Exponential and power-law renormalization in phonon-assisted tunneling

We investigate the spinless Anderson-Holstein model routinely employed to describe the basic physics of phonon-assisted tunneling in molecular devices. Our focus is on small to intermediate electron-phonon coupling; we complement a recent strong coupling study [Phys. Rev. B 87, 075319 (2013)]. The entire crossover from the antiadiabatic regime to the adiabatic one is considered. Our analysis using the essentially analytical functional renormalization group approach backed up by numerical renormalization group calculations goes beyond lowest order perturbation theory in the electron-phonon coupling. In particular, we provide an analytic expression for the effective tunneling coupling at particle-hole symmetry valid for all ratios of the bare tunnel coupling and the phonon frequency. It contains the exponential polaronic as well as the power-law renormalization in the electron-phonon interaction; the latter can be traced back to x-ray edgelike physics. In the antiadiabatic and the adiabatic limit this expression agrees with the known ones obtained by mapping to an effective interacting resonant level model and lowest order perturbation theory, respectively. Away from particle-hole symmetry, we discuss and compare results from several approaches for the zero temperature electrical conductance of the model.

Figure 1

(a) and (b): The ratio of the effective tunneling rate ${\mathrm{\Gamma }}_{\mathrm{eff}}$ to the bare value $\mathrm{\Gamma }$ as a function of $\mathrm{\Gamma }/{\omega }_0$ using different approaches. (c) The ratio ${\left(\lambda /{\omega }_0\right)}^{-2}ln\left({\mathrm{\Gamma }}_{\mathrm{eff}}/\mathrm{\Gamma }\right)$ as a function of $\mathrm{\Gamma }/{\omega }_0$ for different electron-phonon couplings computed within NRG.

Figure 2

(a) $1-({\mathrm{\Gamma }}_{\mathrm{eff}}/\mathrm{\Gamma })$ as a function of $E_{\mathrm{p}}/\mathrm{\Gamma }$ for ${\omega }_0=0.2\mathrm{\Gamma }$ and ${\omega }_0=0.02\mathrm{\Gamma }$ (adiabatic limit). (b) $1-({\mathrm{\Gamma }}_{\mathrm{eff}}/\mathrm{\Gamma })$ as a function of ${\left(\lambda /{\omega }_0\right)}^2$ for ${\omega }_0={10}^3\mathrm{\Gamma }$ (antiadiabatic limit).

Figure 3

Comparison of the different approaches to compute the $T=0$ linear conductance as a function of the gate voltage ${\epsilon }_0-E_{\mathrm{p}}$ for a given phonon frequency ${\omega }_0=2\mathrm{\Gamma }$ in the weak coupling regime ($\lambda =0.5{\omega }_0$).

Figure 4

Comparison of FRG (dashed lines) and NRG (stars) results for the zero temperature linear conductance as a function of the gate voltage ${\epsilon }_0-E_{\mathrm{p}}$ for different strength of electron-phonon couplings in the antiadiabatic limit (${\omega }_0={10}^3\mathrm{\Gamma }$). The inset shows the zero temperature conductance from NRG as a function of the rescaled gate voltage $\left({\epsilon }_0-E_{\mathrm{p}}\right)/{\mathrm{\Gamma }}_{\mathrm{eff}}$ for different strength of the electron-phonon coupling in the antiadiabatic limit (${\omega }_0={10}^3\mathrm{\Gamma }$).

Figure 5

The ratio of the effective tunneling rate ${\mathrm{\Gamma }}_{\mathrm{eff}}$ to the bare value $\mathrm{\Gamma }$ as a function of $N_b$ for different electron-phonon couplings in the antiadiabatic limit (${\omega }_0=100\mathrm{\Gamma }$).

Figure 6

Comparison of the molecular spectral function obtained from perturbation theory (solid lines) and NRG (dashed lines). (a) shows the spectral function at the particle-hole symmetric point ${\epsilon }_0=E_{\mathrm{p}}$ for different $\mathrm{\Gamma }/{\omega }_0$ at a fixed electron-phonon coupling $\lambda =0.7{\omega }_0$ as a function of $\nu /\mathrm{\Gamma }$. Note the logarithmic $x$-axis scale. (b) depicts the spectral function at different gate voltages for ${\omega }_0=1.5\mathrm{\Gamma }$ and $\lambda =0.5{\omega }_0$ as a function of $\nu /{\omega }_0$.