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Using non-equilibrium Green's functions (NEGF), we calculate the currentthrough an interacting region connected to non-interacting leads. The problemis reformulated in such a way that a Landauer-like term appears in the currentas well as extra terms corresponding to non-equilibrium many-body effects. Theinteraction in the central region renormalizes not only the Green's functionsbut also the coupling at the contacts between the central region and the leads,allowing the total current to be further expressed as a generalizedLandauer-like current formula. The general expression for the dynamicalfunctional that renormalizes the contacts is provided. We analyze in detailunder what circumstances Landauer-like approaches to the current, i.e. withoutcontact renormalization, are valid for interacting electron-electron and/orelectron-phonon systems. Numerical NEGF calculations are then performed for amodel electron-phonon coupled system in order to validate our analyticalapproach. We show that the conductance for the off-resonant transport regime isadequately described by Landauer-like approach in the small-bias limit, whilefor the resonant regime the Landauer-like approach results depart from theexact results even at small finite bias. The validity of applying aLandauer-like approach to inelastic electron tunneling spectroscopy is alsostudied in detail.

Generalization and applicability of the Landauer formula for non-equilibrium current in the presence of interactions