Abstract
Spindependent transport through a quantumdot (QD) ring coupled to ferromagnetic leads with noncollinear magnetizations is studied theoretically. Tunneling current, current spin polarization and tunnel magnetoresistance (TMR) as functions of the bias voltage and the direct coupling strength between the two leads are analyzed by the nonequilibrium Green's function technique. It is shown that the magnitudes of these quantities are sensitive to the relative angle between the leads' magnetic moments and the quantum interference effect originated from the interlead coupling. We pay particular attention on the Coulomb blockade regime and find the relative current magnitudes of different magnetization angles can be reversed by tuning the interlead coupling strength, resulting in sign change of the TMR. For large enough interlead coupling strength, the current spin polarizations for parallel and antiparallel magnetic configurations will approach to unit and zero, respectively.
PACS numbers:
Introduction
Manipulation of electron spin degree of freedom is one of the most frequently studied subjects in modern solid state physics, for both its fundamental physics and its attractive potential applications [1,2]. Spintronics devices based on the giant magnetoresistance effect in magnetic multilayers such as magnetic field sensor and magnetic hard disk read heads have been used as commercial products, and have greatly influenced current electronic industry. Due to the rapid development of nanotechnology, recent much attention has been paid on the spin injection and tunnel magnetoresistance (TMR) effect in tunnel junctions made of semiconductor spacers sandwiched between ferromagnetic leads[3]. Moreover, semiconductor spacers of InAs quantum dot (QD), which has controllable size and energy spectrum, has been inserted in between nickel or cobalt leads[46]. In such a device, the spin polarization of the current injected from the ferromagnetic leads and the TMR can be effectively tuned by a gate nearby the QD, and opens new possible applications. Its new characteristics, for example, anomalies of the TMR caused by the intradot Coulomb repulsion energy in the QD, were analyzed in subsequent theoretical work based on the nonequilibrium Green's function method [7].
The TMR is a crucial physical quantity measuring the change in system's transport properties when the angle φ between magnetic moments of the leads rotate from 0 (parallel alignment) to arbitrary value (or to φ in collinear magnetic moments case). Much recent work has been devoted to such an effect in QD coupled to ferromagnetic leads with either collinear[413] or noncollinear[1416] configurations. It was found that the electrically tunable QD energy spectrum and the Coulomb blockade effect dominate both the magnitude and the signs of the TMR[416].
On the other hand, there has been increasing concern about spin manipulation via quantum interference effect in a ringtype or multipath mesoscopic system, mainly relying on the spindependent phase originated from the spinorbit interaction existed in electron transport channels[1720]. Many recent experimental and theoretical studies indicated that the current spin polarization based on the spinorbital interaction can reach as high as 100%[2123] or infinite[2429]. Meanwhile, large spin accumulation on the dots was realized by adjusting external electrical field or gate voltages to tune the spinorbit interaction strength (or equivalently the spindependent phase factor)[2730]. Furthermore, there has already been much very recent work about spindependent transport in a QDring connected to collinear magnetic leads[3134]. Much richer physical phenomena, such as interferenceinduced TMR enhancement, suppression or sign change, were found and analyzed[3134].
