Spin-$\frac{1}{2}$ Kondo effect in an InAs nanowire quantum dot: Unitary limit, conductance scaling, and Zeeman splitting

We report on a comprehensive study of spin-$\frac{1}{2}$ Kondo effect in a strongly coupled quantum dot realized in a high-quality InAs nanowire. The nanowire quantum dot is relatively symmetrically coupled to its two leads, so the Kondo effect reaches the unitary limit. The measured Kondo conductance demonstrates scaling with temperature, Zeeman magnetic field, and out-of-equilibrium bias. The suppression of the Kondo conductance with magnetic field is much stronger than would be expected based on a $g$-factor extracted from Zeeman splitting of the Kondo peak. This may be related to strong spin-orbit coupling in InAs.

Figure 1

(a) SEM image of a typical suspended nanowire-based quantum dot device used in the experiment. The scale bar corresponds to 1 $\mu$m. (b) Schematic representation of the nanowire-based quantum dot device and its experimental setup. (c) The temperature dependence of the nanowire-based quantum dot conductance measured over a wide range of the backgate voltage $V_g$. Five Kondo valleys are labeled I through V here. This identification of valleys will be used throughout the paper. Discontinuities in the temperature dependence in valley II are caused by device instability at this particular range of $V_g$. (d) The gray-scale conductance plot in the $V_g\mathrm{-}{V}_{\text{sd}}$ plane measured in the same range of $V_g$ as in (c) at temperature $T_{\text{base}}=10$ mK. Panels (a) and (b) are adapted with permission from A. V. Kretinin et al., Nano Lett. 10, 3439 (2010). Copyright $\copyright$ 2011 American Chemical Society.

Figure 2

The Kondo effect in its unitary limit. The main plot shows the linear conductance $G$ in valley III, as a function of backgate voltage $V_g$ at different temperatures. The dark blue curve corresponds to the lowest temperature of 10 mK. Inset: the red triangles correspond to the temperature dependence of the conductance at a fixed $V_g=-3.107$ V (marked by the red triangle in the main graph). The blue curve represents the result of approximation with Eq. (4) where $G_0=1.98e^2/h$ and $T_{\mathrm{K}}=1.65$ K.

Figure 3

(a) The detailed measurement of the conductance temperature dependence shown in Fig. 1c, valleys IV and V. The red triangles mark two values of $V_g=-2.835$ and $-$2.680 V for which the conductance as a function of $V_{\mathrm{sd}}$ is plotted in Figs. 4a and 4b, respectively. (b) The gray-scale conductance plot in the $V_g\mathrm{-}{V}_{\mathrm{sd}}$ plane was measured in the same range of $V_g$ as in (a), at temperature $T=10$ mK.

Figure 4

Nonlinear conductance as a function of $V_{\mathrm{sd}}$ around zero bias for different temperatures at $V_g=-2.835$ V (a) and $V_g=-2.680$ V (b), near the centers of Kondo valleys IV and V. The color scale is as in Fig. 3a. (c) and (d) The Kondo temperature $T_{\mathrm{K}}$, plotted on a semi-log scale, as a function of $V_g$ for these same valleys. Panel (c) corresponds to valley IV and panel (d) to valley V. Blue curves in both panels show fits of Eq. (5) to data, with ${\Gamma }_{IV}\approx 176$ $\mu \text{eV}$ for valley IV and ${\Gamma }_V\approx 435$ $\mu \text{eV}$ for valley V.

