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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) P = aT, where P is the Peltier coefficient, a 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 P/a under various applied temperature differences and comparing it to the average T. Results show relative deviations from the KR of |(P/a)/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. 

Advancements in electronic device fabrication with increasing integration levels have resulted in very high device densities. This has led to higher power dissipation and heat fluxes, increasing integrated circuit (IC) operating temperature. High and nonuniform heat generation degrades device and system performance. Therefore, thermal management to keep ICs within prescribed temperature limits is an important challenge for reliable and economic performance. Cooling techniques, including liquid coolants and air conditioning (AC), have been utilized to remove heat at the package and system level. However, these techniques must overcome high thermal impedances and require complex integration, while global cooling is generally wasteful, inefficient, and expensive. To improve thermal management, we have developed Si microthermoelectric coolers (μTECs) with areas as small 1E−5 cm^2 that can be integrated on -chip near local hot spots using the standard fabrication processes. While Si μTECs cannot achieve low base temperatures, they can actively pump relatively high heat fluxes directly to a heat sink, thus reducing local temperature increases and allowing targeted rather than global waste heat removal. We demonstrate μTECs that can pump up to 43 W/cm^2 of locally generated excess heat with no increase in chip temperature. 

The Kelvin relation (KR) connecting the Peltier coefficient Π, the thermopower α, and the absolute temperature T via Π = αT is a cornerstone of thermoelectric (TE) physics. It is also a widely recognized example of an Onsager reciprocal relation, a foundational principle in nonequilibrium irreversible thermodynamics. While the KR is routinely invoked to understand TE systems, it has surprisingly little rigorous empirical verification. Accurate experimental tests of the KR are complicated by several factors, including non-Peltier heat flows such as Joule heating or Fourier thermal conduction, uncharacterized thermal contact impedances, and the need for Peltier and thermopower effects to be measured on the same thermopile at the same temperatures. Most empirical assessments of the KR have either made questionable simplifications or been limited in accuracy to several percent. Here, we present a test of the KR that is free of the difficulties of prior experiments and relies only on conventional voltage, current, and temperature measurements, so that it could be performed on any thermopile. Conducting the test on a Bi 2 Te 3 thermopile, the empirical ratio Π/α is found to equal T within a relative deviation < 0.5% for T in the range of 320–340 K. This result is quantitatively consistent with the KR and justifies the use of the KR in TE applications to reasonably high accuracy.