Heat Transistor: Demonstration of Gate-Controlled Electronic Refrigeration

We present experiments on a superconductor–normal-metal electron refrigerator in a regime where single-electron charging effects are significant. The system functions as a heat transistor; i.e., the heat flux out from the normal-metal island can be controlled with a gate voltage. A theoretical model developed within the framework of single-electron tunneling provides a full quantitative agreement with the experiment. This work serves as the first experimental observation of Coulombic control of heat transfer and, in particular, of refrigeration in a mesoscopic system.

Figure 1

(a) Scanning electron micrograph of the heat transistor, showing also the measurement setup. In the middle, the normal-metal (Cu) island subjected to cooling. Four superconducting probes (Al) with Al-Ox tunnel junctions can be seen above, and truncated capacitor plates (Al) with direct NS contacts on the left and right. (b) Theoretical heat flow from the N island at the optimal bias point at different temperatures as a function of the gate voltage. $T$ denotes the common temperature of the island electrons and quasiparticles in the superconductor, and $T_c=\Delta /(1.764k_B)$ is the superconducting critical temperature.

Figure 2

Measured probe voltage as a function of the drain-source bias voltage for (a) the reference sample with negligibly small charging energy and (b) the heat transistor at different bath temperatures. Inset: gate voltage [gray (red) line] during one period of $V_{\mathrm{DS}}$ sweep (black line). Electron charge corresponds to a gate voltage of 1.6 mV. Panel (b) thus illustrates the range of gate modulation at different bias voltages, whereas there was no observable gate modulation in the reference sample. Superimposed on panel (b) are theoretical curves for the two extremal gate positions. The solid blue curves are calculated assuming an electron-phonon heat load given by Eq. (4) with the experimentally determined value $\Sigma =2.3\times {10}^9\mathrm{W}{\mathrm{K}}^{-5}{\mathrm{m}}^{-3}$, and the dashed red curves given for reference show the expected behavior where the electron temperature is fixed at the bath temperature ($T_{\mathrm{bath}}$) over the full bias range.

Figure 3

(a) Electron temperatures extracted from the data acquired at bath temperature 214 mK with the gate open ($n_g=1/2$, blue squares) and closed ($n_g=0$, red triangles). (b) The theoretical cooling power versus $T_{\mathrm{N}}^5-T_0^5$ for each data point of the left panel, and a linear fit to it. According to Eq. (4), the slope is given by $-\Sigma \mathcal{V}$.