Physics Letters A 360 (2006) 317–322

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Photon-assisted spin-polarized transport in carbon nanotubes with impurities

Hui Pan a,∗, Tsung-Han Lin b

a Department of Physics, Beijing University of Aeronautics and Astronautics, Beijing 100083, Chinab State Key Laboratory for Mesoscopic Physics and Department of Physics, Peking University, Beijing 100871, China

Received 14 April 2006; accepted 9 May 2006

Available online 16 May 2006

Communicated by R. Wu

Abstract

Impurity effects on the photon-assisted spin-polarized transport through armchair carbon nanotubes connected with ferromagnetic leads areinvestigated theoretically. The impurity induces one resonant state whose position depends on the impurity strength, which can break the electron–hole symmetry. Whether the impurity suppresses or enhances the spin-coherent current depends on the nanotube length. When the microwavefields are applied on the nanotube, additional small side peaks caused by the photon-assisted tunneling are found. With increasing the impuritystrength, one new current peak appears under the influence of both the microwave fields and the impurity.© 2006 Elsevier B.V. All rights reserved.

PACS: 72.80.Rj; 73.23.Ad; 73.61.Wp; 73.63.Fg

1. Introduction

Recently, advances in nanotechnology make it possible tocontrol electron transport by means of the spin degree of free-dom in spintronic devices, which may have potential for ap-plications in future nanoelectronics. A carbon-nanotube-basednanoelectronics is possible when finite-sized nanotubes can beefficiently fabricated and coupled to external leads. Carbonnanotubes are considered as promising spin mediator becauseof their ballistic nature of conduction and long spin scatteringlength. Spin-polarized transport may be investigated in carbonnanotubes contacted by two ferromagnetic terminals. Coher-ent spin transport has been observed in multi-walled carbonnanotube [1–3] and single-walled carbon nanotube contactedwith Co electrodes in experiments [4]. It has been found thatcarbon nanotubes reveal quite a considerable giant magnetore-sistance (GMR) effect. The theoretical investigation of trans-port properties of these hybrid nanotube devices is of greatimportance, not only for their basic scientific interest, butalso aiming at the design of novel spintronic devices. The

* Corresponding author.E-mail address: hpan@pku.edu.cn (H. Pan).

0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physleta.2006.05.017

resonant spin polarized current in the ferromagnetic/carbon-nanotube/ferromagnetic (FM/CNT/FM) system have also beenstudied theoretically [5].

Experimental and theoretical studies have indicated that theelectronic and transport properties of carbon nanotubes can besubstantially modified by point defects such as the vacanciesand substitutional impurities [6–12]. Since carbon nanotubesare not strictly one-dimensional (1D) materials but are quasi-1Dones, it is expected that the impurity has unique effects on thesystem. On the other hand, if a microwave (MW) field is appliedto the leads and the central carbon nanotube, one would expectsome interesting features of the current and TMR due to new ef-fective tunneling channels from the photonic sidebands inducedby the MW field. It is found that the photonic sidebands pro-vide new channels for electrons tunneling through the barriers,and give rise to new resonances of the tunneling magnetoresis-tance, which is called photon-assisted spin-dependent tunnel-ing. In this Letter, the impurity effects on the photon-assistedspin-polarized transport in the hybrid FM/CNT/FM system aretheoretically studied. In such a system, the tunneling propertyexhibits special behavior due to the detailed construction andstructure of the nanotubes. The energy of the finite-sized carbonnanotubes is quantized both in the longitudinal and transversedirections, which increases the tunneling channels. By com-

318 H. Pan, T.-H. Lin / Physics Letters A 360 (2006) 317–322

bining standard nonequilibrium Green’s function (NGF) tech-niques [13–15] with a tight-binding model [16,17], we haveanalyzed quantum transport properties of the FM/CNT/FM sys-tem with impurities. Since the impurity changes the energystructure of the carbon nanotube, it has a great influence on thetransport properties of the system, which depends on the tubelength.

The rest of this Letter is organized as follows. In Section 2we present the model and derive the formula of the current I byusing the NGF technique. In Section 3 we study the impurityeffects on the current I on the basis of the theoretical resultsobtained in Section 2. We also discuss the photon-assisted tun-neling. Finally, a brief summary is given in Section 4.

