Local temperature of an interacting quantum system far from equilibrium

A theory of local temperature measurement of an interacting quantum electron system far from equilibrium via a floating thermoelectric probe is developed. It is shown that the local temperature so defined is consistent with the zeroth, first, second, and third laws of thermodynamics, provided the probe-system coupling is weak and broadband. For non-broadband probes, the local temperature obeys the Clausius form of the second law and the third law exactly, but there are corrections to the zeroth and first laws that are higher order in the Sommerfeld expansion. The corrections to the zeroth and first laws are related, and can be interpreted in terms of the error of a nonideal temperature measurement. These results also hold for systems at negative absolute temperature.

Figure 1

Measurements of a floating thermoelectric probe scanned 3.5 Å above the plane of carbon nuclei in a single-molecule junction containing an anthracene molecule. The electronic structure of the molecule is illustrated in the topmost panel, which shows $\text{Tr}\left\{{\mathrm{\Gamma }}^p\left({\mu }_0\right)\right\}$ as a function of the probe's horizontal position. The local temperature ${\overline{T}}_p$ and voltage ${\overline{V}}_p\equiv -{\overline{\mu }}_p/e$ are shown in the left and right panels of the middle row, respectively. The probe is modeled as an atomically sharp Au tip and ${\mathrm{\Gamma }}^p\left(\omega \right)$ was taken as a constant evaluated at the Au Fermi energy (broadband limit). The thermoelectric bias of the junction is applied by two electrodes covalently bonded to the molecule at the points labeled by the blue squares (electrode 1) and red squares (electrode 2), with $T_1=100\text{K}$, $T_2=300\text{K}$, and ${\mu }_2-{\mu }_1=0.2\text{eV}$. The local energy distribution of the system $f_s\left(\omega \right)$ and the Fermi-Dirac distribution of the probe ${\overline{f}}_p\left(\omega \right)$ are shown in the lower two panels for two different probe positions, corresponding to a cold spot and a hot spot, respectively (indicated by circles in the top panels). The zeroth and first moments of the probe's and system's local energy distributions are equal, as described by Eqs. (20, 21) and (35)–(36).