(2+1)-维Calogero-Bogoyavlenskii-Schiff方程的对称约化及其新的类孤子解

固升维生素k2腰酸背痛腿抽筋、身材变矮、驼背、容易骨折中老年人别让单纯补钙耽误骨质疏松“人老骨头松”，你还在忍受骨质疏松的困扰吗？“人老了，骨质就松了”似乎是正常生理情况，长久以来，并不为人们所重视，但在某天也许你会吃惊的发现，自己比年轻时矮了几厘米，并逐渐出现腰背酸痛、腿抽筋，有时轻微外力甚至咳嗽就会骨折，这时你大概患了骨质疏松症。

特别需要提醒的是老年人，当发现背开始慢慢变驼、腰慢慢变弯曲，腿脚不如从前灵便，常常腰酸背疼时，不要以为这些只是老年的现象，还有当心骨质疏松。

在中国，骨质疏松症被列为老年人五大疾病之一，发病总人数逾1.75亿。

其危害在老年人身上表现得更严重。

女性在40岁后，男性在60岁后，进入骨量快速丢失期，使骨质变薄造成中空孔隙现象，这群人骨折或者其他严重骨病的患者几率可能高于正常人。

（以上内容摘自《院士告诉你骨质疏松症防与治》、《中国骨质疏松杂志》2009年09期、搜狐健康医疗常识）光靠补钙不能有效防止骨质疏松“很多人认为骨质疏松就是缺钙，补钙就行了”，这种认识是不科学的。

