Non-equilibrium Green’s function

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. 

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 ... 

To this we follow closely Baym and Baym and Kadanoff in their discussion of perturbation theory for non-uniform systems. They consider non-equilibrium thermal Green s functions at an arbitrary temperature B in the presence of an external, in general non-local potential U that acts in the imaginary time interval O < it < 6. Writing the operators in the interaction representation for the external disturbance, the one-particle Green s function is... 

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