Up to now, the magnetic configurations of the leads coupled to the QDring are limited to collinear (parallel and antiparallel) one. To the best of our knowledge, transport characteristics of a QDring with noncollinear magnetic moments have never studied, which is the motivation of the present paper. As shown in Figure 1 we study the device of a quantum ring with a QD inserted in one of its arms. The QD is coupled to the left and the right ferromagnetic leads whose magnetic moments lie in a common plane and form an arbitrary angle with respect to each other. There is also a bridge between the two leads indicating interlead coupling. It should be noted that such a QDring connected to normal leads has already been realized in experiments[3540]. Considering recent technological development[46], our model may also be realizable.
Figure 1. Schematic picture of singledot ring with noncollinearly polarized ferromagnetic leads.
Model and Method
The system can be modeled by the following Hamiltonian[14,20,30]
where is the creation (annihilation) operator of the electrons with momentum k, spinσ and energy ε_{kβσ }in the βth lead (β = L, R); creates (annihilates) an electron in the QD with spin σ and energy ε_{d}; t_{βd }and t_{LR }describe the dotlead and interlead tunneling coupling, respectively; U is the intradot Coulomb repulsion energy. φ denotes the angle between the magnetic moments of the leads, which changes from 0 (parallel alignment) to π (antiparallel alignment).
The current of each spin component flowing through lead β is calculated from the time evolution of the occupation number , and can be written in terms of the Green's functions as[20,30]
where the Keldysh Green's function G(ε) is the Fourier transform of G(t  t') defined as , . In our present case, it is convenient to write the Green's function as a 6 × 6 matrix in the representation of (L ↑〉, R ↓〉, d ↑〉, L ↓〉, R ↓〉, d ↓〉). Thus the lesser Green's function G^{ < }(ε) and the associated retarded (advanced) Green's function G^{r(a)}(ε) can be calculated from the Keldysh and the Dayson equations, respectively. Detail calculation process is similar to that in some previous works [20,30], and we do not give them here for the sake of compactness. Finally, the ferromagnetism of the leads is considered by the spin dependence of the leads' density of states ρ_{βσ}. Explicitly, we introduce a spinpolarization parameter for lead β of P_{β }= (ρ_{β↑ } ρ_{β↓})/(ρ_{β↑ }+ ρ_{β↓}), or equivalently, ρ_{β↑(↓) }= ρ_{β }(1 ± P_{β}), with ρ_{β }being the spinindependent density of states of lead β .
Result and Discussion
In the following numerical calculations, we choose the intradot Coulomb interaction U = 1 as the energy unit and fix ρ_{L }= ρ_{R }= ρ_{0 }= 1, t_{Ld }= t_{Rd }= 0.04. Then the linewidth function in the case of p_{L }= p_{R }= 0 is Γ_{β }≡ 2πρ_{β}t_{βd}^{2 }≈ 0.01, which is accessible in a typical QD[4143]. The bias voltage V is related to the left and the right leads' chemical potentials as eV = μ_{L } μ_{R}, and μ_{R }is set to be zero throughout the paper.
Bias dependence of electric current J = J_{↑ }+ J↓, where J_{σ }= (J_{Lσ } J_{Rσ})/2 is the symmetrized current for spinσ, current spin polarization p = (J_{↑ } J↓)/(J_{↑ }+ J↓), and TMR=[J(φ = 0)  J(φ)]/J(φ) are shown in Figure 2 for selected values of the angle φ. In the absence of interlead coupling (t_{LR }= 0), the electric current in Figure 2(a) shows typical step configuration due to the Coulomb blockade effect. The current step emerged in the negative bias region occurs when the dot level ε_{d }is aligned to the Fermi level of the right lead (μ_{R }= 0). Now electrons tunnel from the right lead via the dot to the left lead because μ_{L }= eV < εε_{d }= 0. The dot can be occupied by a single electron with either spinup or spindown orientation, which prevents double occupation on ε_{d }due to the Pauli exclusion principle. Since the other transport channel ε_{d }+ U is out of the bias window, the current keeps as a constant in the bias regime of eV < ε_{d }= 0. In the positive bias regime of ε_{d }< eV < ε_{d }+ U a single electron transport sequentially from the left lead through the dot to the right lead, inducing another current step. The step at higher bias voltage corresponds to the case when ε_{d }+ U crosses the Fermi level. Now the dot may be doubly occupied, and no step will emerge regardless of the increasing of the bias voltage.
