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MSc Miloš Dražić
 
 

Microscopic theory of non-equilibrium electronic transport under time-dependent bias through the molecule (or quantum dot) embedded between two semi-infinite metallic electrodes is developed in the non-orthogonal single-particle basis set using ab-initio formalism of Greens functions. The equilibrium zeroth order electron Green’s function and self-energy are corrected by the corresponding time-inhomogeneous dynamical contributions, derived in Hartree approximation in steady-state linear-response regime. It was shown that nonorthogonality contributes to these dynamical contributions by introducing terms related to the central region-electrode interface, appearing only in time-dependent case. The expression for current is also derived, where nonorthogonal-induced dynamical correction gives an additional current, not present in the orthogonal description. It is shown that the obtained expression for current is gauge-invariant and demonstrated that the omission of the additional current violates charge conservation. It is also shown that the additional current term vanishes in orthogonal case.

This approach includes approximations that are to be performed on two particle Green’s functions in order to avoid infinite BBGKY hierarchy and to meet the requirements for calculated self-energy, such as conservation laws, self interaction and self screening free property, as much as it is possible. In order to describe physical system more realistically, we try to go beyond known approximations, or at least to improve them. Once the time dependent potential in dot is calculated, we have the basis to determine alternating current through the dot and its relation with applied voltage in linear and higher orders.

Since the quantum dots are a physical systems where the size and the charge quantization meet together, it is desirable to develop a theories which reduce a self-interaction error, allowing to explore the single charge effects, which is one of the focuses of my work.

In the density functional theory, it is desirable to work with quantities represented in non-orthogonal basis set. The reason is that the nonorthogonality provides the most localized description and the linear-scaling computation. Although the orbitals belonging to the same atom and to nighboring atoms are, without an interaction, expected to be orthogonal, an interatomic interaction leads to non-orthogonal behaviour and the overlap emerging. Any attempt to reintroduce orthogonality by means of a linear combination of non-orthogonal orbitals, would lead to orthogonal functions which have a wider extent than non-orthogonal one, thus spoiling the picture of localized basis set and leading to Hamiltonian with matrix elements which are spread on more distant sites than in non-orthogonal case. Without working with localized orbitals, the Hamiltonian division on separate contributions from electrodes and molecule as well as from the interfaces projections, would be meaningless. Beside the computational advantages and clear physical picture, it is known that non-orthogonality between atomic orbitals produce nontrivial effects on chemical bonds strength, resonance molecular energies, sensitivity of bond orders and charge densities in hetero-molecules. It is also strongly involved in population analysis problem.

In steady state transport, as long as there is no a single molecular orbital that couples appreciably to the electrodes, the non-orthogonality has no influence on final results for transmission or current. The situation significantly changes when we have to deal with time dependent transport where charge starts to pile up in central region consisting of a molecule with additional neighboring parts of electrodes, chosen in such a way to provides complete screening of a molecule. The non-orthogonality introduces a problem due to non-unique definition of the time dependent partial charges associated with electrodes and central region, which is a consequence of non-orthogonality between complementary subspaces, making the corresponding projectors non-Hermitian. Furthermore, it is known that the coupling strength and the current decreases. Additionally, we have obtained that the interfaces nontrivially contributes, giving rise the displacement current, producing the capacitive response only due to non-orthogonality, and its influence on transport properties is one of the topics of present work.

 

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