范文一:石墨烯非传统量子霍尔效应
LETTERS
Unconventional quantum Hall effect and Berry’sphase of 2πin bilayer graphene
K. S. NOVOSELOV 1, E. McCANN 2, S. V. MOROZOV 1, 3, V. I. FAL’KO2, M. I. KATSNELSON 4, U. ZEITLER 4, D. JIANG 1, F . SCHEDIN 1AND A. K. GEIM 1*
12
Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester M139PL, UK Department of Physics, Lancaster University, Lancaster LA14YB, UK 3
Institute for Microelectronics Technology, 142432Chernogolovka, Russia 4
Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525ED Nijmegen, The Netherlands *e-mail:geim@manchester.ac.uk
Published online:26February 2006; doi:10.1038/nphys245
here are two known distinct types of the integer quantum Hall e ?ect. One is the conventional quantum Hall e ?ect, characteristic of two-dimensional semiconductor systems 1,2, and the other is its relativistic counterpart observed in graphene, where charge carriers mimic Dirac fermions characterized by Berry’sphase π, which results in shifted positions of the Hall plateaus 3–9. Here we report a third type of the integer quantum Hall e ?ect. Charge carriers in bilayer graphene have a parabolic energy spectrum but are chiral and show Berry’sphase 2πa ?ecting their quantum dynamics. The Landau quantization of these fermions results in plateaus in Hall conductivity at standard integer positions, but the last (zero-level)plateau is missing. The zero-level anomaly is accompanied by metallic conductivity in the limit of low concentrations and high magnetic ?elds,in stark contrast to the conventional, insulating behaviour in this regime. The revealed chiral fermions have no known analogues and present an intriguing case for quantum-mechanical studies.
Figure 1provides a schematic overview of the quantum Hall e ?ect (QHE)behaviour observed in bilayer graphene by comparing it with the conventional integer QHE. In the standard theory, each ?lledsingle-degenerate Landau level contributes one conductance quantum e 2/h towards the observable Hall conductivity (heree is the electron charge and h is Planck’sconstant). The conventional QHE is shown in Fig. 1a, where plateaus in Hall conductivity σxy make up an uninterrupted ladder of equidistant steps. In bilayer graphene, QHE plateaus follow the same ladder but the plateau at zero σxy is markedly absent (Fig.1b). Instead, the Hall conductivity undergoes a double-sized step across this region. In addition, longitudinal conductivity σxx in bilayer graphene remains of the order of e 2/h , even at zero σxy . The origin of the unconventional QHE behaviour lies in the coupling between two graphene layers, which transforms massless Dirac fermions, characteristic of single-layer graphene 3–9(Fig.1c), into a new type of chiral quasiparticle. Such quasiparticles have an ordinary parabolic spectrum ε(p ) =p 2/2m with e ?ective mass m , but
nature physics ADV ANCE ONLINE PUBLICATION www.nature.com/naturephysics
T
accumulate Berry’sphase of 2πalong cyclotron trajectories (hereεis the energy of quasiparticles and p their momentum). The latter is shown to be related to a peculiar quantization where the two lowest Landau levels lie exactly at zero energy ε, leading to the missing plateau and double step shown in Fig. 1b.
Bilayer ?lmsstudied in this work were made by the micromechanical cleavage of crystals of natural graphite, which was followed by the selection of bilayer ?akesby using a combination of optical microscopy and atomic force microscopy as described in refs 10,11. Multiterminal ?eld-e?ect devices (seethe inset in Fig. 2a) were made from the selected ?akesby using standard microfabrication techniques. As a substrate, we used an oxidized heavily doped Si wafer, which allowed us to apply gate voltage V g between graphene and the substrate. The studied devices showed an ambipolar electric ?elde ?ect such that electrons and holes could be induced in concentrations n up to 1013cm ?2(n =αV g , where α≈7. 3×1010cm ?2V ?1for a 300nm SiO 2layer). For further details about microfabrication of graphitic ?eld-e?ect devices and their measurements, we refer to earlier work 3,4,10,11.