Figure 2. Total current J, current spin polarization p and TMR each as a function of the bias voltage for different values of φ. t_{LR }= 0 in Figs. (a) to (c) and t_{LR }= 0.01 in Figs. (d) to (f). The other parameters are intradot energy level ε_{d }= 0, temperature T = 0.01, and polarization of the leads P_{L }= P_{R }= 0.4.
When the relative angle between the leads' magnetic moments φ. rotates from 0 to π, a monotonous suppression of the electric current appears, which is known as the typical spin valve effect. The suppression of the current can be attributed to the increased spin accumulation on the QD[1416]. Since the linewidth functions of different spin orientations are continuously tuned by the angle variation, a certain spin component electron with smaller tunneling rate will be accumulated on the dot, and furthermore prevents other tunnel processes. As shown in Figure 2(b), the current spin polarizations in the bias ranges of eV < ε_{d }and eV > ε_{d }+ U are constant and monotonously suppressed by the increase of the angle, which changes the spinup and spindown linewidth functions. In the Coulomb blockade region of ε_{d }< eV < ε_{d }+ U, the difference between the current spin polarizations of different values of φ is greatly decreased, which is resulted from the Pauli exclusion principle. The current spin polarizations also have small dips and peaks respectively near eV = ε_{d }and eV = ε_{d }+ U, where new transport channel opens. The most prominent characteristic of the TMR in Figure 2(c) is that its magnitude in the Coulomb blockade region depends much sensitively on the angle than those in other bias ranges. The deepness of the TMR valleys are shallowed with the increasing of the angle. Meanwhile, dips emerge when the Fermi level crosses ε_{d }and ε_{d }+ U. In the antiparallel configuration (φ = π), the magnitude of the TMR is larger than those in other bias voltage ranges.
When the interlead coupling is turned on as shown in Figure 2(d)(f), both the studied quantities are influenced. Since the bridge between the leads serves as an electron transport channel with continuous energy spectrum, the system electric current increases with increasing bias voltage [Figure 2(d)]. For the present weak interlead coupling case of t_{LR }< t_{βd}, the transportation through the QD is the dominant channel with distinguishable Coulomb blockade effect. The current spin polarizations for different angles in the voltage ranges out of the Coulomb blockade one now change with the bias voltage value, but their relative magnitudes somewhat keep constant. The difference between the current spin polarization magnitude of different angle is enlarged by the interference effect brought about by the interlead coupling. Comparing Figure 2(f) with 2(c), the behavior of TMR is less influenced by the bridge between the leads in the present case.
We now fix t_{LR }= 0.01 and the angle φ = π/2, i.e., the magnetic moments of the leads are perpendicular to each other, to examine the bias dependence of these quantities for different values of leads' polarization P_{L }= P_{R }= P . The electric currents in the bias voltage ranges of eV < ε_{d }and eV > ε_{d }+ U. are monotonously suppressed with the increase of P [Figure 3(a)]. This is because the spin accumulation on the dot in these bias ranges is enlarged by the increase of the leads' spin polarization. In the Coulomb blockade region, however, current magnitudes of different P are identical. The reason is that in this region the spin accumulation induced by the Pauli exclusion principle, which was previously discussed, plays a decisive role compared with that brought about by the leads' spin polarization. As is expected, the current spin polarization is increased with increasing P , which is shown in Figure 3(b). The magnitude of the TMR in Figure 3(c) increases with increasing P. For the halfmetallic leads (P_{L }= P_{R }= P = 1), the magnitude of the TMR is much larger than those of usual ferromagnetic leads (P_{β }< 1). All these results are similar to those of a single dot case[1416].
Figure 3. Tunneling current, current polarization and TMR each as a function of the bias voltage for different values of leads' polarization and fixed φ = π/2. The other parameters are as in Fig. 2.