缺钙与骨质疏松有关，但这是原因里面的一部分，不能把缺钙和骨质疏松完全同等起来。

不正确的补钙会耽误治疗时机，影响生活质量。

如有些人钙没有少补，但身体只吸收了很小一部分，补进去的钙大部分通过肠道排出，而且还造成大便干燥。

还有的人补钙后钙吸收的很好，但钙并没有像预期地那样进入骨头来预防骨质疏松等疾病，而是从肾脏、胆囊排出，这样会导致个别老人补钙后出现泌尿感染甚至肾结石。

（摘自搜狐健康医疗知识）你骨头里到底缺什么？骨质是由维生素k2和优质蛋白先形成骨元蛋白，再与钙生成骨质，蓄积骨中，增加骨密度这三大要素缺一不可。

现代人们的饮食中，缺乏的不是钙，而是维生素k2这种活性物质。

（摘自百度百科）调查表明，与摄取量高的人相比，每天维生素K摄入量低于109微克的人，发生髋骨骨质的危险明显增加。

而绝经妇女每天补充1毫克维生素K，能使骨钙素的浓度迅速增加，使骨骼损失减少40%。

metal-organic compoundsm328#2003International Union of CrystallographyDOI:10.1107/S0108270103011703Acta Cryst.(2003).C 59,m328±m330Ferrocene compounds.XXXIX.11-FerrocenylisochromaneMario Cetina,a Senka akovicÂ,b Vladimir Rapic Âb *and Amalija GolobicÏc aFaculty of Textile Technology,University of Zagreb,Pierottijeva 6,HR-10000Zagreb,Croatia,b Laboratory of Organic Chemistry,University of Zagreb,Faculty of Food Technology and Biotechnology,Pierottijeva 6,HR-10000Zagreb,Croatia,and cLaboratory of Inorganic Chemistry,Faculty of Chemistry and Chemical Technology,University of Ljubljana,PO Box 537,SI-1001Ljubljana,Slovenia Correspondence e-mail:*************Received 14April 2003Accepted 27May 2003Online 22July 2003In the title compound,[Fe(C 5H 5)(C 14H 13O)],the plane of the heterocyclic ring is almost perpendicular to the plane of the substituted cyclopentadienyl ring,and the heterocyclic ring adopts a half-chair conformation.The conformation of the nearly parallel cyclopentadienyl (Cp)rings [the dihedral angle between their planes is 2.7(1) ]is almost halfway between eclipsed and staggered,and the rings are mutually twisted by about 19.4(2) (mean value).The mean lengths of the CÐC bonds in the substituted and unsubstituted cyclopentadienylring are 1.420(2)and 1.406(3)AÊ,respectively,and the FeÐC distances range from 2.029(2)to 2.051(2)AÊ.The phenyl and unsubstituted cyclopentadienyl rings are involved in CÐH ÁÁÁ%interactions,with intermolecular H ÁÁÁcentroiddistances of 2.85and 3.14AÊfor CÐH ÁÁÁ%(Ph),and 2.88A Êfor CÐH ÁÁÁ%(Cp).In two of these interactions,the CÐH bond points towards one of the ring bonds rather than towards the ring centroid.In the crystal structure,the CÐH ÁÁÁ%interactions connect the molecules into a three-dimensional framework.CommentOptically active ferrocene derivatives are widely employed as chiral ligands in asymmetric reactions,and there is continuing interest in the development of ef®cient procedures for the preparation of these derivatives in enantiopure forms (Gonsalves &Chen,1995;Bolm et al.,1998;Pioda &Togni,1998;Perea et al.,1999).Ferrocene derivatives exhibiting centro-and planar chirality are very convenient substrates forbiotransformations (KoÈllner et al.,1998;Richards &Locke,1998;Schwink &Knochel,1998;Patti &Nicolosi,1999;akovicÂet al.,2003).In the course of our research on enzyme-catalyzed resolution of centrochiral ferrocene compounds,racemic 2-( -hydroxyferrocenyl)benzenethanol and 1-ferro-cenylisochromane,(I),were prepared by reduction of methyl2-(ferrocenoyl)benzeneacetate ( akovicÂ,2000).The molecular structure of (I)is the ®rst reported structure to contain an isochromanyl group attached to the ferrocenyl moiety (Fig.1).Moreover,the Cambridge Structural Database (Allen,2002)lists only three structures containing an iso-chromanyl group at all (Yamato et al.,1984;Unterhalt et al.,1994;Eikawa et al.,1999).The heterocyclic six-membered ringActa Crystallographica Section CCrystal Structure CommunicationsISSN0108-2701Figure 2Part of the crystal structure of (I),showing the formation of (010)sheets built from C14ÐH14A ÁÁÁCg 1i and C18ÐH18ÁÁÁCg 2ii interactions (Cg 1and Cg 2are the centroids of rings C12/C13/C16±C19and C6±C10,respectively).CÐH ÁÁÁ%interactions are indicated by dashed lines.[Symmetry codes:(i)x ,12Ày ,z À12;(ii)1+x ,y ,1+z.]Figure 1A view of (I),with the atom-numbering scheme.Displacement ellipsoids for non-H atoms are drawn at the 20%probability level.1Part XXXVIII:Cetina et al.