Figure 2a shows a typical QHE behaviour in bilayer graphene at a ?xedV g (?xedn ) and varying magnetic ?eldB up to 30T. Pronounced plateaus are clearly seen in Hall resistivity ρxy in high B , and they are accompanied by zero longitudinal resistivity ρxx . The observed sequence of the QHE plateaus is described by ρxy =h /4N e 2, which is the same sequence as expected for a two-dimensional (2D)free-fermion system with double spin and double valley degeneracy 1,2,12–15. However, a clear di ?erence between the conventional and reported QHE emerges in the regime of small ?llingfactors ν<1(seefigs 2b,c="" and="" 3).="" this="" regime="" is="" convenient="" to="" study="" by="" ?xingb="" and="" varying="" concentrations="" of="" electrons="" and="" holes="" passing="" through="" the="" neutrality="" point="" |n="" |≈0,="" where="" ρxy="" changes="" its="" sign="" and,="" nominally,="" ν="0." also,="" because="" carrier="" mobilities="" μin="" graphitic="" ?lmsare="" weakly="" dependent="" on="" n="" ,="" measurements="" in="" constant="" b="" are="" more="" informative="" 3,4,10.="" they="" correspond="" to="" a="" nearly="" constant="" parameter="" μb="" ,="" which="" de?nesthe="" quality="" of="" landau="" quantization,="" and="" this="" allows="">1(seefigs>
1
LETTERS
ρx y a n d ρx x (k Ω)
b
b
σx y (4e 2/h )
––––n (1012 cm –2)
ρx x (k Ω)
Figure 1Three types of the integer quantum Hall effect. a , b , Schematic
illustration of the conventional integer QHE found in 2D semiconductor systems (a ), incorporated from refs 1,2, and the QHE in bilayer graphene described in the present paper (b ). Plateaus in Hall conductivity σxy occur at values (ge 2/h ) N , where N is an integer, e 2
/h the conductance quantum and g the system degeneracy. The distance between steps along the concentration axis is de?nedby the density of states
gB /φ0on each Landau level, which is independent of a 2D spectrum 1–9. Here, B is the magnetic ?eldand φ0=h /e the ?uxquantum. The corresponding sequences of Landau levels as a function of carrier concentrations n are shown in blue and orange for electrons and holes, respectively. For completeness, c also shows the QHE behaviour for massless Dirac fermions in single-layer graphene.
observation of several QHE plateaus during a single voltage sweep in moderate magnetic ?elds(Fig.2b). The periodicity n of quantum oscillations in ρxx as a function of n is
de?nedby the density of states gB /φ0(whereg is the degeneracy and φ0is the ?uxquantum) on each Landau level 1–10(seeFig. 1). In Fig. 2c, for example, n ≈1. 2×1012cm ?2at B =12T, which yields g =4and con?rmsthe double-spin and double-valley degeneracy expected from band-structure calculations for bilayer graphene 14,15.
2
Figure 2Quantum Hall effect in bilayer graphene. a , Hall resistivities ρxy and ρxx measured as a function of B for ?xedconcentrations of electrons
n ≈2. 5×1012cm ?2induced by the electric ?eldeffect. Inset:Scanning electron micrograph of one of more than ten bilayer devices studied in our work. The width of the Hall bar (darkcentral area) is approximately 1μm. The known geometry of our devices allowed us to convert the measured resistance into ρxx with an accuracy of better than 10%.b , c , σxy and ρxx are plotted as functions of n at a ?xedB and temperature T =4K. Positive and negative n correspond to ?eld-inducedelectrons
2
and holes, respectively. The Hall conductivity σxy =ρxy /(ρ2xy +ρxx ) was calculated directly from experimental curves for ρxy and ρxx . σxy allows the underlying sequences of QHE plateaus to be seen more clearly. σxy crosses zero without any sign of the zero-level plateau that would be expected for a conventional 2D system. The inset shows the calculated energy spectrum for bilayer graphene, which is parabolic at low ε. Carrier mobilities μin our bilayer devices were typically around 3,000cm 2V ?1s ?1, which is lower than for devices made from single-layer graphene 3,4. This is surprising because one generally expects more damage and exposure in the case of single-layer graphene that is unprotected from the immediate environment from both sides.
Figure 2b shows that, although the Hall plateaus in bilayer graphene follow the integer sequence σxy =±(4e 2/h ) N for N ≥1, there is no sign of the zero-N plateau at σxy =0, which is expected for 2D free-fermion systems 1,2(Fig.1a). In this respect, the behaviour resembles the QHE for massless Dirac fermions (Fig.1c),
nature physics ADV ANCE ONLINE PUBLICATION www.nature.com/naturephysics
LETTERS
ρx x (k Ω)
ρx x (k Ω)
V g (V)
T (K)
theory, which attributes the ?nitemetallic conductivity and the absence of localization to the relativistic-like spectrum of single-layer graphene 3. Bilayer graphene has the usual parabolic spectrum, and the observation of the maximum resistivity of approximately h /4e 2and, moreover, its weak dependence on B in this system is most unexpected. Note, however, that the quantization is less accurate than in single-layer graphene, as the peak value varied from 6to 9k for di ?erent bilayer devices.