Finally we study how the interlead coupling strength t_{LR }influence these quantities. In Figure 4 we show their characteristics each as a function of t_{LR }with fixed bias voltage eV = U and ε_{d }= 0.5, which means that we are focusing on the Coulomb blockade region. It is shown in Figure 4(a) that in the case of weak interlead coupling, typical spin valve effect holds true, i.e., the current magnitude is decreased with increasing φ as was shown in Figure 2(a) and 2(d) (see the Coulomb blockade region in them). With the increase of t_{LR}, reverse spin valve effect is found, in other words, current magnitudes of larger angles become larger than those of smaller angles. This phenomenon can be understood by examining the spindependent linewidth function. The basic reason is that in this Coulomb blockade region, the relative magnitudes of the currents through the QD of different angle will keep unchanged regardless of the values of t_{LR }(see Figure 2). But the current through the bridge between the leads, which is directly proportional to the interlead linewidth function , will be drastically varied by the angle. In the parallel configuration, for example, spinup interlead linewidth function is larger than the spindown one since ρ_{L↑ }= ρ_{R↑ }= ρ_{0 }(1 + P_{β}) and ρ_{L↓ }= ρ_{R↓ }= ρ_{0 }(1  P_{β}). So the current polarization will increase with increasing t_{LR }as shown by the solid curve in Figure 4(b). As the polarization of the leads is fixed, both spinup and spindown linewidth functions will be enhanced with increasing t_{LR}, resulting in increased total current as shown in Figure 4(a). For the antiparallel case (φ = π), the current magnitude will also be enhanced for the same reason. But the current spin polarization is irrelevant to the tunnel process through the bridge since ρ_{L↑ }= ρ_{R↓ }= ρ_{0 }(1 + P_{β}) and ρ_{L↓ }= ρ_{R↑ }= ρ_{0 }(1  P_{β}). The interlead linewidth functions of both spin components are equal . The current spin polarization is mainly determined by the transport process through the QD. From the above discussion we also know that the current magnitude of the parallel configuration through the bridge is larger than that of the antiparallel alignment. With the increase of t_{LR}, current through the bridge play a dominant role as compared with that through the dot, and the reverse spin valve effect may emerge accordingly. For the case of 0 < φ < π, the behavior of the current can also be understood with the help of the above discussions. Due to the reverse spin valve effect, the TMR in Figure 4(c) is reduced with increasing t_{LR}, and becomes negative for high enough interlead coupling strength.
Figure 4. Current, current polarization and TMR each as a function of the interlead coupling strength for different values of φ and fixed P_{L }= P_{R }= 0.3. The other parameters are as in Fig. 2.
Conclusion
We have studied the characteristics of tunneling current, current spin polarization and TMR in a quantumdotring with noncollinearly polarized magnetic leads. It is found that the characteristics of these quantities can be well tuned by the relative angle between the leads' magnetic moments. Especially in the Coulomb blockade and strong interlead coupling strength range, the currents of larger angles are larger than those of smaller ones. This phenomenon is quite different from the usual spinvalve effect, of which the current is monotonously suppressed by the increase of the angle. The TMR in this range can be suppressed even to negative, and the current spin polarizations of parallel and antiparallel configurations individually approach to unit and zero, which can then serve as a effective spin filter even for usual ferromagnetic leads with 0 < P_{β }< 1.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
JMM and JZ carried out numerical calculations as well as the establishment of the figures. KCZ, YJP and FC established the theoretical formalism and drafted the manuscript. FC conceived of the study, and participated in its design and coordination.
Acknowledgements
This work was supported by the Education Department of Liaoning Province under Grants No. 2009A031 and 2009R01. Chi acknowledge support from SKLSM under Grant No. CHJG200901.
References

