With the fast development and improvement of experimental techniques 123456789, much important physical properties in QD molecules such as electronic structures, electronic transport, and thermoelectric effects et al have widely attracted academic attention 1011121314151617181920212223242526272829. QDs can be realized by etching a two-dimensional electron gas (2DEG) below the surface of AlGaAs/GaAs heterostructures or by an electrostatic potential. Confinement of particles in all three spatial directions results in the discrete energy levels such like an atom or a molecule. We can therefore think of QDs as artificial atoms or molecules. The small sizes of QDs make the phase-coherent of waves become more important, and quantum interference phenomena emerge when the particles moves along different transport paths. Fano resonances, known in the atomic physics, arise from quantum interference effects between resonant and nonresonant processes 30. The main embodying of the Fano resonances is the asymmetric line profile in the transmission spectrum, which originates from the coexistence the resonant transmission peak and the resonant transmission dip. The first experiment observation of the asymmetrical Fano line shape in the QD system has been reported in a single-electron transistor 31.

The RSOI in the QD can be introduced by an asymmetrical-interface electric field applied to the semiconductor heterostructures 3233. Electron spin, the intrinsic properties of electrons, become more important when electrons transport through the AB interferometer. The RSOI can couple the spin degree of freedom to its orbital motion, which provides a possible method to control the spin of transport electrons. A spin transistor by using the RSOI in a semiconductor sandwiched between two ferromagnetic electrodes has been proposed 34. In spin Hall devices, spin-up and spin-down electrons flow in an opposite direction using the Rashba SOI and a longitudinal electric field such that the spin polarization becomes infinity 353637. Some theoretical and experimental works have also shown that the spin-polarization of current based on the RSOI can reach as high as 100%3839 or infinite 40.

Recently, an experimental measurement of the spin Seebeck effect (the conversion of heat to spin polarization) by detecting the redistribution of spins along the length of a sample of permalloy (NiFe) induced by a temperature gradient was firstly demonstrated 41. The new heat-to-electron spin discovery can be named as "thermo-spintronics". More recently, the spin Seebeck effect was also observed in a ferromagnetic semiconductor GaMnAs 42. Much academic work on spin-dependent thermoelectric effects in single QD attached to ferromagnetic leads with collinear magnetic moments or noncollinear magnetic moments has been reported 43444546. Up to now, we note that most of the spin Seebeck effects are obtained by using ferromagnetic materials such as ferromagnetic thin films, ferromagnetic semiconductors, or ferromagnetic electrodes et al. In our previous work, a pure spin generator consisting of a Rashba quantum dot molecule sandwiched between two non-ferromagnetic electrodes via RSOI instead of ferromagnetic materials has been proposed by the coaction of the magnetic flux 24. It should be noted that charge thermopower of QD molecular junctions in the Kondo regime and the Coulomb blockade regime have been widely investigated 2526272829.

In the present work, we investigate the spin-dependent thermoelectric effects of parallel-coupled double quantum dots embedded in an AB interferometer, in which the RSOI in each QD is considered by introducing a spin-dependent phase factor in the linewidth matrix elements. Due to the quantum destructive interference, an asymmetrical transmission node can be observed in the transmission spectrum in the absence of the RSOI. Using an inversion asymmetrical interface electric field, the RSOI can be introduced in the QDs. The asymmetrical transmission node splits into two spin-dependent asymmetrical transmission nodes in the transmission spectrum and, as a consequence, results in the enhancement of the spin-dependent Seebeck effects near the spin-dependent asymmetrical transmission nodes. We also examine the evolution of spin-dependent Seebeck effects from a symmetrical parallel geometry to a configuration in series. The asymmetrical couplings between QDs and non-ferromagnetic electrodes induce the enhancement of spin-dependent Seebeck effects in the vicinity of spin-dependent asymmetrical transmission nodes. Although the spin-dependent Seebeck effects in the AB interferometer have not been realized experimentally so far, our theoretical study provides a better way to enhance spin-dependent Seebeck effects in the AB interferometer in the absence of the ferromagnetic materials.

