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Room temperature Peierls distortion in small diameter nanotubes D. Connétable,1 G.-M. Rignanese,2,3 J.-C. Charlier,2,3 and X. Blase1 1 Laboratoire de Physique de la Matière Condensée et des Nanostructures (LPMCN), CNRS and Université Claude Bernard Lyon I, Bâtiment Brillouin, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France. 5 2 Unité de Physico-Chimie et de Physique des Matériaux (PCPM), 0 Université Catholique de Louvain, 1 Place Croix du Sud, B-1348 Louvain-la-Neuve, Belgium 0 3 Research Center on Microscopic and Nanoscopic Electronic Devices and Materials (CERMIN), 2 Université Catholique de Louvain, B-1348 Louvain-la-Neuve, Belgium n (Dated: February 2, 2008) a Bymeansofabinitiosimulations,weinvestigatethephononbandstructureandelectron-phonon J couplinginsmall4-˚Adiameternanotubes. WeshowthatboththeC(5,0)andC(3,3)tubesundergo 6 above room temperature a Peierls transition mediated by an acoustical long-wavelength and an ] opticalq=2kF phononsrespectively. Inthearmchairgeometry,weverifythattheelectron-phonon n coupling parameter λ originates mainly from phonons at q = 2kF and is strongly enhanced when o the diameter decreases. These results question the origin of superconductivity in small diameter c nanotubes. - r PACSnumbers: 61.46.+w,61.50.Ah,63.20.Kr,63.22.+m,68.35.Rh,74.25.Kc,74.70.Wz p u s t. Superconductivity (SC) has been recently discovered TCDW temperature of ∼ 240 K. These results suggest a in the 4-˚A diameter carbon nanotubes (CNTs) embed- thatthe C(5,0)tube is the only candidatefor a SC tran- m ded in a zeolite matrix [1] with a transition temperature sitionat15K.However,suchalargediscrepancybetween - TSC= 15 K, much larger than that observed in the bun- the two results call for a unified treatment of both nan- d dles of larger diameter tubes [2]. This has stimulated a otubes within a parameter-free approach. n significantamountofwork atthe theoreticallevelto un- In the present work, we study by means of ab initio o c derstandtheoriginoftheSCtransition[3,4,5,6]. While simulations the phonon band structure and e-ph cou- [ both reduced screening and quantum/thermal fluctua- pling in 4-˚A diameter CNTs. We show that the C(5,0) 1 tions disfavor superconductivity in 1D systems, early and C(3,3) tubes both undergo a Peierls distortion with v work on fullerenes [7] and nanotubes [8] suggested that a critical temperature T larger than room temper- CDW 3 the curvature significantly enhances the strength of the ature. While the distortion is associated with a long- 1 electron-phonon (e-ph) coupling. This is consistent with wavelength acoustical mode in the C(5,0) case, the e-ph 1 the softening of the Raman modes under tube diameter couplingisdominatedbyphononsatq =2k inthecase 1 F 0 reduction [9]. of C(n,n) nanotubes with a coupling parameter λ which 5 In 1D systems, the enhancement of the e-ph deforma- is strongly enhanced with decreasing diameter. 0 tionpotentialshouldalsofavorthe occurrenceofPeierls, The electronic and vibrational properties are studied / t or charge density wave (CDW), transitions driving the withinapseudopotential[14]planewaveapproachtoden- a systemintoaninsulatingphase. Thispossibilityhasbeen sity functional theory (DFT). We adopt the local den- m explored in early works on the basis of model Hamilto- sity approximation (LDA) and a 50 Rydb cut-off to ex- - d nians [10]. The strength of the e-ph coupling was ei- pand the electronic eigenstates. The phonon modes are n ther extrapolated from its value in graphite or taken as calculated at arbitrary q-vectors using the perturbative o an adjustable parameter. The Peierls instability tran- DFT approach [15] (in what follows q- and k-vectors la- c sition temperatures T were estimated to be much bel phonon and electron momenta respectively). Special : CDW v smaller than room temperature: for instance, Huang et careistakeninsamplingtheelectronicstatesaroundthe i al. [10] find T ∼ 9.1 K for a