The unconventional QHE in bilayer graphene originates from peculiar properties of its charge carriers that are chiral fermions with a ?nitemass, as discussed below. First, we have calculated the quasiparticle spectrum in bilayer graphene by using the standard nearest-neighbour approximation 12. For quasiparticles near the corners of the zone known as K-points, we ?nd Brillouin √2
ε(p ) =±(1/2)γ1±1F where v F =(/2)γ0a /h ˉ, a is the lattice periodicity, h ˉ=h /2πand γ0and γ1are the intra layer and inter layer coupling constants, respectively 13. This dispersion relation (plottedin Fig. 2c) is in agreement with the ?rst-principleband-structure calculations 14and, at low energies,
2
becomes parabolic ε=±p 2/2m with m =γ1/2v F (thesign ±
15
refers to electron and hole states). Further analysis shows that quasiparticles in bilayer graphene can be described by using the e ?ective hamiltonian
+ 1 0(π?) 2
??x +i p ?y . ?=p H 2=?where π2
?02m π
Figure 3Resistivity of bilayer graphene near zero concentrations as a function
of magnetic ?eldand temperature. a –d , The peak in ρxx remains of the order of h /4e 2, independent of B (a , b ) and T (c , d ). This yields no gap in the Landau
spectrum at zero energy. b , For a ?xedn ≈0and varying B , we observed only small magnetoresistance. The latter varied for different devices and contact con?gurations(probablyindicating the edge-state transport) and could be non-monotonic and of random sign. However, the observed magnetoresistance (forbilayer devices without chemical doping 10) never exceeded a factor of two in any of our experiments in ?eldsup to 20T.
?2acts in the space of two-component Bloch functions (furtherH
referred to as pseudospins) describing the amplitude of electron waves on weakly coupled nearest sites A1and B2belonging to two nonequivalent carbon sublattices A and B and two graphene layers marked as 1and 2.
For a given direction of quasiparticle momentum p =
?J of a general form (p cos ?,p sin ?), a hamiltonian H +
?) J 0(πJ ?π0
can be rewritten as
where there is also no plateau but a step occurs when σxy passes
the neutrality point. However, in bilayer graphene, this step has a double height and is accompanied by a central peak in ρxx , which is twice as broad as all other peaks (Fig.2c). The broader peak yields that, in bilayer graphene, the transition between the lowest hole and electron Hall plateaus requires twice the number of carriers needed for the transition between the other QHE plateaus. This implies that the lowest Landau level has double degeneracy 2×4B /φ0, which can be viewed as two Landau levels merged together at n ≈0(seethe Landau level charts in Fig. 1).
Continuous measurements through ν=0as shown in Fig. 2b,c have been impossible for conventional 2D systems where the
21,2
from a zero-level plateau in σxy =ρxy /(ρ2xy +ρxx ) is inferred 2
rapid (oftenexponential) increase in ρxx h /e with increasing B and decreasing temperature T for ?llingfactors ν<1, indicating="" an="" insulating="" state.="" to="" provide="" a="" direct="" comparison="" with="" the="" conventional="" qhe="" measurements,="" fig.="" 3shows="" ρxx="" in="" bilayer="" graphene="" as="" a="" function="" of="" b="" and="" t="" around="" zero="" ν.="" bilayer="" graphene="" shows="" little="" magnetoresistance="" or="" temperature="" dependence="" at="" the="" neutrality="" point,="" in="" striking="" contrast="" to="" the="" conventional="" qhe="" behaviour.="" this="" implies="" that="" σxy="" in="" bilayer="" graphene="" does="" not="" vanish="" over="" any="" interval="" of="" νand="" reaches="" zero="" only="" at="" one="" point,="" where="" ρxy="" changes="" its="" sign.="" note="" that="" ρxx="" surprisingly="" maintains="" a="" peak="" value="" of="" approximately="" h="" e="" 2in="" ?eldsup="" to="" 20t="" and="" temperatures="" down="" to="" 1k.="" a="" ?nitevalue="" of="" ρxx="" ≈h="" e="" 2in="" the="" limit="" of="" low="" carrier="" concentrations="" and="" at="" zero="" b="" was="" reported="" for="" single-layer="" graphene="" 3.="" this="" observation="" was="" in="" qualitative="" agreement="">1,>
nature physics ADV ANCE ONLINE PUBLICATION www.nature.com/naturephysics
?J =ε(p ) σ·n (?),H
(1)
where n =?(cos J ?,sin J ?)and vector σis made from Pauli
matrices 15. For bilayer graphene, J =2, but the notation J is useful because it also allows equation (1)to be linked with the case of single-layer graphene, where J =1. The eigenstates
?J correspond to pseudospins polarized parallel (electrons)of H
or antiparallel (holes)to the ‘quantization’axis n . An adiabatic evolution of such pseudospin states, which accompanies the rotation of momentum p by angle ?, also corresponds to the rotation of axis n by angle J ?. As a result, if a quasiparticle encircles a closed contour in the momentum space (thatis ?=2π), a phase shift Φ=J πknown as Berry’sphase is gained by the quasiparticle’swavefunction 16. Berry’sphase can be viewed as arising owing to rotation of pseudospin, when a quasiparticle repetitively moves between di ?erent carbon sublattices (Aand B for single-layer graphene, and A1and B2for bilayer graphene).