2. Physical model and formula

The FM/CNT/FM system under consideration can be de-scribed by the following Hamiltonian [18]

(1)H =∑

α=L,R

Hα + HC + HT ,

with

(2)Hα =∑k,σ

εα,kσ (t)a†α,kσ aα,kσ ,

HC =∑n,σ

(ε0n(t) − evg

)d†nσ dnσ ,

HT =∑

n,kσσ ′

[tLa

†L,kσ dnσ + H.c.

]

(3)

+[tR

(cos

θ

2a

†R,kσ dnσ ′ − σ sin

θ

2a

†R,kσ dnσ ′

)+ H.c.

],

where Hα describes the ferromagnetic leads [18]. HC is theHamiltonian of the central conductor with multiple discrete en-ergy levels ε0

n(t). Vg is the gate voltage which controls theenergy levels in the carbon nanotube. HT denotes the tunnel-ing part of the Hamiltonian, and tL,R are the hopping matrix.Under the adiabatic approximation, the time-dependent MWfield can be reflected in the single-electron energies which canbe separated into two parts: εα,kσ (t) = εα,kσ + Δα(t) for theFM leads, and ε0

n(t) = ε0n + Δ0(t) for the central conductor.

Δα,0(t) is a time-dependent part from the external microwavefields, which can be written as Δα,0(t) = Δα,0 cosωα,0t . Itis know that carbon nanotubes can be well described by thetight-binding model with one π -electron per atom as Htube =∑

〈i,j〉,σ [−γ0c†iσ cjσ + H.c.]+Uc

†0c0, where i, j is restricted to

nearest-neighbor atoms, and the bond potential γ0 = 2.75 eV.This model is known to give a reasonable and qualitative de-scription of the electronic and transport properties of carbonnanotubes [16,17]. Then, the discrete energy levels ε0

n can beobtained by numerically diagonalizing Htube. The impurity isdefined by setting site energy equal to U at one of the sites ofthe unit cell, and various strengths represent typical substitu-tional impurities or vacancy [10–12]. For example, the strengthU = 3, −5 and 106 can simulate the substitutional boron, ni-trogen, and vacancy respectively, according to former tight-

binding and ab initio calculations [10,12]. The conductance andcurrent can be calculated from standard NGF techniques. It isconvenient that the Green’s function can be expressed by

(4)Grn(τ, τ

′) =(

Grn,↑↑(t, t ′) Gr

n,↑↓(t, t ′)Gr

n,↓↑(t, t ′) Grn,↓↓(t, t ′)

),

where Grn,σσ ′(t, t ′) = −i〈T {dn,σ (t), d

†n,σ ′(t ′)}〉. Under the

wide-bandwidth approximation, the current can be derived as[13–15]

Jα(t) = − e

h̄

∫dt ′

∫dε

2πIm

∑n

Trσ{e−iε(t ′−t)e−i

∫dt ′′ Δ(t ′′)

(5)× �α(ε)[G<

n (t, t ′) + 2fα(ε)Grn(t, t

′)]}

,

where fα(ε) is the Fermi distribution function. From theDyson equation Gr = [(gr )−1 − �r ]−1, and the Keldysh equa-tion G< = Gr�<Ga , one can get the retarded and correlatedGreen’s function. gr is the retarded Green’s function of theuncoupled nanotube. The selfenergy �r and �< are givenas �r = − i

2 (�L + �R) and �< = i(fL�L + fR�R). Underthe wide-bandwidth approximation, the linewidth functions areindependent on the energy variable. This means that the trans-porting electrons in the leads are equally coupled to differentenergy levels of CNT. The couplings between the CNT and therespective FM leads are given by

(6)�L =(

ΓL↑(1 + PL) 0

0 ΓL↓(1 − PL)

),

and

(7)�R =(

ΓR↑(1 + PR cos θ) ΓR↑↓PR sin θ

ΓR↓↑PR sin θ ΓR↓(1 − PR cos θ)

