Crossover from adiabatic to antiadiabatic phonon-assisted tunneling in single-molecule transistors

TL;DR

This paper investigates the transition between adiabatic and antiadiabatic phonon-assisted tunneling in single-molecule transistors, revealing a regime where traditional assumptions do not hold and characterizing the low-energy physics through numerical methods.

Contribution

It introduces the concept of an extended antiadiabatic regime governed by polaronic shifts and derives an effective fermionic model for this intermediate regime.

Findings

The polaronic shift governs the crossover more than phonon frequency.

The extended antiadiabatic regime exists where mma exceeds phonon frequency but not the polaronic shift.

Renormalized tunneling rate follows a scaling form across the crossover.

Abstract

The crossover between two customary limits of phonon-assisted tunneling, the adiabatic and antiadiabatic regimes, is studied systematically in the framework of a minimal model for molecular devices: a resonant level coupled by displacement to a localized vibrational mode. Conventionally associated with the limits where the phonon frequency is either sufficiently small or sufficiently large as compared to the bare electronic hopping rate, we show that the crossover between the two regimes is governed for strong electron-phonon interactions primarily by the polaronic shift rather than the phonon frequency. In particular, the perturbative adiabatic limit is approached only as the bare hopping rate \Gamma exceeds the polaronic shift, leaving an extended window of couplings where \Gamma well exceeds the phonon frequency and yet the physics is basically that of the antiadiabatic regime. We term this intermediate regime the extended antiadiabatic regime. The effective low-energy Hamiltonian in the traditional and the extended antiadiabatic regime is shown to be the (purely fermionic) interacting resonant-level model, with parameters that we extract from numerical renormalization-group calculations. The extended antiadiabatic regime is followed in turn by a true crossover region where the polaron gets progressively undressed. The renormalized tunneling rate, which serves as the low-energy scale in the problem and thus sets the width of the tunneling resonance, is found to follow an approximate scaling form on going from the adiabatic to the antiadiabatic regime. Charging properties are governed by two distinct mechanisms at the extended antiadiabatic and into the crossover region, giving rise to distinctive shoulders in the low-temperature conductance as a function of gate voltage.