Abstract
Using the nonequilibrium Green’s function method, we theoretically study the Andreev reflection(AR) in a fourterminal AharonovBohm interferometer containing a coupled double quantum dot with the Rashba spinorbit interaction (RSOI) and the coherent indirect coupling via two ferromagnetic leads. When two ferromagnetic electrodes are in the parallel configuration, the spinup conductance is equal to the spindown conductance due to the absence of the RSOI. However, for the antiparallel alignment, the spinpolarized AR occurs resulting from the crossed AR (CAR) and the RSOI. The effects of the coherent indirect coupling, RSOI, and magnetic flux on the Andreevreflected tunneling magnetoresistance are analyzed at length. The spinrelated current is calculated, and a distinct swap effect emerges. Furthermore, the pure spin current can be generated due to the CAR when two ferromagnets become two half metals. It is found that the strong RSOI and the large indirect coupling are in favor of the CAR and the production of the strong spin current. The properties of the spinrelated current are tunable in terms of the external parameters. Our results offer new ways to manipulate the spindependent transport.
Keywords:
AharonovBohm interferometer; Double quantum dot; Andreev reflection; Rashba spinorbit interaction; Coherent indirect coupling; 73.63.Kv; 73.23.b; 72.25.bBackground
A quantum dot (QD) is an artificially lowdimensional structure that can be filled with electrons (or holes). Two or more QDs can be coupled to form multipleQD systems (i.e., artificial molecules). Because the degrees of freedom of the QDs are well controllable, it is possible to add or remove the electrons in the QDs, and the QD system can be coupled via tunnel barriers to electrodes, in which electrons can be exchanged. Accordingly, the artificial molecule provides an excellent model system in which the thorough investigation of quantum manybody properties in a confined geometry can be performed [16]. Among the various multipleQD systems, an AharonovBohm (AB) interferometer containing double QDs (DQDs) is of particular interest and importance, in which two QDs are embedded in the opposite arms of the AB ring, respectively, and they are coupled to each other via barrier tunneling. As a tunable twolevel system, the parallel DQD system that can become one of the promising candidates for the quantum bit in quantum computation has received more attention [720]. However, in an actual DQD system, the coherent indirect coupling between two QDs via a reservoir is very essential. Kubo et al. introduced the parameter _{1}αcharacterizing the indirect coupling strength, and Gurvitz also indicated the fundamentality of the sign of the coherent indirect coupling parameter [21,22]. Kubo et al. investigated the pseudospin Kondo effect in a lateral DQD system using the slaveboson meanfield method and found that the exotic pseudospin Kondo effect occurs when a coherent indirect coupling is presented through the common reservoirs [23]. Recently, Kubo and coworkers calculated the shot noise and Kondo effect in a DQD structure with the coherent indirect coupling. Their results demonstrate that the coherent indirect coupling can generate a novel antiferromagnetic exchange phenomenon [24]. Trocha and Barnaś studied theoretically the spindependent transport through a DQD coupled to ferromagnetic leads. They observed that the Fano antiresonance of the linear conductance relies on the sign of the indirect coupling in the nondiagonal coupling elements [8]. Furthermore, the transport properties of a DQD system has been considered in the orbital Kondo regime. That the Kondo temperature and Kondo resonances are susceptible to the coherent indirect coupling parameter is also revealed [25]. In addition, if a QD is formed in a semiconductor twodimensional electron gas structure without the inversion symmetry in the growth direction, the Rashba spinorbit interaction (RSOI) will emerge, and the RSOI can induce the spinrelated phase factor in the tunneling matrix elements and the spinflip effect. The RSOI results from a relativistic effect at the low speed limit, and it can couple the electron spin to its orbital motion, thus providing a possible way to control the spin degree of freedom by means of an external electric field. As a consequence, the coherent indirect coupling and the RSOI make the quantum transport through the QD systems rich and varied [2630].
