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Nanoscale Res Lett. 2012; 7(1): 670.
Published online Dec 10, 2012. doi:  10.1186/1556-276X-7-670
PMCID: PMC3575254
Tunable spin-dependent Andreev reflection in a four-terminal Aharonov-Bohm interferometer with coherent indirect coupling and Rashba spin-orbit interaction
Long Bai,corresponding author1 Rong Zhang,1 and Chen-Long Duan2
1College of Science, China University of Mining and Technology, Xuzhou, 221116, China
2School of Chemical Engineering and Technology, China University of Mining and Technology, Xuzhou, 221116, China
corresponding authorCorresponding author.
Long Bai: bailong2200/at/163.com; Rong Zhang: 1979zhangrong/at/163.com; Chen-Long Duan: laoduan/at/126.com
Received June 14, 2012; Accepted November 21, 2012.
Using the nonequilibrium Green’s function method, we theoretically study the Andreev reflection(AR) in a four-terminal Aharonov-Bohm interferometer containing a coupled double quantum dot with the Rashba spin-orbit interaction (RSOI) and the coherent indirect coupling via two ferromagnetic leads. When two ferromagnetic electrodes are in the parallel configuration, the spin-up conductance is equal to the spin-down conductance due to the absence of the RSOI. However, for the antiparallel alignment, the spin-polarized AR occurs resulting from the crossed AR (CAR) and the RSOI. The effects of the coherent indirect coupling, RSOI, and magnetic flux on the Andreev-reflected tunneling magnetoresistance are analyzed at length. The spin-related 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 spin-related current are tunable in terms of the external parameters. Our results offer new ways to manipulate the spin-dependent transport.
Keywords: Aharonov-Bohm interferometer, Double quantum dot, Andreev reflection, Rashba spin-orbit interaction, Coherent indirect coupling, 73.63.Kv; 73.23.-b; 72.25.-b
A quantum dot (QD) is an artificially low-dimensional structure that can be filled with electrons (or holes). Two or more QDs can be coupled to form multiple-QD 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 many-body properties in a confined geometry can be performed [1-6]. Among the various multiple-QD systems, an Aharonov-Bohm (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 two-level system, the parallel DQD system that can become one of the promising candidates for the quantum bit in quantum computation has received more attention [7-20]. 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 1αcharacterizing 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 slave-boson mean-field 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 co-workers 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 spin-dependent 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 two-dimensional electron gas structure without the inversion symmetry in the growth direction, the Rashba spin-orbit interaction (RSOI) will emerge, and the RSOI can induce the spin-related phase factor in the tunneling matrix elements and the spin-flip 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 [26-30].
On the other side, the subgap transport through heterostructures with nano-objects (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 [31-35]. 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 [36-38]. 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 [39-42]. AR (or CAR) in nanoscopic heterostructures gives rise to a rich subgap structure in the current-voltage 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 solid-state 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 two-quantum dot Y-junction [44], which opens a possible route towards a test of the Einstein-Podolsky-Rosen (EPR) paradox and Bell inequalities in solid-state systems. Herrmann et al. used carbon nanotube DQD as Cooper pair beam splitters and realized the quantum optic-like experiments with spin-entangled 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 four-terminal 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 [7-10,13,15,16,19,21-25,44,45], 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 above-mentioned system in this paper. It is found that the RSOI and a nonzero coherent indirect coupling cause the spin-polarized 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 spin-polarized AR conductance. We note that the convex shape of the Andreev-reflected 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 spin-polarized current plot, and the pure spin current can be produced when two ferromagnetic leads are fully polarized. The spin-dependent 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 spin-dependent transport by means of the system parameters.