),

where Pα is the polarization of the αth lead. For the case oftwo same FM leads, we can take P = PL = PR , and Γ↑(↓) =ΓL↑(↓) = ΓR↑(↓). Furthermore, we define two parameters asη = Γ↑/Γ↓ and Γ = Γ↑ + Γ↓. The linewidth functions Γ areset as small values compared with the energy-level spacing forthe symmetric and weak-coupling case. Under the ac bias, theaverage current is

〈J 〉 = e

h

∫dε

∑n,k,l,k′,l′

Trσ{JL

(klk′l′

)fL(ε)�LGr

n(ε)�RGan(ε)

(8)− JR

(klk′l′

)fR(ε)�RGr

n(ε)�LGan(ε)

}.

Next, we consider the case in which the external MW field ineach region has the same frequency, ω0 = ωL = ωR = ω. In thiscase 〈J 〉 reduces to

〈J 〉 = e

h

∫dε

∑n,k

Trσ

{J 2

k

(Δ0 − ΔL

ω

)fL(ε)

× �LGrn(ε)�RGa

n(ε)

(9)− J 2k

(Δ0 − ΔR

ω

)fR(ε)�RGr

n(ε)�LGan(ε)

}.

The energy and the nanotube length in the calculations arescaled by γ0 and the lattice constant a = 0.245 nm, respec-tively. The conductance and current are scaled by G0 = 2e2/h

H. Pan, T.-H. Lin / Physics Letters A 360 (2006) 317–322 319

and I0 = 2eγ0/h, respectively. The linewidth is Γ = 0.02γ0for the weak-coupling case, which have small values comparedwith the energy-level spacing. In the following, the numericalresults are discussed for the (6,6) armchair nanotube in detail.The time-dependent external MW fields are applied only on thenanotube with the magnitude Δ0 = 1. For armchair nanotubes,the length L is measured in terms of unit cells: a unit cell is therepeat unit along the armchair tube consisting of two carbonrings. In this Letter, the length is selected as L = 6 or L = 7,which is shorter than the spin coherence length [4]. η = 2 meansthat Γ↑ > Γ↓, the tunneling probability of electrons from thespin up subband to spin up subband is less than that from downto down subbands.

3. Results and discussion

To clearly show the impurity effects on the time-dependentspin-polarized transport, the average current 〈I 〉 at θ = π/3with (solid line) and without (dash line) the MW field are plot-ted in Fig. 1. First, we look at the dashed lines for the casewithout the MW field. As shown in Fig. 1, the current peaks aresymmetric about the Fermi energy EF = 0 at U = 0 for bothL = 6 and L = 7 because of the electron–hole symmetry in thedefect-free nanotubes [12]. The current at the Fermi energy forL = 6 is much smaller than that for L = 7. In general, resonantstates appear at the Fermi level with the length L = 3N + 1,where N denotes the number of carbon repeat units, becausekF = 2π/3 is now an allowed wave vector [19]. And two res-onant states exist at the Fermi level due to the overlap of theπ and π∗ bands at EF . For other lengths, kF is not an allowedwave vector and no resonant state exists at EF , thus the currentis much smaller due to the energy gap between the resonantstates. When a MW field with ω0 = 0.05 is applied, the main

peaks in the solid lines also show similar behaviour as that with-out the MW field mentioned above. The main resonant peaksare accompanied with two small side resonant peaks. Thesesmall side-peaks are arisen from the photon-assisted tunnelingprocedure such as the photon emission and absorption. Thenthe electrons in the leads can transport through the new chan-nels caused by the MW fields applied on the central conductor.These resonant peaks occur at the positions of the photonicsidebands characterized by k = 0,±1,±2, . . . , which corre-spond to the shifts of the energy levels caused by the MW field.Thus the location of the current peaks is at ε0

n + kω, and theheight is determined by the Bessel function. As Vg changes, themain resonant energy level and the photonic sidebands with dif-ferent k pass through the Fermi level of the leads one by one.It is these additional channels from the photonic sidebands tomake the 〈I 〉 having additional resonant peaks.