Science 1998, 282:1660. PubMed Abstract  Publisher Full Text

Wolf SA, Awschalom DD, Buhrman RA, Daughton JM, von Molnér S, Roukes ML, Chtchelkanova AY, Treger DM: Spintronics: A SpinBased Electronics Vision for the Future.
Science 2001, 294:1488. PubMed Abstract  Publisher Full Text

Jacak L, Hawrylak P, Wójs A: Quantum dots. New York: SpringerVerlag; 1998.

Hamaya K, Masubuchi S, Kawamura M, Machida T, Jung M, Shibata K, Hirakawa K, Taniyama T, Ishida S, Arakawa Y: Spin transport through a single selfassembled InAs quantum dot with ferromagnetic leads.
Appl Phys Lett 2007, 90:053108. Publisher Full Text

Hamaya K, Kitabatake M, Shibata K, Jung M, Kawamura M, Machida T, Ishida S, Arakawa Y: Electricfield control of tunneling magnetoresistance effect in a Ni/InAs/Ni quantumdot spin valve.
Appl Phys Lett 2007, 91:022107. Publisher Full Text

Hamaya K, Kitabatake M, Shibata K, Jung M, Kawamura M, Ishida S, Taniyama T, Hirakawa K, Arakawa Y, Machida T: Oscillatory changes in the tunneling magnetoresistance effect in semiconductor quantumdot spin valves.
Phys Rev B 2008, 77:081302(R). Publisher Full Text

Stefański P: Tunneling magnetoresistance anomalies in a Coulomb blockaded quantum dot.

Bulka BR: Current and power spectrum in a magnetic tunnel device with an atomicsize spacer.
Phys Rev B 2000, 62:1186. Publisher Full Text

Rudziński W, Barnaś J: Tunnel magnetoresistance in ferromagnetic junctions: Tunneling through a single discrete level.

Cottet A, Belzig W, Bruder C: Positive Cross Correlations in a ThreeTerminal Quantum Dot with Ferromagnetic Contacts.
Phys Rev Lett 2004, 92:206801. PubMed Abstract  Publisher Full Text

Weymann I, König J, Martinek J, Barnaś J, Schön G: Metallic Si(111)(7 × 7)reconstruction: A surface close to a MottHubbard metalinsulator transition.
Phys Rev B 2005, 72:115314. Publisher Full Text

Misiorny M, Weymann I, Barnaś J: Spin effects in transport through singlemolecule magnets in the sequential and cotunneling regimes.
Phys Rev B 2009, 79:224420. Publisher Full Text

Weymann I, Barnaś J: Kondo effect in a quantum dot coupled to ferromagnetic leads and sidecoupled to a nonmagnetic reservoir.
Phys Rev B 2010, 81:035331. Publisher Full Text

König J, Martinek J: InteractionDriven Spin Precession in QuantumDot Spin Valves.
Phys Rev Lett 2003, 90:166602. PubMed Abstract  Publisher Full Text

Braun M, König J, Martinek J: Theory of transport through quantumdot spin valves in the weakcoupling regime.
Phys Rev B 2004, 70:195345. Publisher Full Text

Rudziński W, Barnaś J, Świrkowicz R, Wilczyński M: Spin effects in electron tunneling through a quantum dot coupled to noncollinearly polarized ferromagnetic leads.

Li SS, Xia JB: Spinorbit splitting of a hydrogenic donor impurity in GaAs/GaAlAs quantum wells.
Appl Phys Lett 2008, 92:022102. Publisher Full Text

Li SS, Xia JB: Electronic structures of N quantum dot molecule.
Appl Phys Lett 2007, 91:092119. Publisher Full Text

Sun QF, Wang J, Guo H: Quantum transport theory for nanostructures with Rashba spinorbital interaction.
Phys Rev B 2005, 71:165310. Publisher Full Text

Sun QF, Xie XC: Spontaneous spinpolarized current in a nonuniform Rashba interaction system.
Phys Rev B 2005, 71:155321. Publisher Full Text

Chi F, Li SS: Spinpolarized transport through an AharonovBohm interferometer with Rashba spinorbit interaction.
J Appl Phys 2006, 100:113703. Publisher Full Text

Chi F, Yuan XQ, Zheng J: Double Rashba Quantum Dots Ring as a Spin Filter.
Nanoscale Res Lett 2008, 3:343. Publisher Full Text

Chi F, Zheng J: Spin separation via a threeterminal AharonovCBohm interferometers.
Appl Phys Lett 2008, 92:062106. Publisher Full Text

