Enhanced Kondo Effect via Tuned Orbital Degeneracy in a Spin $1/2$ Artificial Atom

A strong Kondo effect is observed for a vertical quantum dot holding an odd number of electrons and spin $1/2$ when an orbital degeneracy is induced by magnetic field. The estimated Kondo temperature for this “doublet-doublet” degeneracy is similar to that for the singlet-triplet degeneracy with an even electron number, indicating that a total of fourfold spin and orbital degeneracy accounts for the enhancement of the Kondo temperature. The experimental observation is qualitatively reproduced by scaling calculations using an SU(4) model at the orbital degeneracy.

Figure 1

Color-scale plot of the linear conductance as a function of the gate voltage and magnetic field. Blue corresponds to zero conductance, white to $G=25\mu \mathrm{S}$, and red to $G=50\mu \mathrm{S}$. All the electrons occupy the first Landau level on the right-hand side of the dashed line (filling factor $\nu =2$). The inset shows a schematic diagram of the dot structure made from AlGaAs/InGaAs/AlGaAs double barrier tunnel diode.

Figure 2

(a) Fock-Darwin states calculated with $\mathrm{\hbar }{\omega }_0=1\mathrm{m}\mathrm{e}\mathrm{V}$. The circle denotes crossing between orbital state $(n,l)=(0,-1)$ (dotted line) and (0, 7) (thick solid line) where the Kondo effect for $N=15$ and 16 is studied in detail. (b) Schematic magnetic field dependence of the electrochemical potential for electron numbers from $N+1$ (odd) to $N+4$ (even) occupying two crossing orbitals. A skew toward the upper-left direction takes into account the effect of decreasing $\mathrm{\hbar }{\omega }_0$ with increasing $N$. (c) Detailed measurement conducted in region A in Fig. 1. The color scale is same as in Fig. 1. S, T, and D denote states with $S=0$ (singlet), $S=1$ (triplet), and $S=1/2$ (doublet).

Figure 3

(a) Temperature dependence of the differential conductance $dI/d{V}_{\mathrm{s}\mathrm{d}}$ versus $V_{\mathrm{s}\mathrm{d}}$ for the S-T and D-D Kondo effect from $T=60\mathrm{m}\mathrm{K}$ (thick solid line) to 1.5 K (dashed line). (b) Temperature dependence of the Kondo peak height normalized by the low temperature limit value. Dashed lines are the theoretical curves fitted to the data.

Figure 4

Color-scale plot of $dI/d{V}_{\mathrm{s}\mathrm{d}}$ in $B\mathrm{\text{−}}{V}_{\mathrm{s}\mathrm{d}}$ plane for (a) the $N=15$ D-D Kondo effect and (b) the $N=16$ S-T Kondo effect with $V_g$ fixed in the center of the respective Coulomb blockade gap. Blue corresponds to $G=10\mu \mathrm{S}$ and red to $G=35\mu \mathrm{S}$. (c) $V_{\mathrm{s}\mathrm{d}}$ values of the conductance peak or step obtained from (a) and (b) as a function of the magnetic field difference $\Delta B=B-B_0$, where $B_0$ is the degeneracy magnetic field. (d) Relative conductance measured from the degeneracy ($\Delta B=0$) at $V_{\mathrm{s}\mathrm{d}}=0$, for the D-D (solid line) and the S-T (dotted line) Kondo effect. (e) Calculated results of $T_K/T_K\left(0\right)$ as a function of $\Delta /T_K\left(0\right)$, on a log-log scale. $T_K\left(0\right)$ is the Kondo temperature at $\Delta =0$. For the S-T Kondo effect (triplet side), $T_K\left(\Delta \right)$ deviates from a power law for large $\Delta$ (see Ref. 10; we choose $J_1=J_2$ and $C_2=0$).