The schematic diagram for the quantum device based on parallel-coupled double quantum dots embedded in an AB interferometer in the present work is illustrated in Figure 1, and two noninteracting QDs embedded in the AB interferometer. QDs can be realized in the two-dimensional electron gas of an AlGaAs/GaAs heterostructure, in which a tunable tunneling barrier between the two dots is formed by using two gate voltages. So we can set t_c as the coupling between the two QDs, which can be modulated by using the gate voltages 1. The RSOI is assumed to exist inside QDs, which can produce two main effects including a spin-dependent extra phase factor in the tunnel matrix elements and interlevel spin-flip term 4748. In the present paper, we only consider the first term because of only one energy level in each QD. When a temperature gradient ΔT between the two metallic electrodes is presented, a spin-dependent thermoelectric voltage ΔV_↑(↓) emerges. The proposed spin-dependent thermoelectric AB interferometer can be described by using the following Hamiltonian in a second-quantized form as,

H t o t a l = ∑ α = L , R ; k σ ϵ α k σ a α k σ † a α k σ + ∑ n = 1 , 2 ; σ ϵ n d n σ † d n σ - t c ( d 1 σ † d 2 σ + H . c . ) + ∑ k , α , σ , n [ V α σ n d n σ † a α k σ + H . c . ] ,

where a α k σ † ( a α k σ ) is the creation(annihilation) operator for an electron with energy ε_αkσ, momentum k and spin index σ in electrode α. The electrode α can be regarded as an independent electron and thermal reservoirs, which can be described by using the Fermi-Dirac distribution such as f_α = 1/{exp[(ε - μ_α)/(k_BT_α) + 1}. Here k_B is the Boltzmann constant. d n σ † ( d n σ ) creates (destroys) an electron with energy ε_n and spin index σ in the nth QD. t_c describes the tunnel coupling between the two QDs, which can be controlled by using the voltages applied to the gate electrodes 1. The tunnel matrix element V_ασn in a symmetric gauge is assumed to be independent of momentum k, and it can be written as V L σ 1 = ∣ V L σ 1 ∣ e i ( ϕ - σ φ R ) ∕ 4 , V L σ 2 = ∣ V L σ 2 ∣ e - i ( ϕ - σ φ R ) ∕ 4 , V R σ 1 = ∣ V R σ 1 ∣ e - i ( ϕ - σ φ R ) ∕ 4 , V R σ 2 = ∣ V R σ 2 ∣ e i ( ϕ - σ φ R ) ∕ 4 , with the AB phase ϕ = 2πΦ/Φ₀ and the flux quantum Φ₀ = h/e. Φ can be calculated by the equation ϕ = B → ⋅ S → , where B is the magnetic field threading the AB interferometer and S is the corresponding area of the quantum ring consisting of the double quantum dots and metallic electrodes. The value S may be obtained in the previous well-known experimental work 1. So the magnitude of the magnetic field B is 16.4mT when ϕ = 2π. In the absence of the RSOI, the work will come back to the previous work 24, in which a 2π-periodic linear conductance is obtained, and it is in good agreement with the experimental work 1. φ_R denotes the difference between φ_R1 and φ_R2, where φ_Ri is the phase factor induced by the RSOI inside the ith QD.

In the steady state, using the Green's functions and Dyson's equations, the electric current with spin index σ through the AB interferometer can be calculated by 49,

I σ ( μ L , T L ; μ R , T R ) = e h ∫ d ε τ σ ( ε ) [ f L ( ε ) - f R ( ε ) ] ,

and the thermal current with spin index σ from the electrode α is calculated by 50,

J σ α = 1 h ∫ d ε ( ε - μ α ) τ σ ( ε ) [ f L ( ε ) - f R ( ε ) ] ,

where τ_σ(ε) is the transmission probability of electron with spin index σ, which can be given by τ σ ( ε ) = T r [ Γ σ L G σ r Γ σ R G σ a ] . The spin-dependent linewidth matrix Γ σ L ( R ) describes the tunnel coupling of the two QDs to the left (right) metallic electrode, which can be expressed as,