Xing YX, Sun QF, Wang J: Nature of spin Hall effect in a finite ballistic twodimensional system with Rashba and Dresselhaus spinorbit interaction.
Phys Rev B 2006, 73:205339. Publisher Full Text

Xing YX, Sun QF, Wang J: Symmetry and transport property of spin current induced spinHall effect.
Phys Rev B 2007, 75:075324. Publisher Full Text

Xing YX, Sun QF, Wang J: Influence of dephasing on the quantum Hall effect and the spin Hall effect.
Phys Rev B 2008, 77:115346. Publisher Full Text

Lü HF, Guo Y: Pumped pure spin current and shot noise spectra in a twolevel Rashba dot.

Chi F, Zheng J, Sun LL: Spinpolarized current and spin accumulation in a threeterminal two quantum dots ring.
Appl Phys Lett 2008, 92:172104. Publisher Full Text

Chi F, Zheng J, Sun LL: Spin accumulation and pure spin current in a threeterminal quantum dot ring with Rashba spinorbit effect.
J Appl Phys 2008, 104:043707. Publisher Full Text

Sun QF, Xie XC: Biascontrollable intrinsic spin polarization in a quantum dot: Proposed scheme based on spinorbit interaction.
Phys Rev B 2006, 73:235301. Publisher Full Text

Trocha P, Barnaś J: Quantum interference and Coulomb correlation effects in spinpolarized transport through two coupled quantum dots.
Phys Rev B 2007, 76:165432. Publisher Full Text

Weymann I: Effects of different geometries on the conductance, shot noise, and tunnel magnetoresistance of double quantum dots.
Phys Rev B 2008, 78:045310. Publisher Full Text

Chi F, Zeng H, Yuan XQ: Fluxdependent tunnel magnetoresistance in parallelcoupled double quantum dots.
Superlatt Microstruct 2009, 46:523. Publisher Full Text

Trocha P, Weymann I, Barnaś J: Negative tunnel magnetoresistance and differential conductance in transport through double quantum dots.
Phys Rev B 2009, 80:165333. Publisher Full Text

Chen JC, Chang AM, Melloch MR: Transition between Quantum States in a ParallelCoupled Double Quantum Dot.
Phys Rev Lett 2004, 92:176801. PubMed Abstract  Publisher Full Text

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

Wang ZhM, Holmes K, Mazur YI, Ramsey KA, Salamo GJ: Selforganization of quantumdot pairs by hightemperature droplet epitaxy.
Nanoscale Res Lett 2006, 1:57. Publisher Full Text

Strom NW, Wang ZhM, Lee JH, AbuWaar ZY, Mazur YuI, Salamo GJ: Selfassembled InAs quantum dot formation on GaAs ringlike nanostructure templates.
Nanoscale Res Lett 2007, 2:112. Publisher Full Text

Lee JH, Wang ZhM, Strm NW, Mazur YI, Salamo GJ: InGaAs quantum dot molecules around selfassembled GaAs nanomound templates.
Appl Phys Lett 2006, 89:202101. Publisher Full Text

Hankea M, Schmidbauer M, Grigoriev D, Stäfer P, Köhler R, Metzger TH, Wang ZhM, Mazur YuI, Jalamo G: Zerostrain GaAs quantum dot molecules as investigated by xray diffuse scattering.
Appl Phys Lett 2006, 89:053116. Publisher Full Text

Hanson R, Kouwenhoven LP, Petta JR, Tarucha S, Vandersypen LMK: Spins in fewelectron quantum dots.
Rev Mod Phys 2007, 79:1217. Publisher Full Text

Li SS, Abliz A, Yang FH, Niu ZC, Feng SL, Xia JB: Electron and hole transport through quantum dots.
J Appl Phys 2002, 92:6662. Publisher Full Text

Li SS, Abliz A, Yang FH, Niu ZC, Feng SL, Xia JB: Electron transport through coupled quantum dots.
J Appl Phys 2003, 94:5402. Publisher Full Text