LIU Yu, ZHOU Xiao-ying, ZHOU Guang-hui

(1. College of Information Science and Engineering, Hunan First Normal University, Changsha 410205; 2. College of Physics and Information Sciences, Hunan Normal University, Changsha 410081, China)



Tunable Magnetic-Resistance for Topological Insulator Thin Film Modulated by a FM/N/FM Junction

LIU Yu1, ZHOU Xiao-ying2, ZHOU Guang-hui2*

(1. College of Information Science and Engineering, Hunan First Normal University, Changsha 410205; 2. College of Physics and Information Sciences, Hunan Normal University, Changsha 410081, China)

The quantum transport for a topological insulator thin film was studied by a ferromagnet/normal/ferromagnet junction with a gate voltage exert on the normal segment. A quantum phase transition occurs due to the competition between the exchange field and the hybridization gap. The conductance for the junction behaves like a conventional spin valve without gate-voltage applied and can be tuned like a spin field-effect transistor via the gate-voltage. Interestingly, a conductance plateau is emerged when the exchange field is twice of the hybridization gap in the absence of voltage. Further, the magnetic-resistance ratio can be up to 100%, and can also be negative due to the anomalous transport.

topological insulator film; ferromagnet/normal/ferromagnet junction; surface state transport

Topological insulators (TIs)[1], possess of unusual phases of quantum matter simultaneously with insulating bulk and conducting edge or surface states, have been extensively studied in recent years[2]. The two-dimensional (2D) TI phase was firstly predicted in a HgTe quantum well[3]and observed by the followed transport measurements[4]. Thereafter, Bi2Se3family of materials have been proposed[5]as three-dimensional (3D) strong TIs. And the single Dirac cone of surface states has been observed by the followed spin- and angle-resolved photoemission spectroscopy measurements[6-7]for Bi2Se3and Bi2Te3, respectively. These results have revealed that electron spins on the surface Dirac cone are locked with their momenta, giving rise to helical Dirac fermions without spin degeneracy[5-7]. The locking of the electron spin to the momentum comes from a combination of strong spin-orbit interaction and the breaking of the inversion symmetry at the surface[5]. Such a spin texture on the surface Dirac cone leads to antilocalization property and plays a central role in inducing exotic quantum phenomena. Moreover, surface states are protected by the time-reversal symmetry[8]and the topology of the bulk gap, and are robust against disorder scattering[9]and electron-electron interactions[10].

Furthermore, 3D TI thin films have been extensively investigated theoretically[11-12]and experimentally[13-14]due to their quite different nature from that with a single surface. Recently, two effective Hamiltonians[11,15]were proposed to describe low energy electrons for 3D TI thin film. Consequently, various interesting properties of 3D TI thin films have been predicted, particularly those relevant to quantum Hall effect[16], Landau levels[15,17], quantum phase transitions[18-20], magnetic-resistive effect[21]and electron-electron interaction[22], etc. However, unlike the 3D TI single surface state, less attention has been paid to theoretical investigation on transport property modulated by ferromagnetic (FM) stripes. It is known that the top and bottom surfaces of a 3D TI thin film are hybridized. When the Fermi level is in the hybridization gap, exotic property such as quantum phase transition may appear[18-20], which is really distinct with single surface states[23-27]. Moreover, such a promising material is vital for device designing in nanoelectronics and spintronics. Therefore, the transport property for 3D TI thin film modulated by FM stripes is an important issue.

In this work, we study the electronic structure and transport for a 3D TI thin film modulated by a ferromagnet/normal/ferromagnet (FM/N/FM) junction with the exchange filed configuration only in thezdirection and a gate voltage on the normal metal segment. A quantum phase transition occurs when the exchange field is equal to the hybridization gap of the film. Normalized conductance is calculated for two phases with the gate is present or absent. We demonstrate that the conductance for the junction behave like a conventional spin value when no gate-voltage applied and can be tuned like a spin field transistor via the gate-voltage. Interestingly, a conductance platform emerged when the exchange field is twice of the hybridization gap with no voltage applied. Furthermore, the magnetic-resistance ratio can be up to 100%, and can also be negative due to the anomalous transport.

The organization of this paper is as follows. In Sec.Ⅰ, we explain the Hamiltonian and present the theoretical formulism for the system. In Sec.Ⅱ, we give some numerical examples with discussions for the analytical calculation. Sec.Ⅲ summarizes our results briefly.

