Abstract
We examine the relation between the phasecoherent processes and spindependent thermoelectric effects in an AharonovBohm (AB) interferometer with a Rashba quantum dot (QD) in each of its arm by using the Green's function formalism and equation of motion (EOM) technique. Due to the interplay between quantum destructive interference and Rashba spinorbit interaction (RSOI) in each QD, an asymmetrical transmission node splits into two spindependent asymmetrical transmission nodes in the transmission spectrum and, as a consequence, results in the enhancement of the spindependent thermoelectric effects near the spindependent asymmetrical transmission nodes. We also examine the evolution of spindependent thermoelectric effects from a symmetrical parallel geometry to a configuration in series. It is found that the spindependent thermoelectric effects can be enhanced by controlling the dotelectrode coupling strength. The simple analytical expressions are also derived to support our numerical results.
PACS numbers: 73.63.Kv; 71.70.Ej; 72.20.Pa
Keywords:
Rashba spinorbit interaction; AharonovBohm interferometer; Quantum dots; Fano effectsIntroduction
With the fast development and improvement of experimental techniques [19], much important physical properties in QD molecules such as electronic structures, electronic transport, and thermoelectric effects et al have widely attracted academic attention [1029]. QDs can be realized by etching a twodimensional 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 phasecoherent 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 singleelectron transistor [31].
The RSOI in the QD can be introduced by an asymmetricalinterface electric field applied to the semiconductor heterostructures [32,33]. 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, spinup and spindown electrons flow in an opposite direction using the Rashba SOI and a longitudinal electric field such that the spin polarization becomes infinity [3537]. Some theoretical and experimental works have also shown that the spinpolarization of current based on the RSOI can reach as high as 100%[38,39] 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 heattoelectron spin discovery can be named as "thermospintronics". More recently, the spin Seebeck effect was also observed in a ferromagnetic semiconductor GaMnAs [42]. Much academic work on spindependent thermoelectric effects in single QD attached to ferromagnetic leads with collinear magnetic moments or noncollinear magnetic moments has been reported [4346]. 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 nonferromagnetic 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 [2529].
In the present work, we investigate the spindependent thermoelectric effects of parallelcoupled double quantum dots embedded in an AB interferometer, in which the RSOI in each QD is considered by introducing a spindependent 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 spindependent asymmetrical transmission nodes in the transmission spectrum and, as a consequence, results in the enhancement of the spindependent Seebeck effects near the spindependent asymmetrical transmission nodes. We also examine the evolution of spindependent Seebeck effects from a symmetrical parallel geometry to a configuration in series. The asymmetrical couplings between QDs and nonferromagnetic electrodes induce the enhancement of spindependent Seebeck effects in the vicinity of spindependent asymmetrical transmission nodes. Although the spindependent Seebeck effects in the AB interferometer have not been realized experimentally so far, our theoretical study provides a better way to enhance spindependent Seebeck effects in the AB interferometer in the absence of the ferromagnetic materials.
Model and method
The schematic diagram for the quantum device based on parallelcoupled 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 twodimensional 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 spindependent extra phase factor in the tunnel matrix elements and interlevel spinflip term [47,48]. 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 spindependent thermoelectric voltage ΔV_{↑(↓) }emerges. The proposed spindependent thermoelectric AB interferometer can be described by using the following Hamiltonian in a secondquantized form as,
Figure 1. (Color online) Schematic diagram for a thermoelectric device based on a double QD AB interferometer in the presence of magnetic flux Φ. A spindependent thermoelectric voltage ΔV_{σ }is generated when a temperature gradient ΔT is presented, where μ is the chemical potential of the metallic electrodes, and T is the temperature of the metallic electrode.
where
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],
and the thermal current with spin index σ from the electrode α is calculated by [50],
where τ_{σ}(ε) is the transmission probability of electron with spin index σ, which can be given by
where
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 spindependent
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
After expanding the FermiDirac distribution function to the first order in ΔT and ΔV_{σ}, we obtain the spindependent Seebeck coefficient by S_{σ }= ΔV_{σ}/ΔT as,
where
The corresponding electronic thermal conductance κ_{el }can be defined by
and the electronic thermal conductance from the Seebeck effects,
The differential conductance with spin index σ may be expressed as
and
respectively, where
Results and discussion
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 (ε_{1 }= ε_{2 }= 0).
In Figure 2, we plot the spindependent transmission probability τ_{σ}, spindependent Seebeck coefficient S_{σ}, and spindependent Lorenz number
Figure 2. (Color online) Spindependent transmission probability τ_{σ}(logarithmic scale), spindependent Seebeck coefficient S_{σ}, and spindependent Lorenz number L_{σ }(in units of
where
where
Table 1. Approximate values of q_{± σ }for various different values of ϕ
In Figure 3, we calculate
Figure 3. (Color online) Spindependent electronic thermal conductance
A detail study of the spindependent 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 spindependent transmission probability τ_{σ }has the following expression as,
Figure 4. (Color online) Spindependent transmission probability τ_{σ}(logarithmic scale) and spindependent Seebeck coefficient S_{σ }as functions of the chemical potential μ in the presence of different values of λ at room temperature (T = 300 K). ϕ_{R }and ϕ have same values as ϕ_{R }= ϕ = π. The black solid lines represents the spinup component, and the red dashed lines represents the spindown component.
where
where + for spin up and  for spin down. Eq. (15) shows the symmetrical spindependent BreitWigner 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 spinindependent due to the absence of the AB phase. The transmission probability can be calculated by the following expression,
Table 2. Approximate values of q_{± σ }for various different values of λ
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 spindependent transmission probability τ_{σ }is composed of near BreitWigner peak and Fano line shapes as shown in Figure 4 and 3. The spindependent transmission probability can be approximated by,
where
where + means spin up case and  represents spindown case. As a result, we find that
the spindependent Seebeck effect is enhanced strongly in the vicinity the spindependent
transmission nodes. The electronic thermal conductance
Figure 5. (Color online) Spindependent electronic thermal conductance
Summary
We investigate the spindependent thermoelectric effects of parallelcoupled DQDs embedded in an AB interferometer in which the RSOI is considered by introducing a spindependent 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 spindependent asymmetrical transmission nodes in the transmission spectrum, which induces that the spindependent Seebeck effects are enhanced strongly at different energy regimes. We also examine the evolution of spindependent 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 spindependent Seebeck effects in the vicinity of the corresponding spindependent asymmetric transmission node in the transmission spectrum.
Abbreviations
2DEG: twodimensional electron gas; AB: AharonovBohm; FOMs: figureofmerits; QD: quantum dot; RSOI: Rashba spinorbit interaction.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors read and approved the final manuscript.
Acknowledgements
The authors thank the support of the National Natural Science Foundation of China (NSFC) under Grants No. 61106126, and the Science Foundation of the Education Committee of Jiangsu Province under Grant No. 09KJB140001. The authors also thank the supports of the Foundations of Changshu Institute of Technology.
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