Abstract
We investigate the spin accumulations of AharonovBohm interferometers with embedded quantum dots by considering spin bias in the leads. It is found that regardless of the interferometer configurations, the spin accumulations are closely determined by their quantum interference features. This is mainly manifested in the dependence of spin accumulations on the threaded magnetic flux and the nonresonant transmission process. Namely, the AharonovBohmFano effect is a necessary condition to achieve the spin accumulation in the quantum dot of the resonant channel. Further analysis showed that in the doubledot interferometer, the spin accumulation can be detailedly manipulated. The spin accumulation properties of such structures offer a new scheme of spin manipulation. When the intradot Coulomb interactions are taken into account, we find that the electron interactions are advantageous to the spin accumulation in the resonant channel.
Keywords:
Spin accumulations; AharonovBohmFano effect; quantum dot; Coulomb interaction; 73.63.Kv; 71.70.Ej; 72.25.bBackground
Quantum dot (QD), especially coupledQD system (i.e., the QD molecule), is of fundamental interest in physics and possesses potential applications, such as quantum logic gates[1,2]. As a result, many experimental and theoretical works have paid so much attention to the electron transport properties of various multiQD systems in the past decades [310]. Besides, the progress of nanotechnology enables researchers to fabricate a variety of coupledQD structures with sizes smaller than the electron coherence length [11]. This also accelerates the development of researches on the coupledQD characteristics.
With respect to the coupledQD structures, the typical one is AharonovBohm (AB) interferometer with one QD or whose individual arm is of one QD, respectively [1238]. In such kind of structure, the AB phase can adjust the quantum interference, leading to abundant interesting results. Kobayashi et al. performed significant work to study the quantum interferences in the AB interferometers with embedded QDs [1820]. According to their conclusions, the Fano effect, which manifests itself in the asymmetric lineshape of the transport spectrum, can be observed in such structures by constructing nonresonant and resonant channels for electron transmission. Moreover, they showed that the orientation of the Fano lineshape changes periodically with the magnetic flux. Due to this reason, in the AB interferometer with QDs, the ABFano interference attracted more attention and was further investigated [22,23]. On the other hand, lots of theoretical investigations about electron transport behaviors of the AB interferometer have been reported. It was found that the interplay between the ABFano effect and the other mechanisms, e.g., Kondo physics and the spinorbit interaction, indeed causes many interesting phenomena [2438].
Electron not only has a charge but also spins with
Model and numerical results
The Hamiltonian that describes the electron motion in the AB interferometer can be written as
H_{α}(α=L,R) is the Hamiltonian in leadα. H_{D} is the Hamiltonian in the QDs, and the last term, H_{T}, denotes electron traveling between the two leads. H_{α}takes a form as
The oneQD AB interferometer
We first focus on the AB interferometer of one QD, whose schematic is shown in Figure
1a. Then in such a case,
Figure 1. The AB interferometer of one QD. (a) Schematic of an AB interferometer with an embedded QD. (b, c, d) The average electron occupation number and spin accumulation in QD affected by the structure parameters. The relevant parameters are taken to be ρ = 1, V_{α}= 0.1, and eV_{s} = 1.0.
The electron properties can be evaluated by using the nonequilibrium Green function technique. In the Green function space, the average electron occupation number of the QD is denoted as [61,62]
G^{<} is the lesser Green function, which can be obtained from the Dyson equation
G^{r} and G^{a} are the retarded and advanced Green functions, respectively. Due to the presence of electron interaction, the Green function is difficult to solve. However, if the system temperature is higher than the Kondo temperature, the electron interaction term can be included by using the HubbardI approximation [6163]. In this work, we would like to consider the case of weak electron correlation; then, the retarded Green function can be analytically solved within the HubbardI approximation, i.e.,
where
Furthermore, by defining
With the help of Equation 5, we investigate the average electron occupation number influenced by the structure parameters in Figure 1b,c,d. The system temperature is fixed at k_{B}T=0.1. For the other parameters, we choose the spin bias eV_{s}=1.0 and the QDlead coupling strength V_{α}=0.1. In Figure 1b, it is observed that at the case of ϕ=0.5Π, a spinup electron can enter the QD only when the QD level decreases to the position of ϵ=−0.5. Instead, the QD is able to confine a spindown electron if ϵ<0.5. By comparing the properties of oppositespin electrons, we might as well consider that the spinup and spindown electrons are both in equilibrium, but they ‘feel’ the different ‘Fermi levels’, with the distance between them being the spin bias magnitude. Therefore, in such a structure, the striking spin accumulation can be realized in the QD. Next, in Figure 1c,d, by assuming ϵ=0, we present the influence of ϕand W on the average occupation number of electron in the QD, respectively. It is observed that with the change of magnetic flux, the average occupation of differentspin electrons show opposite variation features. Different from the result of ϕ=0.5Π, when the magnetic flux is increased to ϕ=1.5Π, the QD confines a spinup electron. Then, the spin accumulation in the QD can be completely adjusted. Alternatively, with W increased to W=0.3, the spin accumulation proportionally enhances; however, the further increase of W will lead to the suppression of spin accumulation.
Since the structure is relatively simple, we try to clarify the numerical result in
an analytical way. Accordingly, we write out the expression of
The underlying physics being responsible for the above results is quantum interference.
It is known that the interference in the QD ring structure is rather complicated.
However, in such a structure, the quantum interference that affects the spin accumulation