Γ σ L ( R ) ( ε ) = Γ 11 L ( R ) Γ 11 L ( R ) Γ 22 L ( R ) e + ( - ) i ϕ σ ∕ 2 Γ 11 L ( R ) Γ 22 L ( R ) e - ( + ) i ϕ σ ∕ 2 Γ 22 L ( R ) ,

where Γ n m α = 2 π ∑ k ∣ V α σ n ∥ V α σ m * ∣ δ ( ε - ε α k σ ) . G σ r ( ε ) is the 2 × 2 matrix of the fourier transform of retarded QD Green's function, and its matrix elements in the time space can be defined as G n σ , m σ r ( t ) = - i Θ ( t ) < { d n σ ( t ) , d m σ † ( 0 ) } > , where Θ(t) is the step function. The advanced dot Green's function can be obtained by the relation G σ a ( ε ) = [ G σ r ( ε ) ] + .

We consider the quantum system in the linear response regime such as an infinitesimal temperature gradient ΔT raised in the right metallic electrode, which will induce an infinitesimal spin-dependent thermoelectric voltage ΔV_σ since the two tunneling channels related to spin are opened. We divide the tunneling current into two parts: one is from the temperature gradient ΔT, which is calculated by Δ I σ T = I σ ( μ , T ; μ , T + Δ T ) ; the other is from the Seebeck effects, which can be calculated by Δ I σ V = I σ ( μ , T ; μ + e Δ V σ , T ) . The spin-dependent Seebeck coefficient S_σ can be calculated by 50,

Δ I σ T + Δ I σ V = 0 .

After expanding the Fermi-Dirac distribution function to the first order in ΔT and ΔV_σ, we obtain the spin-dependent Seebeck coefficient by S_σ = ΔV_σ/ΔT as,

S σ ( μ , T ) = - 1 e T K 1 σ ( μ , T ) K 0 σ ( μ , T ) .

where K ν σ ( μ , T ) = ∫ d ε ( - ∂ f ∂ ε ) ( ε - μ ) ν τ σ ( ε ) ( ν = 0 , 1 , 2 ) . f = {1 + exp[(ε - μ)/(k_BT)]}^-1 denotes the zero bias fermi distribution (μ = μ_L = μ_R) and zero temperature gradient (T = T_L = T_R). The spin-dependent Seebeck effects can be measured in the experiments as the following descriptions. First, the AB interferometer based on DQD molecular junction can be realized by using a two-dimensional electron gas below the surface of an AlGaAs/GaAs heterostructure 1. The RSOI in the QD can be introduced by using an asymmetrical-interface electric field. The temperature of the left electrode is kept at a constant, and that of the right electrode can be heated to a desired temperature by using an electric heater. So a temperature gradient can be generated in the DQD molecular junction. Second, the spin-dependent thermoelectric voltage can be measured by using the spin-detection technique involving inverse-spin-Hall effect 5152. Accompanying the electric charge flowing, the energy of electrons can also be carried from one metallic electrode to the other metallic electrode. In the linear response regime (μ_L = μ_R = μ), we assume that an infinitesimal temperature gradient ΔT is raised in the right metallic electrode, and the heat current Δ J σ ( Δ J σ = Δ J σ α ) is divided into two parts following one from the temperature gradient Δ J σ T and the other from the Seebeck effects Δ J σ V . They can be obtained by the equations Δ J σ T = J σ ( μ , T ; μ , T + Δ T ) and Δ J σ V = J σ ( μ , T ; μ + e Δ V σ , T ) . The total thermal current can be calculated by the sum of two terms as 50,

Δ J σ = Δ J σ T + Δ J σ V .

The corresponding electronic thermal conductance κ_el can be defined by κ e l = Δ J σ Δ T . After expanding the Fermi-Dirac distribution function to the first order in ΔT and ΔV_σ to Eq. (7), we obtain the electronic thermal conductance from the temperature gradient,

κ e l , σ T ( μ , T ) = K 2 σ h T ,

and the electronic thermal conductance from the Seebeck effects,

κ e l , σ V ( μ , T ) = K 1 σ e S σ h .

