Local temperature of out-of-equilibrium quantum electron systems

We show how the local temperature of out-of-equilibrium, quantum electron systems can be consistently defined with the help of an external voltage and temperature probe. We determine sufficient conditions under which the temperature measured by the probe (i) is independent of details of the system-probe coupling, (ii) is equal to the temperature obtained from an independent current-noise measurement, (iii) satisfies the transitivity condition expressed by the zeroth law of thermodynamics, and (iv) is consistent with Carnot's theorem. This local temperature therefore characterizes the system in the common sense of equilibrium thermodynamics, but remains well defined even in out-of-equilibrium situations with no local equilibrium.

Figure 1

Schematic of a mesoscopic conductor connected to two electron reservoirs via transport leads, and to a third reservoir via a weakly coupled probe. The chemical potentials and temperatures in all three reservoirs are indicated.

Figure 2

Local temperature of an armchair graphene nanoribbon probed by an atomically sharp Pt tip scanned 3.5 Å above the graphene plane. Here, ${\mu }_0=-0.15\text{eV}$ from the Dirac point, and a thermal bias $\Delta T=50\text{K}$ is applied between hot and cold electrodes forming an open electrical circuit. (a) $T_{\mathrm{p}}$ calculated from Eq. (3); (b) $T_{\mathrm{p}}^{\left(\mathrm{noise}\right)}=T_{\mathrm{p}}(1+\chi )$ calculated from Eq. (8); (c) $T_{\mathrm{p}}$ and $T_{\mathrm{p}}^{\left(\mathrm{noise}\right)}$ at the three points indicated in (a) as functions of the tip-sample coupling $\text{Tr}\left({\Gamma }^p\right)$, which we vary with an artificial scaling factor multiplying the tunneling-width matrix ${\Gamma }^p$. At this scan height, the tip-sample coupling is mediated by two dominant transmission channels, with an intrinsic $\text{Tr}\left({\Gamma }^p\right)\in [3.6\mu \text{eV},24\text{meV}]$ over the image.

Figure 3

(Top) Two systems sequentially probed by the same probe at local positions ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$. (Bottom) Same systems as in the top panel now connected by a transmission line locally coupled to the systems at ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$.