We consider a hybrid four-terminal AB interferometer including a parallel DQD coupled to two ferromagnetic reservoirs and two superconductors as shown in Figure Figure1.1. The system is described by the following Hamiltonian:
Figure 1
Figure 1
Schematic diagram of a four-terminal AB interferometer (color on line). The AB interferometer contains a coupled DQD with magnetic flux applied perpendicular to rings.
equation M1
(1)
where HF is the Hamiltonian of the left and right ferromagnetic electrodes
equation M2
(2)
Here, equation M3 is the creation (annihilation) operator in the lead νwith energy εν,. HS represents two superconducting reservoirs with chemical potential μs = 0 and the energy gap Δ,
equation M4
(3)
HDQD in Equation 1 denotes the DQD Hamiltonian
equation M5
(4)
in which equation M6 represents the creation (annihilation) operator of the electron with energy εi in the dot i; tc is the coupling strength taken as a real parameter. The last term, HT, in Equation 1 corresponds to the tunneling Hamiltonian between the DQD and four leads,
equation M7
(5)
where the tunneling matrix elements between the DQD and two ferromagnetic leads are equation M8, equation M9, equation M10, and equation M11. The phase shift due to the total magnetic flux threading into the AB ring is assumed to be ϕ = 2ΠL + ΦR)/ϕ0 with the flux quantum ϕ0 = h/e. The phase factor [var phi]Ri comes from the RSOI in dot i, which is tunable in the experiments [46,47]. equation M12 as the tunneling coupling between the DQD and two superconductors is also assumed to be independent of k and σ.
Using the nonequilibrium Green’s function technique, the spin-dependent current through the left ferromagnetic reservoir can be expressed as [48,49]
equation M13
(6)
where Tr is the trace in the spin space; equation M14 is a 4 × 4 matrix with Pauli matrix σz as its diagonal components; Gr,a,<(ε) are retarded, advanced, and lesser Green’s functions in the generalized 4 × 4 Nambu notation.
equation M15
(7)
equation M16
(8)
with the vector equation M17.
After some algebraic manipulations, the spin-dependent current can be derived from Equation 6:
equation M18
(9)
equation M19
(10)
in which equation M20 and equation M21 are the spin-dependent AR and CAR coefficients, respectively. equation M22 represents the single-particle tunneling through FL-DQD-FR or FR-DQD-FL. equation M23 corresponds to the probability of the quasiparticle tunneling among two superconductors and the left ferromagnetic lead. equation M24, equation M25, and fS are Fermi-Dirac distribution functions. The derivation of the spin-dependent current is minutely given in the Appendix.
Since we mainly focus on the AR process at zero temperature limit and set |eVL| = |eVR| < Δ, equation M26 will vanish. In the case of eVL = eVR, the current from the quasiparticle tunneling through FL-DQD-FR or FR-DQD-FL becomes zero; as a consequence, the AR dominates the transport through the four-terminal AB interferometer.
In the following numerical calculations, we mainly elucidate the spin-dependent AR process in the four-terminal AB interferometer with the coherent indirect coupling and the RSOI. We take e = h = kB = 1, and set Δ = 1 as the energy unit. Throughout the paper, the symmetric couplings with equation M27 and |PL| = |PR| are considered as a typical case.
Conductance
Because we mostly investigate the AR within the superconductor gap, in the limit of zero bias VL = VR → 0, the spin-related AR and CAR conductances have the forms
equation M28
(11)
and
equation M29
(12)
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 [var phi]R = Π/2; for the different coherent indirect coupling α, Figure Figure22 shows the total AR conductance ( equation M30 and equation M31) as a function of Fermi energy εF for parallel (P) and antiparallel (AP) configurations. In order to gain the clear physics, the Hamiltonian HDQD is diagonalized, and two energy eigenvalues are given as equation M32; thus, when the Fermi level coincides with the E+ and E, the resonant AR occurs and two peaks of AR conductances are located around the level E± as illustrated in Figure Figure2a,2a, b, c. For α = 0, it is clearly seen that equation M33 is always equal to equation M34 in the P arrangement (PL = PR = 0.4), and the magnitudes of two peaks are equal. However, for the case of the AP configuration (P L= −PR = 0.4), equation M35 appears when α = 0. Because the ferromagnetic leads have majority and minority electrons, the AR and the CAR are governed by the minority electrons for P configuration; thus, the AR and the CAR do not contribute the spin-polarized transport. For AP alignment, although the AR cannot produce the spin-polarized current, equation M36 emerges due to the CAR process, in which the CAR is governed by the majority electrons. This leads to the appearance of the spin-polarized conductance. Since two dots are indirectly coupled via two ferromagnetic leads, which is reflected in the nondiagonal coupled matrix elements (see Equations 15 and 16), α = 0 implies that the off-diagonal matrix elements vanish due to complete destructive quantum interference; thus, two dots are totally decoupled through two ferromagnetic leads. The AR (or the CAR ) can happen only through QD1 and QD2, respectively. This leads to the conductance equation M37 for the P arrangement and the equal height of two peaks ( equation M38 or equation M39) for the AP configuration.
