Solid State Communications 148 (2008) 69–73

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Solid State Communications

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

Magnetically tunable spin-polarization of the current through a double quantumdot deviceHui Pan a,∗, Cong Wang a, Su-Qing Duan b, Wei-Dong Chu b, Wei Zhang b

a Department of Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, Chinab Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

a r t i c l e i n f o

Article history:Received 28 March 2008Received in revised form9 June 2008Accepted 7 July 2008 by S. MiyashitaAvailable online 11 July 2008

PACS:73.23.-b73.63.Kv73.23.Hk

Keywords:A. Coupled quantum dotsD. Coulomb-blockadeD. Spin-dependent transportD. Spin–flip effects

a b s t r a c t

We theoretically study the spin-dependent transport in a double quantum dot device connected withnormal-metal and ferromagnetic leads by using the nonequilibrium Green’s function method. In theabsence of an external magnetic field, the current polarization is zero for a range of positive bias, while itis a relatively large value for a negative bias. In the presence of an external magnetic field perpendicularto the magnetization of the lead, the spin–flip effects make the current spin-polarized for both of thepositive and negative bias. Thus the device can operate as a magnetically tunable spin-polarized currentdiode.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

In the past few years, spin-dependent electronic transportthrough quantum dot (QD) systems have been extensivelystudied both experimentally and theoretically due to its potentialapplication in spintronics [1,2]. The transport through a QDcoupled to ferromagnetic leads strongly depends on the magneticconfiguration of the system [3]. In such QD systems of nanometerlength scale, strong Coulomb interactions play an importantrole on the spin-polarized transport, which leads to a zero-biasanomaly in the Coulomb-blockade valleys [4,5]. When an externalmagnetic field perpendicular to the magnetizations of two leadsis applied on the QD system, the combined effects of spin–flipprocesses and Coulomb correlations result in a new phenomenonof Coulomb promotion of spin-dependent tunneling [6,7]. The QDsystems coupled to normal metal (NM) and ferromagnetic (FM)leads have also been analyzed recently [8,9]. The NM–QD–FMsystem can operate as a spin-current diode, since the currentpolarization is zero or a relative maximum value for positiveor negative bias, respectively. On the other hand, the transport

∗ Corresponding author. Fax: +86 10 82317935.E-mail address: hpan@buaa.edu.cn (H. Pan).

0038-1098/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.ssc.2008.07.003

properties of lateral or vertical double quantum dot (DQD) havealso been extensively studied [10]. The open DQD system makesthe quantum transport phenomena rich and varied. Recently,parallel-coupled DQD have been realized in experiments [11,12],where two quantum dots are respectively embedded into oppositearms of an Aharonov–Bohm ring coupled to two leads. Theseexperiments have inspired a number of theoretical attempts tostudy the coherent and correlated transport through the DQDsystems [13–15]. Furthermore, the DQD is also being consideredfor future applications in quantum computing [16].

However, the spin-resolved transport through the parallel-coupled DQD device attached to NM and FM leads are have notbeen studied intensively. Moreover, the effects of the externalmagnetic field perpendicular to the magnetization of the lead onthe NM–DQD–FM system are not considered in most cases. Themain objective of this paper is to study theoretically the magneticfield effects on the spin-dependent electronic transport throughthe NM–DQD–FM system. The intradot electron correlation istaken into account, and the magnetic field is taken to besmall enough not to affect any lead magnetization. Transportcharacteristics, such as the spin-resolved current associated withthe change of the bias, are calculated by the nonequilibriumGreen’s function (NGF) technique [17]. Since the system cannotbe treated exactly, the Hartree–Fock (HF) approximation schemeis applied to calculate the relevant Green’s functions from the

70 H. Pan et al. / Solid State Communications 148 (2008) 69–73

equations of motion (EOM), which is sufficient to describe thetransport characteristics in the Coulomb blockade regime. It isfound that in the absence of an external magnetic field, thecurrent polarization is zero or a finite value in a range of positiveor negative bias, respectively, which is referred to as the spin-polarized current diode effects. In the presence of an externalmagnetic field, due to the spin–flip effects, the current polarizationhas a finite value in a range of both positive and negative biases.Thus the system can operate as a magnetically tunable spin-polarized current diode.