just occurs between two Feynman paths. This is because
In the following, we incorporate the electron interaction into the calculation. In the case of weak QDlead coupling, 〈n_{σ}〉 can be expressed in an analytical way, i.e.,
in which
In Figure 2, by assuming W=0.3 and
Figure 2. The influence of Coulomb interaction on the properties of 〈n_{σ}〉 (a) and 〈n_{s}〉 (b). The relevant parameters are taken to be W = 0.3, V_{α} = 0.1, and
The doubleQD AB interferometer
The AB interferometer with one QD in each of its arm (see Figure 3a) is another typical structure in studying the electron transport behaviors modified
by the AB phase. For such a structure, both H_{D} and H_{T} have alternative forms as
Figure 3. The AB interferometer with two QDs. (a) Schematic of an AB interferometer with one QD in each of its arm. (b) The spin accumulation in QD2 influenced by the properties of the other arm. (c) The influence of the QD levels on the spin accumulation in QD2. (d) The spin accumulation in QD2 affected by the local magnetic flux. The spin bias is fixed with eV_{s} = 1.0.
Here, we would like to know whether the Fano interference manner is also necessary
to achieve the spin accumulation of such a structure. If so, how do the properties
of nonresonant channel affect the spin accumulation? Based on such an idea, we begin
to analyze the average electron occupation number of QDj by the formula
where
Without loss of generality, we take QD2 as an example to investigate the spin accumulation
behaviors of such a structure. In the presence of magnetic flux, the coupling coefficients
take the following form: V_{L1}=V_{1}e^{iϕ/4},
Next, we choose V_{1}=0.6 and investigate the spin accumulation in QD2 influenced by the change of QD levels, as shown in Figure 3c. We find that similar to the former structure, the spin accumulation occurs only when the corresponding QD level is located in the spin bias window. However, the characteristic of 〈n_{2s}〉 lies where its sign ( + /−) is differentiated by the line of ϵ_{1}=ϵ_{2}, where the spin accumulation disappears. This result indicates that if the spin bias is large enough, at the point of ϵ_{1}=0, the sign of 〈n_{2s}〉 can be altered by the change of ϵ_{2}. On the other hand, in Figure 3d, we investigate 〈n_{2s}〉 as functions of ϕ and ϵ_{2}. The QDlead couplings are taken to V_{1}=5V_{2}=0.5, and the level of QD1 is fixed at ϵ_{1}=1. It is seen that the reversal of the magnetic flux direction can change the sign of spin accumulation, but in such a structure, the level of QD2 tends to affect the maximum of spin accumulation, which appears around the points of ϵ_{2}=0.25 and ϕ=±0.3Π. Thereby, we notice that the properties of the resonant channel, e.g., the level of QD2, are also important factors to change the magnitude of the spin accumulation.
For such a structure, it is difficult for us to write out the analytical expression
of 〈n_{2s}〉. So, we can only present a qualitative discussion to explain the above results by
analyzing the quantum interference that contributes to the spin accumulation. Obviously,
the expression of
where
Despite the complicated quantum interferences among infinite paths, we try to clarify
the quantum interference feature by calculating the phase differences between the
lowestorder paths. This is because the quantum interference among lowestorder paths
contributes mainly to the coupling between QD2 and the leads. For instance, the three
lowestorder paths between QD2 and leadL are
Next, we demonstrate the effect of ϵ_{2}on the value of 〈n_{2s}〉. In Equations 9 to 10, one can find that in the higherorder paths, the two arms of the interferometer are visited repeatedly. Then, the properties of the two arms play an important role in affecting the quantum interference. In the study by Gong et al. [63], our calculations showed that when the levels of the two QDs are the same, the quantum interference between the two arms become weak, but only the nonresonant one determines the electron properties of this structure. As a consequence, in such a case, the interferometer can be considered as a singlechannel structure, and then, the picture of quantum interference disappears. With this viewpoint, we understand the vanishment of the spin accumulation in the case of ϵ_{1}=ϵ_{2}.
In Figure 4, by choosing V_{1}=0.5,V_{2}=0.1, and
Figure 4. The influence of Coulomb interactions on the spectra of 〈n_{2s}〉. The relevant parameters are taken to be V_{1} = 0.5, V_{2} = 0.1, and
Summary
In summary, we have studied the spin accumulation characteristics of two AB interferometers with QDs embedded in their arms by considering spin bias in the leads. It has been found that regardless of the configurations of the interferometers, the spin accumulations are strongly dependent on the quantum interference features of the interferometers. Namely, the nonresonant transmission ability between the leads and the local magnetic flux can efficiently adjust the spin accumulation properties of the QD. By analyzing the quantum interferences among the Feynman paths, it was seen that the quantum interferences can cause the QD in the resonant channel to be decoupled from one of the leads. Accordingly, the spin bias in one lead will drive the spin accumulation in such a QD. So, it is certain that the ABFano effect assists to manipulate the spin accumulation. Further analysis showed that the doubleQD interferometer has advantages in manipulating the spin states in the resonant channel. In view of the obtained results, we propose the AB interferometers with QDs to be alternative candidates for spin manipulation in QD devices.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
WJG designed the theoretical model, deduced the relevant formula, and drafted the manuscript. YH carried out the numerical calculations. GZW participated in the analysis about the results. AD improved the manuscript. All authors read and approved the final manuscript.
Acknowledgements
This work was financially supported by the National Natural Science Foundation of China (grant no. 10904010), the Natural Science Foundation of Liaoning Province (grant no. 201202085), the Fundamental Research Funds for the Central Universities (grant no. N110405010), and China Postdoctoral Science Foundation (grant no. 20100481206).
References

