The Seebeck coefficient of a metal follows from the same zero-current argument as the…

PHYSICS · Multi-part · Fermi University Physics · Condensed Matter, Materials & Soft Matter · Free electrons, energy bands and electronic structure · Electronic transport in solids

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

Problem

The Seebeck coefficient of a metal follows from the same zero-current argument as the Kelvin relation:

S=L^{12}/L^{11}

. Using the Sommerfeld-expansion results

L^{11}=\sigma(E_F)

and

L^{12}=-\dfrac{\pi^2}{3}\dfrac{k_B^2T}{e}\sigma'(E_F)

: ((a)) Derive the Mott formula

S=-\frac{\pi^2}{3}\frac{k_B^2T}{e}\left.\frac{d\ln\sigma}{d\varepsilon}\right|_{E_F}.

((b)) For a free-electron metal with an energy-independent relaxation time,

\sigma(\varepsilon)\propto\varepsilon^{3/2}

. Show that the Mott formula then gives

S=-\frac{\pi^2}{2}\frac{k_B}{e}\frac{k_BT}{E_F}.

((c)) Suppose instead that the relaxation time varies with energy as

\tau\propto\varepsilon^{r}

, so that

\sigma(\varepsilon)\propto\varepsilon^{3/2+r}

. Find how

S

depends on

r

, evaluate it for sodium (

E_F=3.24\,\mathrm{eV}

) at

300\,\mathrm{K}

with

r=-\tfrac12

, and find the range of

r

for which the free-electron sign of

S

would reverse.

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