Universal line shape of the Kondo zero-bias anomaly in a quantum dot

Encouraged by recent real-time renormalization group results, we carried out a detailed analysis of the nonequilibrium Kondo conductance observed in an InAs nanowire-based quantum dot and found them to be in excellent agreement. We show that in a wide range of biases the Kondo conductance zero-bias anomaly is scaled by the Kondo temperature to a universal line shape predicted by the numerical study. The line shape can be approximated by a phenomenological expression of a single argument $e{V}_{\mathrm{sd}}/k_{\mathrm{B}}{T}_{\mathrm{K}}$. The knowledge of an analytical expression for the line shape provides an alternative way to estimate the Kondo temperature in a real experiment, with no need for time-consuming temperature dependence measurements of the linear conductance.

Figure 1

(a) The temperature dependence of the linear conductance observed in the InAs nanowire-based quantum dot. The dotted trace was taken at the lowest temperature (10 mK), and the dashed trace was taken at the highest temperature (620 mK). (b) The two-dimensional grayscale plot of the nonequilibrium conductance measured in a $V_g-V_{\mathrm{sd}}$ plane at $T=T_{\mathrm{base}}$ for the same range of $V_g$ as in (a). The vertical dashed lines mark the conductance traces at fixed values of $V_g$ used for the scaling analysis ($V_g=-2.85$ V, star; $V_g=-2.845$ V, left-pointing triangle; $V_g=-2.84$ V, right-pointing triangle; $V_g=-2.835$ V, downward-pointing triangle; $V_g=-2.83$ V, upward-pointing triangle; $V_g=-2.825$ V, diamond; $V_g=-2.82$ V, square; $V_g=-2.815$ V, circle).

Figure 2

(a) Scaled temperature $G\left(T\right)$ dependence measured at different $V_g$ (symbols) [see Fig. 1b for the symbols code], approximated with Eq. (1) (solid curve). (b) The dependence of the estimated value of $T_{\mathrm{K}}$ on $V_g$. The error bars here represent the 68$\%$ confidence interval. (c) The normalized conductance $G/G_0$ as a function of $V_{\mathrm{sd}}$ measured at different $V_g$ and $T=T_{\mathrm{base}}$. (d) The same normalized conductance traces as in (c), but replotted as a function of $e{V}_{\mathrm{sd}}/k_B{T}_K$ (solid curves). The traces are shown to collapse onto the same universal dependence given by RTRG calculations (Ref. 9) (dashed curve).

Figure 3

(a) The RTRG calculations of the nonequilibrium Kondo conductance as a function of bias taken from Ref. 9 (circles) and its approximation with Eq. (2) (solid curve) and Eq. (3) (dashed curve). (b) Extraction of $T_{\text{K}}$ using two different measurements made at $V_g=-2.83$ V [see the upward-pointing triangle in Fig. 1b], $G\left(T\right)$ (squares) and $G\left(V_{\mathrm{sd}}\right)$ (circles) fitted with Eq. (1) (solid curve) and Eq. (3) (dashed curve), correspondingly. The two extracted values $T_{\mathrm{K}}^{\left(T\right)}$ and $T_{\mathrm{K}}^{\left(V\right)}$ are the same within the statistical error. (c) The experimental Kondo ZBA measured at $V_g=-2.83$ V (circles) and its theoretical approximation made with Eq. (3) (dashed curve). When the bias is such that $e|V_{\mathrm{sd}}|=k_{\mathrm{B}}{T}_{\mathrm{K}}$, the Kondo conductance decreases to 2/3 of its zero-bias value $G_0$.