Figure 2
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 ε1 = ε2 = 0, tc = 0.5, ϕ = 0, and [var phi]R = Π/2.
We also notice that both equation M40 and equation M41 occur with the increase of α for P and AP configurations; thus, equation M42 is nonzero at α ≠ 0, which means the occurrence of the spin-polarized AR for P configuration in the presence of the RSOI and the nonzero parameter α. As a matter of fact, we have found that equation M43 is independent of the parameter α for P configuration in the absence of the RSOI, which is not shown here. In comparison with the case of α = 0, the symmetry of AR conductances with respect to the Fermi energy is significantly broken when α ≠ 0. It is noticeable that amplitudes of conductance peaks near the level E+ decrease, and the magnitude of the right peaks is smaller than that of the left ones. In addition, the positions of peaks for AP alignment are also shifted with α = 1. These results indicate that the coherent indirect coupling and the RSOI play an important role in determining the feature of the AR conductance spectra.
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
equation M44
(13)
In Figure Figure3,3, 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 = 2(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
Figure 3
ARTMR versus the magnetic flux ϕ with different α. Other parameters are ε1 = ε2 = 0, tc = 0.5, and [var phi]R = Π/2
As we know, the RSOI can induce the spin precession and may even cause the inter-dot spin-flip effect. According to [26,27], the spin-dependent phase factor [var phi]R due to the RSOI can be expressed as equation M45, where β is the RSOI strength, m* is the electron effective mass, and Li is the length of dot i. [var phi]R is tunable in experiments. It can reach Π/2 easily or can be larger experimentally [27]. In order to explore further the influence of the coherent indirect coupling and the RSOI on the ARTMR, the ARTMR as a function of the parameter α for different [var phi]R is shown in Figure Figure4.4. We can see from Figure Figure44 that ARTMR versus α exhibits the nonmonotonic features, and there exists the crossing point at α = 0. Since α = 0 means that the coupling off-diagonal terms in Equation 16 are totally suppressed, as a consequence, the AMTMR is independent of the RSOI (see the Appendix). When [var phi]R is relatively small, this corresponds to the weak RSOI strength; thus, the variation of the ARTMR with α is not smart (solid line and dashed line). However, the evolution of the ARTMR is very remarkable as [var phi]R increases (dotted line and dash-dotted line), while the ARTMR first increases and decreases with the increases of α, even the negative ARTMR also emerges, which corresponds to a spin valve effect in the AR process. This reflects that the strong RSOI gives rise to the significant variation of the ARTMR. We also observe that the maximum and minimum values appear in the curves of the ARTMR. These demonstrate that the optimal ARTMR can be tuned by means of the external parameters.
Figure 4
Figure 4
ARTMR versus the coherent indirect coupling α with different [var phi]R. Other parameters are ε1 = ε2 = 0, tc = 0.5, and ϕ = 0.