(2003).adopts a distorted half-chair conformation,in which atoms O1and C15are 0.426(1)and À0.348(2)AÊfrom the plane of the other ring atoms (C11±C14);the C11ÐC12ÐC13ÐC14torsion angle is 2.4(2) .The bond lengths in the heterocyclic and fused phenyl rings (Table 1)mostly agree with the equivalent bond lengths in the structures of 1,1H -oxybis(iso-chromane)(Eikawa et al.,1999)and (S )-1-(phenyl)ethyl-ammonium (S )-isochromane-1-carboxylate (Unterhalt et al.,1994).The exception is the C12ÐC13bond,which is shorter($0.04AÊ)in the latter structure.Heterocyclic ring atoms O1,C11and C15and cyclopentadienyl (Cp)ring atom C1lie in the same plane,the C15ÐO1ÐC11ÐC1torsion angle being 178.95(14) .The dihedral angle between the mean plane of these four atoms and the C1±C5Cp ring is 47.8(1) .Furthermore,the plane of the heterocyclic ring is almost perpendicular to the plane of the C1±C5ring and is parallel to the plane of the fused phenyl ring.The corresponding dihedral angles are 87.3(1)and 4.0(1) .The exocyclic C2ÐC1ÐC11bond angle is larger than the C5ÐC1ÐC11angle (Table 1).The Cp rings are planar and almost parallel to each other [the dihedral angle between their planes is 2.7(1) ],and the FeÐC distances are in the range2.034(2)±2.051(2)AÊfor the substituted (C1±C5)and 2.029(2)±2.049(2)AÊfor the unsubstituted (C6±C10)ring,the average values being 2.042(2)and 2.038(2)AÊ,respectively.The CÐC bonds are slightly longer in the substituted ring than in the unsubstituted ring [1.410(3)±1.429(2)versus 1.397(3)±1.414(3)AÊ],and the bond angles in both rings range from 107.52(15)to 108.33(18) .The geometry of the ferrocenyl moiety agrees well with the structures of ferrocene (Seiler &Dunitz,1979)and of theferrocene derivatives we have reported previously (Cetina et al.,2002,2003).The main conformational difference was observed in the orientation of the Cp rings.In (I),the rings are twisted from an eclipsed conformation by 19.4(2) (mean value).The values of the corresponding CÐCg 3ÐCg 2ÐC pseudo-torsion angles (Cg 3and Cg 2are the centroids of the C1±C5and C6±C10rings,respectively),de®ned by joining two eclipsing Cp C atoms through the ring centroids,range from 19.0(2)to 19.7(2) .The conformation is almost exactly halfway between eclipsed and staggered,as demonstrated by the C1ÐCg 3ÐCg 2ÐC9torsion angle of 163.4(1) .This angle would be 180 for a staggered conformation and 144 for a fully eclipsed conformation.The centroids of the Cp rings are almost equidistant from the Fe atom;the FeÐCg 3and FeÐCg 2distances are 1.647(1)and 1.650(1)AÊ,respectively,while the Cg 3ÐFeÐCg 2angle is 178.2(1) .There are a number of CÐH ÁÁÁ%interactions (Table 2and Fig.2).Atom H14A of the heterocyclic ring is positioned almost perpendicularly above the phenyl-ring centroid (Cg 1)of the adjacent molecule.The six relevant H ÁÁÁC distances fallin the narrow range 3.06±3.28AÊ,and the H ÁÁÁCg i distance is signi®cantly shorter than any of the H ÁÁÁC distances[symmetry code:(i)x ,12Ày ,z À12;Table 2].The CÐH ÁÁÁ%interaction between phenyl atom H18and the unsubstituted Cp ring exhibits a completely different geometry.The H18ÁÁÁC7ii distance is shorter than the H ÁÁÁCg ii distance [symmetry code:(ii)1+x ,y ,1+z ].The second shortest H ÁÁÁC contact is that to atom C6,and the CÐH bond points towards the C6ÐC7bond of the Cp ring rather than towards the ring centroid (Cg 2).Similarly,the longest interaction,C5ÐH5ÁÁÁCg 1iii [symmetry code:(iii)1Àx ,Ày ,2Àz ],points towards the C12ÐC13bond.Both the H5ÁÁÁC12iii and the H5ÁÁÁC13iii contacts are shorter than the H ÁÁÁCg iii distance.The molecules linked by these CÐH ÁÁÁ%interactions build a three-dimensional framework (Fig.3).ExperimentalNaBH 4(253mg,6.7mmol)was added gradually to a solution of methyl 2-(ferrocenoyl)benzeneacetate (326mg,0.9mmol)in a mixture of EtOH and Et 2O (1:1v /v ;5ml).The mixture was re¯uxed for 2h and worked up in the usual manner.Separation by preparative thin-layer chromatography on silica gel (Merck,Kieselgel 60HF 254)yielded 2-( -hydroxyferrocenyl)benzeneethanol (237mg;yield 78%)and orange crystals of 1-ferrocenylisochromane (57mg;yield 20%;m.p.365±366K).Single crystals of the title compound were obtained by slow evaporation from a cyclohexane solution at room tempera-ture.IR (CH 2Cl 2,cm À1):)3081(w )and 3020(w )(CÐH,ferrocene),2942(m )(CÐH,aliphatic),1278(m )(CÐOÐC);1H NMR (DMSO,p.p.m.): 7.18(d ,1H,H16),7.12(d ,1H,H17),7.14(d ,1H,H18),7.16(d ,1H,H19),4.23(s ,5H,unsubstituted ferrocene ring),4.13±4.20(m ,4H,substituted ferrocene ring),3.97(m ,1H,H15A ),3.77(m ,1H,H15B ),2.78(m ,2H,H14),5.58(s ,1H,H11);13C NMR (DMSO,p.p.m.): 137.29(C12),132.91(C13),128.5(C17),126.25(C18),125.98(C16),125.36(C19),90.21(C1),73.63(C11),68.62(unsub-stituted ferrocene ring),68.56±66.31(substituted ferrocene ring),61.55(C15),27.99(C14).Acta Cryst.(2003).C 59,m328±m330Mario Cetina et al.[Fe(C 5H 5)(C 14H 13O)]m329metal-organic compoundsFigure 3Part of the crystal structure of (I),showing the cyclic motif generated by the C5ÐH5ÁÁÁCg 1iii interaction (Cg 1is the centroid of ring C12/C13/C16±C19),which links the (010)sheets into a three-dimensional framework.CÐH ÁÁÁ%interactions are indicated by dashed lines,and the unit-cell box has been omitted for clarity.