Thermal Conductance of a Single-Electron Transistor

We report on combined measurements of heat and charge transport through a single-electron transistor. The device acts as a heat switch actuated by the voltage applied on the gate. The Wiedemann-Franz law for the ratio of heat and charge conductances is found to be systematically violated away from the charge degeneracy points. The observed deviation agrees well with the theoretical expectation. With a large temperature drop between the source and drain, the heat current away from degeneracy deviates from the standard quadratic dependence in the two temperatures.

Figure 1

A single-electron transistor and the setup for the heat transport measurement. (a) False-colored SEM image of the full device. The circuit in red indicates the charge transport setup, while the black one stands for the heat transport setup. (b) Schematic of the device, with the different elements shown in colors. (c) Enlarged view of the central part of the SET. (d) Differential conductance map of the sample $A$ SET at 50 mK against drain-source voltage $V_{\mathrm{SET}}$ and induced charge $n_g$.

Figure 2

Left: Variation of electronic temperature $T_e$ of the sample $B$ source with cooler bias voltage, at gate open ($n_g=0.5$) and gate closed ($n_g=0$) states, at a bath temperature $T_b$ of 152 mK. The full line is a fit of the gate-open state data; see the text. Right: Temperature modulation by the gate voltage expressed in terms of induced charge $n_g$ in the heating regime (top) and in the cooling regime (bottom) at cooler bias points indicated by the blue and red arrows in the left plot.

Figure 3

Top: Thermal (blue dots) and charge (green dots) conductances of the SET at a bath temperature of 132 (left, sample $A$) and 152 mK (right, sample $B$) in units of the conductances in the gate-open state ${\kappa }_0$ and ${\sigma }_0$. The thermal flow through the SET was calculated assuming that the Wiedemann-Franz law is fulfilled at the gate-open state. The charge transport was measured at a bias of 22.4 (sample $A$) and $19.2\mu \mathrm{V}$ (sample $B$). The heat transport data were acquired by cooling the source electronic bath by 30 (sample $A$) and 22 mK (sample $B$) below the bath temperature. Bottom: Lorenz ratio (purple dots) defined as $L/L_0$, where $L=\kappa /(\sigma T_m)$ for sample $A$ (left) and sample $B$ (right). The error bars are related to the uncertainty in the temperature measurement. The Wiedemann-Franz law sets $L=L_0$. The red line is the theoretical prediction based on Ref. [19].

Figure 4

Nonlinear heat flow through the sample $B$ SET at different gate states as a function of the difference of the squared temperatures between the source and the bath (symbols) together with power-law fits (full lines). The slopes are 1.00, 1.10, and 1.14, respectively, at gate positions $n_g=0.5$, 0.3, and 0. The unit slope expected for the linear regime of heat transport is shown as dotted gray lines.