Physica E 43 (2010) 85–88

Contents lists available at ScienceDirect

Physica E

1386-94

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/physe

Charge and spin pumping effects in a single-dot Aharonov–Bohm ring withferromagnetic leads

Hui Pan a,�, Huai-Zhe Xu a, Rong Lu b

a Department of Physics and Key Laboratory of Micro-nano, Measurement-Manipulation and Physics (Ministry of Education), Beijing University of Aeronautics and Astronautics,

Beijing 100191, Chinab Department of Physics, Tsinghua University, Beijing 100084, China

a r t i c l e i n f o

Article history:

Received 26 April 2010

Received in revised form

7 June 2010

Accepted 18 June 2010Available online 23 June 2010

77/$ - see front matter & 2010 Elsevier B.V. A

016/j.physe.2010.06.025

esponding author. Fax: +86 10 82317935.

ail address: hpan@buaa.edu.cn (H. Pan).

a b s t r a c t

The pumping of electrons through a single-dot Aharonov–Bohm ring attached to ferromagnetic leads

are investigated theoretically by using the nonequilibrium Green’s function method. It is found that the

charge and spin pumping effects at zero-bias voltage can be produced by an oscillating electric field

applied to the quantum dot. The pumped charge and spin currents through the single-dot ring are

analyzed for two cases: (i) the single-dot ring is connected with two ferromagnetic leads and (ii) the

single-dot ring is connected with a ferromagnetic and a normal-metal lead. For the former case, a pure

spin current can be generated due to the photon-assisted tunneling effects. For the latter case, both the

charge and spin current can be pumped in opposite directions. The control of the pumped spin and

charge current by using the magnetic flux through the ring is also discussed.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Extensive investigations have been focused on spin-relatedtransport in a mesoscopic quantum dot (QD) system [1,2], since itprovides a way for applications in spintronics and quantuminformation processing [3,4]. In these systems, transport isgoverned not only by the charge current, but also by the spincurrent. The spin current is one of the most important physicalquantities in spintronics [5]. How to generate a pure spin currentwithout net charge current has received more and more attention.Recently, a pure spin current has been reported by direct opticalinjection without generation of a net charge current [6]. A spinsource device has been proposed theoretically to carry pure spinflow based on electron spin resonance [7] in a QD-lead systemwith sizable Zeeman splitting [8,9].

The electron tunneling through a QD system in the presence oftime-varying electric field has also been attracting much atten-tion. An essential feature of the time-dependent transport is thatan electron tunneling through the system can exchange theenergy of no with the ac fields (n is an integer and o is thefrequency of ac fields). This can lead to the opening of newinelastic tunneling channels, which is the well known photon-assisted tunneling (PAT) effects. Experimentally, the observationsof PAT have been reported in the QD systems [10,11]. The photon-electron pump effects in a QD driven by an asymmetric electric

ll rights reserved.

field has also been observed by Kouwenhoven et al. [12,13].Theoretically, many recent works studied the PAT and thepumping effects, which give reasonable results compared withthe experiments [14–16]. Recently, the studies of the chargepumping [17,18] have been extended to the spin pumping[19–23]. In experiments, the electric fields can be generatedmuch more easily, simply by applying a local gate electrode. Inaddition, it allows for greater spatial selectivity, which isimportant for local addressing of spins. It would thus be highlydesirable to control the spin by means of electric fields.

In this paper, we propose a mechanism for the realization ofthe charge and spin pump by using an oscillating electric field. Wecombine the ideas of pumping and spin-dependent transportthrough a QD embedded in a Aharonov–Bohm (AB) ring withferromagnetic leads. Such a single-dot AB ring has been adoptedto study the phase coherence of an electron through a QD, theobserved magnetic oscillation of the current indicates coherenttransport through the QD [24,25]. Due to the microwave (MW)field applied on the QD, the resonant energy level of the QD willshift adiabatically with the time-dependent external field, whichcan lead the photon-assisted tunneling effects. We consider twoscenarios. First, we focus on the situation that both leads areferromagnetic. It is found that a pure spin current can be pumpeddue to the photon-assisted tunneling effects. The spin pumping isrelated to the different coupling of the spin-up and spin-downelectrons, which is a generalization of the pumped spin current.Second, we consider the case that only one of the two leads isferromagnetic. As a result, the pumped spin and charge currentcan be transported in opposite directions. Finally, the magnetic

H. Pan et al. / Physica E 43 (2010) 85–8886

flux effects on the pumped currents are discussed. A particularintriguing result is that both the magnitude and sign of the chargecurrent can be controlled by the magnetic flux, while the sign ofthe spin current does not change by varying the magnetic flux.

