The coupled linear-response equations for the charge current and the heat current are…

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

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

Problem

The coupled linear-response equations for the charge current jj and the heat current jQj_Q are j=L11E′+L12(−∇T),jQ=L21E′+L22(−∇T),j = L^{11}E' + L^{12}(-\nabla T), \qquad j_Q = L^{21}E' + L^{22}(-\nabla T), where E′=E+∇μ/eE' = E + \nabla\mu/e is the electrochemical field a voltmeter senses. Onsager reciprocity gives L21=TL12L^{21}=TL^{12}. ((a)) Starting from the open-circuit condition j=0j=0 that defines the Seebeck coefficient S=−ΔV/ΔTS=-\Delta V/\Delta T, and the uniform-temperature condition (∇T=0\nabla T=0) that defines the Peltier coefficient Π=jQ/j\Pi=j_Q/j, express SS and Π\Pi in terms of the LL coefficients and derive the Kelvin relation Π=ST\Pi=ST. ((b)) A leg of a thermoelectric alloy has S=195 μV K−1S=195\,\mu\mathrm{V\,K^{-1}} at 350 K350\,\mathrm{K}. Use the Kelvin relation to find its Peltier coefficient.

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