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Transport in one-dimensional wires: the role of reservoirs

Abhishek Dhar, B. Sriram Shastry

Published Sep 23, 2002• 0 citations
OpenAlex

Abstract

Using the formalism of Ford-Kac-Mazur we study nonequilibrium transport in fermionic systems connected to general particle and thermal reservoirs. Several exact results on electrical and thermal transport in one-dimensional disordered non-interacting fermionic wires are obtained. Landauer’s formula is shown to follow for a special choice of reservoirs. For general reservoirs the currents in the system depend on the spectral properties of the reservoirs and Landauer’s formula needs modification. Some experimental implications are discussed.

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Using the formalism of Ford-Kac-Mazur we study nonequilibrium transport in fermionic systems connected to general particle and thermal reservoirs. Several exact results on electrical and thermal transport in one-dimensional disordered non-interacting fermionic wires are obtained. Landauer’s formula is shown to follow for a special choice of reservoirs. For general reservoirs the currents in the system depend on the spectral properties of the reservoirs and Landauer’s formula needs modification. Some experimental implications are discussed.
We describe microscopic theory for the quantum transport through finite interacting systems connected to noninteracting leads. It can be applied to small systems such as quantum dots, quantum wires, atomic chain, molecule, and so forth. The Keldysh formalism is introduced to study the nonlinear current-voltage characteristics, and general properties of the nonequilibrium Green's functions are provided. We apply the formulated to an out-of-equilibrium Anderson model that has been used widely for the Kondo effect in quantum dots. In the linear-response regime the Kubo formalism still has an advantage, because it relates the transport coefficients directly to the correlation functions defined with respect to the thermal equilibrium, and it has no ambiguities about a profile of the inner electric field. We introduce a many-body transmission coefficient, by which the dc conductance can be expressed in a Landauer-type form, quite generally. We also discuss transport properties of the Tomonaga-Luttinger liquid, which has also been an active field of research in this decade.
The electrical current through an arbitrary junction connecting quantum wires of spinless interacting fermions is calculated in fermionic representation. The wires are adiabatically attached to two reservoirs at chemical potentials differing by the applied voltage bias. The relevant scale-dependent contributions in perturbation theory in the interaction up to infinite order are evaluated and summed up. The result allows one to construct renormalization group equations for the conductance as a function of voltage (or temperature, wire length). There are two fixed points at which the conductance follows a power law in terms of a scaling variable $\ensuremath{\Lambda}$, which equals the bias voltage VVV, if VVV is the largest energy scale compared to temperature TTT and inverse wire length ${L}^{\ensuremath{-}1}$, and interpolates between these quantities in the crossover regimes.
The transport in a pure one-dimensional quantum wire is investigated for any range of interactions. First, the wire is connected to measuring leads. The transmission of an incident electron is found to be perfect, i.e., the conductance is not renormalized by the interactions. Either Landauer's approach or the Kubo formula can be used as long as the reservoirs impose the boundary conditions. Second, the Kubo formula as a response to the local field is reconsidered in a generic Luttinger liquid: the ``intrinsic'' conductance thus obtained is determined by the same combination of interaction parameters as that which renormalizes the current.