Effects of electron-phonon coupling in the Kondo regime of a two-orbital molecule

We study the interplay between strong electron-electron and electron-phonon interactions within a two-orbital molecule coupled to metallic leads, taking into account Holstein-like coupling of a local phonon mode to the molecular charge as well as phonon-mediated interorbital tunneling. By combining canonical transformations with numerical renormalization-group calculations to address the interactions nonperturbatively and on equal footing, we obtain a comprehensive description of the system's many-body physics in the antiadiabatic regime where the phonons adjust rapidly to changes in the orbital occupancies and are thereby able to strongly affect the Kondo physics. The electron-phonon interactions strongly modify the bare orbital energies and the Coulomb repulsion between electrons in the molecule and tend to inhibit tunneling of electrons between the molecule and the leads. The consequences of these effects are considerably more pronounced when both molecular orbitals lie near the Fermi energy of the leads than when only one orbital is active. In situations where a local moment forms on the molecule, there is a crossover with increasing electron-phonon coupling from a regime of collective Kondo screening of the moment to a limit of local phonon quenching. At low temperatures, this crossover is associated with a rapid increase in the electronic occupancy of the molecule as well as a marked drop in the linear electrical conductance through the single-molecule junction.

Figure 1

Schematic representation of the model studied in this work. A molecule with two active orbitals ($\alpha$ and $\beta$) spans the gap between left and right electrodes. The molecular orbitals are subject to both $e$-$e$ and $e$-ph interactions.

Figure 2

Variation with orbital energy ${\varepsilon }_{\alpha }$ at zero temperature of (left) the charge $\langle n_{\mathrm{mol}}\rangle$ and (right) the phonon occupation $\langle n_b\rangle$, for $U=0$ (top), $U=5$ (middle), and $U=0.5$ (bottom). Data are for ${\varepsilon }_{\beta }=4$, $U^{\prime }=U$, ${\lambda }^{\prime }=\lambda$, and the four values of ${\lambda }^2/{\omega }_0$ listed in the legend.

Figure 3

Zero-temperature conductance $G$ vs orbital energy ${\varepsilon }_{\alpha }$ for (a) $U=0$, (b) $U=5$, and (c) $U=0.5$. Data are for ${\varepsilon }_{\beta }=4$, $U^{\prime }=U$, ${\lambda }^{\prime }=\lambda$, and the four values of ${\lambda }^2/{\omega }_0$ listed in the legend. The inset in (b) shows the data from the main panel replotted as $G$ vs ${\tilde{\varepsilon }}_{\alpha }/\tilde{U}$.

Figure 4

(a) Zero-temperature conductance $G$ vs orbital energy ${\varepsilon }_{\alpha }$ for the same parameters as used in Fig. 3c, except here ${\lambda }^{\prime }=0$. (b) Data for ${\lambda }^{\prime }=\lambda$ from Fig. 3c replotted as $G$ vs ${\tilde{\varepsilon }}_{\alpha }/\tilde{U}$. (c) Data for ${\lambda }^{\prime }=0$ from (a) replotted as $G$ vs ${\tilde{\varepsilon }}_{\alpha }/\tilde{U}$.

Figure 5

Variation with ${\lambda }^2/{\omega }_0$ of (a) the ground-state molecular charge $\langle n_{\mathrm{mol}}\rangle$, (b) the ground-state phonon occupation $\langle n_b\rangle$, (c) the crossover temperature $T^{*}$, and (d) the zero-temperature linear conductance $G$, all calculated for $U^{\prime }=U=0.5$, ${\lambda }^{\prime }=\lambda$, ${\varepsilon }_{\beta }=4$, and the four values of ${\varepsilon }_{\alpha }$ listed in the legend.

Figure 6

Variation with ${\lambda }^2/{\omega }_0$ of the crossover temperature $T^{*}$ calculated for $U^{\prime }=U=0.5$, ${\lambda }^{\prime }=0$, ${\varepsilon }_{\beta }=4$, and the four values of ${\varepsilon }_{\alpha }$ listed in the legend. (Inset) The same $T^{*}$ data plotted vs the ratio ${\tilde{\varepsilon }}_{\alpha }/{\tilde{U}}_{\alpha }$ of phonon-renormalized molecular parameters.

