† Corresponding author. E-mail:
Project supported by the National Natural Science Foundation of China (Grant Nos. 11274208 and 11447170).
We have studied spin-dependent thermoelectric transport through parallel triple quantum dots with Rashba spin–orbital interaction (RSOI) embedded in an Aharonov–Bohm interferometer connected symmetrically to leads using nonequilibrium Green’s function method in the linear response regime. Under the appropriate configuration of magnetic flux phase and RSOI phase, the spin figure of merit can be enhanced and is even larger than the charge figure of merit. In particular, the charge and spin thermopowers as functions of both the magnetic flux phase and the RSOI phase present quadruple-peak structures in the contour graphs. For some specific configuration of the two phases, the device can provide a mechanism that converts heat into a spin voltage when the charge thermopower vanishes while the spin thermopower is not zero, which is useful in realizing the thermal spin battery and inducing a pure spin current in the device.
The efficiency of a material directly converting thermal energy into electrical energy or vice versa can be described by the dimensionless charge figure of merit (FOM)
Recently, the spin Seebeck effect has been observed in a metallic magnet using a spin-detection technique based on the spin Hall effect,[25] in which the temperature difference between the edges of a bulk ferromagnetic slab plays a role of the driving force for the spin voltage and spin current. This spin Seebeck effect is an analogy to the usual charge Seebeck effect, and can be applied directly to the thermal spin generators for driving spin current.[26–28] It is more interesting that a heat flow can also been converted into a spin voltage in the magnetic insulator despite the absence of conduction electrons.[29] Using a similar experiment geometry, one also observed the spin Seebeck effect in ferromagnetic semiconductors.[30] The influence of the ferromagnetic leads on the thermoelectric transport properties has also been investigated in the QD systems with ferromagnetic electrodes.[31–34] In addition, the quantum interference effects such as Fano effect, Dicke resonances, and Aharonov–Bohm oscillations have been proposed as methods of increasing the thermoelectric efficiency in the QD systems.[35–39] Thus many efforts have been devoted to improving the thermoelectric efficiency in the QD systems. In particular, the Rashba spin–orbit interaction (RSOI) in the QD was used to produce the spin-dependent thermoelectric effect. In Refs. [34] (single QD system) and [28] (DQD system), the coexistence of the RSOI in the QD and the leads’ spin polarization enhanced the figure of merit of the spin thermoelectric effects. Reference [27] investigated the thermospin effects in parallel coupled DQD under the coaction of the Rashba spin–orbit interaction and the Zeeman splitting in the DQD, mainly involving the effect of the magnetic field applied to the DQD on the thermoelectric coefficient. However, in multi-QD systems, the research on the spin-dependent thermoelectric properties and the influence of the interference effects remains few. In this paper, we study the spin-dependent thermoelectric transport through parallel triple quantum dots (TQDs) with Rashba spin–orbit interaction (RSOI) embedded in an Aharonov–Bohm interferometer connected symmetrically to the leads, in which the coexistence of a bound state in the continuum (BIC)[40] and a Fano resonance results in wide regions in the space of parameters of large spin thermopower. Due to the presence of the RSOI, both the magnetic flux phase and the RSOI phase affect the thermoelectric transport properties. The spin-dependent electrical and thermal conductances, the thermopower, and the figure of merit as functions of the QD energy level are calculated in the linear-response regime. Under the appropriate configuration of the magnetic flux phase and RSOI phase, the spin figure of merit can be enhanced and even is larger than the charge figure of merit. In particular, the thermopower in both the magnetic flux phase and the RSOI phase is studied when other parameters are given. We find that the device can provide a mechanism of thermal spin battery that converts the heat into a spin voltage and induces a pure spin current.
We consider a TQD molecule with RSOI coupled symmetrically to two leads (see Fig.