For fermions completing cyclotron orbits, Berry’sphase contributes to the semiclassical quantization and a ?ects the phase of Shubnikov–deHaas oscillations (SdHOs).For single-layer graphene, this results in a π-shift in SdHOs and a related 1/2-shift in the sequence of QHE plateaus 3–9, as compared with the conventional 2D systems where Berry’sphase is zero. For bilayer graphene, Φ=2πand there can be no changes in the quasiclassical limit (N 1). One might also expect that phase 2πcannot in?uencethe QHE sequencing. However, the exact analysis
3
LETTERS
(seethe Supplementary Information) of the Landau-level spectra
?J showing Berry’sphase J πshows that there is for hamiltonian H
an associated J -fold degeneracy of the zero-energy Landau level (thatis Berry’sphase of 2πleads to observable consequences in the quantum limit N =0). For the free-fermion QHE systems (noBerry’sphase), the energy is given by εN =h ˉωc (N +1/2) and the lowest state lies at ?niteenergy h ˉωc /2, where cyclotron
ωc =eB /m . For single-layer graphene (J =1, Φ=π), frequency √
5–9
εN =±v F and there is a single state ε0at √zero energy . For bilayer graphene (J =2, Φ=2π), εN =±h ˉωc and
15
the two lowest states ε0=ε1lie at zero energy .
The existence of a double-degenerate Landau level explains the unconventional QHE found in bilayer graphene. This Landau level lies at the border between electron and hole gases and, taking into account the quadruple spin and valley degeneracy, it accommodates carrier density 8B /φ0. With reference to Fig. 1, the existence of such a Landau level implies that there must be a QHE step across the neutrality point, similarly to the case of single-layer graphene 3–9. Owing to the double degeneracy, it takes twice the number of carriers to ?llit (ascompared with all other Landau levels), so that the transition between the corresponding QHE plateaus must be twice as wide (thatis 8B /φ0as compared with 4B /φ0). Also, the step between the plateaus must be twice as high, that is 8e 2/h as compared with 4e 2/h for the other steps at higher carrier densities. This is exactly the behaviour observed experimentally.
In conclusion, bilayer graphene adds a new member to the small family of QHE systems, and its QHE behaviour reveals the existence of massive chiral fermions with Berry’sphase 2π, which are distinct from other known quasiparticles. The observation of a ?nitemetallic conductivity of approximately e 2/h for such fermions poses a serious challenge for theory.
Received 22December 2005; accepted 2February 2006; published 26February 2006. References
1. Prange, R. E. &Girvin, S. M. The Quantum Hall E ?ect (Springer,New Y ork, 1990).
2. Macdonald, A. H. Quantum Hall E ?ect:A Perspective (KluwerAcademic, Dordrecht, 1990). 3. Novoselov, K. S. et al . Two-dimensional gas of massless Dirac fermions in graphene. Nature 438,
197–200(2005).
4. Zhang, Y., Tan, J. W., Stormer, H. L. &Kim, P . Experimental observation of the quantum Hall e ?ect
and Berry’sphase in graphene. Nature 438, 201–204(2005).
5. McClure, J. W. Diamagnetism of graphite. Phys. Rev. 104, 666–671(1956).
6. Haldane, F. D. M. Model for a quantum Hall e ?ect without Landau levels:Condensed-matter
realization of the ‘parityanomaly’. Phys. Rev. Lett. 61, 2015–2018(1988).
7. Zheng, Y. &Ando, T. Hall conductivity of a two-dimensional graphite system. Phys. Rev. B 65,
245420(2002).
