Non-equilibrium electronic structure of interacting single-molecule nanojunctions: vertex corrections and polarization effects for the electron-vibron coupling. L. K. Dash, 1, 2 H. Ness, 1, 2 and R. W. Godby 1, 2 $^{1}$Department of Physics, University of York, York YO10 5DD, UK $^{2}$European Theoretical Spectroscopy Facility (Dated: November 15, 2018) We consider the interaction between electrons and molecular vibrations in the context of electronic transport in nanoscale devices. We present a method based on non-equilibrium Green’s functions to calculate both equilibrium and non-equilibrium electronic properties of a single-molecule junction in the presence of electron-vibron interactions. We apply our method to a model system consisting of a single electronic level coupled to a single vibration mode in the molecule, which is in contact with two electron reservoirs. Higher-order diagrams beyond the usual self-consistent Born approximation (SCBA) are included in the calculations. In this paper we consider the effects of the double-exchange diagram and the diagram in which the vibron propagator is renormalized by one electron-hole bubble. We study in detail the effects of the first- and second-order diagrams on the spectral functions for a large set of parameters and for different transport regimes (resonant and off-resonant cases), both at equilibrium and in the presence of a finite applied bias. We also study the linear response (linear conductance) of the nanojunction for all the different regimes. We find that it is indeed necessary to go beyond the SCBA in order to obtain correct results for a wide range of parameters. PACS numbers: PACS numbers: 71.38.-k, 73.40.Gk, 85.65.+h, 73.63.-b ergy (bias) of the features correspond approximately to the frequency of the vibration, as given by other spectro- scopic data (IR, Raman) obtained on the same molecules in a different environment. Single-molecule electronics has shown significant progress in the recent years. A variety of interesting effects have been observed in the transport properties of single (or a few) conjugated organic molecules in- cluding rectification, negative differential resistance, and switching $^{1–6}$. In these quasi-one dimensional systems, which present well delocalised π -electrons, the electronic current flowing through the quite flexible backbone of the molecule affects the ground state properties of both elec- tronic and mechanical degrees of freedom of the molecule. There have been many theoretical investigations focus- ing on the effects of electron-vibron coupling in molecu- lar and atomic scale wires $^{16–59}$. Most of them have fo- cused on the interpretation of the features in IETS. How- ever, most of these studies have been performed by us- ing the lowest-order expansion possible for treating the effects of the electron-vibron interaction (i.e using the so-called self-consistent Born approximation SCBA). In the language of many-body perturbation theory, it cor- responds to a self-consistent Hartree-Fock calculation for the electron-vibron coupling. The importance of inelastic effects in the transport properties has been demonstrated in several ground- breaking experiments $^{7–11}$; these effects are related to the interaction between electron and mechanical degrees of freedom of the molecule. However, in analogy to what is obtained at the Hartree-Fock level for interacting electrons, there are good reasons to believe that this approximation is not enough to correctly describe the physics of the electron- vibron interacting system, especially beyond the weak electron-vibron coupling regime. For example, the limits of SCBA have already been investigated in Ref.[60] but without introducing remedies to go beyond SCBA. The interaction between an injected charge carrier (electron or hole) and the mechanical degrees of freedom (phonon, vibron) in the molecular junctions is impor- tant in order to understand energy transfer, heating and dissipation in the nanojunction $^{12}$. The electron-vibron interaction is also at the heart of inelastic electron tun- neling spectroscopy (IETS). IETS is a solid-state-based spectroscopy which gives information about the vibration modes (vibrons) of the molecules in the nanojunction $^{13}$. In this paper, we examine this using a true non-equilibrium Green’s-function (NEGF) technique 20,25,26,28,30–34,38,44–46 which allows us to study all the different transport regimes in the presence of electron-vibron interaction. Following the spirit of many-body perturbation theory and Feynman dia- grammatics, we go beyond the commonly-used SCBA approximation by introducing higher-order diagrams for the electron-vibron interaction. We study the simplest possible model system which nonetheless contains the relevant physics of the trans- port properties of the molecular junction $^{39,60}$. Further- It is now possible to measure such vibrational spectra for single molecules by using scanning tunneling microscopy (STM) 9,14 to build IETS maps $^{15}$, or by using other elec- tromigrated junctions or mechanically controlled break junctions $^{7,8,10,11}$. At low applied bias (typically 100-400 meV) the IETS show features (peaks, dips, or peak-dip-like lineshape) which have been attributed to selective excitation of spe- cific vibration modes of the molecule. The position in en- I. INTRODUCTION arXiv:1001.2115v1 [cond-mat.mes-hall] 13 Jan 2010