2. Physical model and formula

The NM–DQD–FM system under consideration is subjected toan external magnetic field B perpendicular to the magnetizationof the right lead. The main effects of the magnetic field is toinduce coherent spin–flip effects [6]. The system Hamiltonian canbe written as

H =

∑α=L,R

Hα + HD + H ′

sf + HT , (1)

with

Hα =

∑kσ

εαkσ aĎαkσ aαkσ , (2)

HD =

∑σ ,i=1,2

εidĎiσdiσ +

∑i=1,2

Uiniσniσ −

∑σ

(tcdĎ1σd2σ + H.c.), (3)

H ′

sf =

∑i=1,2

(rdĎi↑di↓ + H.c.), (4)

HT =

∑αkσ ,i=1,2

(tαiσ aĎαkσdiσ + H.c.). (5)

Hα (α = L, R) describes the α leads. There exists a chemicalpotential imbalance between the left (L) and the right (R) leadsas µL = eV and µR = 0, with the bias voltage V driving thesystem away from equilibrium. HD models the parallel-coupleddouble quantum dots where dĎiσ (diσ ) represents the creation(annihilation) operator of the electron in the dot i (i = 1, 2) withspinσ = ±1 for↑↓ and σ = −σ . The level position εi can be tunedby an external gate voltage. Ui is the on-site Coulomb repulsion ofQD i, and niσ = dĎiσdiσ is the particle number operator. tc denotesthe interdot coupling strength. H ′

sf is the spin–flip term causedby the external magnetic field perpendicular to the magnetizationin the leads. The spin–flip strength is r = µBB with µB theBohr magneton. HT denotes the tunneling part of the Hamiltonian,where tαiσ is the hopping strength between the ith QD and the αlead.

By means of the standard NGF technique, the current can bederived as [17]

Iασ =2eh

∫dε2π

Tr{Re[G<σ (ε)6a

ασ (ε) + Grσ (ε)6<

ασ (ε)]}. (6)

Here, the Green’s function Gr,<σ and the self-energy 6a,<

σ are alltwo-dimensional matrices for the DQD system. The bold-facedletters are used to denote matrices. In the steady state, the totalcurrent is Iσ =

12 (ILσ − IRσ ), and the corresponding current

polarization is defined as PI = (I↑ − I↓)/(I↑ + I↓). In order toobtain the expression of the current, we have to solve the Green’sfunctions. Using the standard equation of motion (EOM) techniqueε〈〈diσ |dĎjσ 〉〉 = 〈{diσ , dĎjσ }〉 + 〈〈[diσ ,H]|dĎjσ 〉〉, we can calculate theGreen’s functions

(ε − εiσ )〈〈diσ |dĎjσ 〉〉 = δij − tc〈〈diσ |dĎjσ 〉〉 +

∑αk

t∗αikσ 〈〈aαkσ |dĎjσ 〉〉

+Ui〈〈niσdiσ |dĎjσ 〉〉, (7)

where i = 2 if i = 1, and vice versa. The new Green’sfunctions appeared on the right side can be determined from thecorresponding EOM as

〈〈aαkσ |dĎjσ 〉〉 =

∑i

tαikσε − εαkσ + i0+

〈〈diσ |dĎjσ 〉〉, (8)

and

(ε − εiσ − Ui)〈〈niσdiσ |dĎjσ 〉〉 = δijniσ − tc〈〈niσdiσ |dĎjσ 〉〉

− tc〈〈dĎiσdiσdiσ |dĎjσ 〉〉 + tc〈〈d

Ď

iσdiσdiσ |dĎjσ 〉〉

+

∑αk

t∗αikσ 〈〈niσ aαkσ |dĎjσ 〉〉 +

∑αk

t∗αikσ 〈〈dĎiσ aαkσdiσ |dĎjσ 〉〉

−

∑αk

tαikσ 〈〈aĎαkσdiσdiσ |dĎjσ 〉〉. (9)

New higher-order Green’s functions appear. To truncate the setof equations, we use the HF approximation for the higher-orderGreen’s functions as the following [18,19]

