Empirical test of the Kelvin relation in thermoelectric nanostructures

Abstract

Thermoelectric (TE) nanostructures with dimensions of ∼100 nm can show substantially better TE properties compared to the same material in the bulk form due to charge and heat transport effects specific to the nanometer scale. However, TE physics in nanostructures is still described using the Kelvin relation (KR) Π = αT, where Π is the Peltier coefficient, α the thermopower, and T the absolute temperature, even though derivation of the KR uses a local equilibrium assumption (LEA) applicable to macroscopic systems. It is unclear whether nanostructures with nanostructures with dimensions on the order of an inelastic mean free path satisfy a LEA under any nonzero temperature gradient. Here, we present an experimental test of the KR on a TE system consisting of doped silicon-based nanostructures with dimensions comparable to the phonon–phonon and electron–phonon mean-free-paths. Such nanostructures are small enough that true local thermodynamic equilibrium may not exist when a thermal gradient is applied. The KR is tested by measuring the ratio Π/α under various applied temperature differences and comparing it to the average T. Results show relative deviations from the KR of |(Π/α)/T − 1| ≤ 2.2%, within measurement uncertainty. This suggests that a complete local equilibrium among all degrees of freedom may be unnecessary for the KR to be valid but could be replaced by a weaker condition of local equilibrium among only charge carriers.