C(5,5) tube. Further, Fermilevelusinganadaptativek-gridtechnique. Theen- X CDW long-wavelength optical [9] or acoustical [11] “opening ergy levels are populated using either a Gaussian broad- r a thegap”phonondistortionshavealsobeenexploredand ening[16]oraFermi-Dirac(FD)distributionwithinMer- the transition temperature was again predicted to be a min’sgeneralization[17]ofDFT to thecanonicalensem- few degrees Kelvin. ble. Thelattertechniqueexplicitlyaccountsfortheeffect The Peierls transition in 4-˚A diameter CNTs has of temperature. We use a periodic cell allowing for 10 ˚A not been addressed explicitly, except for a recent tight- vacuum between the nanotubes. binding exploration of the C(5,0) tube [12] and an ab Westartbystudyingthenanotubesbandstructureaf- initio study of the C(3,3) one [13]. While the inclu- tercarefulstructuralrelaxationwitha25meVFDdistri- sion of electronic screening in the non-self-consistentap- bution broadening (T∼300 K). The first important out- proach[12] for the C(5,0) case yields a negligibleT , come of this calculation is the spontaneous zone-center CDW the ab initio simulations predict for the C(3,3) tube a deformationoftheC(5,0)tube[18]. FromtheD sym- 10h 2 FIG. 1: Symbolic representations of the out-of-plane acoustical and optical modes at Γ for the phonon bands driving thePeierls instability in the(a) C(5,0) and (b) C(3,3) tubes, respectively. FIG. 3: Phonon band structures for the C(n,n) nanotubes with n=3, 4, and 5 from left to right obtained with a Gaus- sian broadening of 136 meV (10 mRy). The dotted line for the C(3,3) nanotube corresponds to the additional explicit calculation of the phonon modes around q = 2kF with a re- duced68meV(5mRy)broadening. Importantbandsappear as thicker lines. The “undressed” ω0 bands (see text) are q indicated with thickdashed lines. thegiant Kohn anomaly [25]at q =2k leading toa dra- F matic softening of a few phonon modes [26]. In particu- lar, the optical band starting at 620 cm−1 at Γ becomes the lowest vibrational state at q = 2k [thick/dotted F FIG. 2: Band structures of (a) the “distorted” D2h C(5,0) low lying band in Fig. 3(a)]. The associated atomic dis- nanotube and (b) the C(3,3) tube. The zero of energy has placements,indicatedinFig.1(b), correspondtoanout- been set to the top of the valence bands and at the Fermi of-plane optical mode in agreement with the results of level, respectively. Ref. 12. This is consistent with the analysis providedby Mahan [27]: the coupling of electrons with out-of-plane opticalmodesshouldbeenhancedawayfromzone-center metry,thestructurerelaxestoformanelliptictube(D andwithdecreasingdiameter. Thepredominanceofsuch 2h symmetry), as indicated by the arrowsin Fig. 1(a), with amode fore-phcouplingis astrongmanifestationofthe principal axes of 3.83 ˚A and 4.13 ˚A. importance of curvature in small diameter tubes. The corresponding band structure is provided in The value of the associated frequency at q = 2k F Fig. 2(a). As a result of the reduction of symmetry, the strongly depends on the energy distribution broadening repulsion of the hybridized σ-π bands [19] at the Fermi used in the phonon calculation, as clearly illustrated in level opens a ∼ 0.2 eV band gap (LDA value). This Fig. 3(a) by the difference between the thick and dot- implies that the C(5,0) nanotube undergoes a metallic- ted lines. Using a 25 meV FD distribution (T∼300 K), semiconducting transition with TCDW larger than room we find a negative phonon mode ω2kF,ν=1 of -200 cm−1. temperature. FollowingRef.9,suchatransformationcan This meansthatthe C(3,3) tube undergoes a Peierls dis- be referred to as a “pseudo-Peierls” transition since it tortion with T larger than room temperature [28]. CDW opens the band gap through a (long-wavelength) struc- Consequently, at the mean-field SC transition tempera- tural deformation. ture measured for the 4-˚A diameter CNTs (T =15 K), SC In the C(3,3) case, we do not observe any zone-center the C(5,0) and C(3,3) tubes should both be semiconduct- deformation