〈〈niσdiσ |dĎjσ 〉〉 ≈ 〈niσ 〉〈〈diσ |dĎjσ 〉〉,

〈〈dĎiσdiσdiσ |dĎjσ 〉〉 ≈ 〈dĎiσdiσ 〉〈〈diσ |dĎjσ 〉〉,

〈〈niσ aαkσ |dĎjσ 〉〉 ≈ 〈niσ 〉〈〈aαkσ |dĎjσ 〉〉,

〈〈dĎiσ aαkσdiσ |dĎjσ 〉〉 ≈ 〈dĎiσ aαkσ 〉〈〈diσ |dĎjσ 〉〉. (10)

The above EOM method is known to be reliable in the Coulombblockade regime, and qualitatively correct for Kondo physics [20,21]. Consequently, the numerical results for a nonvanishingCoulomb interaction are applicable above the Kondo temperature.Moreover, EOM solution becomes exact in the U = 0 limit.Besides, the retarded and lesser self-energies originated from thedot-lead couplings are defined as Σ r

αijσ =∑

k tαikσ grαkσ t

∗

αjkσ andΣ<

αijσ =∑

k tαikσ g<αkσ t

∗

αjkσ , where gαkσ = 1/(ε − εαkσ + i0+)

and g<αkσ = ifα2πδ(ε − εαkσ ) are the retarded and lesser Green’s

function of noninteracting electrons in the α leads. The retardedself-energy originated from the magnetic field is Σ r

cij = r . Underthe wide-band approximation, Σ r

αijσ = −i2Γαijσ , where Γαijσ is

the linewidth function defined by Γαijσ = 2πρασ t∗αiσ tαjσ withρασ being the density of states of the corresponding lead. Withthe definition of the spin polarization of α lead pα = (ρα↑ −

ρα↑)/(ρα↑ + ρα↑), the coupling between the ith QD and the α leadcan be expressed asΓαijσ = Γ0(1±pα)withΓ0 = (Γαij↑+Γαij↓)/2.The lesser self-energy is Σ<

αijσ (ε) = fα(ε)(Σaαijσ − Σ r

αijσ ) wherefα(ε) = 1/(e(ε−µα)/kBT +1) denotes the Fermi distribution functionof electrons in the α lead.

Then, the retarded Green’s function for the DQD system can bewritten in a compact form as Gr

σ = grσ + Gr

σ 6rσg

rσ . The Green’s

function for the isolated DQD at zero field is

(g riσ )−1

=(ε − εiσ )(ε − εiσ − Ui)

ε − εiσ − Ui + Ui〈niσ 〉

+Uitc[〈d

Ďiσdiσ 〉 − 〈dĎ

iσdiσ 〉]

ε − εiσ − Ui + Ui〈niσ 〉. (11)

The retarded self-energy is 6rσ =

∑α 6r

ασ + 6rc , where 6r

ασ iscaused by the α lead and 6r

c by the magnetic field. The advancedself-energy can be obtained from the relation 6a

ασ = (6rασ )Ď. The

lesser Green’s functions is related to the retarded Green’s functionthrough the Keldysh equation G<

σ = Grσ 6<

σ Gaσ , where Ga

σ = (Grσ )Ď

and6<σ =

∑α 6<

ασ . The expectation values of 〈niσ 〉 = 〈dĎiσdiσ 〉 and〈dĎiσdiσ 〉 can be calculated self-consistently by taking advantage ofthe definition of the lesser Green’s function

〈niσ 〉 = −i∫

dε2π

G<iiσ , (12)

H. Pan et al. / Solid State Communications 148 (2008) 69–73 71

Fig. 1. (a) Spin-resolved currents I↑ (solid line) and I↓ (dashed line) vs V . (b) The corresponding current polarization PI vs V . (c) The average occupation number 〈n1↑〉 (solidline), 〈n1↓〉 (dotted line), 〈n2↑〉 (dashed line), and 〈n2↓〉 (dash-dotted line) vs V . (d) the spin accumulation ∆n1 (solid line) and ∆n2 (dashed line) vs V . Here, the externalmagnetic field is r = 0, and ε1 = ε2 = 0.01.

and

〈niiσ 〉 = −i∫

dε2π

G<

iiσ . (13)

The corresponding spin accumulation is defined as ∆ni = 〈ni↑〉 −

〈ni↓〉. In the following, we perform the calculations at zerotemperature in units of h = e = 1. The linewidth function is setas Γ 0 = 0.01 meV, whose typical values in experiments are ofthe order of tens of µeV [22]. The charge energy and the interdotcoupling are set as U1 = U2 = U = 3 meV and tc = 0.01 meV,respectively. Thus the spin–flip strength r = 0.01 corresponds toamagnetic field of 172mT. Furthermore, the spin polarizations areset as pL = 0 for the left normal metal lead, and pR = 0.3 for theright ferromagnetic lead.