[Symmetry code:(iii)1Àx ,Ày ,2Àz .]metal-organic compoundsm330Mario Cetina et al.[Fe(C 5H 5)(C 14H 13O)]Acta Cryst.(2003).C 59,m328±m330Crystal data[Fe(C 5H 5)(C 14H 13O)]M r =318.18Monoclinic,P 21a ca =11.5053(2)A Êb =18.5095(3)A Êc =7.1941(1)AÊ =106.933(1)V =1465.62(4)AÊ3Z =4D x =1.442Mg m À3Mo K radiationCell parameters from 3411re¯ections =2.6±27.5 "=1.02mm À1T =293(2)K Prism,orange0.80Â0.40Â0.15mm Data collectionNonius KappaCCD area-detector diffractometer 9and 3scansAbsorption correction:multi-scan (DENZO±SMN ;Otwinowski &Minor,1997)T min =0.630,T max =0.85716885measured re¯ections3334independent re¯ections 2662re¯ections with I >2'(I )R int =0.064 max =27.4 h =À14314k =À23323l =À939Re®nementRe®nement on F 2R [F 2>2'(F 2)]=0.030wR (F 2)=0.076S =1.033334re¯ections 190parametersH-atom parameters constrainedw =1/['2(F 2o )+(0.0341P )2+0.3545P ]where P =(F 2o +2F 2c )/3(Á/')max =0.001Á&max =0.25e A ÊÀ3Á&min =À0.23e AÊÀ3All H atoms were included in calculated positions as riding atoms,with SHELXL 97(Sheldrick,1997)defaults viz.CÐH =0.93AÊfor aromatic H atoms,0.98AÊfor methine H atoms,and 0.97A Êfor methylene H atoms.For all H atoms,the isotropic displacement parameters were set at 1.2times the equivalent anisotropic displacement parameters of the attached non-H atoms.Data collection:COLLECT (Nonius,2000);cell re®nement:DENZO±SMN (Otwinowski &Minor,1997);data reduction:DENZO±SMN ;program(s)used to solve structure:SHELXS 97(Sheldrick,1997);program(s)used to re®ne structure:SHELXL 97(Sheldrick,1997);molecular graphics:PLATON (Spek,2003);soft-ware used to prepare material for publication:SHELXL 97.The re¯ection data were collected at the Faculty of Chem-istry and Chemical Technology,University of Ljubljana,Slovenia.We acknowledge with thanks the ®nancial contri-bution of the Ministry of Education,Science and Sport of the Republic of Slovenia (grant Nos.X-2000and PS-511-103),which made the purchase of the apparatus possible.Supplementary data for this paper are available from the IUCr electronic archives (Reference:GD1251).Services for accessing these data are described at the back of the journal.ReferencesAllen,F.H.(2002).Acta Cryst.B 58,380±388.Bolm,C.,MunÄiz-Ferna Ândez,K.,Seger,A.,Raabe,G.&Gunther,K.(1998).Chem.63,7860±7867.Cetina,M.,Hergold-BrundicÂ,A.,Nagl,A.,Jukic Â,M.&Rapic Â,V .(2003).Struct.Chem.14,289±293.Cetina,M.,MrvosÏ-Sermek,D.,Jukic Â,M.&Rapic Â,V .(2002).Acta Cryst.E 58,m676±m678.akovicÂ,S.(2000).PhD thesis,University of Zagreb,Croatia. akovicÂ,S.,Lapic Â,J.&Rapic Â,V .(2003).Biocatal.Biotransform.In the press.Eikawa,M.,Sakaguchi,S.&Ishii,Y.(1999).Chem.64,4676±4679.Gonsalves,K.E.&Chen,X.(1995).Ferrocenes ,edited by A.Togni &T.Hayashi,ch.10,pp.497±527.Weinheim:VCH.KoÈllner,C.,Pugin,B.&Togni,A.(1998).J.Am.Chem.Soc.120,10274±10275.Nonius (2000).COLLECT.Nonius BV ,Delft,The Netherlands.Otwinowski,Z.&Minor,W.(1997).Methods in Enzymology ,Vol.276,Macromolecular Crystallography ,Part A,edited by C.W.Carter Jr &R.M.Sweet,pp.307±326.New York:Academic Press.Patti,A.&Nicolosi,G.(1999).Tetrahedron :Asymmetry ,10,2651±2654.Perea,J.J.A.,Lotz,M.&Knochel,P .(1999).Tetrahedron :Asymmetry ,10,375±384.Pioda,G.&Togni,A.(1998).Tetrahedron :Asymmetry ,9,3903±3910.Richards,C.J.&Locke,A.(1998).Tetrahedron :Asymmetry ,9,2377±2407.Schwink,L.&Knochel,P .(1998).Chem.Eur.J.4,950±968.Seiler,P .&Dunitz,J.D.(1979).Acta Cryst.B 35,2020±2032.Sheldrick,G.M.(1997).SHELXS 97and SHELXL 97.University ofGoÈttingen,Germany.Spek,A.L.(2003).J.Appl.Cryst.36,7±13.Unterhalt,B.,Krebs,B.,Lage,M.&Nocon,B.(1994).Arch.Pharm.327,799±804.Yamato,M.,Hashigaki,K.,Kokubu,N.&Nakato,Y.(1984).J.Chem.Soc.Perkin Trans.1,pp.1301±1304.Table 2Hydrogen-bonding geometry (AÊ, ).Cg 1and Cg 2are the centroids of rings C12/C13/C16±C19and C6±C10,respectively.D ÐH ÁÁÁA D ÐH H ÁÁÁA D ÁÁÁA D ÐH ÁÁÁA C14ÐH14A ÁÁÁCg 1i 0.97 2.85 3.627(2)138C18ÐH18ÁÁÁCg 2ii 0.93 2.88 3.660(2)142C18ÐH18ÁÁÁC7ii 0.93 2.86 3.760(3)163C18ÐH18ÁÁÁC6ii 0.93 3.03 3.874(3)151C5ÐH5ÁÁÁCg 1iii 0.93 3.14 3.984(2)152C5ÐH5ÁÁÁC12iii 0.93 2.98 3.840(2)154C5ÐH5ÁÁÁC13iii0.933.023.949(2)177Symmetry codes:(i)x Y 12Ày Y z À12;(ii)1 x Y y Y 1 z ;(iii)1Àx Y Ày Y 2Àz .Table 1Selected geometric parameters (AÊ, ).O1ÐC111.4236(19)O1ÐC15 1.432(2)C1ÐC11 1.503(2)C11ÐC12 1.523(2)C12ÐC16 1.388(2)C12ÐC13 1.397(2)C13ÐC19 1.392(3)C13ÐC14 1.504(3)C14ÐC15 1.501(3)C16ÐC17 1.384(3)C17ÐC18 1.380(3)C18ÐC19 1.379(3)C11ÐO1ÐC15110.39(13)C2ÐC1ÐC11127.61(14)C5ÐC1ÐC11124.78(14)O1ÐC11ÐC1109.37(13)O1ÐC11ÐC12111.64(13)C1ÐC11ÐC12112.14(13)C13ÐC12ÐC11119.77(14)C12ÐC13ÐC14120.38(16)C15ÐC14ÐC13111.24(15)O1ÐC15ÐC14110.04(15)。