1 Model and Hamiltonian

Fig.1 (Color online) (a) Schematic illustration of a 3D TI thin film attached by a FM/N/FM junction, where a gate voltage on the normal metal segment is presented

As shown in Fig.1, we consider a FM/N/FM junction on the surface of a 3D TI thin film with a voltage exerted in the central normal region. The bulk ferromagnetic insulators interacts with electrons in the TI film by the proximity, and the ferromagnetism is induced in two surfaces states[23-27]. The interfaces between ferromagnet (FM) and normal segment are parallel to theydirection, and the normal segment is located betweenx=0 andx=Lwith gate voltageV0exerted on it and we presume the distance between two interfaces is shorter than the mean-free path as well as the spin coherence length for simplicity.

According to the effective low-energy surface Hamiltonian for a clean 3D TI thin film[15], the Hamiltonian for our system reads

(1)

Fig.2 (Color online) Energy (in units of E0) spectrum for a 3D TI thin film with (a) Δt=1, mi=0, Vo=0, (b) Δt=1, mi=0, Vo=-1, (c) Δt=1, mi=1, Vo=0, (d) Δt=1, mi=2, Vo=0. In (a) and (b) the (black) dashed line is for spin-up and (red) solid line for spin-down electrons, respectively, but the slid/dashed line for conduction/valence band in (c) and (d).

In Fig.2, for more intuitive comprehension, the energy (in units ofE0) dispersions in different cases are plotted for the system according to Eq.(2). For a clean film, as seen in Fig.2(a), the energy is degenerated for two spin orientations with a gapΔ=2Δtbetween conduction and valence bands. However, when a gate-voltage is applied both the conduction and valence bands are shifted down form the Fermi level but the degeneracy is still kept [see Fig.2(b)]. Furthermore, when an exchange field is presented, unlike the single surface states[24-26], an interesting spectrum feature emerged: the energy is spilt into four branches and a quantum phase transition occur due to the competition between the exchange field and hybridiztion. As shown in Figs. 2(c) and 2(d), in this case the thin film is conducting whenmi(t)=Δtand semiconducting with a gap Δ=2|mi(t)-Δt|, which has been well explained in Ref.[20].

In order to investigate the transport property for 3D TI thin film modulated by the junction. We now calculate the charge transmission for the system. The thin film is divided into three regions as shown in Fig.1. In the incoming region, the wave function is

(2)

Inthecentralregionwhereagate-voltageisexerted,thewavefunctionis

(3)

wherea(b) is the left (right) going wave amplitude,kx/yis the wavevetor andky=qyfor the momentum conservation in the y direction. IfE=V0+sΔt,

(4)

Andthewavefunctioninthetransmittingregionis[31]

(5)

wheretis the transmission coefficient,Px/yis the wavevetor andpy=kyfor the momentum conservation in theydirection. Therefore, the transmission probability can express as

(6)

Inordertocalculatethetransmissionprobability,weapplythecontinuityconditionsforwavefunctionsatboundariesbetweendifferentregions: ψi(0,y)=ψc(0,y)andψc(L,y)=ψt(L,y).Unlikethesecond-orderderivativeSchrödingerequation,oneonlyneedstomatchthewavefunctionbutnotitsderivative,becausetheHamiltonianemployedhereisafirst-orderlyderivative.ThenwecanobtainthetransmissionprobabilityT(E,θ).Inthispaper,weinvestigatetransportpropertiesforthestateassociatewiths=-1foritsinterestingbandstructure.Asaresult,accordingtotheLandauer-Büttikerformula[32],itisstraightforwardtoobtaintheballisticconductanceatzerotemperature

(7)

whereG0=2e2/hisconductanceunit.Note,weletmi=mt=moinvalueandparallel(P)oranti-parallel(AP)todistinguishtheorientationoftwoFMstripeslaterforconvenientexplanation.

2 Numerical Examples and Discussions

Inwhatfollowsweshowsomenumericalexamplesfora3DTIthinfilmmodulatedbythejunction.

Fig.3 (Color online) Conductance vs transmitting energy with L=2 (a)Δt=1, V=0, mo=1, (b)Δt=1, V=0,mo=2, (c)Δt=1, V=-4, mo=1, (d)Δt=1, V=-4, mo=2, the blue solid line for parallel conductance GP and the red dashed line for the antiparallel conductance GAP.

