Non-equilibrium linear-response transport through quantum dot beyond time homogeneity at Hartree-Fock level

Abstract

The expression for current, induced by finite bias with additional time-dependent (TD) external perturbation, through a molecule/dot is derived using the non-equilibrium Green's function (GF) formalism in the standard two-probe geometry. The GFs as well as self energies (SEs) are split into time-homogeneous (TH) and time-inhomogeneous (TIH) contributions, where the former are obtained as a result of zeroth-order expansion of the full, two-time corresponding quantities and the latter we find as linear corrections. The TD charge in the dot consists of a charge that is injected from the electrodes and the charge that is induced in the dot. The TD potential, induced in the dot due to dot TD charge, was treated self consistently at Hartree–Fock (HF) level as a TIH part of Coulomb interaction related SE. It is assumed that TH quantities are solved either exactly or approximately, i.e., using density functional theory (DFT). The theory is charge conserving and its gauge invariance is explicitly shown. The contribution of TD HF potential to the total Coulomb interaction energy vanishes in the case of one-electron existence, i.e., the self-interaction error (SIE), beyond the one associated with the DFT, was not introduced. Known results in a special case of time homogeneity are recovered and extended to TIH transport. The issues of current partitioning and the displacement current are resolved naturally, without any additional assumptions about any of quantities, due to explicit inclusion of dot potential. The special cases of wide-band limit, zero bias, and zero-bias wide-band limit are also considered and in each case the corresponding expression for the TD current is derived. The theory is particularly suitable for use in connection with DFT when it provides a first-principle microscopic linear-response description of the non-equilibrium TIH quantum transport useful for calculation of TD current through quantum dots, molecules, junctions, or devices at the nano-scale.