Inelastic Kondo-Andreev tunneling in a vibrating quantum dot

Phonon-assisted electronic tunneling through a vibrating quantum dot embedded between normal and superconducting leads is studied in the Kondo regime. In such a hybrid device, with the bias applied to the normal lead, we find a series of Kondo sidebands separated by half a phonon energy in the differential conductance, which are distinct from the phonon-assisted sidebands previously observed in conventional Andreev tunneling and in systems with only normal leads. These Kondo sidebands originate from the Kondo-Andreev cooperative cotunneling mediated by phonons, which exhibit an interesting Kondo transport behavior due to the interplay of the Kondo effect, the Andreev tunneling, and the mechanical vibrations. Our result could be observed in a recent experiment setup [J. Gramich et al., Phys. Rev. Lett. 115, 216801 (2015)], provided that their carbon nanotube device reaches the Kondo regime at low temperatures.

Figure 1

Schematics of elastic [(a), (d)] and inelastic [(b), (c), (e), (f)] electronic tunneling in an N-QD-S system. (a)–(c) represent Kondo-Andreev tunneling through an interacting QD with the numbers $1,2,3$ denoting the tunneling sequences. (d)–(f) show conventional AT through a noninteracting QD. The solid lines in the QD indicate the two ABSs with energies $\pm E_A$, while the wavy arrows represent the emission of phonons during the inelastic tunneling.

Figure 2

(a) Differential conductance in the Kondo regime with different EPI strengths for ${\tilde{\varepsilon }}_d=-2.5, {\tilde{\mathrm{\Gamma }}}_{\text{S}}=4, \tilde{U}=10$, and ${\varepsilon }_{\text{ph}}=0.1$. The curve for $g=1.5$ is offset by 0.025. (b) LDOS at different bias voltages $V=0$ (top), $-{\varepsilon }_{\text{ph}}/4, -{\varepsilon }_{\text{ph}}/2, -3{\varepsilon }_{\text{ph}}/4$, and $-{\varepsilon }_{\text{ph}}$ (bottom). The curves are offset for clarity. The dashed lines guide the shift of the Kondo satellites, while the circles mark their mergence. (c) Illustrations of the Kondo cotunneling processes corresponding to the Kondo peaks in (b). (d) Currents vs ${\tilde{\varepsilon }}_d$ at different bias voltages ($V$ decreases from $-{\varepsilon }_{\text{ph}}/4$ to $-2{\varepsilon }_{\text{ph}}$ by a step ${\varepsilon }_{\text{ph}}/4$, from top to bottom). The inset shows the corresponding differential conductance.

Figure 3

Left panel: The conductance of a noninteracting ($\tilde{U}=0$) N-QD-S system with finite EPI $g=0.8$ for different ${\tilde{\mathrm{\Gamma }}}_{\text{S}}$, as indicated. Right panel: The corresponding LDOS at four different bias voltages $V=0, {\varepsilon }_{\text{ph}}/2, {\varepsilon }_{\text{ph}}$, and $3{\varepsilon }_{\text{ph}}/2$ (from top to bottom in each figure). The curves are offset for clarity. The horizontal distance between the two dashed lines represents the bias window $\omega \in [-V,V]$. Other parameters are ${\tilde{\varepsilon }}_d=0$ and ${\varepsilon }_{\text{ph}}=30$.