Fig.4 (Color online) Corresponding magnetic-resistance ratio(MR) for Fig.3

Fig.3presentsthetunnelingconductanceGpandGAPv.s.energywithL=2and(a)Δt=1, V=0, mb=1, (b)Δt=1, V=0, m0=2, (c)Δt=1, V=-4, m0=1, (d)Δt=1, V=-4, m0=2,wherethe(blue)solidlineforparallelconductanceGpandthe(red)dashedlinefortheantiparallelconductanceGAP.Whennogate-voltageapplied,theconductanceinparallelconfigurationisalwayslargethanthatinantiparallelconfigurationasintheconventionalspinvalve[33]anditscounterpartingraphene[34]andtheconductanceisanoscillatedevenfunctionofEwhichmeanselectronsandholescontributetoconductanceequally(seeFig.3(a)and3(b)).InFig.3(a),theparallelconductanceGp(the(blue)solidline)showsanonzeroplatformatsmalltransmittingenergybecauseoftheevanescentwavesthoughatransmissiongap[-1,1]formedinthecentralregion(seeFig.2(a))fortheincomingregionisinmetalphase(seeFig.2(c)),while,theantiparallelconductanceGAP(the(red)dashedline)isvanishedwhenthetunnelingenergylocatesinthetunnelinggap[-2,2]whichisdeterminedbythebandstructureoftransmittingregionaccordingtoEq. (2).Interestingly,inFig.3(b), Gp(the(blue)solidline)isalwaysG0whenelectronenergyishigherthanacriticalvaluewhichseemsquitetooursurprise.Actually,thiscanbeunderstoodasfollow.ForFig.3(c)theFMstripesareinPalignmentwithm0=2andtheincomingregionisinsemiconductorphasewithagapΔ=1,sodothetransmittingregion,andthenormalregionisalsoasemiconductorwithaΔ=1inlinewiththebandstructure(seeFig.2(a)),whicheliminatethedistinguishesinthreedifferentregionsfromtheviewofbandstructureleadingtoaperfectwavfunctionmatchinthreedifferentregions.Moregeneralconclusionisthatwhentheexchangefiledistwiceofthehybridizationgap,thereisnodifferenceinthreedistinctregions,electronsmovefreelywhentheirenergyishigherthanthetunnelinggap.However,itseemsquitedifferentfortheantiparallelconductanceGAP(the(red)dashedline)forthetransmittingregionisanisotropicwiththeothersandtheexplanationissimilarwiththatinFig.3(a).However,whennogate-voltageapplied,theparallelconductanceGpcanbelessthantheantiparallelconductanceGAPwhichissimilarwiththeconductancefeatureinaspin-fieldtransistorandatopologicaljunction[27](seeFigs.3(c)and3(d)).Meanwhile,theconductanceisasymmetrywithtunnelingenergywhichmeanselectronsandholescontributeunequallytoconductanceduetotheasymmetrybandstructureincentralregion(seeFig.2(b)).InFig.3(c),noconductinggapformedforparallelconductanceGpowingtoanegativegate-voltagepushedtheconductingbandbelowtheFermienergy(seeFig.2(a)wesetEF=0).Yet,conductingisalwaysblockedforGAPwhenthetunnelingenergylocatesintheconductinggap[-2,2]determinedbythetransmittingregion.AsforFig.3(d),theconductingfeatureissimilartoFig.3(c)apartfromaconductinggapformedforbothGpandGAP.

AfterobtainingtheconductanceGP(GAP)fortheparallel(antiparallel)configuration,wecangetthemagneticresistance(MR)directly,whichisdefinedasMR=(GP-GAP)/GP.Fig.4plottedthecorrespondingMRv.s.energyforFig.3.TheMRcanapproach100%inallcasesfordifferentconductinggapinPandAPalignment.Moreover,thegate-voltageinfluencedtheMRgreatlyfortheMRisalwayspositiveandsymmetrywithEwhenthegate-voltageisabsent,however,theMRisasymmetrywithEandcanbenegativeowingtoanomalouselectronictransport[23,26].ThebignegativeMRalsomeansabigvariationinconductancebetweentheparallelandantiparallelconfigurations.OnecanunderstandotherfeaturesaboutMRfromFig.3.