Spin-dependent current
Above, we analyze the properties of the AR conductances. In the following discussions, we will explore the spin-dependent 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 spin-related current, Figure Figure55 displays the images of the spin-polarized current Is = IL↑ IL↓ as a function of the energy levels ε1 and ε2 of the DQD. The blue regions correspond to zero current, namely, IL↑ = IL↓ in these regimes. In the diagram, it is found that the spin-polarized current is symmetrical about the line of ε1 = ε2 and is asymmetrical with respect to the line of ε1 = −ε2 as illustrated in Figure Figure5a,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 (off-resonance). Is is relative small. This is because one QD is in the on-resonance state and the other is in the off-resonance state. When both ε1 and ε2 are close to the Fermi level by tuning the gate voltage, the maximal Isappears since DQD is in the on-resonance states. We also observe that, for α = 0.5 and α = −0.5, the spin-polarized 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 spin-polarized current.
Figure 5
Figure 5
Images of spin-polarized AR current current as a function of QD levels ε1 and ε2 (Color on line). (a) α = −0.5 and (b) α = 0.5. Other parameters are ϕ=0, [var phi]R = 0, tc = 0.5, and PL = PR = 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 PL = −PR = 1.0 or PL = −PR = −1.0, i.e., two ferromagnetic leads become two half metals; the normal AR vanishes due to equation M46 (see Equations 21 and 25). However, CAR dominates the transport through the four-terminal AB interferometer for AP alignment. As a consequence, we can obtain the pure spin current via CAR and two half-metal reservoirs. Thus, this device may be used as a pure spin-current injector even in the absence of the RSOI. In Figure Figure6,6, we also depict AB oscillations of the spin current for different α. For the case of α = 0.5, the magnitudes of the resonant peaks and valleys are enhanced with the increase of the RSOI strength, and positions of peaks and valleys are also shifted to the left, as illustrated in Figure Figure6a,6a, b. Since the RSOI gives rise to an extra spin-related phase factor [var phi]R (see Equation 16), the curves of the spin current versus magnetic flux ϕ move towards the left with the emergence of the RSOI phase, and the shifted magnitude of peaks (or valleys) is equal to [var phi]R as shown in Figure Figure6a,6a, b. Physically, the increase of [var phi]R corresponds to the strong RSOI, which also favors the CAR process and the generation of the large spin current. When the DQD is fully coupled via two ferromagnetic reservoirs (α = 1.0), in comparison with the case of α = 0.5, it is noted from Figure Figure6b6b that not only the positions of peaks and valleys are altered, but also the amplitudes of those are remarkably enhanced. This originates from the fact that the reduction of the destructive interference enhances the spin current for the case of α = 1.0. These results indicate that the variation of the spin current is sensitive to the parameter α and the strength of the RSOI, and the interplay between them determines the nature of the spin current.
Figure 6
Figure 6
The spin current versus the magnetic flux ϕ for different [var phi]R . (a) α = 0.5 and (b) α = 1.0. Other parameters are ε1 = ε2 = 0, tc = 0.5, PL = −PR = 1.0
In this paper, we have analyzed the AR of a four-terminal 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 spin-polarized AR can occur, stemming from the RSOI and a nonzero coherent indirect coupling. On the contrary, for AP configuration, the spin-polarized 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 spin-polarized current. A pure spin current can be generated via the CAR and two half-metal 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 spin-current 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 spin-dependent transport via the four-terminal AB setup.
In this appendix, we present the derivation of the current formulas in detail.
