + W^{(2)}. Let γ\gamma be uxu_x, where u\mathbf{u} is the solution to (16.38). Let σ(1)\boldsymbol{\sigma}^{(1)} and σ(2)\boldsymbol{\sigma}^{(2)} be the conductivities one would have if only W(1)W^{(1)} or W(2)W^{(2)} were present. Deduce from (16.45), as it applies to σ(1)\boldsymbol{\sigma}^{(1)} and σ(2)\boldsymbol{\sigma}^{(2)}, that 1σxx≥1σxx(1)+1σxx(2).(16.46)\frac{1}{\sigma_{xx}} \geq \frac{1}{\sigma_{xx}^{(1)}} + \frac{1}{\sigma_{xx}^{(2)}}. \tag{16.46}

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