The paper is organized as follows. In Section 2, we present theHamiltonian and the general formulation to calculate thepumping current within the nonequilibrium Green’s function(NGF) framework. In Section 3, we discuss the numerical resultson the pumped charge and spin currents, and the control of thepumped currents is discussed. Finally, we give a conclusion inSection 4.

2. Physical model and formula

The device considered here is a QD embedded in a ring whichis coupled to two ferromagnetic leads. In second quantized formthis device is described by the following Hamiltonian

H¼Xaks

eaksayaksaaksþXsedðtÞd

ysds

þXaks½taayaksdsþH:c:�

þXkkus½tLRe�ifayLksaRkusþH:c:�: ð1Þ

The first term in Eq. (1) describes the ferromagnetic leads withayaks (aaks) being the creation (annihilation) operator of anelectron with momentum k and spin s in the a (a¼ L,R) lead.eaks ¼ eakþsM, where M is the magnetization of the two leads. Inreality, M shows the difference of density of states between spin-up and spin-down electrons in the electrodes. The second term isfor the QD embedded in the ring. edðtÞ ¼ edþVaccosot, where Vac

and o are the amplitude and frequency of the MW field applied tothe QD, respectively. The third term is the coupling between theleads and the QD, and ta is the tunneling coefficients between thea lead and the QD. The fourth term is for the arm of the ringwithout the QD, and tLR is the tunneling coefficients between thetwo leads with f being a spin-independent phase factor caused bythe magnetic flux through the AB interferometer.

The spin-dependent charge current flowing from the left leadinto the AB ring can be calculated by using the standard KeldyshNGF method

ILsðtÞ ¼2e

‘Re

ZdtuTr½Q LsR

rðt,tuÞGo

ðtu,tÞ�, ð2Þ

where

Q Ls ¼

0 0 0

0 qs 0

0 0 0

0B@

1CA, ð3Þ

with

qm ¼1 0

0 0

� �and qk ¼

0 0

0 1

� �: ð4Þ

The Keldysh Green’s function Goðt,tuÞ is defined as

Goðt,tuÞ ¼ i/WyðtuÞWðtÞS with the operators Wy ¼ ðDy,AyL ,AyRÞ,

Dy ¼ ðdym,dykÞ, and Aya ¼ ðP

kayakm,P

kayakkÞ. In the following, we take

the double Fourier transform for Green’s functions

Gðt,tuÞ ¼P

neinotRðde=2pÞe�ieðt�tuÞ ~GnðeÞ, and take the definition

GmnðeÞ ¼ ~Gn�mðeþmoÞ. The time-dependent current becomes

ILsðtÞ ¼2e

‘ReX

l

eilot

Zde2p

Tr½Q LsRrGoðeÞ�l0

� �: ð5Þ

To solve Go , we first calculate the retarded Green functions Gr

using the Dyson equation,

Grmn ¼ gr

mnþX

l

GrmlR

rllg

rln, ð6Þ

where Green’s function Gr is a 6�6 matrix defined as

Gr¼

GrDD Gr

DL GrDR

GrLD Gr

LL GrLR

GrRD Gr

RL GrRR

0B@

1CA: ð7Þ

In Eq. (6), gr is Green’s function of the system without couplingbetween the leads and the QD (tLR¼tL¼tR¼0). It can be obtainedexactly as

grmnðeÞ ¼

grd,mn 0 0

0 grL,mn 0

0 0 grR,mn

0B@

1CA, ð8Þ

where

grd,mnðeÞ ¼

Pl

Jl�mðaÞJl�nðaÞe�edm�loþ i0þ

0

0P

l

Jl�mðaÞJl�nðaÞe�edk�loþ i0þ

0BBBB@

1CCCCA, ð9Þ

and

gra,mnðeÞ ¼�ipdmn

ram 0

0 rak

!: ð10Þ

In Eq. (10), ram (rak) is the density of states for the spin-up (spin-

down) subband of the a lead. With the definition of the spinpolarization of the a lead pa ¼ ðram�rakÞ=ðramþrakÞ, we have

ras ¼ rað1þspaÞ in the parallel magnetization configuration,

and rLs ¼ rLð1þspLÞ and rRs ¼ rRð1�spRÞ in the antiparallel

magnetization, where ra ¼ ramþrak. The retarded self-energy in

Eq. (6) is

Rrmn ¼ dmn

0 t�L t�RtL 0 tLR

tR t�LR 0

0B@

1CA: ð11Þ

The corresponding linewidth function is defined as Gas ¼

2prast2a ¼Gað1þspaÞ with Ga ¼ 2prat2

a .

