Thermal drag in electronic conductors

TL;DR

This paper investigates thermal drag effects in Coulomb-coupled electronic systems, revealing how heat currents and trans-resistivity behave under various conditions using analytical and numerical methods.

Contribution

It provides new analytical expressions for thermal drag and trans-resistivity in Coulomb-coupled conductors, including metallic islands and quantum wires, across different regimes.

Findings

Heat current is quadratic in bias or temperature difference.

Thermal trans-resistivity scales with temperature as T or 1/T depending on regime.

Replacing an electrode with a superconductor enables heat extraction from the normal electrode.

Abstract

We study the electronic thermal drag in two different Coulomb-coupled systems, the first one composed of two Coulomb blockaded metallic islands and the second one consisting of two parallel quantum wires. The two conductors of each system are electrically isolated and placed in the two circuits (the drive and the drag) of a four-electrode setup. The systems are biased, either by a temperature $\Delta T$ or a voltage $V$ difference, on the drive circuit, while no biases are present on the drag circuit. In the case of a pair of metallic islands we use a master equation approach to determine the general properties of the dragged heat current $I^{\rm (h)}_{\rm drag}$, accounting also for co-tunneling contributions and the presence of large biases. Analytic results are obtained in the sequential tunneling regime for small biases, finding, in particular, that $I^{\rm (h)}_{\rm drag}$ is quadratic in $\Delta T$ or $V$ and non-monotonous as a function of the inter-island coupling. Finally, by replacing one of the electrodes in the drag circuit with a superconductor, we find that heat can be extracted from the other normal electrode. In the case of the two interacting quantum wires, using the Luttinger liquid theory and the bosonization technique, we derive an analytic expression for the thermal trans-resistivity $\rho^{\rm (h)}_{12}$, in the weak-coupling limit and at low temperatures. $\rho^{\rm (h)}_{12}$ turns out to be proportional to the electric trans-resistivity $\rho^{\rm (c)}_{12}$, in such a way that their ratio (a kind of Wiedemann-Franz law) is proportional to $T^3$. We find that the thermal trans-resistivity is proportional to $T$ for low temperatures and decreases like $1/T$ for intermediate temperatures or like $1/T^3$ for high temperatures. We complete our analyses by performing numerical simulations that confirm the above results and allow to access the strong coupling regime.