van der Wiel WG, De Franceschi S, Elzerman JM, Fujisawa T, Tarucha S, Kouwenhoven LP: Electron transport through double quantum dots.
Rev Mod Phys 2002, 75:1. Publisher Full Text

Amlani I, Orlov AO, Toth G, Bernstein GH, Lent CS, Snider GL: Digital logic gate using quantumdot cellular automata.
Science 1999, 284:289. PubMed Abstract  Publisher Full Text

Waugh FR, Berry MJ, Mar DJ, Westervelt RM, Campman KL, Gossard AC: Singleelectron charging in double and triple quantum dots with tunable coupling.
Phys Rev Lett 1995, 75:705. PubMed Abstract  Publisher Full Text

Aguado R, Langreth DC: Outofequilibrium Kondo effect in double quantum dots.
Phys Rev Lett 2000, 85:1946. PubMed Abstract  Publisher Full Text

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

Hayashi T, Fujisawa T, Cheong HD, Jeong YH, Hirayama Y: Coherent manipulation of electronic states in a double quantum dot.
Phys Rev Lett 2003, 91:226804. PubMed Abstract  Publisher Full Text

Saraga DS, Loss D: Spinentangled currents created by a triple quantum dot.
Phys Rev Lett 2003, 90:166803. PubMed Abstract  Publisher Full Text

Jiang Z, Sun Q, Wang Y: Kondo transport through serially coupled triple quantum dots.

Ladrón de Guevara ML, Orellana PA: Electronic transport through a parallelcoupled triple quantum dot molecule: Fano resonances and bound states in the continuum.

