Thermoelectric Characterization of the Kondo Resonance in Nanowire Quantum Dots

We experimentally verify hitherto untested theoretical predictions about the thermoelectric properties of Kondo correlated quantum dots (QDs). The specific conditions required for this study are obtained by using QDs epitaxially grown in nanowires, combined with a recently developed method for controlling and measuring temperature differences at the nanoscale. This makes it possible to obtain data of very high quality both below and above the Kondo temperature, and allows a quantitative comparison with theoretical predictions. Specifically, we verify that Kondo correlations can induce a polarity change of the thermoelectric current, which can be reversed either by increasing the temperature or by applying a magnetic field.

Figure 1

(a) SEM image of device QD3. An $\mathrm{InAs}/\mathrm{InP}$ nanowire containing a QD is contacted to metallic leads for electrical biasing with voltage $V$ (see the SM [27] for details on the circuitry). Additional heater leads (lighter gray) enable the application of a thermal bias $\mathrm{\Delta }T$ to the QD by running a current $I_H$ resulting from a heater bias $V_H=V_1-V_2$. Only one heater is used in the experiment. (Inset) Scanning transmission electron microscope enlargement with a high angle annular dark field image of an $\mathrm{InAs}/\mathrm{InP}$ nanowire from the same growth. (b) Sketch of an unbiased spin-$1/2$ QD tunnel coupled to two leads ($h$ and $c$).

Figure 2

(a) $g$ as a function of $V$ and $V_G$, measured at $T=T_0<100\mathrm{mK}$. (b) The corresponding $g_0$ as a function of $V_G$ measured at three different $T$. (c) $I_{\mathrm{th}}$ normalized by $\mathrm{\Delta }T$ as a function of $V_G$ measured at three different $T$. The horizontal $V_G$ axis is the same in (a)–(c). Vertical dashed lines refer to $V_G$ values in Fig. 3.

Figure 3

(a) Dots are the measured values of $g_0(T)$ (left panel) and of $g(V)$ at $T_0$ (right panel), both for four different values of $V_G$, also indicated by vertical dashed lines in Fig. 2 with the corresponding colors. The solid lines for $g_0(T)$ are fits to Eq. (1), whereas the solid lines for $g(V)$ are fits to Eq. (2) in the SM [27]. The curves and data points at different $V_G$ are offset by 0.5 $e^2/h$ in $g$ from each other. (b) $T_K$ as a function of $V_G$ determined from fitting the measured $g_0(T)$ to Eq. (1) (dark blue points) and the measured $g(V)$ to Eq. (2) in the SM [27] (light blue points). The error bars represents a 95% confidence interval for $T_K$ as a fit parameter. (Inset) The corresponding estimates of $\mathrm{\Gamma }$ as a function of $V_G$ using Eq. (2).

Figure 4

(a)–(d) Measured $I_{\mathrm{th}}$ normalized by $\mathrm{\Delta }T$ for different $T$ as indicated in the figure. The black arrows indicate $V_G$ positions around which the values of $\sigma$ in (f) are determined. The magnetic field is increased from $B=0$ to $B=2\mathrm{T}$ from (a)–(d) as indicated in the figures. The dashed lines in (a) indicate the $V_G$ range that corresponds to the Kondo regime. (e) Charge stability diagrams showing $g$ as a function of $V$ and $V_G$, measured at the base temperature $T_0<100\mathrm{mK}$ for values of $B$ corresponding to those used in (a)–(d). (f) Thermocurrent slope, $\sigma =(d{I}_{\mathrm{th}}/d{V}_G)/\mathrm{\Delta }T$, as a function of $T$ at $B$ field values corresponding to those used in (a)–(d).