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

TL;DR

This work reveals how a concealed resonance in a carbon nanotube quantum dot emerges from multiple Kondo transitions when a four-level, SU(4)/SU(2)×SU(2) symmetric framework is driven out of equilibrium. The authors develop a Keldysh effective action (KEA) approach to compute the tunneling density of states for each channel, incorporating a complex expansion parameter 𝔈 and self-energies that depend on digamma functions, enabling a description of Kondo resonances, P-transitions, and the influence of bias and tunneling asymmetries. They show that asymmetric couplings between the Kramers doublets and the leads amplify pseudospin-non-preserving transitions at specific biases, giving rise to experimentally observable resonances at the pseudospin-preserving transition bias. The analysis matches experimental data by extracting T_K, Δ, and Γ, and demonstrates how charge-transfer peaks shape the low-energy Kondo features, elucidating transport in non-equilibrium Kondo systems with hidden symmetries. The findings identify microscopic mechanisms governing non-equilibrium Kondo transport and highlight the role of lead- and bias-induced asymmetries in revealing concealed resonances.

Abstract

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 SU(2)$\otimes$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-non-preserving 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.

Figures (8)

• Figure S-1: Mechanism of resonances in the TDOS. Resonances in the Kondo regime are related to the low temperature behavior of the constituent digamma functions in the self-energies. These occur at $\varepsilon=\varepsilon_0 -{\rm Im}{\cal E}$, where the real part has a dip while the imaginary part has a phase change of $\pi$. The resonance features are sharp for small ${\cal E}/2\pi k_{\rm B}T$. For the simulation we choose $\varepsilon_0/2\pi k_{\rm B}T=1$, corresponding to the location of the vertical dashed line in panels (a) and (b).
• Figure S-2: TDOS signatures of the $P$-resonance. Channel density of states $\nu_j$, (a)-(d), and self-energy $\Sigma_2$ (e), (f), evaluated at bias drops around the energy $\Delta_P$ of the $P$-resonance, for the case $N=1$, and $N=3$ case. The gray stripe indicates the integration range set by the lead chemical potentials. At $eV=\Delta_P$ the channel density of states $\nu_2$ and $\nu_3$ are maximal. This is due to a resonance of the associated self-energy, as illustrated in (e), (f) on the example of $\Sigma_2$. The magnetic field is $B=8.05$ T, and we set $\Gamma_u=\Gamma_d=\Gamma$.
• Figure S-3: Differential conductance for the $N=1$ case for (a) $B=0$ T and (b) $B=8.05$ T for various coupling asymmetries. The remaining parameters are the one used to match the $N=1$ experimental data. The red curve in (b) with $\gamma_{L{\rm u}}>\gamma_{L{\rm d}}$ shows a $P$-transition at $\mu_L-\mu_R>0$.
• Figure S-4: Interplay of bias asymmetries and charge transfer peak on the differential conductance. The $N=1$ case at $B=0$ T is shown for two distinct values of the asymmetry parameter $\eta$ for (a) $\mu_0-\varepsilon_d=4.8389\Gamma$, and (b) $\mu_0-\varepsilon_d=8\Gamma$. The remaining dot parameters are the ones used to match the $N=1$ experimental data. Notice the negligible effect of bias asymmetries on the Kondo resonances for the parameter set in (b).
• Figure S-5: Extracting the contribution from the charge transfer peaks. (a) Experimentally obtained $dI/dV$ data. The green lines in valley $N=1$, $N=3$ mark the bias required to reach the border of the Coulomb diamond from the middle of the valley; the distance between the blue lines yields the addition energy corrected by the level arm $\alpha_{\rm g}$. (b) Lorentzian fit (red) of the experimental gate trace (green) at $V=-3.95$ mV. (c) Contribution to the differential conductance according to the Lorentzian fit (red dashed line) and experimental data (blue) as a function of the bias voltage.