The impurity, such as the substitutional boron (nitrogen) andthe vacancy in the infinite-length carbon nanotube can lead to aquasibound state near the lower or upper subbands [12]. How-ever, it is quite different for finite-length carbon nanotubes asshown in Fig. 1. Compared with the case of U = 0, one reso-nant state appears below or above the Fermi level at U = 3 orU = −5 for both L = 6 and L = 7. Furthermore, for the case ofL = 7, the original resonant states at the Fermi level is split intotwo ones with one half of the original value each. One of thetwo peaks is still at the Fermi level, and the other one is belowor above the Fermi level according to positive or negative U , re-spectively. Because the resonant state induced by the impurityis near below or above the Fermi level, the electron–hole sym-metry in perfect nanotubes within the π -band approximation isthen broken by the impurity [9]. The position of the resonantstate is related to the impurity strength [10,12]. With increas-ing the impurity strength, the position approaches to the Fermi

Fig. 1. The average current 〈I 〉 versus Vg with a MW field applied on the central (6,6) nanotube of the length (a) L = 6 and (b) L = 7 with different U . Here,θ = π/3.

320 H. Pan, T.-H. Lin / Physics Letters A 360 (2006) 317–322

Fig. 2. 〈I 〉 with (solid lines) and without (dash lines) the MW fields versus U for (6,6) nanotube of the length L = 6 (a) and L = 7 (b). (c) and (d) are thecorresponding 〈I 〉 versus Vg .

level. The single vacancy with very strong impurity strengthU = 106 induces a resonant state just located at the Fermi level,which can result in the recovery of the electron–hole symme-try.

The impurity strength effects are clearly shown in Fig. 2. Thesolid and dash lines are for the current with and without the MWfield. When there are no MW fields, 〈I 〉 increases with increas-ing U for the nanotube with L = 6. With U → ∞, 〈I 〉 finallyapproaches to a constant, because the impurity induces a reso-nant state at the Fermi level. While for the nanotube with L = 7,〈I 〉 decreases rapidly with increasing U . With U → ∞, 〈I 〉 be-comes about one half of the original value, because the tworesonant states are split and only one is left at the Fermi leveldue to the impurity. It means that whether the impurity increasesor decreases the current depends on the nanotube length. Underthe influence of both the MW fields and the impurity, some newphenomena can appear. It is noted that there is one new cur-rent peak located at U = 9 for L = 6 and at U = 3.5 for L = 7with MW fields applied in the central region. The reason is re-lated to both the photon-assisted tunneling caused by the MWfields and the resonant state induced by the impurity. To clearlyshow this, 〈I 〉 vs Vg at U = 9 for L = 6 and at U = 3.5 forL = 7 are plotted in Fig. 2(c) and (d). For the case of L = 6,the small side peak caused by the photon-assisted tunnelingis just located at the Fermi level at U = 9, which leads to thepeak in the curve of 〈I 〉 vs U . While for the case of L = 7, theoriginal main resonant state are split into two ones due to theimpurity, one of which is located at the Fermi level. The dif-ference between the two main resonant states depends on theimpurity strength U . Since the difference just equals the ω0at U = 3.5, the electron can tunnel through the resonant state

above the Fermi level under the assistance of photons. Thus〈I 〉 is not only contributed by the resonant state at the Fermilevel, but also by the one above the Fermi level, which resultsin the new peak in the curve of 〈I 〉 vs U . With increasing U ,〈I 〉 becomes small again, because the difference between thetwo split resonant states becomes larger than ω0 and only oneresonant state left at the Fermi level contributes to the averagecurrent.

In Fig. 3, the current and TMR versus the magnetic mo-ment orientation θ are plotted for different nanotube length L

and impurity strength U . A clear spin-valve effect is observedsuch that TMR varies smoothly with θ . In agreement with theTMR experimental results [1,2]. The current has a maximumvalue at θ = 0, when the magnetic moments are parallel. Theminimum is at θ = π , when they are antiparallel. This varia-tion of I is due to the difference in the parameters Γ↑ and Γ↓,which reflect he differences in the majority and minority carrierconcentration of the FM materials. The θ dependence of TMRshows the normal spin-valve effect. The TMR minimum is dueto quantum resonance. The impurity can decrease the TMR forthe off-resonance case, and the decrease depends on the impu-rity strength. However, the impurity has less influence on TMRfor the on-resonance case.