Gaudreau L, Studenikin SA, Sachrajda AS, Zawadzki P, Kam A, Lapointe J, Korkusinski M, Hawrylak P: Stability diagram of a fewelectron triple dot.
Phys Rev Lett 2006, 97:036807. PubMed Abstract  Publisher Full Text

Kiravittaya S, Rastelli A, Schmidt OG: Advanced quantum dot configurations.
Rep Prog Phys 2009, 72:046502. Publisher Full Text

Yacoby A, Heiblum M, Mahalu D, Shtrikman H: Coherence and phase sensitive measurements in a quantum dot.
Phys Rev Lett 1995, 74:4047. PubMed Abstract  Publisher Full Text

Yacoby A, Schuster R, Heiblum M: Phase rigidity and h/2e oscillations in a singlering AharonovBohm experiment.
Phys Rev B 1996, 53:9583. Publisher Full Text

Buks E, Schuster R, Heiblum M, Mahalu D, Umansky V: Dephasing in electron interference by a ‘whichpath’ detector.
Nature (London) 1998, 391:871. Publisher Full Text

Schuster R, Buks E, Heiblum M, Mahalu D, Umansky V, Shtrikman H: Phase measurement in a quantum dot via a doubleslit interference experiment. Nature.
(London) 1997, 385:417. Publisher Full Text

AvinunKalish M, Heiblum M, Zarchin O, Mahalu D, Umansky V: Crossover from ‘mesoscopic’ to ‘universal’ phase for electron transmission in quantum dots.
Nature (London) 2005, 436:529. Publisher Full Text

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

Kobayashi K, Aikawa H, Katsumoto S, Iye Y: Tuning of the Fano effect through a quantum dot in an AharonovBohm interferometer.
Phys Rev Lett 2002, 88:256806. PubMed Abstract  Publisher Full Text

Kobayashi K, Aikawa H, Sano A, Katsumoto S, Iye Y: Fano resonance in a quantum wire with a sidecoupled quantum dot.

Sigrist M, Ihn T, Ensslin K, Reinwald M, Wegscheider W: Coherent probing of excited quantum dot states in an interferometer.
Phys Rev Lett 2007, 98:036805. PubMed Abstract  Publisher Full Text

Aharony A, EntinWohlman O, Otsuka T, Katsumoto S, Aikawa H, Kobayashi K: Breakdown of phase rigidity and variations of the Fano effect in closed AharonovBohm interferometers.

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

Hatano T, Kubo T, Tokura Y, Amaha S, Teraoka S, Tarucha S: AharonovBohm oscillations changed by indirect interdot tunneling via electrodes in parallelcoupled vertical double quantum dots.
Phys Rev Lett 2011, 106:076801. PubMed Abstract  Publisher Full Text

Yeyati AL, Bütiker M: AharonovBohm oscillations in a mesoscopic ring with a quantum dot.
Phys Rev B 1995, 52:R14360. Publisher Full Text

Hackenbroich G, Weidenmüler HA: Transmission through a quantum dot in an AharonovBohm ring.
Phys Rev Lett 1996, 76:110. PubMed Abstract  Publisher Full Text

Weidenmüler HA: Transmission phase of an isolated Coulomb blockade resonance.

Hofstetter W, König J, Schoeller H: Kondo correlations and the Fano effect in closed AharonovBohm interferometers.
Phys Rev Lett 2001, 87:156803. PubMed Abstract  Publisher Full Text

Pala MG, Iannaccone G: Effect of dephasing on the current statistics of mesoscopic devices.
Phys Rev Lett 2004, 93:256803. PubMed Abstract  Publisher Full Text

Bułka BR, Stefański P: Fano and Kondo resonance in electronic current through nanodevices.
Phys Rev Lett 2001, 86:5128. PubMed Abstract  Publisher Full Text

Urban D, König J: Coulombinteraction effects in full counting statistics of a quantumdot AharonovBohm interferometer.

Osawa K, Kurihara S, Yokoshi N: Fano effect in a Josephson junction with a quantum dot.

Aharony A, EntinWohlman O, Imry Y: Measuring the transmission phase of a quantum dot in a closed interferometer.
Phys Rev Lett 2003, 90:156802. PubMed Abstract  Publisher Full Text

Lim JS, López R, Platero G, Simon P: Transport properties of a molecule embedded in an AharonovBohm interferometer.

Fang TF, Wang SJ, Zuo W: Fluxdependent shot noise through an AharonovBohm interferometer with an embedded quantum dot.

Moldoveanu V, Tolea M, Gudmundsson V, Manolescu A: Fano regime of onedot AharonovBohm interferometers.

Hiltscher B, Governale M, König J: Spindependent transport through quantumdot AharonovBohm interferometers.