3 Summary and Conclusion

Insummary,wehavestudiedtheelectronicstructureandchargetransportforatopologicalinsulatorthinfilmmodulatedbyaferromagnet/normal/ferromagnetjunctionwithagatevoltageexertonthenormalsegment.Aquantumphasetransitionoccursowingtothecompetitionbetweentheexchangefieldandthehybridizationgap.Normalizedconductanceiscalculatedfortwophaseswiththegateispresentorabsent.Wedemonstratethattheconductanceforthejunctionbehavelikeaconventionalspinvaluewhennogate-voltageappliedandcanbetuninglikeaspinfieldtransistorviathegate-voltage.Interestingly,aconductanceplatformemergedwhentheexchangefieldistwiceofthehybridizationgapwithnovoltageapplied.Furthermore,themagnetic-resistanceratiocanbe100%,andcanalsobenegativeduetoanomaloustransport.TheseinterestingfindingsfortheFMmodulatednanostructurebasedonthe3DTIthinfilmmaybetestableinthepresentexperimentaltechnique[7,37],andmayprovideafurtherunderstandingthenatureof3DTIthinfilm.

[1] KANE C L, MELE E J. A New Spin on the Insulating State [J]. Science, 2006,314(11):1692-1693.

[2] HASAN M Z, KANE C L. Topological insulators [J]. Rev Mod Phys, 2010,82(1):3045-3057.

[3] BERNEVIG B A, HUGHES T L, ZHANG S C. Quantum spin hall effect and topological phase transition in HgTe quantum wells [J]. Science, 2006,314(4):1757-1761.

[4] KONIG M, WIEDMANN S, BRUNE C,etal. Quantum spin hall insulator state in HgTe quantum wells [J]. Science, 2007,318(3):766-770.

[5] ZHANG H, LIU C X, QI X L,etal. Topological insulators in Bi2Se3and Sb2Te3with a single Dirac cone on the surface [J]. Nature Phys, 2009,5(2):483-442.

[6] XIA Y, QIAN D, HSIEH D,etal. Observation of a large-gap topological-insulator class with a single Dirac cone on the surface [J]. Nature Phys, 2009,5(2):398-402.

[7] CHEN Y L, ANALYTIS J G, CHU J H,etal. Experimental realization of a three- dimensional topological insulator Bi2Te3[J]. Science, 2009,325(1):178-181.

[8] FU L, KANE C L, MELE E J. Topological insulators in three dimensions [J]. Phys Rev Lett, 2007,98(4):106803.

[9] JIANG H, CHENG S G, SUN Q F,etal. Topological insulator:a new quantized spin hall resistance robust to dephasing [J]. Phys Rev Lett, 2009,103:036803.

[10] EGGER R, ZAZUNOV A, YEYATI A L. Helical luttinger liquid in topological insulator nanowires [J]. Phys Rev Lett, 2010,105:136403.

[11] LU H Z, SHAN W Y, YAO W,etal. Massive dirac fermions and spin physics in an ultrathin film of topological insulator [J]. Phys Rew B, 2010,81:115407.

[12] BIHLMAYER G, KOROTEEV Y M, CHULKOV E V,etal. Surface- and edge-states in ultrathin Bi-Sb films [J]. New J Phys, 2010,12:065006.

[13] ZHANG Y, HE K, CHANG C Z,etal. Crossover of the three-dimensional topological insulator Bi2Se3to the two-dimensional limit [J]. Nature Phys, 2010,6(4):584-588.

[14] PLUCINSKI L, MUSSLER G, KRUMRAIN J,etal. Robust surface electronic properties of topological insulators:Bi2Te3films grown by molecular beam epitaxy [J]. Appl Phys Lett, 2011,98:222503.

[15] ZYUZIN A A, BURKOV A A. Thin topological insulator film in a perpendicular magnetic field [J]. Phys Rev B, 2011,83:195413.

[16] LI H, SHENG L, XING D Y. Quantum hall effect in thin films of three-dimensional topological insulators [J]. Phys Rev B, 2011,84:035310.

[17] YANG Z, HAN J H. Landau level states on a topological insulator thin film [J]. Phys Rev B, 2011,83:045415.

[18] LI H, SHENG L, XING D Y. Quantum phase transitions in ultrathin films of three-dimensional topological insulators in the presence of an electrostatic potential and a Zeeman field [J]. Phys Rev B, 2012,85:045118.

[19] ZYUZIN A A, HOOK M D, BURKOV A A. Parallel magnetic field driven quantum phase transition in a thin topological insulator film [J]. Phys Rev B, 2011,83:245428.