Figure 7

Variation with ${\lambda }^2/{\omega }_0$ of (a) the ground-state molecular charge $\langle n_{\mathrm{mol}}\rangle =\langle n_e+n_o\rangle$, (b) the ground-state phonon occupation $\langle n_b\rangle$, (c) the crossover temperature $T^{*}$, and (d) the zero-temperature linear conductance $G$, all calculated for $U^{\prime }=U=0.5$, ${\lambda }^{\prime }=\lambda$, and the four values of $\delta ={\varepsilon }_{\beta }=-{\varepsilon }_{\alpha }$ listed in the legend. In the case $\delta =0.05$, the orbital energy splitting is in resonance with the phonon energy, i.e., ${\varepsilon }_{\beta }-{\varepsilon }_{\alpha }=2\delta ={\omega }_0$.

Figure 8

Occupation of individual molecular orbitals vs ${\lambda }^2/{\omega }_0$ for $U^{\prime }=U=0.5$, ${\lambda }^{\prime }=\lambda$, and the four values of $\delta ={\varepsilon }_{\beta }=-{\varepsilon }_{\alpha }$ listed in the legend: (a) $\langle n_{\alpha }\rangle$ (open symbols) and $\langle n_{\beta }\rangle$ (solid symbols); (b) $\langle n_e\rangle$ (open symbols) and $\langle n_o\rangle$ (solid symbols).

Figure 9

Crossover temperature $T^{*}$ vs scaled $e$-ph coupling ${(\lambda /{\lambda }_x)}^2$. The left panels show different ratios $U^{\prime }/U$ for ${\lambda }^{\prime }=\lambda$ while the right panels show different ${\lambda }^{\prime }/\lambda$ for $U^{\prime }=U$. The top panels (a),(b) correspond to $\delta =0.05$, and the bottom panels (c),(d) treat $\delta =0.1$. All data are for $U=0.5$. Vertical dashed lines at $\lambda ={\lambda }_x$ [calculated via the condition $E_{\min }^{\left(2\right)}(\delta =0)=E_{\min }^{\left(1\right)}(\delta =0)$] separate the Kondo regime from the phonon-dominated regime.

Figure 10

Temperature dependence of (a) the molecular entropy, (b) temperature times the molecular susceptibility, $T{\chi }_{\mathrm{mol}}\equiv {\left|{\mathit{\mu }}_{\mathrm{mol}}\right|}^2/3$, where ${\mathit{\mu }}_{\mathrm{mol}}$ is the molecule's magnetic moment, and (c) the phonon occupation. Data are for $U^{\prime }=U=0.5$, $\delta =0.1$, ${\lambda }^{\prime }=\lambda$, and the four values of ${\lambda }^2/{\omega }_0$ listed in the legend. In (a), the horizontal dashed lines mark $S_{\mathrm{mol}}=\ln 2$, $\ln 3$, and $\ln 5$. In (c), the dashed line shows the occupation of a free phonon mode of energy ${\omega }_0=0.1$.

Figure 11

Temperature times the molecular susceptibility $T{\chi }_{\mathrm{mol}}$ vs scaled temperature $T/T^{*}$ for $U^{\prime }=U=0.5$, ${\lambda }^{\prime }=\lambda$, $\delta =0.1$, and values of ${\lambda }^2/{\omega }_0$ spanning the crossover from the Kondo regime to the doubly occupied regime. The collapse over the range $T\lesssim 10T^{*}$ of all curves corresponding to ${\lambda }^2/{\omega }_0\le 0.0391$ demonstrates the universal physics of the Kondo regime. No such universality is present in the boson-dominated limit.

Figure 12

Zero-temperature conductance $G$ as a function of ${\varepsilon }_{\alpha }=-\delta -V_g$ (where $V_g$ is an applied gate voltage) for the five values of ${\lambda }^2/{\omega }_0$ listed in the legend and (a) $\delta =0.025$, (b) $\delta =0.05$, and (c) $\delta =0.1$. The other parameters are $U^{\prime }=U=0.5$ and ${\lambda }^{\prime }=\lambda$.

Figure 13

(a) Ground-state molecular charge and (b) ground-state phonon occupation as functions of ${\varepsilon }_{\alpha }=-\delta -V_g$ (where $V_g$ is an applied gate voltage) for $\delta =0.1$ and the values of ${\lambda }^2/{\omega }_0$ listed in the legend. The other parameters are $U^{\prime }=U=0.5$ and ${\lambda }^{\prime }=\lambda$.