Two-Dimensional Gas of Massless Dirac Fermions in Graphene K.S. Novoselov1, A.K. Geim1, S.V. Morozov2, D. Jiang1, M.I. Katsnelson3, I.V. Grigorieva1, S.V. Dubonos2, A.A. Firsov21Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester, M13 9PL, UK2Institute for Microelectronics Technology, 142432, Chernogolovka, Russia3Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, the NetherlandsElectronic properties of materials are commonly described by quasiparticles that behave as nonrelativistic electrons with a finite mass and obey the Schrödinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective “speed of light” c∗ ≈106m/s. Our studies of graphene – a single atomic layer of carbon – have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at halfinteger filling factors; b) graphene’s conductivity never falls below a minimum value corresponding to the conductance quantum e2/h, even when carrier concentrations tend to zero; c) the cyclotron mass mc of massless carriers with energy E in graphene is described by equation E =mcc∗2; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of π due to Berry’s phase.Graphene is a monolayer of carbon atoms packed into a dense honeycomb crystal structure that can be viewed as either an individual atomic plane extracted from graphite or unrolled single-wall carbon nanotubes or as a giant flat fullerene molecule. This material was not studied experimentally before and, until recently [1,2], presumed not to exist. To obtain graphene samples, we used the original procedures described in [1], which involve micromechanical cleavage of graphite followed by identification and selection of monolayers using a combination of optical, scanning-electron and atomic-force microscopies. The selected graphene films were further processed into multi-terminal devices such as the one shown in Fig. 1, following standard microfabrication procedures [2]. Despite being only one atom thick and unprotected from the environment, our graphene devices remain stable under ambient conditions and exhibit high mobility of charge carriers. Below we focus on the physics of “ideal” (single-layer) graphene which has a different electronic structure and exhibits properties qualitatively different from those characteristic of either ultra-thin graphite films (which are semimetals and whose material properties were studied recently [2-5]) or even of our other devices consisting of just two layers of graphene (see further). Figure 1 shows the electric field effect [2-4] in graphene. Its conductivity σ increases linearly with increasing gate voltage Vg for both polarities and the Hall effect changes its sign at Vg ≈0. This behaviour shows that substantial concentrations of electrons (holes) are induced by positive (negative) gate voltages. Away from the transition region Vg ≈0, Hall coefficient RH = 1/ne varies as 1/Vg where n is the concentration of electrons or holes and e the electron charge. The linear dependence 1/RH ∝Vg yields n =α·Vg with α ≈7.3·1010cm-2/V, in agreement with the theoretical estimate n/Vg ≈7.2·1010cm-2/V for the surface charge density induced by the field effect (see Fig. 1’s caption). The agreement indicates that all the induced carriers are mobile and there are no trapped charges in graphene. From the linear dependence σ(Vg) we found carrier mobilities µ =σ/ne, whichreached up to 5,000 cm2/Vs for both electrons and holes, were independent of temperature T between 10 and 100K and probably still limited by defects in parent graphite. To characterise graphene further, we studied Shubnikov-de Haas oscillations (SdHO). Figure 2 shows examples of these oscillations for different magnetic fields B, gate voltages and temperatures. Unlike ultra-thin graphite [2], graphene exhibits only one set of SdHO for both electrons and holes. By using standard fan diagrams [2,3], we have determined the fundamental SdHO frequency BF for various Vg. The resulting dependence of BF as a function of n is plotted in Fig. 3a. Both carriers exhibit the same linear dependence BF = β·n with β ≈1.04·10-15 T·m2 (±2%). Theoretically, for any 2D system β is defined only by its degeneracy f so that BF =φ0n/f, where φ0 =4.14·10-15 T·m2 is the flux quantum. Comparison with the experiment yields f =4, in agreement with the double-spin and double-valley degeneracy expected for graphene [6,7] (cf. caption of Fig. 2). Note however an anomalous feature of SdHO in graphene, which is their phase. In contrast to conventional metals, graphene’s longitudinal resistance ρxx(B) exhibits maxima rather than minima at integer values of the Landau filling factor ν (Fig. 2a). Fig. 3b emphasizes this fact by comparing the phase of SdHO in graphene with that in a thin graphite film [2]. The origin of the “odd” phase is explained below. Another unusual feature of 2D transport in graphene clearly reveals itself in the T-dependence of SdHO (Fig. 2b). Indeed, with increasing T the oscillations at high Vg (high n) decay more rapidly. One can see that the last oscillation (Vg ≈100V) becomes practically invisible already at 80K whereas the first one (Vg <10V) clearly survives at 140K and, in fact, remains notable even at room