On the other side, the subgap transport through heterostructures with nanoobjects (such as QDs, molecules, nanowires, etc.) coupled to one conductor and another superconducting lead has attracted a great deal of attention over the past years due to the fundamental physics and its potential applications [3135]. Andreev reflection (AR) usually occurs in the hybrid systems, in which two electrons with opposite spins enter the superconductor from the normal metal region, leading to the formation of a Cooper pair in the superconducting region [3638]. In comparison with the standard mechanism of normal AR, the crossed AR (CAR) is a nonlocal dynamics process which occurs at the contact between a superconductor and two normal leads, where two subgap electrons from different metals enter into the superconductor and generate a Cooper pair there [3942]. AR (or CAR) in nanoscopic heterostructures gives rise to a rich subgap structure in the currentvoltage characteristics. Accordingly, understanding the AR and CAR has attracted theoretical and experimental attention mainly because the AR (or CAR) may create the entangled electrons in a solidstate device, and CAR can be readily probed by spin selection using ferromagnetic electrodes. This approach is almost unrealized for entangler devices, since projecting the spin will cause the destruction of entanglement [43]. Based on the CAR, the controlled Cooper pair splitting has been realized in terms of a twoquantum dot Yjunction [44], which opens a possible route towards a test of the EinsteinPodolskyRosen (EPR) paradox and Bell inequalities in solidstate systems. Herrmann et al. used carbon nanotube DQD as Cooper pair beam splitters and realized the quantum opticlike experiments with spinentangled electrons [45]. These results show that the CAR has an important application in testing a fundamental property of quantum mechanics.
To our knowledge, the AR in the DQD with a maximum coupling α = 1 has been studied widely. However, the quantum transport through a fourterminal AB interferometer including a DQD in the presence of the AR, the coherent indirect coupling, and RSOI is less explored. Motivated by recent theoretical and experimental advances in the DQD systems [710,13,15,16,19,2125,44,45], one may expect that the interplay of the coherent indirect coupling and the RSOI in the presence of the AR will add new physics to hybrid quantum systems, which may have practical applications for future spintronics. Consequently, we investigate the AR in the abovementioned system in this paper. It is found that the RSOI and a nonzero coherent indirect coupling cause the spinpolarized AR when the polarizations of two ferromagnetic leads are parallel, but for antiparallel (AP) arrangement of the polarizations of two ferromagnetic leads, the CAR can contribute the spinpolarized AR conductance. We note that the convex shape of the Andreevreflected tunneling magnetoresistance (ARTMR) versus the magnetic flux relies on the sign of the coherent indirect coupling parameter, and there are extreme values in the plot of the ARTMR versus the coherent indirect coupling parameter. Even the negative ARTMR also occurs. This is a spin valve effect in the AR process. It is interesting to note that the sign of the coherent indirect coupling parameter leads to the swap effect in the spinpolarized current plot, and the pure spin current can be produced when two ferromagnetic leads are fully polarized. The spindependent AR current can be controlled by means of the gate voltage, RSOI, magnetic flux, and so on. These results provide the ways to manipulate the spindependent transport by means of the system parameters.
Methods
We consider a hybrid fourterminal AB interferometer including a parallel DQD coupled to two ferromagnetic reservoirs and two superconductors as shown in Figure 1. The system is described by the following Hamiltonian:
Figure 1. Schematic diagram of a fourterminal AB interferometer (color on line). The AB interferometer contains a coupled DQD with magnetic flux applied perpendicular to rings.
