arXiv:1001.3423v2 [cond-mat.str-el] 4 Nov 2010 The Kelvin Formula for Thermopower Michael R. Peterson Condensed Matter Theory Center, Department of Physics, University of Maryland, College Park, MD 20742, USA B. Sriram Shastry Physics Department, University of California, Santa Cruz, CA 95064, USA (Dated: November 1, 2018) Thermoelectrics are important in physics, engineering, and material science due to their useful applications and inherent theoretical difficulty. Recent experimental interest has shifted to strongly correlated materials, where the calculations become particularly difficult. Here we reexamine the framework for calculating the thermopower, inspired by ideas of Lord Kelvin from 1854. We find an approximate but concise expression, which we term as the Kelvin formula for the the Seebeck coefficient. According to this formula, the Seebeck coefficient is given as the particle number N derivative of the entropy S , at constant volume V and temperature T , S Kelvin = 1 q e { ∂ S $_{∂N}$}$_{V,T}$ . This formula is shown to be competitive compared to other approximations in various contexts including strongly correlated systems. We finally connect to a recent thermopower calculation for non-Abelian fractional quantum Hall states, where we point out that the Kelvin formula is exact . PACS numbers: 72.15.Jf, 71.27.+a, 73.43.Cd A complete understanding of thermoelectric effects is important in the physical sciences where wide rang- ing applications utilize materials with large thermoelec- tric power S (Seebeck coefficient). Thermoelectrics of strongly correlated materials are of fundamental interest since they present an important and challenging prob- lem. Recent experiments have revealed that some ma- terials, such as sodium cobalt oxide Na$_{x}$CoO$_{2}$ (NCO), possess unusually large thermopower $^{1}$, due in part to strong electron interactions $^{2}$. Frustrated systems $^{3}$, like NCO, might produce further surprises in enhanced ther- mopower in some situations $^{2,4}$. In addition, emerging work 5 from the fractional quantum Hall effect (FQHE) is revitalizing thermopower as a tool to investigative the topological non-Abelian quasiparticles 6 thought to exist at filling factor 5/2 $^{7}$. S$_{Kelvin}$ is obtained by completing Shastry’s argument 11 for the “absolute thermopower”, i.e. S of an isolated sys- tem. Kelvin originally studied 8 this object using the then available techniques, later he and others emphasized rela- tive thermopower between two materials. Let us revert to the absolute thermopower as a starting point and imag- ine a long isolated cylinder of material of length GLYPH<suppress>L subject to a time dependent electric field −∇ Φ and temperature gradient ∇ T . −∇ Φ couples to the dipole moment and ∇ T couples to the moment of the energy density (cf. Luttinger $^{13}$). These fields individually generate a dipole moment linear in the fields to lowest order, and the con- dition for the cancellation of the two contributions, i.e., the zero dipole moment (or zero current) condition, leads to the thermopower S for a finite system size GLYPH<suppress>L at finite frequencies ω as S (GLYPH<suppress>L , ω ) = ∇ Φ ∇ T { GLYPH<suppress>L , ω } . Here we present the Kelvin formula for thermopower, S$_{Kelvin}$ . This is a formula inspired by Lord Kelvin’s ther- modynamic treatment of this variable in 1854 $^{8}$. It is found by reconsidering the sequence of taking the ther- modynamic and uniform limits, and is a valuable approx- imation to the exact, but computationally intractable re- sult, obtained via Onsager and Kubo’s treatments $^{9,10}$. The thermodynamic limit, GLYPH<suppress>L → ∞ , and the static limit, ω → 0, must both be taken, as known from On- sager 9 and others $^{13,14}$. Kubo’s exact formulas obtain in the fast or transport limit , where GLYPH<suppress>L → ∞ before ω → 0. Taking the static limit ω → 0 before GLYPH<suppress>L → ∞ leads to the slow , where Kelvin’s approximate formula arises and is expressible solely in terms of equilibrium thermodynamic variables. We transcribe this discussion to a more convenient periodic system, by trading the length scale GLYPH<suppress>L for a wave vector q$_{x}$ = 2 π/ GLYPH<suppress>L and the GLYPH<suppress>L → ∞ limit by the uniform limit q$_{x}$ → 0. The slow limit corresponds to lim { q$_{x}$ → 0 , ω → 0 } , and the fast limit corresponds to lim { ω → 0 , q$_{x}$ → 0 } . The thermopower measures the in- duced thermoelectric voltage due to a temperature gra- dient and, as such, a useful and general formula for ther- mopower is given by the ratio between the thermoelec- For strongly correlated systems, such as the t - J model, S$_{Kelvin}$ is found to possess an accuracy between the rather coarse Mott-Heikes formulation, and a better argued high frequency limit formulation due to Shastry 4 and studied in Refs. 2 and 11. For intermediate couplings, such as the Hubbard model, we argue that S$_{Kelvin}$ provides one of the best available approximations, it is better than the high frequency limit. In certain dissipation-less situa- tions, such as the FQHE, S$_{Kelvin}$ is exact, thereby provid- ing an elegant and simple derivation for the thermopower formula used in Ref. 5 (derived originally in Refs. 12.) I. INTRODUCTION