temperature. To quantify this behaviour we measured the T-dependence of SdHO’s amplitude at various gate voltages and magnetic fields. The results could be fitted accurately (Fig. 3c) by the standard expression T/sinh(2π2kBTmc/heB), which yielded mc varying between ≈ 0.02 and 0.07m0 (m0 is the free electron mass). Changes in mc are well described by a square-root dependence mc ∝n1/2 (Fig. 3d). To explain the observed behaviour of mc, we refer to the semiclassical expressions BF = (h/2πe)S(E) and mc =(h2/2π)∂S(E)/∂E where S(E) =πk2 is the area in k-space of the orbits at the Fermi energy E(k) [8]. Combining these expressions with the experimentally-found dependences mc ∝n1/2 and BF =(h/4e)n it is straightforward to show that S must be proportional to E2 which yields E ∝k. Hence, the data in Fig. 3 unambiguously prove the linear dispersion E =hkc∗ for both electrons and holes with a common origin at E =0 [6,7]. Furthermore, the above equations also imply mc =E/c∗2 =(h2n/4πc∗2)1/2 and the best fit to our data yields c∗ ≈1⋅106 m/s, in agreement with band structure calculations [6,7]. The employed semiclassical model is fully justified by a recent theory for graphene [9], which shows that SdHO’s amplitude can indeed be described by the above expression T/sinh(2π2kBTmc/heB) with mc =E/c∗2. Note that, even though the linear spectrum of fermions in graphene (Fig. 3e) implies zero rest mass, their cyclotron mass is not zero. The unusual response of massless fermions to magnetic field is highlighted further by their behaviour in the high-field limit where SdHO evolve into the quantum Hall effect (QHE). Figure 4 shows Hall conductivity σxy of graphene plotted as a function of electron and hole concentrations in a constant field B. Pronounced QHE plateaux are clearly seen but, surprisingly, they do not occur in the expected sequence σxy =(4e2/h)N where N is integer. On the contrary, the plateaux correspond to half-integer ν so that the first plateau occurs at 2e2/h and the sequence is (4e2/h)(N + ½). Note that the transition from the lowest hole (ν =–½) to lowest electron (ν =+½) Landau level (LL) in graphene requires the same number of carriers (∆n =4B/φ0 ≈1.2·1012cm-2) as the transition between other nearest levels (cf. distances between minima in ρxx). This results in a ladder of equidistant steps in σxy which are not interrupted when passing through zero. To emphasize this highly unusual behaviour, Fig. 4 also shows σxy for a graphite film consisting of only two graphene layers where the sequence of plateaux returns to normal and the first plateau is at 4e2/h, as in the conventional QHE. We attribute this qualitative transition between graphene and its two-layer counterpart to the fact that fermions in the latter exhibit a finite mass near n ≈0 (as found experimentally; to be published elsewhere) and can no longer be described as massless Dirac particles. 2The half-integer QHE in graphene has recently been suggested by two theory groups [10,11], stimulated by our work on thin graphite films [2] but unaware of the present experiment. The effect is single-particle and intimately related to subtle properties of massless Dirac fermions, in particular, to the existence of both electron- and hole-like Landau states at exactly zero energy [912]. The latter can be viewed as a direct consequence of the Atiyah-Singer index theorem that plays an important role in quantum field theory and the theory of superstrings [13,14]. For the case of 2D massless Dirac fermions, the theorem guarantees the existence of Landau states at E=0 by relating the difference in the number of such states with opposite chiralities to the total flux through the system (note that magnetic field can also be inhomogeneous). To explain the half-integer QHE qualitatively, we invoke the formal expression [9-12] for the energy of massless relativistic fermions in quantized fields, EN =[2ehc∗2B(N +½ ±½)]1/2. In QED, sign ± describes two spins whereas in the case of graphene it refers to “pseudospins”. The latter have nothing to do with the real spin but are “built in” the Dirac-like spectrum of graphene, and their origin can be traced to the presence of two carbon sublattices. The above formula shows that the lowest LL (N =0) appears at E =0 (in agreement with the index theorem) and accommodates fermions with only one (minus) projection of the pseudospin. All other levels N ≥1 are occupied by fermions with both (±) pseudospins. This implies that for N =0 the degeneracy is half of that for any other N. Alternatively, one can say that all LL have the same “compound” degeneracy but zeroenergy LL is shared equally by electrons and holes. As a result the first Hall plateau occurs at half the normal filling and, oddly, both ν = –½ and +½ correspond to the same LL (N =0). All other levels have normal degeneracy 4B/φ0 and, therefore, remain shifted by the same ½ from the standard sequence. This explains the QHE at ν =N + ½ and, at the same time, the “odd” phase of SdHO (minima in ρxx correspond to plateaux in ρxy and, hence, occur at half-integer ν; see Figs. 2&3), in agreement with theory [9-12]. Note however that from another perspective the