8. Gusynin, V . P . &Sharapov, S. G. Unconventional integer quantum Hall e ?ect in graphene. Phys. Rev.
Lett. 95, 146801(2005).
9. Peres, N. M. R., Guinea, F. &Castro Neto, A. H. Electronic properties of two-dimensional carbon.
Preprint at 10. Novoselov, K. S. et al . Electric ?elde ?ect in atomically thin carbon ?lms.Science 306, 666–669(2004).11. Novoselov, K. S. et al . Two dimensional atomic crystals. Proc. Natl Acad. Sci. USA 102, 10451–10453(2005). 12. Wallace, P . R. The band theory of graphite. Phys. Rev. 71, 622–634(1947). 13. Dresselhaus, M. S. &Dresselhaus, G. Intercalation compounds of graphite. Adv. Phys. 51, 1–186(2002).14. Trickey, S. B., M¨u ller-Plathe, F., Diercksen, G. H. F. &Boettger, J. C. Interplanar binding and lattice relaxation in a graphite delayer. Phys. Rev. B 45, 4460–4468(1992). 15. McCann, E. &Falko, V. I. Landau level degeneracy and quantum Hall e ?ect in a graphite bilayer. Preprint at 16. Berry, M. V . Quantal phase factor accompanying adiabatic change. Proc. R. Soc. Lond. A 392, 45–57(1984). Acknowledgements We thank the High Field Magnet Laboratory (Nijmegen)for their hospitality. U.Z. and K.S.N. were partially supported by EuroMagNET of the 6th Framework ‘Structuringthe European Research Area, Research Infrastructures Action’and by the Leverhulme Trust. S.V .M. acknowledges support from the Russian Academy of Sciences. This research was funded by the EPSRC (UK).Correspondence and requests for materials should be addressed to A.K.G. Supplementary Information accompanies this paper on www.nature.com/naturephysics. Competing ?nancialinterests The authors declare that they have no competing ?nancialinterests. Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/ 4nature physics ADV ANCE ONLINE PUBLICATION www.nature.com/naturephysics 前沿评述 铁磁石墨烯体系的 C T 不变量子自旋霍尔效应 1 ,2 ,?1 谢心澄孙庆丰 ( )1 中国科学院物理研究所 北京 100190 ( )2 美国俄克拉荷马州立大学物理系 俄克拉荷马州 74078 摘 要 文章作者在垂直磁场作用下的铁磁石墨烯体系里预言了一种新类型的量子自旋霍尔效应 . 这量子自旋霍 ( 尔效应与自旋轨道耦合无关 ,体系也不具有时间反演不变性 ;但是有 C T 不变 C 为电子 - 空穴变换 、T 为时间反演变 ) 换. 由于量子自旋霍尔效应 ,体系的纵向电阻和自旋霍尔阻出现量子化平台 . 