where H_{F }is the Hamiltonian of the left and right ferromagnetic electrodes
Here,
H_{DQD} in Equation 1 denotes the DQD Hamiltonian
in which
where the tunneling matrix elements between the DQD and two ferromagnetic leads are
Using the nonequilibrium Green’s function technique, the spindependent current through the left ferromagnetic reservoir can be expressed as [48,49]
where Tr is the trace in the spin space;
with the vector
After some algebraic manipulations, the spindependent current can be derived from Equation 6:
in which
Since we mainly focus on the AR process at zero temperature limit and set eV_{L} = eV_{R} < Δ,
Results and discussions
In the following numerical calculations, we mainly elucidate the spindependent AR
process in the fourterminal AB interferometer with the coherent indirect coupling
and the RSOI. We take e = h = k_{B }= 1, and set Δ = 1 as the energy unit. Throughout the paper, the symmetric couplings
with
Conductance
Because we mostly investigate the AR within the superconductor gap, in the limit of zero bias V_{L} = V_{R }→ 0, the spinrelated AR and CAR conductances have the forms
and
It is well known that a DQD system with the maximum coupling α = 1 has already been investigated. Indeed, such case is very special, and most experimental conditions correspond to α < 1; as a result, α characterizing the coherent indirect coupling between two QDs via two ferromagnetic electrodes is introduced (see the Appendix). α < 1 comes from the various factors, such as imperfections in the ferromagnetic reservoirs producing the destructive quantum interference, the geometrical structure of the system, and so forth.
Let us begin with the case of ϕ = 0 and φ_{R }= Π/2; for the different coherent indirect coupling α, Figure
2 shows the total AR conductance (
Figure 2. The AR conductance versus Fermi energy for P and AP configurations. (a) α = 0, (b) α = 0.5, and (c) α = 1.0. Other parameters are ε_{1 }= ε_{2 }= 0, t_{c }= 0.5, ϕ = 0, and φ_{R }= Π/2.
We also notice that both
To elucidate better the properties of the AR under P and AP configurations, in analogy with the conventional tunneling magnetoresistance (TMR) effect of ferromagnetic tunnel junctions, the ARTMR is introduced and defined as
In Figure 3, we present the ϕ dependence of ARTMR for different α. The oscillation period of the ARTMR versus magnetic flux ϕ is 2Π, and the sign of the ARTMR does not change. It is interesting to note that the convex shape of the ARTMR at ϕ = 2nΠ (n is an integer) relies on the sign of the coherent indirect coupling parameter α. In comparison to the case of α = 0.5, the magnitudes of ARTMR are considerably increased for α = 1.0. This is because the reduction of the destructive interference results in the enhancement of ARTMR.
Figure 3. ARTMR versus the magnetic flux ϕ with different α. Other parameters are ε_{1 }= ε_{2 }= 0, t_{c }= 0.5, and φ_{R }= Π/2
As we know, the RSOI can induce the spin precession and may even cause the interdot
spinflip effect. According to
[26,27], the spindependent phase factor φ_{R} due to the RSOI can be expressed as
Figure 4. ARTMR versus the coherent indirect coupling α with different φ_{R}. Other parameters are ε_{1 }= ε_{2 }= 0, t_{c }= 0.5, and ϕ = 0.
Spindependent current
Above, we analyze the properties of the AR conductances. In the following discussions, we will explore the spindependent current in the AR process with the help of the current formulas (Equations 9 and 10). To gain a full physical picture on the DQD levels’ influence on the spinrelated current, Figure 5 displays the images of the spinpolarized current I_{s }= I_{L↑ }− I_{L↓} as a function of the energy levels ε_{1} and ε_{2} of the DQD. The blue regions correspond to zero current, namely, I_{L↑ }= I_{L↓ }in these regimes. In the diagram, it is found that the spinpolarized current is symmetrical about the line of ε_{1 }= ε_{2} and is asymmetrical with respect to the line of ε_{1 }= −ε_{2 }as illustrated in Figure 5a, b. It is interesting to note that one level is aligned to the Fermi level, and the other is far from the Fermi one (offresonance). I_{s }is relative small. This is because one QD is in the onresonance state and the other is in the offresonance state. When both ε_{1} and ε_{2} are close to the Fermi level by tuning the gate voltage, the maximal I_{s}appears since DQD is in the onresonance states. We also observe that, for α = 0.5 and α = −0.5, the spinpolarized current shows the opposite feature, which is a swap effect originating from the different sign of the parameter α. This indicates that the sign of the coherent indirect coupling parameter has a remarkable impact on the spinpolarized current.