After solving Gr, Keldysh Green’s function Go can be obtainedstraightforwardly from the standard Keldysh equation,

Gomn ¼

Xl1 l2

Grml1½gr�1go ga�1�l1 l2

Gal2n: ð12Þ

For the present case, gr�1d go

d ga�1d ¼ 0 and gr�1

a goa ga�1

a ¼

gr�1a faðga

a�graÞg

a�1a , where fa is the Fermi distribution function in

the a lead. With these Green’s functions Gr and Go , the averagecurrent can be expressed as

ILs ¼/ILsðtÞS

¼2e

‘Re

Zde2pTr½Q LsR

rGoðeÞ�l0

� �: ð13Þ

Similarly, we can obtain IRs. The total charge and spin current are,respectively, Ic ¼ Imþ Ik and Is ¼ Im�Ik, where Is ¼ 1

2 ðILs�IRsÞ [23].The following numerical calculations are performed in units of‘¼ e¼ 1 at zero temperature and zero bias voltage. We take G asthe energy unit and I0 ¼ eG=‘ as the current unit.

Fig. 1. Ic (dashed line) and Is (solid line) versus ed for pL¼pR¼0.5. Other

parameters are V¼0, f¼ 0, o¼ 1, Vac¼2.5, tLR¼0.01 and G¼ 1.Fig. 2. Ic (dashed line) and Is (solid line) versus ed for pL¼0.3 and pR¼0. Other

parameters are the same as those in Fig. 1.

H. Pan et al. / Physica E 43 (2010) 85–88 87

3. Numerical results and discussions

In the following studies, we consider two ceases: (i) the ring isconnected with two ferromagnetic leads; (ii) the ring is connectedwith a ferromagnetic and a normal-metal lead. First, we focus onthe case that both leads are ferromagnetic with pL¼pR. Then, thecoupling between the dot and the leads has the relation GLs ¼GRsfor the parallel magnetization configuration and GLs ¼GRs(s ¼�s) for the antiparallel magnetization configuration. It isfound that the spin polarizations of the leads have a greatinfluence on the pumped current through the system. If the twoferromagnetic leads are in the parallel magnetization configura-tion, there is no pump current due to the same tunnelingprobability for spin-up or spin-down electrons. However, in theantiparallel magnetization configuration, pumped current may begenerated. To elucidate this, in Fig. 1, we plot the pumped charge(Ic) and spin (Is) currents as a function of the QD energy level ed.There are two remarkable features in the plot: (i) The chargecurrent (dashed line) is zero. (ii) The spin current (solid line)contains a positive resonant peak at ed ¼�o and a negative one ated ¼o. It is noted that feature (ii) is absent when the system isconnected with two normal-metal leads, since such a system issymmetrical for the spin-up and spin-down electrons flowing inthe opposite directions. The reason for the generation of thepumped pure spin current can be explained in the picture ofphoton-assisted tunneling effects as follows. A spin-up electroncan tunnel into the QD state ed ¼�o from the left lead, and due tothe MW field it absorbs a photon with frequency o and transits tothe virtual state at the Fermi level EF¼0. This spin-up electronthen tunnels out of the scattering region with certain probabilitiesto the left and right leads. Due to the antiparallel magnetizationconfiguration, the coupling strengthes have the relationGLm ¼GRk4GLk ¼GRm. Thus, the probabilities for the spin-upelectrons tunneling out from the QD to the left or right leads aredifferent, which results in a positive Im. Similar procedureshappens for the spin-down electrons in the right lead, resultinga negative Ik. Since Im and Ik have the same magnitude and flowtoward the opposite directions, a pure spin current is generatedwithout any charge current. If the QD state lies at ed ¼o, theemission of a photon enables the dot electron to tunnel to thelead, which results in a reversed spin current. As a result, a purespin current with two resonant peaks can be generated in theantiparallel magnetization configuration. In the presentcalculation, we use the pL¼pR¼0.5 as a prototype, which iswithin the typical range of most well known and used

ferromagnetic leads [26]. If G¼ 1 meV, o¼ 1 corresponds tohundreds of GHz and is in the microwave range. The pumped spincurrent in Fig. 1 with a value of 0.05 is about tens of nA, which ismeasurable in experiments.