特别是 ,自旋霍尔阻的量子化平台有很 强的抗杂质干扰能力 . ,拓扑绝缘体 量子自旋霍尔效应 关键词 CT invariant quantum spin Hall eff ect in f erromagnetic gra phene 11 ,2 ,?SU N Qi ng2Feng XI E Xi n2Cheng ( )1 I nst i t ute o f Ph y sics , Chi nese A ca de m y o f S ciences , B ei j i n g 100190 , Chi na ( )2 De p a rt ment o f P h y si cs , O k l ahom a S t ate U ni ve rsi t y , S t i l l w ate r , Ok l a hom a 74078 , U S A ( ) Abstract We p redict a qua nt um spi n Hall eff ect Q S H Ei n f er ro magnetic grap hene under a magnetic fiel d . U nli ke t he p revio usly repo rt ed eff ect , t hi s Q S H E app ear s i n t he absence of any spi n2o r bit i nt eractio n , ( ) a nd it s syst e m does not po sse ss ti me2rever sal Ti nvariance but ha s C T i nvariance , where C i s t he char ge co nj ugatio n op eratio n . Due to t hi s Q S H E , t he lo ngit udi nal a nd spi n Hall resi st a nces exhi bit quant um plat2 eaus . In particular , t he plat eau of t he spi n Hall re si st a nce i s ver y ro bust agai nst di so r der . Key words quant um spi n Hall eff ect , topolo gical i nsulato r 近几年来 ,一种全新的量子物质态 ———拓扑绝 人们最早发现的具有拓扑序的体系是量子霍尔 [ 1 ,2 ] 体系. 在这体系 ,霍尔电阻出现量子化平台 ;特别是 , 缘体 ,已经蓬勃兴起. 拓扑绝缘体有很迷人的特 由于它的拓扑不变性 ,这量子化电阻平台不会被外 异性质 ,这吸引了很多科学家们的注意 . 拓扑绝缘体 界各种干扰所影响 ,它能非常精确地保持它的量子 与传统的普通绝缘体有本质的不同 . 它的体材料是 有能隙的绝缘体 ,但其表面或边缘是无能隙的金属 化平台值 . 最近人们在一些有自旋轨道耦合的材料 [ 3 —6 ] 态 . 这种金属边界态的存在完全由材料的体能带的 里发现另一类拓扑绝缘体. 这类拓扑绝缘体是 () 全局性质 拓扑序所决定. 由于这全局拓扑序 ,如果 由内在的自旋轨道耦合结合特 殊能 带结 构 所引 起 我们把一块拓扑绝缘体材料劈开 ; 那劈开面在劈开 的 ,是材料的固有性质 ,而不像量子霍尔效应需要外 前是材料的体内 ,它是绝缘的 . 但劈开后变成材料的 加磁场. 这类拓扑绝缘体既可以是二维体系也可以 [ 3 —6 ] 表面时 ,它也自动的转变为金属态. 这种性质是以往 是三维体系. 对于二维拓扑绝缘体 ,自旋相反的 普通绝缘体和金属所不具有的 . 更重要的是 ,这种金 边缘态电子运动沿相反的方向 ,出现量子自旋霍尔 属边界态受时间反演不变所保护 ,并且它有很强的 抗外界各种干扰能力 . 即使体系中存在一定的缺陷 、 杂质 、退相干等 ,边界态还能很好保持. 2010 - 05 - 04 收到 ? 通讯联系人. Email : xi ncheng. xie @o kst at e . edu + (效应 即加纵向电压将引起横向自旋流 ,且自旋霍尔 其中 aσ和 a 是 在格 点 i 上的 电子 消灭 和 产生 算i ,i ,σ ) (σε阻是 量 子 化 的. 在 实 验 方 面 , 最 近 人 们 在 Cd Te/ 符 , 是自旋指标 , 0 是同位置能量 也就是在 M = 0 ) H g Te/ Cd Te 量子阱体系已观察到量子自旋霍尔效 时的狄拉克 点 能 量, t = 2 . 75eV 是 最 近 邻 的 跃 迁[ 7 ] 应. 再者 ,通过角分辨光电子谱测量也已确定一些 φ是由垂直磁场引起的磁通 . 能 , M 是铁磁交换能 , ij [ 8 ] () 三维材料具有拓扑序. 我们考虑整个体系 包括中心区和六根导线都是铁 [ 9 ] 磁石墨烯构成和 加有 磁 场. 用 L a nda ue r - B ut ti ke r 近几年 ,另一个被广泛研究的课题是石墨烯. 公式并结合非平衡格林函数方法 ,每个端口的电流 所谓石墨烯 ,即由单层碳原子组成六角蜂窝状结构 () 能被计算 具体计算公式见参考文献 [ 10 ]. 