Quantum Thermal Conductance of Electrons in a One-Dimensional Wire

We use an electron thermometer to measure the temperature rise of approximately $2\times {10}^5$ electrons in a two-dimensional box, due to heat flow into the box through a ballistic one-dimensional (1D) constriction. Using a simple model we deduce the thermal conductance $\kappa (V_g)$ of the 1D constriction, which we compare to its electrical conductance characteristics; for the first four 1D subbands the heat carried by the electrons passing through the wire is proportional to its electrical conductance $G(V_g)$. In the vicinity of the 0.7 structure this proportionality breaks down, and a plateau at the quantum of thermal conductance ${\pi }^2{k}_B^{2}T/3h$ is observed.

Figure 1

Schematic of the device and setup for thermal conductance measurements. Electrons in the heating channel are heated to $T_H=T_L+\Delta T$ by a current $I_H$; hot electrons pass through the sample constriction ($A$) into the box, where the temperature $T_{\mathrm{box}}=T_L+\delta T$ is determined by the thermovoltage $V_{\mathrm{th}}^{\mathrm{box}}$ generated across the thermometer and reference constrictions ($B$ and $C$, respectively). Electrons can only pass through the constrictions labeled $A$, $B$, and $C$; the much narrower channels are pinched off.

Figure 2

The thermovoltage $V_{\mathrm{th}}^{\mathrm{box}}$ of the electron box together with the conductance $G_A$, measured at a lattice temperature $T_L=0.27\mathrm{K}$ using heating currents of (a) $I_H=0.2\mu \mathrm{A}$, (b) $0.3\mu \mathrm{A}$, (c) $0.5\mu \mathrm{A}$, and (d) $1\mu \mathrm{A}$. The corresponding temperature rise $\Delta T=15$, 30, 55, and 100 mK, respectively, were estimated using a source-drain calibration of the subbands. At $T_L=0.27\mathrm{K}$ the thermovoltage approximately scales with $I_H^2$ up to $I_H=1\mu \mathrm{A}$.

Figure 3

(a) The thermally derived conductance ${\tilde{G}}_A(V_g)$ and electrical conductance $G_A(V_g)$ of constriction $A$ at $T_L=0.27\mathrm{K}$. Left solid line: conductance $G_A(V_g)$. Measurements of ${\tilde{G}}_A(V_g)$ were obtained with $I_H=1\mu \mathrm{A}$, $G_B=1.5G_0$, and $G_C=5G_0$ [trace (i)], $G_C=4G_0$ [trace (ii)], and $G_C=3G_0$ [trace (iii)]. For clarity successive traces have been shifted by 0.3 V to the right. (b) Close-up of the thermal and electrical conductances near pinch off, showing that at the gate voltage corresponding to the 0.7 structure, ${\tilde{G}}_A$ exhibits a half plateau. Far left traces: $G_A(V_g)$ at $T_L=0.3\mathrm{K}$ (solid line) and $T_L=1.2\mathrm{K}$ (dashed line). Traces $A$–$E$: ${\tilde{G}}_A(V_g)$ at $T_L=0.27\mathrm{K}$ for heating currents $I_H=0.2$, 0.4, 1, 2, and $3\mu \mathrm{A}$; the corresponding $\Delta T$ are calculated to be 26, 66, 133, 193, and 234 mK. The ${\tilde{G}}_A(V_g)$ traces $A$ and $B$ were obtained with $G_B=1.5G_0$ and $G_C=3G_0$; traces $C$–$E$ were obtained with $G_B=1.5G_0$ and $G_C=2G_0$. The small spikes in traces $D$ and $E$ close to pinch off (marked with a $*$) only occur at high $I_H$, and are due to a capacitative coupling that can be avoided if dc measurements are performed.