When only one of the two leads is ferromagnetic, the pumpedcurrent through such a system can be directly obtained by settingpR¼0. We show in Fig. 2 the charge and spin currents as afunction of the QD energy level. The solid and dashed linerepresent the spin and charge current, respectively. Note thatboth the charge and spin current can exist. A particular intriguingresult is that the pumped spin and charge currents can betransported in opposite directions. The reason is related to thecoupling strength between the QD and the leads. Since the rightlead is nonmagnetic, the relation between the coupling strengthesbecome GLm4GRm ¼GRk4GLk. With the help of the photon-assisted tunneling, we have Im40 and Iko0. However, due to thedifferent coupling strengthes, the magnitude of Im and Ik becomesdifferent. Therefore, both the spin and charge current can begenerated and flow in the opposite directions. This featuredepends on the relative coupling strength between QD and theleads.

In the following, we discuss the AB effects on the pumpedcurrent. Due to the existence of magnetic flux, the electronsflowing through different arms of the AB ring will acquire a spin-independent phase factor in the tunnel-coupling strengths. Thisphase factor will induce various interesting interference phenom-ena. To make the physics clear about how the phase f influencethe transport properties, a qualitative analysis about the effectivetunneling strength can be given before the detailed numericalcalculations. Consider an electron tunneling to the a lead from theQD, it has two interference paths. One path is through the directtunneling. The other is to first travel to the a (a ¼ R when a¼ L

and vice reversed) lead and then tunnel to the a lead. Forsimplicity, only the leading tunneling process is considered. Thetotal effective tunneling strength TLs between the QD and theleft lead is TLs ¼ jtLþtLRe�ifð�iprsÞtRj

2 ¼ jtLj2þjprstLRtRj

2�2prsjtLRtRtLjsinf. Similarly, the total effective tunneling strength TRsbetween the QD and the right lead is TRs ¼ jtLþtLReifð�iprsÞtLj

2 ¼

jtRj2þjprstLRtLj

2þ2prsjtLRtRtLjsinf. It is seen the tunnelingstrength depends on the magnetic flux f distinctly. Whetherthe charge and spin current can be pumped depends on thedifference between the tunneling strength for spin-up and spin-down electrons. To elucidate this, in Fig. 3, we plot Is and Ic as afunction of f at ed ¼�o. Both the spin and charge currents showthe AB oscillations with a normal 2p period. Furthermore, Ic and Is

Fig. 3. Ic (dashed line) and Is (solid line) versus f at ed ¼�1 for (a) pL¼pR¼0.5, and (b) pL¼0.5 and pR¼0. Other parameters are the same as those in Fig. 1.

H. Pan et al. / Physica E 43 (2010) 85–8888

can flow in the same or oppositive directions, which dependsdistinctly on f. For example, when f¼ 0:5p, Ic o0 and Is40;while when f¼�0:5p, Ic 40 and Is40. What is more interestingis that for some special magnetic flux f, the charge current can bezero while the spin current has a finite value. This means that thedirection and the magnitude of the pumped current are easilycontrolled and tuned by changing f, which enables one to controlthe charge and spin current in a nontrivial way by varying themagnetic flux. The proposed device to generate the pumped spinand charge currents can be realized in experiments by using thepresent technology. The microwave radiation on the QD deviceand the microwave-pumped quantum transport measurementshave already been reported experimentally [10,11]. Furthermore,the single-dot Aharonov–Bohm ring structure has already beenfabricated in laboratories to probe the coherent quantumtransport features [24,25].

4. Conclusions

In conclusion, we have investigated theoretically the pumpedcurrents at zero bias through a single-dot Aharonov–Bohm ringwith ferromagnetic leads by means of the NGF method. It is foundthat the charge and spin pumps can be produced by applying amicrowave field to the QD. The pumped charge and spin currentsthrough the single-dot ring are analyzed for two cases. For thefirst case, the single-dot ring is connected with two ferromagneticleads in the antiparallel magnetization configuration. The purespin current without any net charge current can be generated dueto the photon-assisted tunneling effects. For the second case, thesingle-dot ring is connected with a ferromagnetic and a normal-metal lead. Both the charge and spin current can be pumped inopposite directions. For both cases, the spin and charge currentsshow the AB oscillations with increasing the magnetic flux.Furthermore, the sign of the spin current does not change, whilethe sign of the charge current oscillates from positive to negative.For some special magnetic flux, the charge current can be zerowhile the spin current has a finite value. This enables one to

control the charge and spin current in a nontrivial way by varyingthe magnetic flux.

Acknowledgments

This project is supported by NSFC (Grant nos. 10704005,10974011 and 60776067) and BMSTC (Grant no. 2007B017).

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