3. Results and discussion

In this section, the numerical results of Iσ , PI , 〈niσ 〉, and ∆niare discussed in detail. Due to the inter-dot coupling, the formedbonding and antibonding states of the DQD are at ε± =

12 [ε1+ε2±√

(ε1 − ε2)2 + 4t2c ]. Since the intra-dot Coulomb interaction cancause an effective charging energy on the bonding and antibondingstates, the integer number of electrons confined to the DQD regionoccurs approximately at the following energies: ε−, ε+, ε− + U ,and ε+ + U [18]. We first consider the case that the two QD levelsare the same as ε1 = ε2 = tc . Then, the bonding and antibondingstate are located at ε− = 0 and ε+ = 2tc , respectively. As shownin Fig. 1(a) and (b), the current is spin-independent I↑ = I↓ andthe corresponding current polarization is PI = 0 for ε− < V <ε− +U , while the current becomes spin-dependent |I↑| > |I↓| andPI > 0 for V < ε− or V > ε− + U . The current polarization iszero for direct bias (current flows from NM lead to the FM lead)but nonzero for reversed bias (current flows from FM lead to theNM lead). This phenomena are referred to as the spin-polarized

current diode effects,which are caused by the interplay of Coulombinteraction and spin accumulation in the DQD [9]. The electronoccupation and the corresponding spin accumulation as a functionof the bias are presented in Fig. 1(c) and (d). In these curves, ni↑and ni↓ are separated dramatically and ∆ni < 0 at the plateauswithin ε− < V < ε− + U . The negative spin accumulation canbe understood in terms of the tunneling rates between dots andleads. Due to the ferromagnetism of the right lead, the tunnelingrates become asymmetric: ΓL↑ = ΓL↓ but ΓR↑ > ΓR↓. For positivebias, the spin-up electrons can tunnel out the DQD faster thanthey come into it, whereas the spin-down electrons leave the DQDslower than they come into it. On average, the spin-down electronsspend more time in the DQD than the spin-up ones. Furthermore,due to the Coulomb blocked effects, the spin-up electrons tend tobe more blockade than the spin down ones. For negative bias, thespin accumulation inverts its sign as ∆ni > 0, which enhances thedifference between I↑ and I↓.

However, when a perpendicular magnetic field is applied to theDQD, the spin-diode effects can be destroyed. The current becomesspin-polarized even in the Coulomb blockade regime as shown inFig. 2(a). The spin accumulation in the DQD can be destroyed bythe spin–flip effects caused by the magnetic field. With the helpof the spin–flip effects, the accumulated spin-down electron canflip its spin and tunnel from the DQD to the leads, which make thecurrent be spin-polarized as I↑ > I↓. Thus the current becomesspin-dependent both at positive and negative bias, though thequalitative behavior of Iσ remains unchanged. The correspondingcurrent polarization PI becomes a finite value within the range ofε− < V < ε− + U as shown in Fig. 2(b). The enhancement of thePI is due to the spin–flip effects induced by the magnetic field. Togain a more detailed understanding about this, ni↑ and ni↓ are alsoplotted in Fig. 2(c) and (d). It is seen that ni↑ and ni↓ coincide atthe plateaus within the range of ε− < V < ε− + U , because thespin accumulation is greatly suppressed by the spin–flip effects,

72 H. Pan et al. / Solid State Communications 148 (2008) 69–73

Fig. 2. (a) Spin-resolved currents I↑ (solid line) and I↓ (dashed line) vs V . (b) The average occupation number 〈n1↑〉 (solid line), 〈n1↓〉 (dotted line), 〈n2↑〉 (dashed line), and〈n2↓〉 (dash-dotted line) vs V . Here, the external magnetic field is r = 0.01, and ε1 = ε2 = 0.01.

