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

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 Γ exceeds the polaronic shift, leaving an extended window of couplings where Γ 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 lowenergy 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. In this latter region, the phonon configuration strongly deviates from a simple superposition of just one or two coherent states. 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 characteristic shoulders in the low-temperature conductance as a function of gate voltage. These shoulders serve as a distinct experimental fingerprint of phonon-assisted tunneling when the electron-phonon coupling is strong.

Neutrons on a surface of liquid helium

“…Similar problem appears in other condensed-matter systems and requires a special theoretical study (see, e.g., Refs. [42][43][44][45][46]). …”

Quantum transport through a molecular level: a scattering states numerical renormalization group study

“…For modeling realistic situations, the restriction to a single phonon and a single electronic level must be lifted. In spite of a lot of theoretical progress [5] this model has only been accurately solved in equilibrium [14,20,25], while its non-equilibrium dynamics has only be perturbatively investigated in lowest order of the coupling constants [5].…”

“…(1) after identifying ε = E d + g 2 ω 0 and performing a linear shift of the bosonic operators [14,21]. Figure 4 shows the evolution of the current for a symmetric junction Γ L /Γ R = 1 from medium to strong coupling at T /Γ 0 = 0.2, ε = 0 and ω 0 /Γ 0 = 2.…”

“…The equilibrium physics of two extreme limits have been well understood in a model containing only a single vibrational mode [5,9,14]. In the adiabatic limit, the phonon frequency is the smallest energy scale of the problem and a small electron-phonon coupling yields a reduction of the phonon frequency by particle-hole excitations.…”

Quantum quenches and driven dynamics in a single-molecule device

“…Note that these expressions for ω and τ coincide with second-order perturbation theory in g when applied directly to the electronic Hamiltonian of Eq. (1), 38 thus validating the cutoff scheme used in bosonization. The expression for τ can be further improved by going to the next order in ω 0 /D d , i.e., by including one more order in ξ in the expansion of Σ (+) (z).…”