![](cpb_25_11_117307/cpb160511ieqn2.gif)
![](cpb_25_11_117307/cpb160511ieqn3.gif)
![](cpb_25_11_117307/cpb160511ieqn4.gif)
![](cpb_25_11_117307/cpb160511ieqn5.gif)
By using the nonequilibrium Green’s function technique, the electronic and heat currents flowing in the system can be written in the forms
Here
In the linear response regime, the chemical potentials and the temperatures of the two leads are set to be μL = μR = μ and TL = TR = T. In this system, the chemical potential difference between the two leads is related to the spin channel. One can introduce a spin-dependent voltage ΔVσ = ΔV + δσΔVs with charge bias ΔV and spin bias ΔVs. The spin bias originates from spin accumulation. Corresponding, the spin-resolved electric and heat currents can be expressed as
![](cpb_25_11_117307/cpb160511ieqn12.gif)
![](cpb_25_11_117307/cpb160511ieqn28.gif)
![](cpb_25_11_117307/cpb160511ieqn13.gif)
![](cpb_25_11_117307/cpb160511ieqn14.gif)
![](cpb_25_11_117307/cpb160511ieqn15.gif)
![](cpb_25_11_117307/cpb160511ieqn16.gif)
![](cpb_25_11_117307/cpb160511ieqn17.gif)
![](cpb_25_11_117307/cpb160511ieqn18.gif)
For simplicity, we take V1 = V3 = V and V2 = βV with both V and β real, at the same time assume ϕR1 = −ϕR3 = ΔϕR/2. Thus, in the new subspace these coupling coefficients can be expressed as
![](cpb_25_11_117307/cpb160511ieqn19.gif)
![](cpb_25_11_117307/cpb160511ieqn20.gif)
![](cpb_25_11_117307/cpb160511ieqn21.gif)
![](cpb_25_11_117307/cpb160511ieqn22.gif)
![](cpb_25_11_117307/cpb160511ieqn23.gif)
Now we investigate numerically the spin-depended thermoelectric transport through the system. For convenience, Γ = 2πρ(0)V2 is used as the energy unit with ρ(0) the density of states in the leads. The system temperature T is fixed to be 0.026Γ (the constant kB = 1). The other parameters are chosen as μL = μR = 0, t = 2Γ, and β = 1. In the presence of the RSOI and the magnetic flux, the electron transmission depends on the phase factors, which is presented in the transmission coefficient
The dependence of the transmission on the phase factor can be used to tune the thermoelectric coefficients via
![]() | Fig. 2. Thermoelectric coefficients (a) G, (b) κ, (c) S, and (d) ZT as functions of the QD level ε for t = 2Γ. Here the phase factors are set to be φ = 1.25π and ΔϕR = 0.25π. |
In order to observe comprehensively the modulation of the configuration of the magnetic flux phase and the RSOI phase on the thermoelectric coefficients, we present the thermopower as a function of both the magnetic flux phase φ and the RSOI phase ΔϕR in Fig.
![]() | Fig. 4. Thermopower S as a function of the QD level for phase factor φ = −1.68π and ΔϕR = −1.96π. The other parameters are the same as those in Fig. |
For a given configuration of the magnetic flux phase and the RSOI phase, one can obtain maximal thermopower by choosing the appropriate interdot coupling. In Fig.
We have studied the spin-dependent thermoelectric transport through a parallel triple-dot system with RSOI embedded in an Aharonov–Bohm interferometer connected symmetrically to leads using the nonequilibrium Green’s function method in the linear response regime. Because the influences of the magnetic flux phase and the RSOI phase on the thermoelectric transport are not exactly the same, the thermopower can be considered as a function of both the magnetic flux phase φ and the RSOI phase ΔϕR when other parameters are given. The contour graphs of spin-dependent thermopower Sσ present double-peak structures in the specific (φ,ΔϕR) region. While the charge thermopower Sc and the spin thermopower Ss present quadruple-peak structures. For some specific configurations of the magnetic flux phase and the RSOI phase, one can obtain the spin thermopower while the charge thermopower vanishes, which is useful in realizing the thermal spin battery that would allow to convert heat into a spin voltage and induce a pure spin current in the device. Moreover, under the appropriate configuration of the phase and parameters, one can obtain a relatively large spin thermopower and the spin figure of merit ZsT is even larger than the charge figure of merit ZcT.
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