Vernek E, Sandler N, Ulloa SE: Kondo screening suppression by spinorbit interaction in quantum dots.

Heary RJ, Han JE, Zhu L: Spincharge filtering through a spinorbit coupled quantum dot controlled via an AharonovBohm interferometer.

Wolf SA, Awschalom DD, Buhrman RA, Daughton JM, von Molnr S, Roukes ML, Chtchelkanova AY, Treger DM: A spinbased electronics vision for the future.
Science 2001, 294:1488. PubMed Abstract  Publisher Full Text

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

Zutic I, Fabian J, Das Sarma S: Spintronics: Fundamentals and applications.
Rev Mod Phys 2004, 76:323. Publisher Full Text

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

Recher P, Sukhorukov EV, Loss D: Quantum dot as spin filter and spin memory.
Phys Rev Lett 2000, 85:1962. PubMed Abstract  Publisher Full Text

Paillard M, Marie X, Renucci P, Amand T, Jbeli A, Gérard JM: Spin relaxation quenching in semiconductor quantum dots.
Phys Rev Lett 2001, 86:1634. PubMed Abstract  Publisher Full Text

Cortez S, Krebs O, Laurent S, Senes M, Marie X, Voisin P, Ferreira R, Bastard G, Gérard JM, Amand T: Optically driven spin memory in ndoped InAsGaAs quantum dots.
Phys Rev Lett 2002, 89:207401. PubMed Abstract  Publisher Full Text

Besombes L, Lĺę ger Y, Maingault L, Ferrand D, Mariette H, Cibert J: Probing the spin state of a single magnetic ion in an individual quantum dot.
Phys Rev Lett 2004, 93:207403. PubMed Abstract  Publisher Full Text

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

Wang B, Wang J, Wang J, Xing DY: Spin current carried by magnons.

Cummings AW, Akis R, Ferry DK: Electron spin filter based on Rashba spinorbit coupling.
Appl Phys Lett 2006, 89:172115. Publisher Full Text

Li J, Shen SQ: Spincurrentinduced charge accumulation and electric current in semiconductor nanostructures with Rashba spinorbit coupling.

Zhang P, Xue QK, Xie XC: Spin current through a quantum dot in the presence of an oscillating magnetic field.
Phys Rev Lett 2003, 91:196602. PubMed Abstract  Publisher Full Text

Lu HZ, Shen SQ: Using spin bias to manipulate and measure spin in quantum dots.

Lu HZ, Shen SQ: Detecting and switching magnetization of Stoner nanograin in nonlocal spin valve.

Lu HZ, Zhou B, Shen SQ: Spinbias driven magnetization reversal and nondestructive detection in a single molecular magnet.

Xing YX, Sun QF, Wang J: Spin bias measurement based on a quantum point contact.
Appl Phys Lett 2008, 93:142107. Publisher Full Text

Jedema FJ, Heersche HB, Filip AT, Baselmans JJA, van Wees BJ: Electrical detection of spin precession in a metallic mesoscopic spin valve.
Nature 2002, 416:713. PubMed Abstract  Publisher Full Text

Kobayashi T, Tsuruta S, Sasaki S, Fujisawa T, Tokura Y, Akazaki T: Kondo Effect in a semiconductor quantum dot with a spinaccumulated lead.
Phys Rev Lett 2010, 104:036804. PubMed Abstract  Publisher Full Text

Frolov SM, Lüscher S, Yu W, Ren Y, Folk JA, Wegscheider W: Ballistic spin resonance.
Nature 2009, 458:868. PubMed Abstract  Publisher Full Text

Katsura H: Nonequilibrium Kondo problem with spindependent chemical potentials: exact results.
J Phys Soc Jpn 2007, 76:054710. Publisher Full Text

Bao YJ, Tong NH, Sun QF, Shen SQ: Conductance plateau in quantum spin transport through an interacting quantum dot.
Europhys Lett 2008, 83:37007. Publisher Full Text

Meir Y, Wingreen NS: Landauer formula for the current through an interacting electron region.

Souza FM, Jauho AP, Egues JC: Spinpolarized current and shot noise in the presence of spin flip in a quantum dot via nonequilibrium Green’s functions.

Gong W, Xie X, Wei G: Coulombmodified equilibrium and nonequilibrium properties in a double quantum dot AharonovBohmFano interference device.
J Appl Phys 2010, 107:073702. Publisher Full Text