Fig. 4 shows the spin-coherent current as a function ofthe bias voltage V . For the (6,6) nanotube of length L = 6with U = 0, the spin-coherent current is symmetric about theFermi level and nearly zero at V = 0, which shows the off-resonance behaviour. While at U = 3,−5, the current is asym-metric about the Fermi level and is nonzero at V = 0 due tothe nonzero tunneling caused by the impurity. The amplitudeof the current increases with increasing U . However, the cur-

H. Pan, T.-H. Lin / Physics Letters A 360 (2006) 317–322 321

Fig. 3. The average current 〈I 〉 versus the magnetic moment orientation θ for (6,6) nanotube of the length (a) L = 6 and (b) L = 7 with different U . (c) and (d) arethe corresponding TMR.

Fig. 4. The average current 〈I 〉 versus the bias voltage V (6,6) nanotube of the length (a) L = 6 and (b) L = 7 with different U .

rent is nonzero even at U = 0 for the nanotube with L = 7,which shows the on-resonant behaviour. If there is an impurityin the nanotube, the current is suppressed and the amplitudeof the current decreases to about one half of the original one,because only one resonant state contribute to the current. Thecurrent is still exist even when the impurity strength U → ∞because of the existence of one resonant state at the Fermilevel.

In addition, the resonant magnitude of the current is sensi-tive to the ratio of the strength and the frequency of the MWfield as α = D0/ω0, which appears as an argument of the Besselfunction. Fig. 5 clearly shows the α dependence of the 〈I 〉 andthe TMR for different U . The 〈I 〉 oscillates damply with in-creasing α, and a larger α may suppress the transmission. TheTMR oscillates with increasing α. Since a resonant state ap-

proaches to the Fermi level with increasing U for L = 6, thepeaks are sharper and more obvious. While for L = 7, the res-onant state at the Fermi level is split into two ones, the firstresonant peak of the current and the TMR moves towards thedirection of larger α. These characters are the consequencesof the photonic sidebands induced by the MW field. The res-onant level is modulated not just by a displacement by emittingor absorbing photons, but is modulated in terms of a proba-bility amplitude that is characterized by the square of the nthBessel function Jn(Δ/ω) for each level to be displaced in en-ergy by kω0. The peaks are sharper and more obvious at U = 0,because the there is resonant energy level at the Fermi level.While at U = 3,−5, the energy level leaves the Fermi level, thepeaks are dampened, and the first resonant peak moves towardslarger α.

322 H. Pan, T.-H. Lin / Physics Letters A 360 (2006) 317–322

Fig. 5. The conductance G at θ = π/3 versus D0ω0 for (6,6) nanotube of the length L = 6 (a) and L = 7 (b) with different U . (c) and (d) are the corresponding TMR.

4. Conclusion

In summary, we have investigated the impurity effects onthe time-dependent current in the FM/CNT/FM system theo-retically. The system shows the on-resonance behaviour for theperfect armchair carbon nanotube with length L = 3N + 1 andshows the off-resonance behaviour with other lengths, respec-tively. The impurity suppresses the spin-coherent current for theon-resonance case, while it enhances the spin-coherent currentfor the off-resonance case. The position of the resonant state in-duced by the impurity depends on the impurity strength. Whenthe resonant state is near the Fermi level, the electron–holesymmetry can be broken. While for the very strong impuritystrength, the electron–hole symmetry can be recovered. WhenMW fields are applied on the central conductor, photonic side-bands are formed in the central region. When the sidebandsmeet with the Fermi energy of the terminals, photon-assistedtunneling will occur. With increasing the impurity strength, onenew current peak appears due to both the photon-assisted tun-neling caused by the MW fields and the resonant state inducedby the impurity. It is also found that the resonant peaks of boththe current and the TMR depend on the ratio of the magnitudeand the frequency of the MW field. The present study mightopen a way to control the spin-dependent transport in a spin-tronic device by applying a time-dependent electrical field.

Acknowledgements

This project is supported by NSFC under Grant No.10547102.

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