The differential conductance with spin index σ may be expressed as G σ ( μ , T ) = e 2 h K 0 σ ( μ , T ) . In the linear response regime, the charge and spin figure-of-merits (FOMs) can be defined as,

Z C T = S c 2 G c T ∑ σ κ e l , σ T + ∑ σ κ e l , σ V ,

and

Z S T = S S 2 G s T ∑ σ κ e l , σ T + ∑ σ κ e l , σ V ,

respectively, where G c = e 2 h [ K 0 ↑ ( μ , T ) + K 0 ↓ ( μ , T ) ] and G s = e 2 h [ K 0 ↑ ( μ , T ) - K 0 ↓ ( μ , T ) ] . In this study, the phonon thermal conductance of the junction, which is typically limited by the QDs-electrode contact, has been ignored in the case of the poor link for phonon transport.

In the following numerical calculations, we set Г = 1ev as the energy unit in this paper. For simplicity, the energy levels of QDs are identical (ε₁ = ε₂ = 0).

In Figure 2, we plot the spin-dependent transmission probability τ_σ, spin-dependent See-beck coefficient S_σ, and spin-dependent Lorenz number L σ = h ( κ e l , σ T + κ e l , σ V ) ∕ ( e 2 τ σ T ) as functions of the chemical potential μ under several different values of ϕ at room temperature (T = 300 K). The phase factor ϕ_R due to the RSOI inside the QD is fixed at π 2 , which is reasonable in semiconductor heterostructures 51525354. We first consider the case of the AB interferometer with symmetrical parallel geometry λ = 1 and a magnetic flux ϕ threading through the AB interferometer. When the interdot tunnel coupling is considered (t_c = Г₀), the transmission probability τ_σ has an exact expression,

τ σ = ( t c - μ cos ϕ σ 2 ) 2 Ω ( μ ) ,

where Ω ( μ ) = [ ( μ 2 - t c 2 ) ∕ ( 2 Γ 0 ) - Γ 0 2 sin 2 ϕ σ 2 ] 2 + ( μ - t c cos ϕ σ 2 ) 2 . After a simple derivation, the transmission probability τ_σ has an approximate expression as,

