_{1}

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 occasions in the interacting case, and also compare them with the results in the noninteracting case.

With advantage of top-down and bottom-up fabrication techniques for nanometer scale structures, it becomes possible to create quantum wire (QW) with the diameter of the order of the Fermi wavelength, and to experimentally study the quantum transport properties through them [

In this study, the rigorous formulas of conductance, transport current and charge distributions for the three-site QW model are provided within Hartree-Fock approximation, based on the nonequilibrium transport theory (Keldysh formalism) [

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 _{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.

H ^ = ∑ k σ ( ε k σ , L c ^ k σ , L + c ^ k σ , L + ε k σ , R c ^ k σ , R + c ^ k σ , R ) + ε 1 σ d ^ 1 σ + d ^ 1 σ + ε 2 σ d ^ 2 σ + d ^ 2 σ + ε 3 σ d ^ 3 σ + d ^ 3 σ + ∑ σ ( t 12 σ d ^ 1 σ + d ^ 2 σ + t 12 σ * d ^ 2 σ + d ^ 1 σ + t 23 σ d ^ 2 σ + d ^ 3 σ + t 23 σ * d ^ 3 σ + d ^ 2 σ ) + U 1 d ^ 1 ↑ + d ^ 1 ↑ d ^ 1 ↓ + d ^ 1 ↓ + U 2 d ^ 2 ↑ + d ^ 2 ↑ d ^ 2 ↓ + d ^ 2 ↓ + U 3 d ^ 3 ↑ + d ^ 3 ↑ d ^ 3 ↓ + d ^ 3 ↓ + ∑ k σ ( V L , k σ c ^ k σ , L + d ^ 1 σ + V L , k σ * d ^ 1 σ + c ^ k σ , L ) + ∑ k σ ( V R , k σ c ^ k σ , R + d ^ 3 σ + V R , k σ * d ^ 3 σ + c ^ k σ , R ) (1)

We consider a system consisting of three regions, a left electrode, a right electrode and a intermediate wire, they uncouple and each one maintains its noninteracting thermal equilibrium when t = − ∞ , then turn on the perturbation coupling between the wire and electrodes adiabatically with a route of t = − ∞ → 0 → + ∞ → − ∞ (Keldysh contour). According to the quantum statistical theory (perturbation expansion), any nonequilibrium observable physical quantity at time t can be expressed exactly by Keldysh Green’s Functions (GF), such as correlation function G i j < ( t , t 0 ) = i 〈 c ^ i + ( t ) c ^ j ( t 0 ) 〉 and retarded/advanced GF G i j r ( a ) ( t , t 0 ) = ∓ i θ [ ± ( t − t 0 ) ] 〈 c ^ i ( t ) c ^ j + ( t 0 ) + c ^ j + ( t 0 ) c ^ i ( t ) 〉 (Keldysh formalism) [

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:

[ Σ p q ( t ) − − ( ε ) Σ p q ( t ) < ( ε ) Σ p q ( t ) > ( ε ) Σ p q ( t ) + + ( ε ) ] = ∑ k σ V L , k σ * δ p , 1 σ δ q , k σ − L [ 1 0 0 − 1 ] + ∑ k σ V L , k σ δ p , k σ − L δ q , 1 σ [ 1 0 0 − 1 ] + ∑ k σ V R , k σ * δ p , N σ δ q , k σ − R [ 1 0 0 − 1 ] + ∑ k σ V R , k σ δ p , k σ − R δ q , N σ [ 1 0 0 − 1 ] (2a)

Σ p q ( t ) − − ( ε ) = ∑ k σ V L , k σ * δ p , 1 σ δ q , k σ − L + ∑ k σ V L , k σ δ p , k σ − L δ q , 1 σ + ∑ k σ V R , k σ * δ p , N σ δ q , k σ − R + ∑ k σ V R , k σ δ p , k σ − R δ q , N σ ,

Σ p q ( t ) + + ( ε ) = − ∑ k σ V L , k σ * δ p , 1 σ δ q , k σ − L − ∑ k σ V L , k σ δ p , k σ − L δ q , 1 σ − ∑ k σ V R , k σ * δ p , N σ δ q , k σ − R − ∑ k σ V R , k σ δ p , k σ − R δ q , N σ ,

Σ p q ( t ) < ( ε ) = 0 , Σ p q ( t ) > ( ε ) = 0 (2b)

Σ n 1 σ , n 2 σ ' ( i ) − − ( ε ) = U n σ ¯ ρ n σ ¯ , n σ ¯ ( ε ) / e , Σ p q ( i ) > ( ε ) = 0 ,

Σ n 1 σ , n 2 σ ' ( i ) + + ( ε ) = ( − U n σ ¯ ) ρ n σ ¯ , n σ ¯ ( ε ) / e , Σ p q ( i ) < ( ε ) = 0 (3)

Corresponding retarded/advanced self-energies are given by

i Γ α σ r ( ε ) = − i π ν α , σ ( ε ) | V α , k σ | 2 , i Γ α σ a ( ε ) = i π ν α , σ ( ε ) | V α , k σ | 2 (4)

where ν α , σ ( ε ) is the density-of-states (DOS) in the electrodes, α = L, R.

