Magnetic field dependence of the spin-$\frac{1}{2}$ and spin-1 Kondo effects in a quantum dot

We study the magnetic field evolution of the spin-$\frac{1}{2}$ and spin-1 Kondo effects in a quantum dot formed from a single-walled carbon nanotube. In the spin-$\frac{1}{2}$ case, the energy of spin-resolved Kondo conductance peaks is proportional to magnetic field at high fields, contrary to recent reports. At lower fields, the energy falls below this linear dependence, in qualitative agreement with theoretical expectations. For even electron occupancy, we observe a zero-bias Kondo peak due to the degeneracy of the spin-1 triplet ground states. Tuning gate voltage within the same Coulomb diamond drives a transition to a singlet ground state. We also independently tune the energy difference between singlet and triplet states with a magnetic field. The Zeeman splitting thus measured confirms the value of the $g$ factor deduced from the high-field behavior of spin-$\frac{1}{2}$ Kondo.

Figure 1

(a) Differential conductance $G$ as a function of bias voltage $V_b$, showing zero-bias features for odd electron occupancy. Data to the right of the location marked “×” have been shifted to account for a random charging event. (b) Linear conductance as a function of voltage $V_g$ on a back gate, at $T\approx 317\mathrm{mK}$ (black line), $1.8\mathrm{K}$ (fine black line), and $6\mathrm{K}$ (gray line). (Insets) Zero-bias features in valleys ii and iii at the same temperatures.

Figure 2

(a) Kondo diamond at zero magnetic field. $T\approx 352\mathrm{mK}$. Black is low conductance and white high. (b) The same diamond at $4\mathrm{T}$. (c) Evolution of the features in the middle of the valley in (a) and (b) with magnetic field. (d) Slices of the data in (a) in $0.5\mathrm{T}$ steps. (Our data are 20 times denser.) Successive curves are offset downward by $0.02\times 2e^2∕h$. Peak locations for the central Kondo peaks were obtained by fitting the data from (c) with two Lorentzians plus a field-independent background (+ symbols). A slightly different background is assumed for a similar fit at intermediate field points (× symbols). Between 4 and $6\mathrm{T}$, the two versions of our fitting procedure produce nearly indistinguishable results [see also Figs. 3a, 3b, 3c, 3d].

Figure 3

(a) Peak positions in Fig. 2c obtained as described in Fig. 2d. Black (gray) dots mark results from the high (intermediate) field fit. (b) Energy difference between peaks from (a). The line is a fit to the high-field points. (c) Splitting from (a) normalized by the naïve Zeeman energy—$\delta ∕2{\mu }_{B}B$ (black and gray dots). The gray line is the Moore-Wen prediction, the dashed line is the Costi prediction, the solid black line is the predicted high-field limit, and the gray dot marks the Logan-Dickens low-field prediction, all for $g=2.07$. (d) The data from (b) are reproduced (black and gray dots). The gray line is the Moore-Wen prediction and the dashed line is the Costi prediction, now using unrealistically large Zeeman coupling $g=2.9$ and 2.35, respectively, to attempt to match the data.

Figure 4

(a) Zero-bias feature in an even Coulomb diamond (parity determined by careful study of ten diamonds on either side). $T\lesssim 250\mathrm{mK}$. (b) Higher resolution scan of the region in (a) bounded by the rectangular box. The data indicate a gate-induced transition between singlet and triplet ground states for the dot. (c) Magnetic field dependence of the conductance at the point marked by the white triangle in (a).

Figure 5

[(a) and (b)] Magnetic field evolution of conductance versus bias voltage on the left (triplet) and right (singlet) sides of the diamond in Fig. 4a. The gate voltages chosen have been marked with dashed lines in Fig. 4a. [(c) and (d)] Schematic of singlet $\left(s\right)$ and triplet $\left(t\right)$ energy levels as a function of magnetic field in the configurations corresponding to (a) and (b), respectively. Double-headed arrows indicate transitions seen in our data. (e) Peak positions in (a) obtained by fitting two Lorentzians, corresponding to the transition shown in (c). The lines are $\pm g{\mu }_{B}B$, with $g=2.07$. (f) Analysis of data in (b). The black circles are peaks obtained using a simple peak-finding function. The gray triangles are the locations of steepest slope for the step. The lines are guides to the eye and have slope $\pm g{\mu }_{B}B$, with $g=2.07$, and mark the transitions shown in (d).