phase shift can be viewed as the direct manifestation of Berry’s phase acquired by Dirac fermions moving in magnetic field [15,16]. Finally, we return to zero-field behaviour and discuss another feature related to graphene’s relativistic-like spectrum. The spectrum implies vanishing concentrations of both carriers near the Dirac point E =0 (Fig. 3e), which suggests that low-T resistivity of the zero-gap semiconductor should diverge at Vg ≈0. However, neither of our devices showed such behaviour. On the contrary, in the transition region between holes and electrons graphene’s conductivity never falls below a well-defined value, practically independent of T between 4 and 100K. Fig. 1c plots values of the maximum resistivity ρmax(B =0) found in 15 different devices, which within an experimental error of ≈15% all exhibit ρmax ≈6.5kΩ, independent of their mobility that varies by a factor of 10. Given the quadruple degeneracy f, it is obvious to associate ρmax with h/fe2 =6.45kΩ where h/e2 is the resistance quantum. We emphasize that it is the resistivity (or conductivity) rather than resistance (or conductance), which is quantized in graphene (i.e., resistance R measured experimentally was not quantized but scaled in the usual manner as R =ρL/w with changing length L and width w of our devices). Thus, the effect is completely different from the conductance quantization observed previously in quantum transport experiments. However surprising, the minimum conductivity is an intrinsic property of electronic systems described by the Dirac equation [17-20]. It is due to the fact that, in the presence of disorder, localization effects in such systems are strongly suppressed and emerge only at exponentially large length scales. Assuming the absence of localization, the observed minimum conductivity can be explained qualitatively by invoking Mott’s argument [21] that mean-free-path l of charge carriers in a metal can never be shorter that their wavelength λF. Then, σ =neµ can be re-written as σ = (e2/h)kFl and, hence, σ cannot be smaller than ≈e2/h per each type of carriers. This argument is known to have failed for 2D systems with a parabolic spectrum where disorder leads to localization and eventually to insulating behaviour [17,18]. For the case of 2D Dirac fermions, no localization is expected [17-20] and, accordingly, Mott’s argument can be used. Although there is a broad theoretical consensus [18-23,10,11] that a 2D gas of Dirac fermions should exhibit a minimum 3conductivity of about e2/h, this quantization was not expected to be accurate and most theories suggest a value of ≈e2/πh, in disagreement with the experiment. In conclusion, graphene exhibits electronic properties distinctive for a 2D gas of particles described by the Dirac rather than Schrödinger equation. This 2D system is not only interesting in itself but also allows one to access – in a condensed matter experiment – the subtle and rich physics of quantum electrodynamics [24-27] and provides a bench-top setting for studies of phenomena relevant to cosmology and astrophysics [27,28].1. Novoselov, K.S. et al. PNAS 102, 10451 (2005). 2. Novoselov, K.S. et al. Science 306, 666 (2004); cond-mat/0505319. 3. Zhang, Y., Small, J.P., Amori, M.E.S. & Kim, P. Phys. Rev. Lett. 94, 176803 (2005). 4. Berger, C. et al. J. Phys. Chem. B, 108, 19912 (2004). 5. Bunch, J.S., Yaish, Y., Brink, M., Bolotin, K. & McEuen, P.L. Nanoletters 5, 287 (2005). 6. Dresselhaus, M.S. & Dresselhaus, G. Adv. Phys. 51, 1 (2002). 7. Brandt, N.B., Chudinov, S.M. & Ponomarev, Y.G. Semimetals 1: Graphite and Its Compounds (North-Holland, Amsterdam, 1988). 8. Vonsovsky, S.V. and Katsnelson, M.I. Quantum Solid State Physics (Springer, New York, 1989). 9. Gusynin, V.P. & Sharapov, S.G. Phys. Rev. B 71, 125124 (2005). 10. Gusynin, V.P. & Sharapov, S.G. cond-mat/0506575. 11. Peres, N.M.R., Guinea, F. & Castro Neto, A.H. cond-mat/0506709. 12. Zheng, Y. & Ando, T. Phys. Rev. B 65, 245420 (2002). 13. Kaku, M. Introduction to Superstrings (Springer, New York, 1988). 14. Nakahara, M. Geometry, Topology and Physics (IOP Publishing, Bristol, 1990). 15. Mikitik, G. P. & Sharlai, Yu.V. Phys. Rev. Lett. 82, 2147 (1999). 16. Luk’yanchuk, I.A. & Kopelevich, Y. Phys. Rev. Lett. 93, 166402 (2004). 17. Abrahams, E., Anderson, P.W., Licciardello, D.C. & Ramakrishnan, T.V. Phys. Rev. Lett. 42, 673 (1979). 18. Fradkin, E. Phys. Rev. B 33, 3263 (1986). 19. Lee, P.A. Phys. Rev. Lett. 71, 1887 (1993). 20. Ziegler, K. Phys. Rev. Lett. 80, 3113 (1998). 21. Mott, N.F. & Davis, E.A. Electron Processes in Non-Crystalline Materials (Clarendon Press, Oxford, 1979). 22. Morita, Y. & Hatsugai, Y. Phys. Rev. Lett. 79, 3728 (1997). 23. Nersesyan, A.A., Tsvelik, A.M. & Wenger, F. Phys. Rev. Lett. 72, 2628 (1997). 24. Rose, M.E. Relativistic Electron Theory (John Wiley, New York, 1961). 25. Berestetskii, V.B., Lifshitz, E.M. & Pitaevskii, L.P. Relativistic Quantum Theory (Pergamon Press, Oxford, 1971). 26. Lai, D. Rev. Mod. Phys. 73, 629 (2001). 27. Fradkin, E. Field Theories of Condensed Matter Systems (Westview Press, Oxford, 1997). 