Vibrationally Mediated Control of Single Electron Transmission in Weakly Coupled Molecule-Metal Junctions Thomas Olsen ∗ and Jakob Schiøtz Danish National Research Foundation’s Center for Individual Nanoparticle Functionality (CINF), Department of Physics, Technical University of Denmark, DK–2800 Kongens Lyngby, Denmark (Dated: November 15, 2018) We propose a mechanism which allows one to control the transmission of single electrons through a molecular junction. The principle utilizes the emergence of transmission sidebands when molecular vibrational modes are coupled to the electronic state mediating the transmission. We will show that if a molecule-metal junction is biased just below a molecular resonance one may induce the transmission of a single electron by externally exciting a vibrational mode of the molecule. The analysis is quite general but requires that the molecular orbital does not hybridize strongly with the metallic states. As an example we perform a density functional theory (DFT) analysis of a benzene molecule between two Au(111) contacts and show that exciting a particular vibrational mode can give rise to transmission of a single electron. PACS numbers: 82.53.St, 34.35.+a Several experiments have established that vibrational excitations can have a significant effect on the I - V char- acteristics of single-molecule junctions $^{1–5}$. In particular, the emergence of peaks in the differential conductance corresponding to vibrational frequencies, shows that tun- neling electrons interact with certain vibrational states of the molecule, possibly providing a means for con- trolling the transmission of electrons. A considerable amount of theoretical work have been dedicated to elu- cidating the effect of phonons on electronic transport in mesoscopic systems. Analysis of model Hamiltonians 6–9 have given qualitative insight into the interaction of tun- neling electrons with molecular vibrations, while DFT based studies in conjunction with a non-equilibrium Green function approach show quantitative agreement with experiments $^{10–12}$. Most efforts so far have been di- rected towards the influence of vibrations on transmission functions and I - V characteristics. In the present paper we will take a slightly different point of view and parti- tion the electronic transmission function into pieces that involve different vibrational excitations. We then show that the molecular junction may be put in a configura- tion where the vibrationally excited molecule allows the transmission of a single electron, while transmission is forbidden in the vibrational ground state. Controlling the vibrational state of the molecule, e.g. by means of a laser, then implies control of single electron transmission. The system under consideration is a molecule sand- wiched between two metallic leads. We assume that there is a single unoccupied molecular state which obtains a fi- nite lifetime due to hybridization with metallic states. This state will be referred to as the resonance and its position may be tuned by applying a gate voltage. If a bias voltage is applied to the contacts and the resonance is positioned in the bias window, electrons may tunnel through the resonance into the downstream contact and one will observe a current. If the molecule is weakly inter- acting with the metal such that the resonance is well lo- calized in energy, one can apply a gate voltage which sit- uates the resonance above the upstream chemical poten- tial and no current will be observed. However, if the res- onance couples to molecular vibrations, an off-resonant enhancement of transmission known as transmission side- bands may be observed $^{6}$. In particular, if the molecule is initially vibrationally excited, off-resonant electrons be- low the resonance may tunnel though the molecule by absorbing a vibrational quantum of energy. This is il- lustrated in Fig. 1. Since transmission is only allowed combined with a downward vibrational transition, only one or a few electrons may tunnel through the contact and the transmission channel will be closed once the vi- brational mode reaches the ground state. Inspired by these considerations, we perform a quanti- tative analysis based on the model Hamiltonian 6,13 where c † a is the creation operator for the lowest unoccu- pied molecular orbital (LUMO), c † Lk and c † Rk are creation operators for metallic states in the left and right lead respectively and b † i are creation operators for the vibra- tional normal modes of the molecule with frequencies ω$_{i}$ . Thus, the electronic states of the left and right contacts are coupled through the molecular resonance which is coupled to molecular vibrations with coupling strengths λ$_{i}$ . We impose the wide band limit in which the contact density of states is constant in the region of the resonance and the resonance hybridization with metallic states is determined by the parameters H = ε$_{0}$c † $_{a}$c$_{a}$ + ∑ i ℏ ω$_{i}$b † $_{i}$b$_{i}$ + ∑ i λ$_{i}$c † $_{a}$c$_{a}$ ( b † i + b$_{i}$ ) (1) + ∑ k ϵ$_{Lk}$c † $_{Lk}$c$_{Lk}$ + ∑ k ( V$_{Lk}$c † $_{a}$c$_{Lk}$ + V ∗ $_{Lk}$c † $_{Lk}$c$_{a}$ ) + k ϵ$_{Rk}$c † $_{Rk}$c$_{Rk}$ + k V$_{Rk}$c † $_{a}$c$_{Rk}$ + V ∗ $_{Rk}$c † $_{Rk}$c$_{a}$ , Γ$_{L}$ = 2 π ∑ k | V$_{Lk}$ | $^{2}$δ ( ε$_{0}$ − ϵ$_{Lk}$ ) , (2) arXiv:1001.0455v2 [cond-mat.mes-hall] 25 Mar 2010