Fig. 3. (a) Spin-resolved currents I↑ (solid line) and I↓ (dashed line) vs V . (b) The corresponding current polarization PI vs V . (c) The average occupation number 〈n1↑〉 (solidline), 〈n1↓〉 (dotted line), 〈n2↑〉 (dashed line), and 〈n2↓〉 (dash-dotted line) vs V . (d) the spin accumulation ∆n1 (solid line) and ∆n2 (dashed line) vs V . Here, the externalmagnetic field is r = 0, and ε1 = 0.01 and ε2 = 1.0.

resulting a spin-polarized current even within ε− < V < ε− + U .Thus, whether the device shows the spin-polarized current diodeeffects can be magnetically tunable.

We next consider the case that the two QD levels are quitedifferent: ε1 = 0.01 and ε1 = 1.0. As shown in Fig. 3(a), thecurrent is also spin-independent as I↑ = I↓ for a range of positivebias of ε− < V < ε− + U . However, there appear four steps atε−, ε+, ε− + U , and ε+ + U . When the bias is within the rangeof V < ε− or V > ε− + U , the current becomes spin-dependentas |I↑| > |I↓|. The corresponding current polarization in Fig. 3(b)is zero only for ε− < V < ε− + U . The suppression and theenhancement of the PI are also due to the interplay of Coulombinteraction and spin accumulation in the DQD asmentioned above.

〈niσ 〉 and∆ni as a function of the bias are presented in Fig. 3(c) and(d). In these curves, n1(2)↑ is lower than n1(2)↓ for ε−(+) < V <ε−(+) + U , resulting a negative spin accumulation. In the presenceof a magnetic field applied to the DQD, since the spin–flip effectsgreatly suppress the spin accumulation, the current becomes spin-polarized for both positive and negative bias as shown in Fig. 4.However, the qualitative behavior of Iσ remains unchanged. Thecurrent polarization versus the spin–flip strength is plotted inFig. 5. With increasing r from 0 to 0.01, the current polarizationincreases from 0 to a finite value of 0.1. With further increasingr , the current polarization increases slowly. It means that even avery small magnetic field can make the current be spin-polarizedin the Coulomb blockade regime. Thus the system can operate as amagnetically tunable spin-polarized current diode.

H. Pan et al. / Solid State Communications 148 (2008) 69–73 73

Fig. 4. (a) Spin-resolved currents I↑ (solid line) and I↓ (dashed line) vs V . (b) The average occupation number 〈n1↑〉 (solid line), 〈n1↓〉 (dotted line), 〈n2↑〉 (dashed line), and〈n2↓〉 (dash-dotted line) vs V . Here, the external magnetic field is r = 0.01, and ε1 = 0.01 and ε2 = 1.0.

Fig. 5. The current polarization PI vs r for ε1 = ε2 = 0.01 at V = 0.5.

4. Summary

In summary, based on the NGF method, we have studiedthe spin-resolved current through double quantum dots coupledto one nonmagnetic and one ferromagnetic lead. We apply theHartree–Fock approximation scheme to calculate the relevantGreen’s functions from the equation of motion, which is sufficientto describe the transport characteristics in the Coulomb blockaderegime. In the absence of external magnetic field, the currentpolarization PI is zero for the bias of ε− < V < ε− + U ,while it becomes a relative large value for the bias of V < ε− orV > ε− + U . This spin-polarized current diode effects come froman interplay between the Coulomb blockade effects and the spinaccumulation in the double quantum dots. In the presence of anexternal magnetic field perpendicular to the magnetization of thelead, the spin–flip effects greatly suppress the spin accumulation

in the Coulomb blockade regime, which make the current be spin-polarized. The corresponding PI has a relative large value in a rangeof both positive and negative bias. Thus the system can operate asa magnetically tunable spin-polarized current diode.

Acknowledgments

This work is supported by the National Natural ScienceFoundation of China (Grant Nos. 10704005 and 10574017), theBeijing Municipal Science and Technology Commission (Grant No.2007B017).

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