Vibrational Andreev bound states in magnetic molecules

We predict the existence of vibrational Andreev bound states in deformable magnetic molecules on superconducting surfaces. We discuss the Anderson impurity model with electron-phonon coupling to a realistic anharmonic vibrational mode that modulates the tunneling barrier and show that the vibronic features are spectroscopically most visible near the transition point between the Kondo-screened singlet and the unscreened doublet ground state. We find competing tendencies between phonon hardening due to anharmonicity and softening due to coupling to electrons, contrary to the Anderson-Holstein model and other models with harmonic local phonon mode where the vibrational mode is always softened. In addition, we find that the singlet and doublet many-body states may experience very different effective phonon potentials.

Figure 1

(a) Energy of the subgap singlet ($S$) and doublet ($D$) states as a function of the hybridization strength ${\Gamma }_0$. (b) Subgap part of the spectral function. Subfigures show cross sections at particular values of ${\Gamma }_0$, indicated by dashed lines. (c) Visibility of the first and second vibronic side peaks. Parameters are $g=0.05$, $D_e=0.5$, and $\delta =0.1$.

Figure 2

(a) Energy of the subgap singlet ($S$) and doublet ($D$) states as a function of the electron-phonon coupling $g$. (b) Subgap part of the spectral function. Parameters are ${\Gamma }_0=0.1$, $D_e=0.1$, and $\delta =0$.

Figure 3

Top panel: bare (nonrenormalized) Morse potential for different parameters of the anharmonicity. Bottom panel: (a)–(c) Energy of the subgap singlet ($S$) and doublet ($D$) states versus electron-phonon coupling $g$ for different parameters of the anharmonicity. Parameters: ${\Gamma }_0=0.1$, $\delta =0$. (d)–(f) Effective phonon frequencies of $D$ and $S$ states. (g)–(i) Displacement and fluctuations (insets) of the displacement. (j)–(l) Visibility of first vibronic side peak.

Figure 4

(a) Effective (renormalized) potential of $S$ and $D$ subgap states and bare (nonrenormalized) Morse potential. (b),(c) Oscillator distribution $\rho \left(x\right)$ for ABS in the effective potential for $S$ and $D$ states. The probability is not normalized. (d) Overlap of the phonon part of the wave function between different subgap states. Parameters are ${\Gamma }_0=0.1$, $D_e=0.1$, $\delta =0$.