Single-Quantum-Dot Heat Valve

We demonstrate gate control of electronic heat flow in a thermally biased single-quantum-dot junction. Electron temperature maps taken in the immediate vicinity of the junction, as a function of the gate and bias voltages applied to the device, reveal clearly defined Coulomb diamond patterns that indicate a maximum heat transfer at the charge degeneracy point. The nontrivial bias and gate dependence of this heat valve results from the quantum nature of the dot at the heart of device and its strong coupling to leads.

Figure 1

A single-quantum-dot transistor integrating a local thermometer and a heater for heat transport measurement. (a) False-colored SEM image of a typical device. The source is colored in red, the drain in green, and the superconducting leads in blue. The circuit diagram shows the heat transport setup. The longer ($2.5\mu \mathrm{m}$) SNS junction is used as a heater driven by a constant d.c. battery and the shorter (700 nm) SNS junction is used as a thermometer. (b) Enlarged view of the nanogap between the “source” and “drain” created by electromigration and the nanoparticles made by Au evaporation. (c) Schematic of the device, with the different heat flows to and from the source. (d) Differential conductance map of the device measured at 70 mK against the drain-source bias voltage $V_b$ and the gate voltage $V_g$ with no additional heating applied.

Figure 2

Characterization of the SNS thermometer. (a) d.c. IV characteristics of the SNS thermometer junction at different bath temperature $T_b$, the current bias value at which the voltage exceeds a threshold $V_0\simeq 0.5\mu \mathrm{V}$ defining the switching current. (b) The critical current $I_C$ as a function of the bath temperature, the axes being normalized. It is defined as the most probable switching current extracted from the histograms. The calibration curve (red solid line) is a fit with the theory [45]. (c) Histogram of the stochastic switching current of the SNS junction at different bath temperatures with a fitted Gaussian envelope for each.

Figure 3

Competition between thermal transport and dissipation across the QD junction. (a) Experimental map of the source electronic temperature in the $V_b-V_g$ plane. (b) Individual gate traces of the source temperature at two different bias values. (c) Schematic energy diagram of the heat flows in and out of the source in various conditions as indicated by labels in (b): (1) away from charge degeneracy and at zero bias (left), (2) at a charge degeneracy point $V_g=V_g^0$ but still at zero bias (middle), or (3) at nonzero bias in a conducting region (right). The gray profile depicts the quantum level spectral density. The ratio between the level broadening $\hslash \mathrm{\Gamma }$, the bias $V_b$, and the thermal energy $k_{B}T$ are in correspondence with panel (b) conditions. The arrows indicate the applied heating power ${\dot{Q}}_H$, the Joule power ${\dot{Q}}_J$, the electron-phonon coupling power ${\dot{Q}}_{\mathrm{e}\text{−}\mathrm{ph}}$ and the power flow through the QD ${\dot{Q}}_D$.

Figure 4

Quantum dot heat valve. (a) A highly resolved map of the source electronic temperature at the same experimental condition as in Fig. 3 and around a charge degeneracy point defined by $V_g=V_g^0$. (b) Calculated temperature map obtained with the inbedding technique with $\mathrm{\Gamma }=0.25\mathrm{meV}$, ${\mathrm{\Gamma }}_L/{\mathrm{\Gamma }}_R=3/17$, and $T_d=85\mathrm{mK}$. (c) Experimental and (d) theoretical variation of the temperature in the region where crossing from cooling to heating is observed; each curve refers to a given applied bias $V_b$: (blue) $20\mu \mathrm{V}$, (orange) $22\mu \mathrm{V}$, (red) $24\mu \mathrm{V}$. (e) Schematics describing the crossover between the heat flow ${\dot{Q}}_D$ and the Joule heat ${\dot{Q}}_J$ as a function of the gate at a fixed bias, resulting in temperature decrease at $V_g-V_g^0=-0.12\mathrm{mV}$ (case 1, left) or increase at 0.46 mV (3, right). At 0.16 mV (2, middle), the two flows are equilibrated. The electron-phonon heat ${\dot{Q}}_{\mathrm{e}\text{−}\mathrm{ph}}$ as well as the injected heat ${\dot{Q}}_H$ are omitted for clarity. The widths of the arrows indicate their relative strengths.