28. Volovik, G.E. The Universe in a Helium Droplet (Clarendon Press, Oxford, 2003).Acknowledgements This research was supported by the EPSRC (UK). We are most grateful to L. Glazman, V. Falko, S. Sharapov and A. Castro Netto for helpful discussions. K.S.N. was supported by Leverhulme Trust. S.V.M., S.V.D. and A.A.F. acknowledge support from the Russian Academy of Science and INTAS.43µ (m2/Vs)0.8c4P0.4 22 σ (1/kΩ)10K0 0 1/RH(T/kΩ) 1 2ρmax (h/4e2)1-5010 Vg (V) 50 -10ab 0 -100-500 Vg (V)50100Figure 1. Electric field effect in graphene. a, Scanning electron microscope image of one of our experimental devices (width of the central wire is 0.2µm). False colours are chosen to match real colours as seen in an optical microscope for larger areas of the same materials. Changes in graphene’s conductivity σ (main panel) and Hall coefficient RH (b) as a function of gate voltage Vg. σ and RH were measured in magnetic fields B =0 and 2T, respectively. The induced carrier concentrations n are described by [2] n/Vg =ε0ε/te where ε0 and ε are permittivities of free space and SiO2, respectively, and t ≈300 nm is the thickness of SiO2 on top of the Si wafer used as a substrate. RH = 1/ne is inverted to emphasize the linear dependence n ∝Vg. 1/RH diverges at small n because the Hall effect changes its sign around Vg =0 indicating a transition between electrons and holes. Note that the transition region (RH ≈ 0) was often shifted from zero Vg due to chemical doping [2] but annealing of our devices in vacuum normally allowed us to eliminate the shift. The extrapolation of the linear slopes σ(Vg) for electrons and holes results in their intersection at a value of σ indistinguishable from zero. c, Maximum values of resistivity ρ =1/σ (circles) exhibited by devices with different mobilites µ (left y-axis). The histogram (orange background) shows the number P of devices exhibiting ρmax within 10% intervals around the average value of ≈h/4e2. Several of the devices shown were made from 2 or 3 layers of graphene indicating that the quantized minimum conductivity is a robust effect and does not require “ideal” graphene.ρxx (kΩ)0.60 aVg = -60V4B (T)810K12∆σxx (1/kΩ)0.4 1ν=4 140K 80K B =12T0 b 0 25 50 Vg (V) 7520K100Figure 2. Quantum oscillations in graphene. SdHO at constant gate voltage Vg as a function of magnetic field B (a) and at constant B as a function of Vg (b). Because µ does not change much with Vg, the constant-B measurements (at a constant ωcτ =µB) were found more informative. Panel b illustrates that SdHO in graphene are more sensitive to T at high carrier concentrations. The ∆σxx-curves were obtained by subtracting a smooth (nearly linear) increase in σ with increasing Vg and are shifted for clarity. SdHO periodicity ∆Vg in a constant B is determined by the density of states at each Landau level (α∆Vg = fB/φ0) which for the observed periodicity of ≈15.8V at B =12T yields a quadruple degeneracy. Arrows in a indicate integer ν (e.g., ν =4 corresponds to 10.9T) as found from SdHO frequency BF ≈43.5T. Note the absence of any significant contribution of universal conductance fluctuations (see also Fig. 1) and weak localization magnetoresistance, which are normally intrinsic for 2D materials with so high resistivity.75 BF (T) 500.2 0.11/B (1/T)b5 10 N 1/2025 a 0 0.061dmc /m00.04∆0.02 0c0 0 T (K) 150n =0e-6-3036Figure 3. Dirac fermions of graphene. a, Dependence of BF on carrier concentration n (positive n correspond to electrons; negative to holes). b, Examples of fan diagrams used in our analysis [2] to find BF. N is the number associated with different minima of oscillations. Lower and upper curves are for graphene (sample of Fig. 2a) and a 5-nm-thick film of graphite with a similar value of BF, respectively. Note that the curves extrapolate to different origins; namely, to N = ½ and 0. In graphene, curves for all n extrapolate to N = ½ (cf. [2]). This indicates a phase shift of π with respect to the conventional Landau quantization in metals. The shift is due to Berry’s phase [9,15]. c, Examples of the behaviour of SdHO amplitude ∆ (symbols) as a function of T for mc ≈0.069 and 0.023m0; solid curves are best fits. d, Cyclotron mass mc of electrons and holes as a function of their concentration. Symbols are experimental data, solid curves the best fit to theory. e, Electronic spectrum of graphene, as inferred experimentally and in agreement with theory. This is the spectrum of a zero-gap 2D semiconductor that describes massless Dirac fermions with c∗ 300 times less than the speed of light.n (1012 cm-2)σxy (4e2/h)4 3 2 -2 1 -1 -2 -3 2 44Kn7/ 5/ 3/ 1/2 2 2 210 ρxx (kΩ)-4σxy (4e2/h)0-1/2 -3/2 -5/2514T0-7/2 -4 -2 0 2 4 n (1012 cm-2)Figure 4. Quantum Hall effect for massless Dirac fermions. Hall conductivity σxy and longitudinal resistivity ρxx of graphene as a function of their concentration at B =14T. σxy =(4e2/h)ν is calculated from the measured dependences of ρxy(Vg) and ρxx(Vg) as σxy = ρxy/(ρxy + ρxx)2. The behaviour of 1/ρxy is similar but exhibits a discontinuity at Vg ≈0, which is avoided by plotting σxy. Inset: σxy in “two-layer graphene” where the quantization sequence is normal and occurs at integer ν. The latter shows that the half-integer QHE is exclusive to “ideal” graphene.。