然后我 的体系. 石墨烯有独特的能带结构 :在费米面附近的 们考虑如下的端口条件 : 在纵向端口 1 和 4 之间加 色散关系是线性的. 由于这独特的能带结构 ,使得石 () 很小的电压 V ,4 个横向端 2 、3 、5 和 6作为电压探 测端和它们的电流是 0 . 从它们端电流为 0 ,可计算 墨烯有很多新奇性质 : 例如它的低能准粒子服从狄 出端电压 V 、V 、V 和 V . 最后 ,纵向电阻是 R2 3 5 6 14 ,23 拉克方程 、具有类似于相对论粒子的性质 ;它的霍尔 ( ) ) ( ?V - V / I= V - V / I和 霍 尔 电 阻 是2 3 14 6 5 14 电导平台在半整数位置 ;等等 . () ) ( R?V - V / I= V- V/ I.14 ,26 2 6 14 3 5 14 最近 ,我们在铁磁石墨烯体系中预言了一种新 [ 10 ] 类型的拓扑绝缘体和量子自旋霍尔效应. 与原来 的由自旋轨道耦合所引起的拓扑绝缘体比较 ,本文 的拓扑绝缘体与自旋轨道耦合无关 ,体系也不具有 ( 时间反演不变性. 但具有 C T 不变性 C 为电子 - 空 ) 穴变换 , T 为时间反演变换. 再者 ,在平衡时 ,原来 拓扑绝缘体的边态只携带持续自旋流. 而本文的边 态携带持续自旋流和电流. 在加电压时 ,我们体系可 以同时拥有量子自旋霍尔效应和量子霍尔效应. 所 以这是一种新类型的拓扑绝缘体 . 首先让我们设想一种系统 ,它的载流子有以下 () () 三点性质 : 1载流子既有电子又有空穴 ; 2电子和 () 空穴都是完全自旋极化的 ; 3并且电子和空穴的自 旋极化方向正好完全相反. 然后给这系统加上垂直 外磁场 ,电子携带着向上自旋沿样品边缘的顺时针 方向运动 ,但自旋向下的空穴沿逆时针方向运动 . 所 以这系统将出现量子自旋霍尔效应和具有全局的拓 扑序. 尽管绝大多数体系不能同时满足上面提到的 三点要求 . 幸运的 ,铁磁石墨烯对这三点要求恰好都 满足. 在铁磁石墨烯 ,自旋向上和自旋向下的狄拉克 ( ) 点是分开的. 如图 1 a所示 ,把体系的费米面调节 到两狄拉克点之间时 ,体系的载流子有自旋向上的 ( ) ( ) 图 1 a铁磁石墨烯的能带图 ; b六端铁磁石墨烯装置示意 电子和自旋向下的空穴 ,正好满足上面三点要求 . 最 ( ) () ε图 ; c和 d显示电阻 R和 R对能量和磁交换 M 的 14 ,23 14 ,26 0 近 ,几个理论工作已建议一些方法来实现铁磁石墨 关系. 每六角格子的磁通是 0 . 005 和体系宽度 N = 80 烯 [ 11 ] ,例如把石墨烯生长在铁磁绝缘体材料上 ,通 [ 11 ] 过邻近效应可以在石墨烯上诱导出磁距. 再者 , () ( ) 图 1 c和 1 d显示纵向电阻 R14 ,23 和霍尔电阻[ 12 ] 在实验上 ,也已成功地向石墨烯注入自旋. εR对能量和磁交换 M 的关系. 因为体系存在 14 ,26 0 下面我们开始研究铁磁石墨烯体系的新型量子上面提到的量子自旋霍尔效应以及也可能有量子霍 自旋霍尔效应和拓扑序. 我们主要研究电阻的量子 尔效应 ,所以纵向和霍尔电阻 R和 R都呈现 14 ,23 14 ,26 ( ) 化平台. 考虑如图 1 b所示的六端体系 ,它的哈密 νν平台结构 . 当自旋向上和自旋向下的填充因子和 ??顿量是 : ε固定时 ,电阻 R和 R不会随和 M 而变14 ,23 14 ,26 0 φ+ i + ij (εσ) H = ρ - M aa σ -ρ tea a σ i ,σ i ,σ 0i ,j ,i ,σ 〈i , j〉,σ ( ( ) ) ν端体系 如 图 1 b 但 去 掉 端 口 3 和 5 和 同 位 置 化 ,所以 R和 R的平台值仅仅与填充因子 14 ,23 14 ,26 ? A nde r so n无序. 无序只存在于中间区域 . 由于无序 , ν和有关 . 考虑体系的载流子是沿着边态传输和结 ?σσεε中间区的同位置能- M 变为- M + W ,其中 0 0 i 合 L a nda uer - B ut ti ke r 公式 ,能分析地获得电阻平 W 随机均匀地分布在区间 [ - W / 2 , W / 2 ] 和 W 是 i 2 台值 : R= 0 和 Rνν) ( 14 ,23 14 ,26= [ 1/ +] h/ e, 当? ? 无 序 强 度. 图 2 显 示 自 旋 霍 尔 阻 R[ R? S 3 (νν) ( ) ( νν) , - 时 ; R= [ | |? ,?= + , + 或 -14 , 23 ?? ( ) ( ) εV - V / I= - V - V / I] 与能量2 ? 2 ?14 4 ? 4 ?14 0 3 3 2 以及无序强度 W 的关系. 结果显示自旋霍尔阻 RS ) ν ν ( R | +| | ] h/ e和= / | ?14 , 26? 2 23 也有平台 . 