Figure 5. Images of spinpolarized AR current current as a function of QD levels ε_{1 }and ε_{2 }(Color on line). (a) α = −0.5 and (b) α = 0.5. Other parameters are ϕ=0, φ_{R }= 0, t_{c }= 0.5, and P_{L }= P_{R }= 0.4.
As we know, when ferromagnets are fully polarized, two ferromagnets become half metals
where all electrons have the same spin. AR is usually suppressed at the ferromagnet/superconductor
interface. However, AR still can occur, and the pure spin current can be generated
in the present system. For P_{L }= −P_{R }= 1.0 or P_{L }= −P_{R }= −1.0, i.e., two ferromagnetic leads become two half metals; the normal AR vanishes
due to
Figure 6. The spin current versus the magnetic flux ϕ for different φ_{R }. (a) α = 0.5 and (b) α = 1.0. Other parameters are ε_{1 }= ε_{2 }= 0, t_{c }= 0.5, P_{L }= −P_{R }= 1.0
Conclusions
In this paper, we have analyzed the AR of a fourterminal AB interferometer containing a coupled DQD with with the RSOI and the coherent indirect coupling via two ferromagnetic leads. The formulas of the transmission coefficients are derived based on the framework of the nonequilibrium Green’s function technique. For P configuration, the spinpolarized AR can occur, stemming from the RSOI and a nonzero coherent indirect coupling. On the contrary, for AP configuration, the spinpolarized AR always happens because of the CAR mechanism. Under the introduction of the ARTMR, we find that the sign of the ARTMR versus the magnetic flux keeps invariable for different parameter α, but the convex shape of the ARTMR depends distinctly on the sign of the parameter α. With the increase of the RSOI strength, the ARTMR versus the parameter α exhibits the more significant nonmonotonic features, and there exist the extreme values in the ARTMR plot, even the negative ARTMR also emerges. Since the energy levels of the DQD can be manipulated via the gate voltage, we can obtain the optimal spinpolarized current. A pure spin current can be generated via the CAR and two halfmetal leads. Moreover, the strong RSOI and the reduction of the destructive interference (α = 1) favor the enhancement of the spin current. Thus, this device may become an effective spincurrent generator, and the pure spin current is tuned in terms of the magnetic flux, the RSOI strength, and so forth. These results offer the ways to manipulate the spindependent transport via the fourterminal AB setup.
Appendix
In this appendix, we present the derivation of the current formulas in detail.
Let g^{r}(ε) and G^{r}(ε) denote the retarded Green’s function of the DQD without and with the coupling to the external reservoirs. In the Nambu space, g^{r}(ε) can be given as
Based on the following Dyson equation, the retarded Green’s function of the system
can be written as G^{r}(ε)]^{−1 }= g^{r}(ε)^{−1}−Σ^{r}, in which
where
in which f_{ν}(ε−eV_{ν}) = f_{ν},
Thus, we can investigate the quantum transport through our model system based on the abovementioned equations.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
LB established the physical model and the theoretical formalism. RZ and CLD carried out the numerical calculations and the establishment of the figures. LB performed the physical analysis and revised the manuscript. All the authors read and approved the final manuscript.
Acknowledgements
We thank the reviewer for the useful comments. This work is supported by the National Natural Science Foundation of China (grant no. 11104346) and the Fundamental Research Funds for the Central Universities (grant nos. 2011QNA28 and 2010NQB26). ChenLong Duan gratefully acknowledges the financial support from the Research Fund for the Doctoral Program of Higher Education of China (grant no. 20100095120003) and China Postdoctoral Science Foundation (grant no. 20090461153).
References