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 [

I σ = e ℏ ∑ k σ , n ∫ − ∞ + ∞ d ε 2 π [ V k σ − L , n σ G n , k σ − L < ( ε ) − V k σ − L , n σ * G k σ − L , n < ( ε ) ] (5)

ρ n σ = e ( − i ) 1 2 π ∫ − ∞ + ∞ d ε G n σ , n σ < ( ε ) (6)

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

J σ ( μ L σ , μ R σ ) = e h ∫ − ∞ + ∞ d ε ( f μ L σ − f μ R σ ) 4 Γ L σ r Γ R σ r | t 12 σ t 23 σ | 2 | B σ r | 2 (7)

the spin conductance is

G σ ( μ L ( R ) σ ) = e 2 h ∫ − ∞ + ∞ d ε { 4 Γ L σ r Γ R σ r | t 12 σ t 23 σ | 2 | B σ r | 2 ( 1 4 k B T 1 cosh 2 ( ε − μ L ( R ) σ 2 k B T ) ) } (8)

and the up-spin electron charges on each site of the wire are expressed by

ρ 1 ↑ , 1 ↑ ( μ L ↑ , μ R ↑ ) = e π ∫ − ∞ + ∞ d ε { f μ L ↑ Γ L ↑ r [ ( ( x 2 ↑ − U 2 ρ 2 ↓ , 2 ↓ / e ) ( x 3 ↑ − U 3 ρ 3 ↓ , 3 ↓ / e ) − | t 23 ↑ | 2 ) 2 + ( Γ R ↑ r ) 2 ( x 2 ↑ − U 2 ρ 2 ↓ , 2 ↓ / e ) 2 ] + f μ R ↑ Γ R ↑ r | t 12 ↑ t 23 ↑ | 2 | B ↑ r | 2 }

ρ 2 ↑ , 2 ↑ ( μ L ↑ , μ R ↑ ) = e π ∫ − ∞ + ∞ d ε { f μ L ↑ Γ L ↑ r [ ( x 3 ↑ − U 3 ρ 3 ↓ , 3 ↓ / e ) 2 + ( Γ R ↑ r ) 2 ] | t 12 ↑ | 2 + f μ R ↑ Γ R ↑ r [ ( x 1 ↑ − U 1 ρ 1 ↓ , 1 ↓ / e ) 2 + ( Γ L ↑ r ) 2 ] | t 23 ↑ | 2 | B ↑ r | 2 } (9)

ρ 3 ↑ , 3 ↑ ( μ L ↑ , μ R ↑ ) = e π ∫ − ∞ + ∞ d ε { f μ R ↑ Γ R ↑ r | t 12 ↑ t 23 ↑ | 2 + f μ R ↑ Γ R ↑ r [ ( ( x 1 ↑ − U 1 ρ 1 ↓ , 1 ↓ / e ) ( x 2 ↑ − U 2 ρ 2 ↓ , 2 ↓ / e ) − | t 12 ↑ | 2 ) 2 + ( Γ L ↑ r ) 2 ( x 2 ↑ − U 2 ρ 2 ↓ , 2 ↓ / e ) 2 ] | B ↑ r | 2 }

where x 1 σ = ε − ε 1 σ , x 2 σ = ε − ε 2 σ , x 3 σ = ε − ε 3 σ . Value B is given by

B ↑ r = [ ε − ( ε 1 ↑ + U 1 ρ 1 ↓ , 1 ↓ / e ) + i Γ L ↑ r ] [ ε − ( ε 2 ↑ + U 2 ρ 2 ↓ , 2 ↓ / e ) ] × [ ε − ( ε 3 ↑ + U 3 ρ 3 ↓ , 3 ↓ / e ) + i Γ R ↑ r ] − [ ε − ( ε 1 ↑ + U 1 ρ 1 ↓ , 1 ↓ / e ) + i Γ L ↑ r ] | t 23 ↑ | 2 − [ ε − ( ε 3 ↑ + U 3 ρ 3 ↓ , 3 ↓ / e ) + i Γ R ↑ r ] | t 12 ↑ | 2 (10)

The electron charge formulas for down-spin can be obtained by exchanging the subscript ↑ and ↓ in the up-spin formulas above.

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 ε i σ = 0 , t 12 ↑ ↓ = t 23 ↑ ↓ = t , V L , k ↑ ↓ = V R , k ↑ ↓ = V , Γ L ↑ ↓ ( ε ) = Γ R ↑ ↓ ( ε ) = Γ and U 1 = U 2 = U 3 = U All of the energies are normalized by the transfer integral t. Especially, the normalized self-energy is defined as γ = Γ / t in the following numerical calculations.

The numerical results of conductance, current and electron charges in the three sites of the wire as a function of electrochemical potential μ for several values of γ are illustrated in Figures 2-4.

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 _{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.

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

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

In

of μ_{L} (μ_{R} = −5) for γ = 0.2 and 1 when U has a large value of 4. Especially, in the case of γ = 0.2, when μ_{L} approaches some positions where the conductance has peak values in

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.

This research was supported in part by Grants-in-Aid for the basic research and development of Mitsubishi Electric (China) Company Limited.

The author declares no conflicts of interest regarding the publication of this paper.

Zheng, Y.D. (2019) Analysis of Nonequilibrium Transport Properties of Interacting Quantum Wire Models. Journal of Applied Mathematics and Physics, 7, 1677-1685. https://doi.org/10.4236/jamp.2019.78114