Basing on the structural peculiarity of two-layer graphene which consists of translational

and energetical nonequivalency of carbon atoms from different sublattices, it is shown that the

density of long-wave electronic states at the Fermi level is finite (in contrast to the monolayer

graphene). It is suggested that the same may be the reason why the critical temperature of

superconducting transition in multiwall nanotubes more than ten times higher than in single-wall

nanotubes

Gaididei, Yu.B.

Loktev, V.M.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Fizika Nizkikh Temperatur, 2006, v. 32, No. 11, p. 1458–1462On the possible reason for superconductivitystrengthening in multiwall carbon nanotubesYu.B. Gaididei and V.M. LoktevBogolyubov Institute for Theoretical Physics, 14 B Metrologichna Str., Kiev 01413, UkraineE-mail: vloktev@bitp.kiev.uaReceived April 17, 2006Basing on the structural peculiarity of two-layer graphene which consists of translationaland energetical nonequivalency of carbon atoms from different sublattices, it is shown that thedensity of long-wave electronic states at the Fermi level is finite (in contrast to the monolayergraphene). It is suggested that the same may be the reason why the critical temperature ofsuperconducting transition in multiwall nanotubes more than ten times higher than in single-wallnanotubes.PACS: 71.23.–k, 71.21.Ac, 81.05.UwKeywords: graphene, quantum Hall effect, unconventional Hall effect.1. IntroductionSince the discovery of fullerene [1] (see also reviewpapers [2–4]) there is an increasing interest in theinvestigation of fullerene-like structures which be-sides carbon nanotubes (CNT) [5] include also single-and multilayer graphene [6]. Carbon nanotubes arevery perspective both from fundamental and appliedview points. CNT may be used as the elements ofdifferent nanoelectronic devices, constant-force nano-springs [7], containers to hold in storage of light gases(in particular hydrogen), pressure transducers and soon. On the other hand, CNT are known as modelsystems for the study of one-dimensional electronictransport. Depending on their diameter and helicity,carbon nanotubes are either semiconducting or me-tallic [4]. It is well known that the usual Fermi liquidwhich is described by the Landau–Silin theory isexpected to be unstable in one-dimensional (1D)metals with the formation of a correlated Tomona-ga–Luttinger liquid state where the electron—elect-ron repulsive interaction prevents the appearance ofsuperconducting state to a greater extent than in con-ventional 3D metals.Investigations of the magnetic and transport pro-perties of ropes of single-walled small-diameter carbonnanotubes (SWNT) [8] and SWNT embedded in azeolite matrix [9] revealed that a few angstrom tubesexhibit superconducting behavior manifest as an an-isotropic Meissner effect, with a superconducting gapand fluctuation supercurrent. It turned out that thesuperconducting transition in SWNT occurs at verylow temperature, less than 0.55 K [8]. It is quitenatural to connect such a low critical temperaturewith the low-dimensional character of CNT*, withstrong quantum and thermal fluctuations and withproximity to Peierls instabilities which drive thesystem to an insulating state. Therefore the report[10] that multiwalled carbon nanotubes (with up to 9shells) exhibit superconductivity with an unexpect-ably high transition temperature Tc  12 K, which isapproximately 30 times greater than Tc reported forSWNT [8] has attracted considerable attention. Thevalue of the superconducting gap   115. meV which© Yu.B. Gaididei and V.M. Loktev, 2006* It is worth to emphasize that carbon nanotubes can not be considered as genuine one-dimensional objects. In 1D systemsthe processes of electron—electron, electron—phonon and electron—impurity scattering involve only one, «longi-tudinal» component of the electron wave vector while in nanotubes these processes include also the «transversal» ororbital component of the quasimomentum.is in excellent