Statistics of Temperature Fluctuations in an Electron System out of Equilibrium

We study the statistics of the fluctuating electron temperature in a metallic island coupled to reservoirs via resistive contacts and driven out of equilibrium by either a temperature or voltage difference between the reservoirs. The fluctuations of temperature are well defined provided that the energy relaxation rate inside the island exceeds the rate of energy exchange with the reservoirs. We quantify these fluctuations in the regime beyond the Gaussian approximation and elucidate their dependence on the nature of the electronic contacts.

Figure 1

Setup and limit considered in this Letter: A conducting island is coupled to reservoirs via electrical contacts characterized by the transmission eigenvalues $\{T_n^{\alpha }\}$, which, for example, yield the conductances $G_{\alpha }$ of the contacts. The temperature fluctuates on a time scale ${\tau }_E$ characteristic for the energy transport through the junctions. We assume the limit ${\tau }_{e-e}\ll {\tau }_E$ where the internal relaxation within the island is much faster than the energy exchange with reservoirs. In this limit both the temperature and its fluctuations are well defined.

Figure 2

Time line of a huge fluctuation. The measurement is made at $t\approx {\tau }_0$ with the result $T_e=2.5T_a$, and thereby the time lines here are conditioned to give this (very unprobable) outcome at $t={\tau }_0$. For $t<{\tau }_0$, the temperature follows the “antirelaxation” branch $A$, whereas after the measurement, it relaxes as predicted by a “classical” equation in branch $B$. (a) and (b) show the time dependence of the fluctuation for $T_I(t)$ and ${\xi }_I(t)$, respectively. The heat current into the island corresponding to this fluctuation is plotted in (c), and (d) shows the charge current flowing through the island. The statistical fluctuations of these curves are small (with amplitude $\sim \sqrt{T_a{\delta }_I/k_B}$) on the plot scale $\sim T_a$.

Figure 3

Logarithm of temperature fluctuation statistics probability $P(T_e)$ in a few example cases. Solid lines from top to bottom: temperature bias with symmetric tunneling contacts, $T_a=T_1/\sqrt{2}$, $T_2=0$ (magenta); Gaussian equilibrium fluctuations (black), nonequilibrium fluctuations with $T_a=\sqrt{3}|eV|/(2\pi k_B)$, $T_1=T_2=0$ for symmetric tunneling and ballistic contacts (blue and red lines, respectively). The dashed lines are Gaussian fits to small fluctuations $(T_e-T_a)\ll T_a$, described by the heat current noise $S_Q$ at $T_e\approx T_a$.