[20] CHO G Y, MOORE J E. Quantum phase transition and fractional excitations in a topological insulator thin film with Zeeman and excitonic masses [J]. Phys Rev B, 2011,84:165101.

[21] ZHANG H B, YU H L, BAO D H,etal. Magnetoresistance swich effect of a Sn-doped Bi2Te3topological insulator [J]. Adv Mater, 2012,24(1):132-136.

[22] WANG J, DASILVA A M, CHANG C Z,etal. Evidence for electron-electron interaction in topological insulator thin films [J]. Phys Rev B, 2011,83:245438.

[23] YOKOYAMA T, ZANG J, NAGAOSA N. Theoretical study of the dynamics of magnetization on the topological surface [J]. Phys Rev B, 2010,81:241410(R).

[24] ZHU J J, YAO D X, ZHANG S C,etal. Electrically controllable surface magnetism on the surface of topological insulators [J]. Phys Rev Lett, 2011,106:097201.

[25] ZHAI F, WU P. Tunneling transport of electrons on the surface of a topological insulator attached with a spiral multiferroic oxide [J]. Appl Phys Lett, 2011,98:022107.

[26] WU Z, PEETERS F M, CHANG K. Spin and monentum filtering of electrons on the surface of a topological insulator [J]. Appl Phys Lett, 2011,98:162101.

[27] ZHANG K H, WANG Z C, ZHENG Q R,etal. Gate-voltage controlled electronic transport through a ferromagnet/normal/ferromagnet junction on the surface of a topological insulator [J]. Phys Rev B, 2012,86:174416(R).

[28] HAUGEN H, HERNANDO D H, BRATAAS A. Spin transport in proximity-induced ferromagnetic graphene [J]. Phys Rev B, 2008,77:115406.

[29] CHAKHALIAN J, FREELAND J W, SRAJER G,etal. Magnetism at the interface between ferromagnetic and superconducting oxides [J]. Nature Phys, 2006,2(1):244-248.

[30] PERSHOGUBA S S, YAKOVENKO V M. Spin-polarized tunneling current through a thin film of a topological insulator in a parallel magnetic field [J]. Phys Rev B, 2012,86:165404(R).

[31] KATSNELSON M I. Zitterbewegung, chirality, and minimal conductivity in graphene [J]. Eur Phys J B, 2006,51:157-160.

[32] DATTA S. Electronic transport in mesoscopic systems [M]. Cambridge: Cambridge University Press, 1995.

[33] ZUTIC I, FABIAN J, SARMA S D. Spintronics: fundamentals and applications [J]. Rev Mod Phys, 2004,76(2):323-410.

[34] BAI C, ZHANG X. Large oscillating tunnel magnetoresistance in ferromagnetic graphene single tunnel junction [J]. Phys Lett A, 2009,372(3):725-729.

[35] DATTA S, DAS B. Electroic analog of the electro-optic modulator [J]. Appl Phys Lett, 1990,56(3):665.

[36] SOODCHOMSHOM B. Magneto transport on the surface of a topological insulator spin valve [J]. Phys Lett A, 2010,374(9):2894-2899.

[37] PAN Z H, VESCOVO E, FEDOROV A V,etal. Electronic structure of the topological insulator Bi2Se3using angle-resolved photoemission spectroscopy: evidence for a nearly full surface spin polarization [J]. Phys Rev Lett, 2011,106:257004.

(编辑 CXM)

2016-09-18

国家自然科学基金资助项目(11274108)

O441.6

A

1000-2537(2016)06-0061-07

铁磁/正常/铁磁结调制的拓扑绝缘体薄膜表面输运性质

刘 宇1, 周小英2, 周光辉2*

(1.湖南第一师范学院信息科学与工程学院, 中国 长沙 410205; 2.湖南师范大学物理与信息科学学院, 中国 长沙 410081)

研究了拓扑绝缘体薄膜表面态在铁磁/正常/铁磁结调制下的电子自旋相关输运. 发现由于交换场与杂化带隙的竞争而产生量子相变, 在结无门电压时电导行为类似于自旋阀, 加门电压后为自旋场效应管. 有趣的是, 无门电压且交换场能是杂化带隙的两倍时出现一个电导平台, 磁阻比率可达100%.

拓扑绝缘体薄膜; 铁磁/正常/铁磁结; 表面态输运

10.7612/j.issn.1000-2537.2016.06.011

*通讯作者,E-mail：ghzhou@hunnu.edu.cn