Analysis of Nonequilibrium Transport Properties of Interacting Quantum Wire Models

We analyze nonequilibrium electronic transport properties of a typical interacting three-site quantum wire model within Hartree-Fock approximation making use of Keldysh formalism. Some rigorous formulas are provided for direct calculations when Coulomb repulsion is present. According to numerical calculations using above formulas, we investigate the conductance, transport currents, and on site electronic charges of the wire on some special occa-sions in the interacting case, and also compare them with the results in the noninteracting case.

[4] [5]. From these formulas, the relevance between the transport properties and the temperature or the parameters in the Hamiltonian is expressed clearly and can be investigated in detail. In the noninteracting case (U = 0), we focus on the resonant tunneling transport and conductance quantization phenomenon.
While in the interacting case (U > 0), we investigate Coulomb blockade and metal-insulator transition, as well as spin transport properties in the QW. It is reasonable to consider that the results of our study are available in the complicated case of actual QWs which are probably longer and thicker containing a larger number of atoms (sites) having multiple levels.

Model
We consider a one-dimensional QW with three lattice sites which are mutually coupled by tunneling barriers. They are combined with two external electrodes as shown in Figure 1. The tight-binding Hamiltonian of such system is described by Equation (1). Here denote the tunnel combination integrals between those boundary sites and the electrodes. The on-site Coulomb repulsion energies are denoted by U i . When bias voltage V is applied to the wire, it can be regarded as electrochemical potentials, μ L and μ R , associate with the left and right electrode, respectively (eV = μ L − μ R ). We assume that the electrodes are electric reservoirs, the capacities of which are large enough that μ L and μ R are not perturbed by the transport current. In the case of μ L > μ R , electrons will flow from the left electrode to the right electrode. (1)

Formulation
We These GFs can be solved from Dyson-equation.
The self-energies resulting from wire-electrode coupling and Coulomb repulsion are derived within Hartree-Fock approximation and are shown in Equation (2) and Equation (3), respectively: Corresponding retarded/advanced self-energies are given by where , ( ) α σ ν ε is the density-of-states (DOS) in the electrodes, α = L, R. Journal of Applied Mathematics and Physics The single spin current flowing in the wire and the spin electron charge on the site n are given by Equation (5) and Equation (6), respectively [5]: From Equation (5) and Equation (6), the following transport formulas can be obtained by correlation functions calculations straightforwardly (f μσ is the Fermi distribution function).
The spin current is the spin conductance is   The electron charge formulas for down-spin can be obtained by exchanging the subscript ↑ and ↓ in the up-spin formulas above.

Numerical Results and Interpretations
In this section, we calculate the transport properties of the three-site QW in some special cases applying the formulas in previous section. We assume that    The behavior of conductance and transport current changes dramatically when the value of γ crosses unity. When γ < 1, the conductance has three maximums at μ/t = 0 and μ/t = 2 ± , and the corresponding current increases intermittently with a step shape. These phenomena imply that resonant tunneling and conductance quantization take place easily in this case. Whereas when γ ≥ 1, these quantum effects in transport will disappear gradually with the increase of γ. In the case of T > 0 K, the line shapes of the transport characteristics become not to change so much and become all smoother than those in T = 0 K due to the thermal fluctuations. The charges distributions shown in Figure 4 results in the fact that in the area of μ L < 0, a minus charge barrier will be formed at the boundary of the wire, whereas in the area of μ L > 0, a plus charge barrier will be formed.

Interacting Case (U > 0)
We select comparative small value of U (U < 5) to investigate Coulomb interaction effects in transport due to the limits of Hartree-Fock approximation. The transport properties are computed by self-consistent calculations concern with site charges ρ n ↑↓. The initial site charges are decided by the ground state of the three-site QW with half-filling (N = 3) assumption, which is an antiferromagnet state with total spin of +1/2.
The numerical results of spin conductance in the case of γ = 0.2 and 1 as a function of μ for several values of U are illustrated in Figure 5(1) and Figure  5(2), respectively. Compared with the case of U = 0, the conductance curves shift to right and peaks are broadened with the increase of U. When U > 2γ, the peaks of conductance start to split into two corresponding to up or down spin conductance. These phenomena all result from the changes of spin orbits in the wire due to the Coulomb repulsion between the up and down spin electrons on sites. The series of peaks and valleys in the conductance characteristics can be considered a synthetic effect of resonant tunneling and Coulomb blockade. One of the valley happen to shift on the Fermi-level of the wire (μ = 0), the metal-insulator transition (Mott transition) will takes place. We show the spin conductance as a function of U for several values of γ when μ = 0 in Figure 6. This result indicates that, if the self-energy γ has small value compared with Journal of Applied Mathematics and Physics  U, generally the spin conductance will rapidly decrease with the increase of U, and the wire becomes an insulator from a metal (Mott transition).
We illustrate the spin current as a function of left electrode potential μ L (μ R = −5) for several values of U in Figure 7(1) when γ = 0.2. Because the self-energy γ has a small value, the nonequilibrium spin current gradually decreases with the increase of U. Meanwhile, the line shape of the up spin current separates from that of the down spin current.
In Figure 7(2), we demonstrate the up and down spin current as a function Y. D. Zheng

Summary
Based on the Keldysh formalism, we provided some rigorous formulas of nonequilibrium electronic transport for a typical interacting three-site QW model within Hartree-Fock approximation when Coulomb repulsion is present. According to numerical calculations, we investigated the conductance, transport current and electronic charge distribution of the three-site QW in some special occasions. In the noninteracting case, when self-energy γ < 0, the resonant tunneling transport and the conductance quantization can be easily observed. The transport properties of up-spin are identical with those of down-spin. While if the Coulomb interaction is present, the conductance curves shift to right and the peaks are broadened with the increase of U because of electron-electron repulsions. When U > 2γ, the peaks of conductance split into two. The Coulomb blockade and metal-insulator transition (Mott transition) phenomena are obvious if γ has a small value compared with U. The conductance and transport current of the up-spin also become quite different from those of the down-spin indicating that the spin polarization takes place in the wire.