Let gr(ε) and Gr(ε) denote the retarded Green’s function of the DQD without and with the coupling to the external reservoirs. In the Nambu space, gr(ε) can be given as
equation M47
(14)
Based on the following Dyson equation, the retarded Green’s function of the system can be written as Gr(ε)]−1 = gr(ε)−1Σr, in which equation M48. The lesser Green’s function G<(ε) = Gr(ε)Σ<Ga(ε), where Ga(ε) = Gr(ε)]and equation M49 In the wide-band limit approximation, the retarded self-energy can be derived from the definition
equation M50
(15)
equation M51
(16)
equation M52
(17)
equation M53
(18)
where equation M54 with ρνσ being the density of states of the spin σ band in the lead ν. We calculate the tunneling matrix element by means of the Bardeen’s formula, i.e., equation M55, where me is the effective mass, S is the region of the integration, equation M56 is the wave function of evanescent mode of the lead ν, and equation M57 is the wave function of an electron localized in the QD i. Considering the propagation of electrons in the reservoir ν, this propagation process (the wave number dependence of equation M58) induces the coherent indirect coupling via the reservoir ν between two QDs, which is characterized with the parameter αν. We assume that (Xi, Yi, 0) is the center position of the ith QD, X1 = X2 = XD and L = |Y1Y2|. thus, αν is given by equation M59 based on [21]. We find |α| ≤ 1 and decreases with L, and |α| = 1 corresponds to L = 0. We define equation M60, in which equation M61, equation M62. With the definition of the spin polarization equation M63 in the lead ν, the tunneling matrix element can be written as equation M64 and equation M65 with equation M66. equation M67, Nγσ is the density of states when the superconductor is the normal state, and ργ(ε) is the modified BCS density of states equation M68. With the RSOI phase factor [var phi]R = [var phi]R1[var phi]R2, the spin-dependent phase factor is given by ϕσ = ϕ + 2σ[var phi]R. We mainly take account of the case of the symmetric coupling between two superconducting electrodes and DQD, that is, Γγ = Γs. According to the fluctuation-dissipation theorem, the lesser self-energy can be given as equation M69 and equation M70, where equation M71. Fν and Fγ are, respectively,
equation M72
(19)
equation M73
(20)
in which fν(εeVν) = fν, equation M74, and fs(ε) are the Fermi distribution functions. By substituting Equations 15 to 20) into Equation 6, we can obtain the spin-related current as shown in Equations 9 and 10. The AR (CAR) coefficients ( equation M75 and equation M76) and the probability of the quasiparticle tunneling ( equation M77 and equation M78) can be calculated as
equation M79
(21)
equation M80
(22)
equation M81
(23)
equation M82
(24)
equation M83
(25)
equation M84
(26)
equation M85
(27)
equation M86
(28)
Thus, we can investigate the quantum transport through our model system based on the above-mentioned 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). Chen-Long 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).
  • Reimann SM, Manninen M. Electronic structure of quantum dots. Rev Mod Phys. 2002;74:1283. doi: 10.1103/RevModPhys.74.1283. [Cross Ref]
  • Wang ZM. Self-Assembled Quantum Dots. New York: Springer; 2008.
  • Hanson R, Kouwenhoven LP, Petta JR, Tarucha S, Vandersypen LMK. Spins in few-electron quantum dots. Rev Mod Phys. 2007;79:1212.
  • Andergassen S, Meden V, Schoeller H, Splettstoesse J, Wegewijs MR. Charge transport through single molecules, quantum dots and quantum wires. Nanotechnology. 2010;21:272001. doi: 10.1088/0957-4484/21/27/272001. [PubMed] [Cross Ref]
  • Aleiner IL, Brouwer PW, Glazman LI. Quantum effects in Coulomb blockade. Phys Rep. 2002;358:309. doi: 10.1016/S0370-1573(01)00063-1. [Cross Ref]
  • Dubi Y, Di Ventra M. Heat flow and thermoelectricity in atomic and molecular junctions. Rev Mod Phys. 2011;83:131. doi: 10.1103/RevModPhys.83.131. [Cross Ref]
  • Lu HZ, Lü R, Zhu BF. Tunable Fano effect in parallel-coupled double quantum dot system. Phys Rev B. 2005;71:235320.
  • Trocha P, Barnas J. Quantum interference and Coulomb correlation effects in spin-polarized transport through two coupled quantum dots. Phys Rev B. 2007;76:165432.
  • Kubala B, Konig BK. Flux-dependent level attraction in double-dot Aharonov-Bohm interferometers. Phys Rev B. 2002;65:245301.