agreement with the BCS canonical ratio2 3 5/k TB c  . , the observed behavior of the criticalcurrent as a function of the temperature and externalmagnetic field is also in good qualitative agreementwith the Ginzburg–Landau critical current behavior.It can be therefore said with confidence that theobserved superconductivity in multiwalled carbon na-notubes is of the BCS type and that the multishellstructure of the nanotubes plays a crucial role.The aim of this work is to show that the bigdifference of the critical temperature of single- andmultiwalled nanotubes may be due to a qualitativelydifferent behavior of the density of states of low-energy charge carriers. As a model we consider mono-and bi-layer graphenes whose structure is similar tothe structure of graphit sheets rolled-up in nanotubes.It is a great honour and pleasure for us to devotethis work to the 100 year jubilee of outstandingphysicist-experimentalist and marvellous person, An-tonina Fedorovna Prikhot’ko, who did so much forour formation as scientists. It is worth to note alsothat being a bright and leading figure in the low tem-perature physics, she always supported new lines ofinquiry in this area and was one of devotees of thequest of new superconductors with unusual and un-expectable features.2. Hamiltonian and electron spectrumWe employ the tight-binding model to describe theband structure of single- and bi-layer graphenes.Basing on the well developed theory of molecularcrystals with several molecules per unit cell (see, e.g.,[11]). the electronic Hamiltonian of bi-layer graphenecan be repsented in the formH a a b b      ( )n n n nn   12t a a b bn m n m n mn m      ,,( )  12t a b b a| |( )( )n n n nn     , (1)where  is the on-site energy of -electron at the siten (the vector n gives the position of the elementarycell, the index ( , ) 1 2 characterizes the sublattices);the matrix element tn m  describes an intra-layerelectron hopping (tunneling) between the sites nand m, and t| | characterizes an inter-layer electrontransfer; the Fermi-operators an and bn describe acreation of an electron at the site n of the first andsecond layer, respectively (the index of the electronspin is omitted for brevity). It is worth noting thatthe last term of the Hamiltonian (1) is chosen in theform which takes into account that the sheets in thebi-layer graphene are shifted with respect to eachother by the vector ( )a a1 2 2 / , where a j are theprimitive vectors of graphene sheet [12]. In con-sequence of such an unusual geometry the atoms ineach sheet become energetically (as well as trans-lationally) nonequivalent*.By applying the Fourier transformaNa i ( )knnkn1e ,bNb i ( )knnkn1e , (2)where k is the wave vector and N is the number ofcells in the sheet, the Hamiltonian (1) can be pre-sented in the formH t a a b b   { ( )[ ( ) ( ) ( ) ( )]1 1 2 1 2k k k k kkh.c.} +   t a a b b212( )[ ( ) ( ) ( ) ( )], ,k k k k kk     t a b b a| | [ ( ) ( ) ( ) ( )]1 2 2 1k k k kk, (3)wheret t i i1 1 1 1 1 1 2( ) ( ); ( )k k kka ka     e e ; (4)t t2 2 2( ) ( )k k  ;2 1 2 1 2( ) cos cos cos ( )k ka ka k a a   	(5)are the Fourier-components of so-called resonanceinteractions which form the dispersion of electro-nic bands. The functions t1( )k and t2( )k given byEqs. (4) and (5) represent the intra- and inter-sublattice electron dispersion, respectively. The latterleads to a splitting which in the theory of molecularexcitons is called the Davydov splitting (see, e.g.,[11]). The parameters t1 and t2 in Eqs. (4) and (5)are the hopping integrals between nearest and nextnearest neighbors, respectively; a1 3 0 ( , ), a2  ( , )3 2 3 2/ / are the primitive vectors of the rhombiclattice with the distance between the nearest neigh-On the possible reason for superconductivity strengthening in multiwall carbon nanotubesFizika Nizkikh Temperatur, 2006, v. 32, No. 11 1459* Note that the same phenomenon takes place in multi-wall carbon nanotubes consisting of several rolled-up graphit sheetsin the cylindrical form. Here the atoms