Non-equilibrium Green function

In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle-hole propagators, which have been at the heart of Jens electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9-15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions ... 

Superoperator many-body theory of molecular currents non-equilibrium Green functions in real time... 

Keywords Density functional theory (DFT) Green s functions Keldysh non-equilibrium Green s functions (NEGF) linear combination of atomic orbitals (LCAO) tunnel junction metal-fullerene-metal junction density of states (DOS) transmission function Landauer formula renormalized molecular levels (RMLs) I-V curves. 

For equilibrium problems, the imaginary part of the retarded Green s function determines the charge density p [46]. For transport problems, p(r) can still be computed by Green s functions — the Keldysh non-equilibrium Green s functions [51]. The formula is [51] ... 

The most comprehensive description of the tunneling problem is based either on a self-consistent solution of the Lippman-Schwinger equation [3] or on the non-equilibrium Green s function approach [4-8]. Inelastic effects within e.g. a molecule-surface interface can be included by considering multiple electron paths from the vacuum into the surface substrate [9], The current between two leads with the chemical potentials /ja and hb is given by the energy integral ... 

The other type of theory uses non-equilibrium Green s functions. Green s functions are more tractable in a localized basis set, such as the one... 

A computational approach was not only very successful in fullerene research, but advances in these studies have created the demand for the development of new theoretical methods. The last part of this chapter describes the application of the non-equilibrium Green s function formalism to the investigation of the current-voltage dependence of the fullerene molecule. This method can be also q>plied to a wide range of nanomolecular devices. 

If vibrons are noninteracting, in equilibrium, and non-dissipative, then the vibronic Green functions write ... 

In this section we will consider the case of a multi-level electronic system in interaction with a bosonic bath [288,289], We will use unitary transformation techniques to deal with the problem, but will only focus on the low-bias transport, so that strong non-equilibrium effects can be disregarded. Our interest is to explore how the qualitative low-energy properties of the electronic system are modified by the interaction with the bosonic bath. We will see that the existence of a continuum of vibrational excitations (up to some cut-off frequency) dramatically changes the analytic properties of the electronic Green function and may lead in some limiting cases to a qualitative modification of the low-energy electronic spectrum. As a result, the I-V characteristics at low bias may display metallic behavior (finite current) even if the isolated electronic system does exhibit a band gap. The model to be discussed below... 

We emphasize that the density matrix calculated from Eq. (6) is equivalent to that from Eq. (4), but Eq. (6) is much easier to compute for open systems. To see why this is so, let us consider zero temperature and assume ftL — ftR = eV], > 0. Then, in the energy range -oo < E < pR the Fermi functions = fR = 1. Because the Fermi functions are equal, no information about the non-equilibrium statistics exists and the NEGF must reduce to the equilibrium Green s function GR. In the range pR < E < pR, fL 7 fR and NEGF must be used in Eq. (6). A more careful mathematical manipulation shows that this is indeed true [30], and Eq. (6) can be written as a sum of two terms ... 

However, this is still a non-equilibrium formulation of the problem, since the chemical potentials p account for the non-equilibrium condition of nonzero bias voltage. The only additional assumption in this formulation of the tunneling problem, compared to the general formulation above, is that the leads remain in thermal equilibrium. The expression can be calculated using a standard eigenvector expansion of surface and tip Green s functions ... 

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