Reimann SM, Manninen M: Electronic structure of quantum dots.
Rev Mod Phys 2002, 74:1283. Publisher Full Text

Wang ZM: SelfAssembled Quantum Dots. New York: Springer; 2008.

Hanson R, Kouwenhoven LP, Petta JR, Tarucha S, Vandersypen LMK: Spins in fewelectron quantum dots.

Andergassen S, Meden V, Schoeller H, Splettstoesse J, Wegewijs MR: Charge transport through single molecules, quantum dots and quantum wires.
Nanotechnology 2010, 21:272001. PubMed Abstract  Publisher Full Text

Aleiner IL, Brouwer PW, Glazman LI: Quantum effects in Coulomb blockade.
Phys Rep 2002, 358:309. Publisher Full Text

Dubi Y, Di Ventra M: Heat flow and thermoelectricity in atomic and molecular junctions.
Rev Mod Phys 2011, 83:131. Publisher Full Text

Lu HZ, Lü R, Zhu BF: Tunable Fano effect in parallelcoupled double quantum dot system.

Trocha P, Barnas J: Quantum interference and Coulomb correlation effects in spinpolarized transport through two coupled quantum dots.

Kubala B, Konig BK: Fluxdependent level attraction in doubledot AharonovBohm interferometers.

Chi F, Yuan XQ, Zheng J: Double Rashba quantum dots ring as a spin filter.
Nanoscale Res Lett 2008, 3:343. Publisher Full Text

Liu YS, Chen H, Yang XF: Transport properties of an AharonovBohm ring with strong interdot Coulomb interaction.
J Phys Condens Matter 2007, 19:246201. PubMed Abstract  Publisher Full Text

Zitko R, Mravlje J, Haule K: Ground state of the parallel double quantum dot system.
Phys Rev Lett 2012, 108:066602. PubMed Abstract  Publisher Full Text

Fang TF, Luo HG: Tuning the Kondo and Fano effects in double quantum dots.

Krause T, Schaller G, Brandes T: Incomplete current fluctuation theorems for a fourterminal model.

Loss D, Sukhorukov EV: Probing entanglement and nonlocality of electrons in a doubledot via transport and noise.
Phys Rev Lett 2000, 84:1035. PubMed Abstract  Publisher Full Text

Smirnov AY, Horing NJM, Mourokh LG: AharonovBohm phase effects and inelastic scattering in transport through a parallel tunnelcoupled symmetric doubledot device.

Sukhorukov EV, Burkard G, Loss D: Noise of a quantum dot system in the cotunneling regime.

Mourokh LG, Horing NJM, Smirnov AY: Electron transport through a parallel doubledot system in the presence of AharonovBohm flux and phonon scattering.

Holleitner AW, Decker CR, Qin H, Eberl K, Blick RH: Coherent coupling of two quantum dots embedded in an AharonovBohm interferometer.
Phys Rev Lett 2001, 87:256802. PubMed Abstract  Publisher Full Text

Holleitner AW, Blick RH, Huttel AK, Eberl K, Kotthaus JP: Probing and controlling the bonds of an artificial molecule.
Science 2002, 297:70. PubMed Abstract  Publisher Full Text

Kubo T, Tokura Y, Hatano T, Tarucha S: Electron transport through AharonovBohm interferometer with laterally coupled double quantum dots.

Gurvitz SA: Quantum interference in resonant tunneling single spin measurements.
IEEE Trans Nanotechol 2005, 4:45. Publisher Full Text

Kubo T, Tokura Y, Hatano T, Tarucha S: Exotic pseudospin Kondo effect in laterally coupled double quantum dots.

Kubo T, Tokura Y, Hatano T, Tarucha S: Kondo effects and shot noise enhancement in a laterally coupled double quantum dot.