which belong to different cylindrical surfaces are energetically nonequivalent.bors equal to unity. By introducing the «even» and«odd» linear combinations of electron operators1 1 212( ) [ ( ) ( )]k k k a b ;1 1 212( ) [ ( ) ( )]k k k 	a b ;2 2 112( ) [ ( ) ( )]k k k a b ;2 2 112( ) [ ( ) ( )]k k k 	a b(6)and applying the unitary transformations   ( ) ( ) ( ),k k ku12;   ( ) ( ) ( ),k k kv12;(7)with| ( )|( )| ( )| | ( )|( )( )vt122122kkk k;| ( )|| ( )|( ) | ( )| | ( )|( ) ( )vtt22 122122kkk k k ,(8)one can diagonalize the Hamiltonian (3) and obtainthe electron eigenenergies in the form 	   	  ( ) | | | |( ) ( ) | ( )|k k ktt tt22122 2, (9) 	     ( ) | | | |( ) ( ) | ( )|k k ktt tt22122 2. (10)In the mono-layer case which can be obtained fromEqs. (9) and (10) by neglecting the interlayer hopp-ing integral: t| |  0, the electron spectrum consists oftwo symmetric branches which in fact, are the Da-vydov bands. They intersect and have the same energy  	 3 22t / at six points of the Brillouin zonewhich can be obtained by rotations of the vectorsK12 2 9 3 3, ( )( , ) / by  /3. By virtue of the factthat the number of states is the same below and abovethis energy, in graphene there exists a rather rare forsolid state physics situation (first apparently noticedin Ref. 13) when the two-dimensional metal has not aFermi line (as usually in two-dimensional metals) buta set of Fermi points. As a result of such a pecularityof the spectrum, in the vicinity of Fermi points chargecarriers are characterized by a massless Dirac-likedispersion law which in particularly manifests itself inan unconventional Hall effect [14,15].In the bi-layer graphene structure the electronspectrum consists of four bands, two of them touch atthe points K j ( ... )j  1 6 (they do not intersect as inthe mono-layer case) and two others are nondege-nerate for all wave vectors k. Assuming for examplethat t2 0 , and t| |  0, it is easy to see that the bandwith the energy ( )( )k is the conduction band ofbi-layer graphene.Note also that the expressions similar to given byEqs. (9) and (10) (but without intra-sublattice hopp-ing) was obtained in Ref. 16 (see also [17]). It followsfrom them, in particularly that the Hall effect ac-quires some new features. However their discussion isbeyond the scope of the paper.3. Superconducting gapLet us now consider the superconducting phasetransition in mono- and bi-layer graphene. To this endwe augment the Hamiltonian (3) with the operator ofattractive electron-electron interaction. There is astrong belief (see, e.g., [18]) that phonon exchangeholds responsible for attractive electron-electron in-teraction in metallic carbon nanotubes. The derivationof the operator of the phonon mediatated electron-electron interaction is well known (see, e.g., [19,20]).We take this operator in the formHNVint,( , ) ( ) ( ) ( ) ( ) 	 	 1k q k k q qk q    , (11)where the parameter V( , )k q characterizes the strengthof the electron—electron interaction;  ( )k (( )k )is the creation (annihilation) operator of a carrierwith the wave vector k and the spin    , . By usingthe standart procedure [19,20] it is easy to obtainthat the order parameter( ) ( , ) ( ) ( )k k q q qq  	   N V1  (the gap in the electron spectrum) is determined bythe equation( ) ( , )( )[ ( ) ] | ( )|( )k k qqq qq	 12 2 2NVF ,(12)where the symbol    denotes a thermodynamicmean value and F is the Fermi energy which in ourcase is doping controlled.In the simplest approximation when V V( , )k q the gap is k independent  ( )k  , and Eq. (12) takesthe form1460 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11Yu.B. Gaididei and V.M. Loktev181 12 2 2  dVFkk	 [ ( ) ] | |cond , (13)wherecond( ) | | | |k k k 	  9812123222212t t t t (14)is the energy of charge carriers in the vicinity of thenode points K j [12,17]. By introducing the density ofquasiparticle states  ( ) (see Appendix for details)Eq. (14) can be transformed to the formdVFF  ( )( ) | |	  2 22,which for a given  ( ) (see Eq. (A.2)) gives theexpression  	 !"