τ σ ≃ τ - σ ( μ ) + τ + σ ( μ ) ,

where τ - σ ( μ ) = 1 1 + q - σ 2 [ ( μ - t c ) + q - σ Γ - σ ] 2 ( μ - t c ) 2 + Γ - σ 2 and τ + σ ( μ ) = 1 1 + q + σ 2 [ ( μ + t c ) + q + σ Γ + σ ] 2 ( μ + t c ) 2 + Γ + σ 2 . The parameter, q_{± σ} = ± t_c/Г_∓σ, describes the degree of electron phase coherence between two different paths. For example, one is the path through the bonding molecular state, and the other is the path through the antibonding molecular state. Г_{± σ} is the expanding function due to the coupling between the bonding (antibonding) molecular state and metallic electrodes, which is given by Γ ± σ = Γ 0 ± Γ 0 cos ( ϕ σ 2 ) . When the spin-dependent electron phase is considered, the transmission spectrum is composed of four resonant peaks, and their asymmetrical degrees can thus be marked by the parameter q_{± σ}. In the absence of the interdot tunnel coupling (t_c = 0), a symmetrical transmission node (q_{± σ} = 0) arising from the quantum destructive interference is obtained. In the presence of the interdot tunnel coupling (t_c = Г₀) and absence of the magnetic flux (ϕ = 0), the relation between the spin-up and spin-down phase factors owns ϕ_↑ = -ϕ_↓. The transmission probability τ_σ, Seebeck coefficient S_σ and Lorenz number L_σ become spin-independent as shown in Figure 2), 1), and 1), respectively. In this case, the transmission probability τ_σ as a function of the chemical potential displays a near symmetrical Breit-Wigner peak centered at the bonding molecular state and an asymmetrical Fano line shape centered at the antibonding molecular state. The degree of the asymmetry of the Fano-Like peak can be attributed to the electron phase coherence. In the table 1, we calculate the approximate values of q_{± σ} of four resonate peaks for different AB phase ϕ with ϕ_R = 0.5π. For ϕ = 0, we find q_+↑ = q_+↓≃ 6.8 (near symmetrical Breit-Wigner peak at energy -t_c) and q_-↑ = q_-↓≃ -1.2 (Fano-Like peak at energy t_c). According to Eq. (12), an asymmetrical transmission node centered at energy t_c/cos(ϕ_R/2) can be found as shown in Figure 2 (a1). So we find that Seebeck coefficient S_↑ = S_↓ is enhanced strongly in the vicinity of the asymmetrical transmission node, and the corresponding value of Lorenz number L_↑ = L_↓ in units of L_WF at the asymmetrical transmission node approaches to a temperature-independent value of 4.2 55. Once the AB phase ϕ is presented, the asymmetrical transmission node splits into two spin-dependent asymmetrical transmission nodes at energies t_c/cos(ϕ_σ/2). S_↑ and L_↑ are enhanced strongly in the vicinity of energy t_c/cos(ϕ_↑/2), and S_↓ and L_↓ are enhanced strongly in the vicinity of energy t_c/cos(ϕ_↓/2). Some interesting features in table 1 and Figure 2 should be noted as the following expressions. First, q_{± σ} has a negative value when the spin-dependent molecular states are located at the high energy region, while q_{± σ} has a positive value when they are located at the low energy region. We also find that the region of the enhanced thermoelectric effects appears at the molecular states with the lower value of |q_{± σ}|. For example, when ϕ = 0.25π and ϕ_R = 0.5π, S_↑ is enhanced strongly in the vicinity of the molecular states with q_-↑ = -1.0, and S_↓ can be enhanced strongly in the vicinity of the molecular states with q_-↓ = -1.4. Second, S_σ always has a larger positive value when q_{± σ} < 0, and S_σ has a smaller negative value when q_{± σ} > 0. The last feature is that one spin component of Seebeck effects can be tuned while the other spin component is retained. The behind reason is that the behavior of the spin-dependent transmission as a function of the chemical potential is dominated by the level expanding functions Г_{± σ}, which gives rise to a similar behavior of the Seebeck effects as a function of the chemical potential.

In Figure 3, we calculate κ e l , σ V ( T ) , Z_CT and Z_ST as functions of the chemical potential for the different values of ϕ. The results show that κ e l , σ T and τ_σ has a similar behavior due to κ e l , σ T ∝ τ σ in the lower temperature region. κ e l , σ V has a negative value for the whole energy region due to κ e l , σ V ∞ - ( τ ′ ( μ ) ) 2 , and it should be noted that κel,σV has an obvious negative value in the vicinity of transmission peak with |q_{± σ}| ≃ 1.0 as shown in Figure 3 (a2), (a3), (a4), (a5), and (a6). Z_CT and |Z_ST| are enhanced strongly in the vicinity of transmission peaks with |q_{± σ}| ≃ 1.0 and |q_{± σ}| ≃ 1.4. The magnitude of |Z_ST| can approach to that of Z_CT in the vicinity of transmission peaks with |q_{± σ}| ≃ 1.4. The results indicate that a near pure spin thermoelectric generator can be obtained by tuning the AB phase ϕ with a fixed value of ϕ_R.