  • Chi F, Yuan XQ, Zheng J. Double Rashba quantum dots ring as a spin filter. Nanoscale Res Lett. 2008;3:343. doi: 10.1007/s11671-008-9163-z. [Cross Ref]
  • Liu YS, Chen H, Yang XF. Transport properties of an Aharonov-Bohm ring with strong interdot Coulomb interaction. J Phys Condens Matter. 2007;19:246201. doi: 10.1088/0953-8984/19/24/246201. [PubMed] [Cross Ref]
  • Zitko R, Mravlje J, Haule K. Ground state of the parallel double quantum dot system. Phys Rev Lett. 2012;108:066602. [PubMed]
  • Fang TF, Luo HG. Tuning the Kondo and Fano effects in double quantum dots. Phys Rev B. 2010;81:113402.
  • Krause T, Schaller G, Brandes T. Incomplete current fluctuation theorems for a four-terminal model. Phys Rev B. 2011;84:195113.
  • Loss D, Sukhorukov EV. Probing entanglement and nonlocality of electrons in a double-dot via transport and noise. Phys Rev Lett. 2000;84:1035. doi: 10.1103/PhysRevLett.84.1035. [PubMed] [Cross Ref]
  • Smirnov AY, Horing NJM, Mourokh LG. Aharonov-Bohm phase effects and inelastic scattering in transport through a parallel tunnel-coupled symmetric double-dot device. Appl Phys Lett. 2578;77:2000.
  • Sukhorukov EV, Burkard G, Loss D. Noise of a quantum dot system in the cotunneling regime. Phys Rev B. 2001;63:125315.
  • Mourokh LG, Horing NJM, Smirnov AY. Electron transport through a parallel double-dot system in the presence of Aharonov-Bohm flux and phonon scattering. Phys Rev B. 2002;66:085332.
  • Holleitner AW, Decker CR, Qin H, Eberl K, Blick RH. Coherent coupling of two quantum dots embedded in an Aharonov-Bohm interferometer. Phys Rev Lett. 2001;87:256802. [PubMed]
  • Holleitner AW, Blick RH, Huttel AK, Eberl K, Kotthaus JP. Probing and controlling the bonds of an artificial molecule. Science. 2002;297:70. doi: 10.1126/science.1071215. [PubMed] [Cross Ref]
  • Kubo T, Tokura Y, Hatano T, Tarucha S. Electron transport through Aharonov-Bohm interferometer with laterally coupled double quantum dots. Phys Rev B. 2006;74:205310.
  • Gurvitz SA. Quantum interference in resonant tunneling single spin measurements. IEEE Trans Nanotechol. 2005;4:45. doi: 10.1109/TNANO.2004.840151. [Cross Ref]
  • Kubo T, Tokura Y, Hatano T, Tarucha S. Exotic pseudospin Kondo effect in laterally coupled double quantum dots. Phys Rev B. 2008;77:041305(R).
  • Kubo T, Tokura Y, Hatano T, Tarucha S. Kondo effects and shot noise enhancement in a laterally coupled double quantum dot. Phys Rev B. 2011;83:115310.
  • 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. doi: 10.1088/0953-8984/24/5/055303. [PubMed] [Cross Ref]
  • Sun QF, Xie XC. Bias-controllable intrinsic spin polarization in a quantum dot: proposed scheme based on spin-orbit interaction. Phys Rev B. 2006;73:235301.
  • Sun QF, Wang J, Guo H. Quantum transport theory for nanostructures with Rashba spin-orbital interaction. Phys Rev B. 2005;71:165310.
  • Tserkovnyak Y, Akhanjee S. Spin-selective localization due to intrinsic spin-orbit coupling. Phys Rev B. 2009;79:085114.
  • Wu MW, Jiang JH, Weng MQ. Spin dynamics in semiconductors. Phys Rep. 2010;493:61. doi: 10.1016/j.physrep.2010.04.002. [Cross Ref]
  • Stepanenko D, Rudner M, Halperin BI, Loss D. Singlet-triplet splitting in double quantum dots due to spin-orbit and hyperfine interactions. Phys Rev B. 2012;85:075416.