Trocha P: The role of the indirect tunneling processes and asymmetry in couplings in orbital Kondo transport through double quantum dots.
J Phys Condens Matter 2012, 24:055303. PubMed Abstract  Publisher Full Text

Sun QF, Xie XC: Biascontrollable intrinsic spin polarization in a quantum dot: proposed scheme based on spinorbit interaction.

Sun QF, Wang J, Guo H: Quantum transport theory for nanostructures with Rashba spinorbital interaction.

Tserkovnyak Y, Akhanjee S: Spinselective localization due to intrinsic spinorbit coupling.

Wu MW, Jiang JH, Weng MQ: Spin dynamics in semiconductors.
Phys Rep 2010, 493:61. Publisher Full Text

Stepanenko D, Rudner M, Halperin BI, Loss D: Singlettriplet splitting in double quantum dots due to spinorbit and hyperfine interactions.

Sun QF, Wang J, Lin TH: Resonant Andreev reflection in a normalmetalquantum dotsupercoductor system.
Phys Rev B 1999, 59:3831. Publisher Full Text

Koerting V, Andersen BM, Flensberg K, Paaske J: Nonequilibrium transport via spininduced subgap states in superconductor/quantum dot/normal metal cotunnel junctions.

Baranski J, Domanski T: Fanotype interference in quantum dots coupled between metallic and superconducting leads.

Xing YX, Wang J: Universal conductance fluctuations in mesoscopic systems with superconducting leads: beyond the Andreev approximation.

Whitney RS, Jacquod P: Controlling the sign of magnetoconductance in Andreev quantum dots.
Phys Rev Lett 2009, 103:247002. PubMed Abstract  Publisher Full Text

Skadsem HJ, Brataas A, Martinek J, Tserkovnyak Y: Ferromagnetic resonance and voltageinduced transport in normal metalferromagnetsuperconductor trilayers.

Golubov AA, Tanaka Y, Mazin II, Dolgov OV, Brinkman: Andreev spectra and subgap bound states in multiband superconductors.

Annunziata G, Cuoco M, Noce C, Sudbo A, Linder J: Spinsensitive longrange proximity effect in ferromagnet/spintripletsuperconductor bilayers.

Morten JP, Brataas A, Belzig W: Circuit theory of crossed Andreev reflection.

Golubev DS, Zaikin AD: Nonlocal Andreev reflection in superconducting quantum dots.

Sothmann B, Futterer D, Governale M, Konig J: Probing the exchange field of a quantumdot spin valve by a superconducting lead.

Futterer D, Governale M, Pala MG, Konig J: Nonlocal Andreev transport through an interacting quantum dot.

Brauer J, Hubler F, Smetanin M, Beckmann D, Lohneysen HV: Nonlocal transport in normalmetal/superconductor hybrid structures: role of interference and interaction.

Hofstetter L, Csonka S, Nygardand C, Schonenberger S: Cooper pair splitter realized in a twoquantumdot Yjunction.
Nature 2009, 461:960. PubMed Abstract  Publisher Full Text

Herrmann LG, Portier F, Roche P, Yeyati AL, Kontos T, Strunk C: Carbon nanotubes as Cooperpair beam splitters.
Phys Rev Lett 2010, 104:026801. PubMed Abstract  Publisher Full Text

Nitta J, Akazaki T, Takayanagi H, Enoki T: Gate control of spinorbit interaction in an inverted In0.53Ga0.47As/In0.52Al0.48As heterostructure.
Phys Rev Lett 1997, 78:1335. Publisher Full Text

Matsuyama T, Kursten R, Meissner C, Merkt U: Rashba spin splitting in inversion layers on ptype bulk InAs.
Phys Rev B 2000, 61:15588. Publisher Full Text

Jauho AP, Haug H: Quantum Kinetics in Transport and Optics of Semiconductors. Berlin: Springer; 2008.

Jauho AP, Wingreen NS, Meir Y: Timedependent transport in interacting and noninteracting resonanttunneling systems.
Phys Rev B 1994, 50:5528. Publisher Full Text