特别是 ,这平台有很强的抗无序能力. 即 ) ( (ν) ( ν | / | si gn [ | | νν| ?? - | + ??使无序强度 W 到 2 t ,自旋霍尔阻的平台还能保持 . 3 2 ν) (νν) ( ) ( ) | | ] h/ e,当 ,= + , - 或 - , + 时 .? ? ?由于这平台有如此好的稳定性 ,它的平台值可以作 在图 1 中 ,一些较小的填充因子的电阻平台值已被 为自旋霍尔阻的标准值 . 再者 ,即使无序增大到非常 ( ) 标出. 下面我们分区域再深入讨论这一体系 : 1在 大或者说趋向无穷大 ,这时尽管自旋霍尔阻的平台 (νν) ( ( ) ) ,= + , +或 - , - 区域 ,自旋向上和自 ? ?可能被破坏 ,但是自旋霍尔阻依然保持在一定的有 旋向下的载流子同为电子型或空穴型 ,这时体系只 限值. 也就是说 ,尽管量子自旋霍尔效应可能被很强 有量子霍尔效应 ,所以纵向电阻是 0 ,而霍尔电阻呈 的无序所破坏 ,但自旋霍尔效应始终存在 . ν() νν现平台结构 . 2在= - ?区域 ,自旋向上和 ? ? 自 总之 ,本文中我们预言了在铁磁石墨烯体系中 旋向下的载流子分别为电子型和空穴型 ,并且它 们 的边态数目相同 ,这时体系具有 C T 不变性 ,以及 出一种新类型的量子自旋霍尔效应. 这量子自旋霍尔 现量子自旋霍尔效应 ,但没有量子霍尔效应 ,从而 效应与自旋轨道耦合无关 . 另外 ,除了量子自旋霍尔 2 (ν) 霍尔电阻是 0 ,纵向电阻是 1/ 2h/ e. 注意 ,尽管现 效应之外 ,该体系也可能有量子霍尔效应 ,从而纵向 在出现的物理现象与由自旋轨道耦合所引起的量子 电阻和霍尔电阻都呈现平台结构 . 特别是 ,由于体系 自旋霍尔效应的现象完全相同 ,但是它们物理机制 受 C T 不变所保护 ,自旋霍尔阻的量子化平台有很 强的抗杂质干扰能力 . (νν)( ) ( ) ( = + , - 或 - , 和起因是不同的 . 3在 ? ,? ) ( ) νν + 区域 但? ?? , 电子 和 空穴 的边 态数 目 不 等 ,这时量子自旋霍尔效应和量子霍尔效应共同存 参考文献 在 ,两个电阻 R和 R都是非 0 .14 ,23 14 ,26 Moo re J M. Nat ure , 2010 , 464 : 104 [ 1 ] Day C. Physics To day , 2008 , 61 : 19 ; Nagao sa N . Science , [ 2 ] 2007 , 318 :758 Ka ne C L , Mele E J . Phys. Rev. L et t . , 2005 , 95 : 146802 [ 3 ] Ka ne C L , Mele E J . Phys. Rev. L et t . , 2005 , 95 : 226801 [ 4 ] Ber nevig B A , Hughes T L , Zha ng S C. Science , 2006 , 314 : [ 5 ] 1757 Fu L , Ka ne C L , Mele E J . Phys. Rev. L et t . , 2007 , 98 : [ 6 ] 106803 ; Zha ng H et al . Nat . Phys. , 2009 , 5 : 438 Ko nig M et al . Science , 2007 , 318 : 766 [ 7 ] ( ) ( ) ε图 2 a和 b分别显示自旋霍尔阻 R 与能量和无序强度 H sie h D et al . Nat ure , 2008 , 452 : 970 ; 2009 , 460 : 1101 S 0 [ 8 ] W 的关系. 体 系 参 数 是 : 每 六 角 上 的 磁 通 为 0 . 007 , 体 系 宽 度 Beena kker C W. J . Rev. Mo d . Phys. , 2008 , 80 : 1337 ; Ca s2 [ 9 ] N = 40和 M = 0 . 05t t ro Neto A H et al . Rev. Mo d . Phys. , 2009 , 81 :109 Sun Q F , Xie X C. Phys. Rev. L et t . , 2010 ,104 : 066805 [ 10 ] Ha ugen H et al . Phys. Rev. B , 2008 , 77 : 115406 [ 11 ] 下面我们研究这 C T 不变的量子自旋霍尔效应 So n Y W et al . Nat ure , 2006 , 444 : 347 [ 12 ] ( ) 的 强壮性 既抗杂质干扰的能力. 我们考虑一个四 前沿评述 铁磁石墨烯体系的CT不变量子自旋霍尔效应 孙庆丰
转载请注明出处范文大全网 » 石墨烯非传统量子霍尔效应范文二:铁磁石墨烯体系的CT不变量子自旋霍尔效应
范文三:铁磁石墨烯体系的CT不变量子自旋霍尔效应_孙庆丰(1)