!#$!%!29 12DFtV texp( )| |, (15)for the superconducting gap in the one-electron spect-rum. Here D is the Debye frequency.Comparing the superconducting gap  & bi in thebi-layer graphene which is given by Eq. (15) with thesuperconducting gap  & mono in the mono-layergraphene which can be also obtained from Eq. (15) byputting t  0 one can see that the ratiobimono !"!#$!%!exp9112tV F, (16)where  & t/ in the low-doping regime may be ex-ponentially big. This result is a direct consequence ofthe qualitative difference between the spectra ofcharge carriers in single- and bi-layer graphene.While in the mono-layer graphene the spectrum ismassless and the density of of states vanishes whenthe concentration of carriers decreases ( ' 0) in thebilayer graphene the carriers have a finite effectivemass and the density of states in the limit ( ' 0) isfinite.4. SummaryWe have shown that the interlayer carrier transferin graphene may be of crucial importance in creationof superconducting condensate. It is interesting tonotice that in the same way the low-energy beha-vior of the density of states plays a pivotal rolein electron-hole pairing and forming the exciton di-electric phase graphene. In the mono-layer graphenesuch a phase can form only in the presence of strongmagnetic field which effectively reduces to 1D thecharge carrier motion and in this way enhances Cou-lomb attraction (magnetic catalysis [21]). In thebi-layer graphene the presence of interlayer hoppingmakes inessential applying the magnetic field.In our opinion, the strengthening of supercon-ductivity in multiwalled nanotubes may be also due tothe interlayer electron hopping. The charge carriers inmultiwalled nanotubes have a finite effective massand as a result the density of states is finite as  ' F .It follows also from our model that the criticaltemperature in a mono-layer graphene (singlewallednanotube) increases when the doping increases. More-over the critical temperatures of mono- and bi-layergraphenes (as well as the critical temperatures ofsingle- and multiwalled CNT) equalize when F t | | .The above developed theory does not take intoaccount the Coulomb interaction between electrons. Itis well known, however, that this interaction it de-creases the critical temperature of conventional 3Dsuperconductors. The decisive factor here is a screen-ing of Coulomb interaction in metals. In the caseunder the consideration we meet with a qualitativelynew situation when not only the low-dimensionalcharacter of charge transport (quasi 1D in CNT and2D in graphene) is important but the size and theshape of the objects (radius of CNT, relative shift ofsheets in graphene and so on) as well. Quite recentstudies of the screening of Coulomb interaction innon-polar systems [22] showed that in low dimensionsand small finite-size systems this screening deviatesstrongly from conventionally assumed. For example,in one dimension at small and intermediate distancesthe Coulomb interaction is anti-screened therebystrongly reducing the gradient of the Coulomb in-teraction. It is also shown in [22] that in finite-sizesystems due to weak dependence of the effectiveCoulomb interaction on the distance between theelectrons the correlation effects are reduced and themean-field approach works well because the corre-lation effects are reduced and the mean-field approachworks well. Note also that increasing the number ofwalls in CNT (number of layers in graphene) is tosome extent equivalent to increasing of their di-mensionality. So the superconducting effects are en-hanced in these systems.Appendix AThe aim of this Appendix is to calculate the densityof quasiparticle states (  ( ) [ ( )] 	14 2dk kcond (A.1)On the possible reason for superconductivity strengthening in multiwall carbon nanotubesFizika Nizkikh Temperatur, 2006, v. 32, No. 11 1461for the case of bi-layer graphene. Here cond( )k isgiven by Eq. (14) and the two-dimensional characterof the system is taken into account. From Eq. (A.1)we get   ( )max   	 	14 298 23202 2212dkt tkt tk 22k,where the maximal wave vector kmax is determin-ed from the condition that the total number of statesin the conduction band is equal to the numberof elementary cells N. By using the new variablek x t / / t / 	[ ( )] ( )| |2122 3 2 , we obtain (( ) [( )( )]||max 	 	 49 22tdxx x x x xt /x,wherex / t /t	  	 ( )[ ( )1 2 2 122   ( ) ( )( ) ]| | | |4 8 214221222t /t t /t t / tare the roots of the equation cond( )k  0 and thenotation x t / tmax | | max( ) 2212 2k is introduced. Asa result we get  ( )( / ) ( / )| | 	 )*+++,-...29111 2 42 2 122 12t t t t t t.