For elastic scattering the Boltzmann equation gives each transport coefficient as a…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Electronic transport in solids · Electronic transport in solids

Solid State Physics, Ch Two currents, two forces: Wiedemann-Franz and thermoelectricity

Problem

For elastic scattering the Boltzmann equation gives each transport coefficient as a moment of the energy-dependent conductivity σ(ε)=13e2τ(ε)v(ε)2D(ε)\sigma(\varepsilon)=\tfrac13 e^2\tau(\varepsilon)v(\varepsilon)^2D(\varepsilon) (with D(ε)D(\varepsilon) the density of states per unit volume, both spins counted), weighted by −∂f/∂ε-\partial f/\partial\varepsilon and powers of (ε−μ)(\varepsilon-\mu): L11=L(0),L12=−1eTL(1),L22=1e2TL(2),L^{11}=L^{(0)},\qquad L^{12}=-\frac{1}{eT}L^{(1)},\qquad L^{22}=\frac{1}{e^2T}L^{(2)}, where L(n)=∫dε(−∂f∂ε)σ(ε)(ε−μ)nL^{(n)}=\int d\varepsilon\left(-\dfrac{\partial f}{\partial\varepsilon}\right)\sigma(\varepsilon)(\varepsilon-\mu)^n. For kBT≪EFk_BT\ll E_F the Sommerfeld expansion gives L(0)=σ(EF),L(1)=π23(kBT)2σ′(EF),L(2)=π23(kBT)2σ(EF).\begin{aligned}L^{(0)}&=\sigma(E_F),\qquad L^{(1)}=\frac{\pi^2}{3}(k_BT)^2\sigma'(E_F),\\L^{(2)}&=\frac{\pi^2}{3}(k_BT)^2\sigma(E_F).\end{aligned} ((a)) Show that the electronic thermal conductivity, defined at zero charge current by κel=L22−T(L12)2/L11\kappa_{\rm el}=L^{22}-T(L^{12})^2/L^{11}, reduces at leading order in kBT/EFk_BT/E_F to κel=π23(kBe)2σT\kappa_{\rm el}=\dfrac{\pi^2}{3}\left(\dfrac{k_B}{e}\right)^2\sigma T, and show that the term dropped is smaller by a factor of order (kBT/EF)2(k_BT/E_F)^2. ((b)) Write down the Lorenz number L0=κel/σTL_0=\kappa_{\rm el}/\sigma T, note which material properties have cancelled, and evaluate L0L_0 numerically.

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