Kondo temperature evaluated from linear conductance in magnetic fields

We theoretically and experimentally study the universal scaling property of the spin-$\frac{1}{2}$ Kondo state in the magnetic field dependence of bias-voltage linear conductance through a quantum dot at low temperatures. We discuss an efficient and reliable procedure to evaluate the Kondo temperature defined at the ground state from experimental or numerical data sets of the magnetic field dependence of the linear conductance or the magnetization of the quantum dot. This procedure is helpful for quantitative comparison of the theory and the experiment, and useful in Kondo-correlated systems where temperature control over a wide range is difficult, such as for cold atoms. We demonstrate its application to experimentally measure electric current through a carbon nanotube quantum dot.

Figure 1

Magnetization $\langle m_{\mathrm{d}}^{}\rangle$ at particle-hole symmetric point (${\epsilon }_{\mathrm{d}}^{}=-U/2$) as a function of magnetic field $B$ at absolute zero for (a) small $U/\left(\pi \mathrm{\Gamma }\right)(\le 1)$ and (b) large $U/\left(\pi \mathrm{\Gamma }\right)(\le 1.5)$. The resulting linear conductance $G$ for (c) small $U/\left(\pi \mathrm{\Gamma }\right)(\le 1)$ and (d) large $U/\left(\pi \mathrm{\Gamma }\right)(\le 1.5)$. The broken lines and the data points are calculated by the NRG approach. The thin (blue) solid lines are the conductance at absolute zero for the noninteracting limit $U=0$, which are calculated by the exact solution (ES). The thick solid (black) lines are the conductance for the Kondo limit ($U\gg \mathrm{\Gamma }$), which are calculated by the Bethe ansatz exact solution. In this figure, we use the Kondo temperature $T_{\mathrm{K}}^{}$ of the local Fermi liquid theory, given by Eq. (8) at $B=0$. The arrow in (b) and (d) indicates the magnetic field at which the conductance and the magnetization takes half of the zero field value.

Figure 2

The linear conductance calculated by the empirical formula for the magnetic field dependence, given in Eq. (19) (the solid line), and comparison with that of the exact solution of the Kondo model (dashed line for ${\mu }_{\mathrm{B}}^{}B<T_K$ and dotted line for ${\mu }_{\mathrm{B}}^{}B>T_K$). The parameter $t$ of the empirical formula is determined to fit the universal part (${\mu }_{\mathrm{B}}^{}B<T_{\mathrm{K}}^{}$) given by the exact solution. The dashed-dotted line and the dashed double-dotted line are asymptotes given by Eq. (14) for ${\mu }_{\mathrm{B}}^{}B\ll T_{\mathrm{K}}^{}$ and Eq. (15) for ${\mu }_{\mathrm{B}}^{}B\gg T_{\mathrm{K}}^{}$, respectively.

Figure 3

Experimentally measured linear conductance $G$ as a function of gate voltage $V_{\mathrm{g}}^{}$ (a) at 16 mK with magnetic fields from 0.08 T to 4.08 T in 0.25 T steps, and (b) at 0.08 T and for several temperatures (16, 80, 180, 280, 380, 480, 580, 630, 680, 730, and 780 mK). The left conductance ridge around $V_{\mathrm{g}}^{}\sim 21.1$ V and right one around $V_{\mathrm{g}}^{}\sim 22.6$ V are labeled as Ridges A and B, respectively. (c and d) Evaluated Kondo temperature $T_{\mathrm{K}}^{}$ for Ridges A and B, respectively. The Kondo temperature is evaluated by using Eq. (16) for the magnetic field dependence of the conductance ($•$) and the empirical formula given in Eq. (1) ($▴$) with $\eta \left(U\right)=1.1$ for the temperature dependence of the conductance. The solid and dashed lines are the fitting curves given by Eq. (B1).

Figure 4

The linear conductance as a function of the applied magnetic field scaled by the Kondo temperature for several choices of gate voltage $V_{\mathrm{g}}^{}$ corresponding to the Kondo regime. (a) Ridge A with an asymmetric lead-dot connection and (b) Ridge B with a symmetric lead-dot connection. The magnetic field is scaled by the Kondo temperature given in Fig. 3. The solid line is the conductance calculated by the Bethe ansatz exact solution for the Kondo model, which is normalized by the value at zero fields $B=0$, $G\left(0\right)=G_0^{}$. The data points are the experimentally observed conductance (a) for $V_{\mathrm{g}}^{}$ from 20.98 V to 21.18 V in 0.02 V steps in Ridge A and (b) for $V_{\mathrm{g}}^{}$ from 22.48 V to 22.68 V in 0.02 V steps in Ridge B. The experimental conductance is normalized by its value at the lowest field, 0.08T, instead of its zero field value. The values of the Kondo temperature, determined by the magnetic field dependence for each gate voltage, are used. The dashed lines are the formula for the low fields, given by Eq. (14).