A detail study of the spin-dependent thermoelectric effects is presented in Figure 4 when the configuration of the AB interferometer evolves from a symmetrical parallel geometry to a series. The AB phase ϕ and ϕ_R are chosen an identical value ϕ = ϕ_R = π. The spin-dependent transmission probability τ_σ has the following expression as,

τ σ = [ 1 + λ 2 t c ∓ λ μ ] 2 Ω ( μ ) ,

where Ω ( μ ) = [ μ 2 - t c 2 2 Γ 0 - ( 1 - λ ) 2 Γ 0 8 ] 2 + [ 1 + λ 2 μ ∓ λ t c ] 2 . When λ = 1, we have a simple expression for τ_σ as,

τ σ = 4 Γ 0 2 ( μ ± t c ) 2 + 4 Γ 0 2 ,

where + for spin up and - for spin down. Eq. (15) shows the symmetrical spin-dependent Breit-Wigner peaks centered at ± t_c as shown in Figure 4). The corresponding q_-↑ and q_+↓ become infinity (see table 2). When λ = 0, the two QDs in a serial configuration are sandwiched between two metallic electrodes, in the case, the linear transmission probability become spin-independent due to the absence of the AB phase. The transmission probability can be calculated by the following expression,

τ ↑ = τ ↓ = t c 2 Γ 0 ( μ 2 - t c 2 - Γ 0 2 4 ) 2 + μ 2 Γ 0 2 .

We note that the transmission probability vanishes when t_c = 0, which means the full reflection for electrons happening in this AB interferometer. When 0 < λ < 1, the spin-dependent transmission probability τ_σ is composed of near Breit-Wigner peak and Fano line shapes as shown in Figure 4 and 3. The spin-dependent transmission probability can be approximated by,

τ σ ≃ τ + σ ( μ ) + τ - σ ( μ ) ,

where τ - σ ( μ ) = 1 1 + q - σ 2 [ ( μ - t c ) + q - σ Γ - σ ] 2 ( μ - t c ) 2 + Γ - σ 2 and τ + σ ( μ ) = 1 1 + q + σ 2 [ ( μ + t c ) + q + σ Γ + σ ] 2 ( μ + t c ) 2 + Γ + σ 2 with Γ ± σ = ( 1 + λ ) Γ 0 ∕ 2 ± λ Γ 0 . From Eq. (14), we can see clearly that there are two asymmetrical transmission nodes centered at,

μ = ± 1 + λ 2 λ t c ,

where + means spin up case and - represents spin-down case. As a result, we find that the spin-dependent Seebeck effect is enhanced strongly in the vicinity the spin-dependent transmission nodes. The electronic thermal conductance κ e l V ( T ) , Z_CT and Z_ST as functions of the chemical potential under different values of λ are displayed in Figure 5. κel,σT has a similar behavior with the transmission probability as the chemical potential changes. κel,σV has an obvious negative values in the vicinity of the spin-dependent transmission node. Similarly, Z_CT and Z_ST are enhanced strongly in the vicinities of the transmission nodes. As λ increases from 0 to 1, we find the maximum values of Z_CT and Z_ST become larger. The corresponding q_+↓ and |q_-↑| decrease, while q_+↑ and |q_-↓| increase as λ increases (see table 2).

We investigate the spin-dependent thermoelectric effects of parallel-coupled DQDs embedded in an AB interferometer in which the RSOI is considered by introducing a spin-dependent phase factor in the linewidth matrix elements. Due to the interplay between the quantum destructive interference and RSOI in the QDs, an asymmetrical transmission node can be observed in the transmission spectrum in the absence of the RSOI. Using an inversion asymmetrical interface electric field, we can induce the RSOI in the QDs. We find that the asymmetrical transmission node splits into two spin-dependent asymmetrical transmission nodes in the transmission spectrum, which induces that the spin-dependent Seebeck effects are enhanced strongly at different energy regimes. We also examine the evolution of spin-dependent Seebeck effects from a symmetrical parallel geometry to a configuration in series. The asymmetrical couplings between the QDs and metallic electrodes induce the enhancement of spin-dependent Seebeck effects in the vicinity of the corresponding spin-dependent asymmetric transmission node in the transmission spectrum.

2DEG: two-dimensional electron gas; AB: Aharonov-Bohm; FOMs: figure-of-merits; QD: quantum dot; RSOI: Rashba spin-orbit interaction.

The authors declare that they have no competing interests.

All authors read and approved the final manuscript.