(A.2)From Eq. (A.2) we obtain that for the mono-layergraphene (when the interlayer hopping is absent,t| | ) 0 the density of states takes the form  12 2 120 1229111 2Lt t /t t( )( ) 	)*++,-.. .(A.3)In the two-layer case (when one can neglect theintra-sublattice hopping t2) we have the followingexpression for the density of states  212 012492Ltttt( ) ( )|| || |  / . (A.4)Comparing Eqs. (A.3) and (A.4), we see that theratio    1 20L L/ t( ) ( )| / | | /is small as long as  0 F t| | .Similar enhancement of density of states one canexpect, as it was mentioned above, under the transi-tion from singlewalled to multiwalled nanotubes.Indeed, it is easy to see that due to a masslesscharacter of the carrier dispersion in the singlewallednanotube the density of states  NTSW /t( ) / 1 1 whilein the multiwalled nanotube the carriers have a finiteeffective mass and the density of states  NTMW /( ) / 1 , fastly growing for small .1. H. Kroto, J.-R. Health, S.C. O’Brian, R.F. Curl, andR. Smalley, Nature 318,162 (1985).2. V.M. Loktev, Fiz. Nizk. Temp. 18, 217 (1992) [LowTemp. Phys. 18, 149 (1992)].3. A.V. Eletskii, Usp. Fiz. Nauk 174, 1191 (2004).4. Carbon Nanotubes: Synthesis, Structure, Propertiesand Application, M.S. Dresselhaus, G. Dresselhaus,and P. Avouris (eds.), Springer, Berlin (2000).5. S. Iijima, Nature 354, 56 (1991).6. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang,I.V. Grigorieva, S.V. Dubonos, and A.A. Firsov,Science 306, 666 (2004).7. J. Cumings and A. Zell, Science 289, 602 (2000).8. M. Kociak, A.Yu. Kasumov, S. Guron, B. Reulet, I.I.Khodos, Yu.B. Gorbatov, V.T. Volkov, l. Vaccarini,and H. Bouchard, Phys. Rev. Lett. 86, 2416 (2001).9. Z.K. Tang, L. Zhang, N. Wang, X.X. Zhang, G.H.Wen, G.D. Li, J.N. Wang, C.T. Chan, and P. Sheng,Science 292, 2462 (2001).10. I. Takesue, J. Haruyama, N. Kobayashi, S. Chiashi,S. Maruyama, T. Sugai, and H. Shinohara, Phys. Rev.Lett. 96, 057001 (2006).11. Cryocrystals, B.I. Verkin and A.F. Prokhot’ko (eds.),Naukova Dumka, Kiev (1983).12. E. McCann and V.I. Falko, E-print archives: cond-mat/0510237 (2005); Phys. Rev. Lett. 96, 086805(2006).13. G.W. Semenoff, Phys. Rev. Lett. 53, 2449 (1984).14. V.P. Gusynin and S.G. Sharapov, Phys. Rev. Lett.95, 146801 (2005).15. K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang,M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, andA.A. Firsov, Nature 438, 197 (2005).16. K.S. Novoselov, E. McCann, S.V. Morozov, V.I.Fal‘ko, M.I. Katsnelson, U. Zeitler, D. Jiang, S. Sche-din, and A.K. Geim, Nature Phys. 2, 177 (2006).17. Yu.B. Gaididei and V.M. Loktev, Fiz. Nizk. Temp.32, 923 (2006) [Low Temp. Phys. 32, 703 (2006)].18. J. Gonzalez, Phys. Rev. Lett. 87, 2415 (2002); ibid.88, 076403 (2002).19. J.R. Schrieffer, Theory of Superconductivity, Ben-jamin, N.Y. (1964).20. A.A. Abrikosov, L.P. Gor‘kov, and I.E. Dzyaloshin-ski, Methods of Quantum Field Theory in StatisticalPhysics, Dover, N.Y. (1975).21. V.P. Gusynin, V.A. Miransky, and I.A. Shovkovy,Phys. Rev. D67, 107703 (2003).22. J. van der Brink and G.A. Sawatzky, Europhys. Lett.50, 447 (2000).1462 Fizika Nizkikh Temperatur, 2006, v. 32, No. 11Yu.B. Gaididei and V.M. Loktev

On the possible reason for superconductivity strengthening in multiwall carbon nanotubes

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On the possible reason for superconductivity strengthening in multiwall carbon nanotubes

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