FanoRashba effect in thermoelectricity of a double quantum dot molecular junction
 YS Liu^{1},
 XK Hong^{1},
 JF Feng^{1} and
 XF Yang^{1, 2}Email author
DOI: 10.1186/1556276X6618
© Liu et al; licensee Springer. 2011
Received: 25 January 2011
Accepted: 7 December 2011
Published: 7 December 2011
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 [1–9], much important physical properties in QD molecules such as electronic structures, electronic transport, and thermoelectric effects et al have widely attracted academic attention [10–29]. 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 [35–37]. 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 [43–46]. 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 [25–29].
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
where ${a}_{\alpha k\sigma}^{\u2020}\left({a}_{\alpha k\sigma}\right)$ 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 FermiDirac distribution such as f _{ α } = 1/{exp[(ε  μ _{ α })/(k _{ B } T _{ α }) + 1}. Here k _{ B } is the Boltzmann constant. ${d}_{n\sigma}^{\u2020}\left({d}_{n\sigma}\right)$ 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 ${V}_{L\sigma 1}=\phantom{\rule{2.77695pt}{0ex}}\mid {V}_{L\sigma 1}\mid {e}^{i\left(\varphi \sigma {\phi}_{R}\right)\u22154}$, ${V}_{L\sigma 2}=\phantom{\rule{2.77695pt}{0ex}}\mid {V}_{L\sigma 2}\mid {e}^{i\left(\varphi \sigma {\phi}_{R}\right)\u22154}$, ${V}_{R\sigma 1}=\phantom{\rule{2.77695pt}{0ex}}\mid {V}_{R\sigma 1}\mid {e}^{i\left(\varphi \sigma {\phi}_{R}\right)\u22154}$, ${V}_{R\sigma 2}=\phantom{\rule{2.77695pt}{0ex}}\mid {V}_{R\sigma 2}\mid {e}^{i\left(\varphi \sigma {\phi}_{R}\right)\u22154}$, with the AB phase ϕ = 2π Φ/Φ_{0} and the flux quantum Φ_{0} = h/e. Φ can be calculated by the equation $\varphi =\overrightarrow{B}\cdot \overrightarrow{S}$, 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 wellknown 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 2πperiodic linear conductance is obtained, and it is in good agreement with the experimental work [1]. φ _{ R } denotes the difference between φ _{ R 1}and φ _{ R 2}, where φ _{ Ri } is the phase factor induced by the RSOI inside the ith QD.
where ${\Gamma}_{nm}^{\alpha}=2\pi {\sum}_{k}\mid {V}_{\alpha \sigma n}\parallel {V}_{\alpha \sigma m}^{*}\mid \delta \left(\epsilon {\epsilon}_{\alpha k\sigma}\right)$. $\phantom{\rule{2.77695pt}{0ex}}{G}_{\sigma}^{r}\left(\epsilon \right)$ 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 ${G}_{n\sigma ,m\sigma}^{r}\left(t\right)=i\Theta \left(t\right)<\left\{{d}_{n\sigma}\left(t\right),\phantom{\rule{2.77695pt}{0ex}}{d}_{m\sigma}^{\u2020}\left(0\right)\right\}>$, where Θ(t) is the step function. The advanced dot Green's function can be obtained by the relation ${G}_{\sigma}^{a}\left(\epsilon \right)={\left[{G}_{\sigma}^{r}\left(\epsilon \right)\right]}^{+}$.
respectively, where ${G}_{c}=\frac{{e}^{2}}{h}\left[{K}_{0\uparrow}\left(\mu ,\phantom{\rule{2.77695pt}{0ex}}T\right)+{K}_{0\downarrow}\left(\mu ,\phantom{\rule{2.77695pt}{0ex}}T\right)\right]$ and ${G}_{s}=\frac{{e}^{2}}{h}\left[{K}_{0\uparrow}\left(\mu ,\phantom{\rule{2.77695pt}{0ex}}T\right){K}_{0\downarrow}\left(\mu ,\phantom{\rule{2.77695pt}{0ex}}T\right)\right]$. In this study, the phonon thermal conductance of the junction, which is typically limited by the QDselectrode contact, has been ignored in the case of the poor link for phonon transport.
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).
Approximate values of q _{ ± σ } for various different values of ϕ
ϕ  q _{+↑ }  q _{+↓ }  q _{ ↑ }  q _{ ↓ } 

0  6.8  6.8  1.2  1.2 
0.25π  26.3  3.2  1.0  1.4 
0.75π  26.3  1.4  1.0  3.2 
1.25π  3.2  1.0  1.4  26.3 
1.75π  1.4  1.0  3.2  26.3 
2.25π  1.0  1.4  26.3  3.2 
Approximate values of q _{ ± σ } for various different values of λ
λ  q _{+↑ }  q _{+↓ }  q _{ ↑ }  q _{ ↓ } 

1  +∞  No  No  ∞ 
0.6  78.7  1.3  1.3  78.7 
0.3  19.6  1.7  1.7  19.6 
0  4  4  4  4 
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.
Declarations
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.
Authors’ Affiliations
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