Figure 5

The Zeeman splitting of the Kondo ZBA measured at $T=10$ mK. (a) The gray-scale conductance plot of Kondo valley IV [see Fig. 3a] measured at $B=0$. (b) The same as in (a) but at $B=100$ mT. (c) Gray-scale conductance plot in the $V_{\mathrm{sd}}\mathrm{-}B$ plane measured at fixed $V_g=-2.835$ V denoted by the cross in panel (a). The red dashed lines represent the result of the fitting with expression $V_{\mathrm{sd}}=\pm \left|g\right|{\mu }_{\mathrm{B}}B/e$, where $\left|g\right|=$ 7.5 $\pm$ 0.2. Vertical blue dashed line marks magnetic field value 0.5$k_B{T}_{\mathrm{K}}/\left|g\right|{\mu }_{\mathrm{B}}$ as a reference for the onset of Zeeman splitting (here $T_{\mathrm{K}}=300$ mK). While $\left|g\right|=7.5$ gives the best match to linear Zeeman splitting, $\left|g\right|=18$ (green dotted lines) could account for the fact that Zeeman splitting is resolved at very low field. (d) Conductance at $V_{\mathrm{sd}}=0$ as a function of $T$ (blue squares) and as a function of the effective temperature $T_B\equiv \left|g\right|{\mu }_{\mathrm{B}}B/k_{\mathrm{B}}$ (red triangles). The solid blue curve shows $G\left(T\right)$ from NRG, the solid red curve $G\left(B\right)$ from NRG, and the dashed black curve $G\left(B\right)$ from exact Bethe ansatz (BA) calculations for the Kondo model.[44, 45] These assume $\left|g\right|=7.5$. For NRG and BA calculations of magnetic field dependence, additional curves (solid green and dashed brown) are plotted for $\left|g\right|=18$, showing better match to linear conductance data—though not to the differential conductance in (c) above.

Figure 6

(a) and (b) The equilibrium conductance of Kondo valleys IV (a) and V (b) at different $V_g$, scaled as a function of a single argument $T/T_{\mathrm{K}}$ (blue squares) and $T_B/T_{\mathrm{K}}$ (red triangles), where $T_B\equiv \left|g\right|{\mu }_{\mathrm{B}}B/k_{\mathrm{B}}$. The dashed curve shows the universal function described by Eq. (4). The dotted line represents the low-energy limit of Eq. (1) with $c_A=c_T=5.38$. The dash-dotted line shows the theoretically predicted low-field scaling of $G\left(B\right)$ with $c_B=0.55$. The values of $G_0$ and $T_{\mathrm{K}}$ were found by fitting the data with Eq. (4), see Sec. 3b1. For values of $V_g$ refer to Figs. 4c, 4d, and 5d. (c) and (d) The scaled conductance $\Delta G/\tilde{\alpha }=[1-G(T,V_{\mathrm{sd}})/G(T,0)]/\tilde{\alpha }$, where $\tilde{\alpha }=c_T\alpha /[1+c_T(\gamma /\alpha -1)]{(T/T_{\mathrm{K}})}^2$, versus ${(e{V}_{\mathrm{sd}}/k_{\mathrm{B}}{T}_{\mathrm{K}})}^2$ taken at several $V_g$ along Kondo valleys IV (c) and V (d). For valley IV, the backgate voltage was chosen from the range $V_g=-2.82$ to $-2.85$ V with 5 mV step and for valley V from the range $V_g=-2.6$8 to $-2.72$ V with 20 mV step. Different colors of the data points represent different temperatures (9.5, 12.9, 22.4, 32.6, 46.1, and 54.2 mK). The dashed line shows the corresponding scaling function given by Eq. (7) with $\alpha =0.18$ and $\gamma =1.65$.

Figure 7

(a) The nonequilibrium Kondo conductance as a function of $V_{\mathrm{sd}}$ for several values of $B$ (open blue squares). The solid red curves represent the approximation of the data made with the sum of two Fano-shaped peaks and a cubic background. (b) The normalized Zeeman splitting $\Delta /\left[2\right|g\left|{\mu }_{\mathrm{B}}B\right]$ as a function of $B$ data acquired from the peak maximum search (blue squares) and after fitting with two asymmetric peak shapes (red triangles). The vertical blue and green dashed lines denote magnetic field of 0.5$k_{\mathrm{B}}{T}_{\mathrm{K}}/\left|g\right|{\mu }_{\mathrm{B}}$ and $k_{\mathrm{B}}{T}_{\mathrm{K}}/\left|g\right|{\mu }_{\mathrm{B}}$ correspondingly (here $\left|g\right|=7.5$ and $T_{\mathrm{K}}=300$ mK).