  • Sun QF, Wang J, Lin TH. Resonant Andreev reflection in a normal-metal-quantum dot-supercoductor system. Phys Rev B. 1999;59:3831. doi: 10.1103/PhysRevB.59.3831. [Cross Ref]
  • Koerting V, Andersen BM, Flensberg K, Paaske J. Nonequilibrium transport via spin-induced subgap states in superconductor/quantum dot/normal metal cotunnel junctions. Phys Rev B. 2010;82:245108.
  • Baranski J, Domanski T. Fano-type interference in quantum dots coupled between metallic and superconducting leads. Phys Rev B. 2011;84:195424.
  • Xing YX, Wang J. Universal conductance fluctuations in mesoscopic systems with superconducting leads: beyond the Andreev approximation. Phys Rev B. 2010;82:245406.
  • Whitney RS, Jacquod P. Controlling the sign of magnetoconductance in Andreev quantum dots. Phys Rev Lett. 2009;103:247002. [PubMed]
  • Skadsem HJ, Brataas A, Martinek J, Tserkovnyak Y. Ferromagnetic resonance and voltage-induced transport in normal metal-ferromagnet-superconductor trilayers. Phys Rev B. 2011;84:104420.
  • Golubov AA, Tanaka Y, Mazin II, Dolgov OV. , Brinkman. Andreev spectra and subgap bound states in multiband superconductors. Phys Rev Lett 2009. p. 077003. [PubMed]
  • Annunziata G, Cuoco M, Noce C, Sudbo A, Linder J. Spin-sensitive long-range proximity effect in ferromagnet/spin-triplet-superconductor bilayers. Phys Rev B. 2011;83:060508(R).
  • Morten JP, Brataas A, Belzig W. Circuit theory of crossed Andreev reflection. Phys Rev B. 2006;74:214510.
  • Golubev DS, Zaikin AD. Non-local Andreev reflection in superconducting quantum dots. Phys Rev B. 2007;76:184510.
  • Sothmann B, Futterer D, Governale M, Konig J. Probing the exchange field of a quantum-dot spin valve by a superconducting lead. Phys Rev B. 2010;82:094514.
  • Futterer D, Governale M, Pala MG, Konig J. Nonlocal Andreev transport through an interacting quantum dot. Phys Rev B. 2009;79:054505.
  • Brauer J, Hubler F, Smetanin M, Beckmann D, Lohneysen HV. Nonlocal transport in normal-metal/superconductor hybrid structures: role of interference and interaction. Phys Rev B. 2010;81:024515.
  • Hofstetter L, Csonka S, Nygardand C, Schonenberger S. Cooper pair splitter realized in a two-quantum-dot Y-junction. Nature. 2009;461:960. doi: 10.1038/nature08432. [PubMed] [Cross Ref]
  • Herrmann LG, Portier F, Roche P, Yeyati AL, Kontos T, Strunk C. Carbon nanotubes as Cooper-pair beam splitters. Phys Rev Lett. 2010;104:026801. [PubMed]
  • Nitta J, Akazaki T, Takayanagi H, Enoki T. Gate control of spin-orbit interaction in an inverted In0.53Ga0.47As/In0.52Al0.48As heterostructure. Phys Rev Lett. 1997;78:1335. doi: 10.1103/PhysRevLett.78.1335. [Cross Ref]
  • Matsuyama T, Kursten R, Meissner C, Merkt U. Rashba spin splitting in inversion layers on p-type bulk InAs. Phys Rev B. 2000;61:15588. doi: 10.1103/PhysRevB.61.15588. [Cross Ref]
  • Jauho AP, Haug H. Quantum Kinetics in Transport and Optics of Semiconductors. Berlin: Springer; 2008.
  • Jauho AP, Wingreen NS, Meir Y. Time-dependent transport in interacting and noninteracting resonant-tunneling systems. Phys Rev B. 1994;50:5528. doi: 10.1103/PhysRevB.50.5528. [PubMed] [Cross Ref]
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