Unraveling a concealed resonance by multiple Kondo transitions in a quantum dot

Kondo correlations are responsible for the emergence of a zero-bias peak in the low temperature differential conductance of Coulomb blockaded quantum dots. In the presence of a global $\mathrm{SU}\left(2\right)\otimes \mathrm{SU}$(2) symmetry, which can be realized in carbon nanotubes, they also inhibit inelastic transitions which preserve the Kramers pseudospins associated to the symmetry. We report on magnetotransport experiments on a Kondo correlated carbon nanotube where resonant features at the bias corresponding to the pseudospin-preserving transitions are observed. We attribute this effect to a simultaneous enhancement of pseudospin-nonpreserving transitions occurring at that bias. This process is boosted by asymmetric tunneling couplings of the two Kramers doublets to the leads and by asymmetries in the potential drops at the leads. Hence, the present work discloses a fundamental microscopic mechanisms ruling transport in Kondo systems far from equilibrium.

Figure 1

The $\mathrm{SU}\left(2\right)\otimes \mathrm{SU}$(2) Kondo effect. (a) A CNT longitudinal shell possesses two Kramers doublets separated by a splitting $\mathrm{\Delta }$. The upper (u) and lower (d) doublets are coupled to left ($L$) and right ($R$) leads by tunneling couplings ${\gamma }_{Lp}$ and ${\gamma }_{Rp}$, where $p=\mathrm{u},\mathrm{d}$. (b) Experimental differential conductance as a function of gate, $V_{\mathrm{g}}$, and bias, $V_{}$, voltages for different occupation of a longitudinal shell. In the valleys with odd occupancy a zero-bias Kondo ridge is clearly seen. Two additional ridges, symmetrically located with respect to the central Kondo peak, are observed at finite bias. The asymmetric response implies the presence of asymmetries in the pro- blem. (c) Bias traces taken at gate voltages corresponding to the center of the $N=1$ and $N=3$ valleys show inelastic peaks with roughly the same spacing $\mathrm{\Delta }$.

Figure 2

Behavior in perpendicular magnetic field. (a) Evolution of energy levels in magnetic field; to each level is associated a Kramers pseudospin. (b) Addition spectrum obtained from (a) with inelastic excitations ${\mathrm{\Delta }}_T,{\mathrm{\Delta }}_C$, and ${\mathrm{\Delta }}_P$ associated to $\mathcal{T}$, $C$, and $P$ transitions being indicated. The magnetic field is scaled by the characteristic fields $B_{K_{N1}}=1.425\mathrm{T}$ and $B_{K_{N3}}=1.036\mathrm{T}$ to emphasize universal behavior. (c), (d) Experimentally measured differential conductance vs scaled magnetic field. The $P$-like resonances are clearly visible and indicated by an arrow. (e), (f) Experimental $dI/dV$ vs bias voltage for various values of the magnetic field for (e) the one-electron and (f) the three-electron valley.

Figure 3

Second derivative of the current, $d^{2}I/d{V}_{}^2$, for the (a) $N=1$ valley and (b) $N=3$ valley as a function of magnetic field. The left subpanels show analytical predictions for the asymmetric four-level Anderson model, and the right subpanels the experimental observations. Kondo peaks in the differential conductance are manifested as zeros of $d^{2}I/d{V}^2$ as it changes from positive (red) to negative (blue) with increasing bias. Arrows point to the $P$ resonance.

Figure 4

Channel density of states ${\nu }_j$, (a)–(d), and self-energy ${\mathrm{\Sigma }}_2$ of channel 2, panels (e) and (f), evaluated at bias drops $e{V}_{}\simeq {\mathrm{\Delta }}_P$, where ${\mathrm{\Delta }}_P$ is the addition energy of the $P$ resonance, for the $N=1$ case at $B=8.05T$. The gray stripe indicates the bias window set by the lead chemical potentials when $e{V}_{}={\mathrm{\Delta }}_P$. At $e{V}_{}={\mathrm{\Delta }}_P$ the channel density of states ${\nu }_2$ and ${\nu }_3$ are maximal and of the order of ${\nu }_1$. This is due to a resonance of the associated self-energy, as illustrated in (e), (f) on the example of ${\mathrm{\Sigma }}_2$. On the other hand, ${\nu }_4$ is negligible.