leading the band structure away from the ing, seriously questioning the origin of superconductivity well-documented [20] zone-folding picture of two linear in small diameter tubes. bands crossing at the Fermi level E . However, com- OurresultsareingoodagreementwiththoseofRef.13 F pared to the zone-folding analysis, the Fermi wavevec- concerning the C(3,3) tube, but contrast significantly tor k is no longer at 2/3 ΓX but at ∼ 0.58 ΓX so that withthosebasedonamodelfortheinteractionHamilto- F 2π/2k isnolongercommensuratewiththeunperturbed niansandtheelectronicsusceptibility[12]. Weemphasize F lattice (or commensurate with a very large supercell). thatourself-consistenttreatmentautomaticallyincludes This is a well documented effect of the curvature. therenormalizationofboththe e-phande-einteractions WefurtherpresentinFig.3(a)thephononbandstruc- by electronic screening at the DFT level. ture of the C(3,3) tube [21]. The most salient feature is To study the influence of tube diameter on the soften- 3 ing of the modes at q =2k , we represent in Fig. 3(b,c) F the phononbandstructureoftheC(n,n) tubeswithn=4 TABLEI:Contributions(λν)tothecouplingconstantforall the q-vectors in the neighborhood of Γ (q ⊂ {Γ}) and 2kF and5[21]. Again,weobservethatthesameopticalband (q ⊂ {2kF}) for the C(n,n) tubes with n=3, 4, and 5. The (thick lowlying band)is softenedat2k , butclearlythe F correspondingfrequencies(ων)atΓ and2kF areindicatedin effect is strongly reduced with increasingdiameter. This cm−1. For the C(3,3) tube, the values underlined for q ⊂ confirms that TCDW will quickly lower with increasing {2kF} correspond to the “undressed” frequencies (see text). diameter. The large amount of sp3 character in ultra- Theglobal couplingconstant(λ)isindicated at thelast line. smalltubesinvitetodrawacomparisonwiththeSCtran- C(3,3) C(4,4) C(5,5) sition in doped carbon clathrates [22] and diamond [23]. ων λν ων λν ων λν Surprisingly, the softening of another mode at higher q⊂{Γ} energy (starting as an optical mode around 1450 cm−1, RBM 536 0.009 416 0.005 338 0.001 also indicated by a thick line) seems to be less sensitive ZO 692 0.009 799 0.004 848 0.001 tothediameter. Thissofteningisthesignatureofalarge A1(L) 1354 0.007 1468 0.009 1538 0.002 e-ph coupling which might thus play an important role A1(T) 1431 0.012 1505 0.009 1570 0.002 q⊂{2kF} in the transport properties of larger CNTs. [24] 520 0.074 414 0.022 497 0.010 To further quantify the strength of the e-ph interac- 499 0.026 509 0.010 533 0.004 tion, we cancalculate the e-ph coupling matrix elements 1320 0.018 1013 0.010 1154 0.009 ′ g (knn) and the coupling constant λ: 1144 0.001 1180 0.014 1237 0.011 qν 0.156 0.083 0.040 λ= λ =2N(E ) <|g |2 >/~ω (1) q,ν f qν qν X X q,ν q,ν <|gqν|2 >=Z LdBkZ nX,n′|gqν(knn′)|2δ(ǫkN+q(,En′F)δ)(2ǫkn(2)) maboodveesTaCnDdWp,rwovhiidcihngartehtehovaselutehsatfomr aλt2tkeFr,νfo=r1torabntaspinoerdt measurements in the normal state. ~ δV gqν(knn′)=(2Mω )1/2 <ψn0k|ˆǫqν · δuˆ |ψn0′k+q(>3) While the g2kF parameters are rather stable with qν qν temperature above T , the phonon frequencies still CDW Due to conservation of energy and momentum, only vi- vary significantly. Following Ref. 12, we provide an up- brationalstatesaroundq =Γand±2kF contributetoλ, perbound for ω2kF,ν by removing the Kohn’s anomaly. as the nesting factor [29] n(q) quickly decays away from Namely, we assume that around q =2k , F these points. At Γ, only four modes contribute to λ due to symmetry considerations. They arethe radialbreath- ing mode (RBM), the optical out-of-plane (ZO) mode, ωq2,ν =(ωq0,ν)2+A×ln (q−2kF)/(q+2kF) (4) (cid:12) (cid:12) and the in-plane optical longitudinal A (L) and trans- (cid:12) (cid:12) 1 (cid:12) (cid:12) verseA (T) (G-band)modes (seeRef. 9) locatedrespec- and fit the ω2 bands by a cubic polynomial for (ω0 )2 1 q,ν q,ν tivelyat536cm−1,692cm−1,1354cm−1 and1431cm−1 plus the logarithmic term that we subtract to obtain in the C(3,3) tube. the bare phonon frequency. The important “undressed” In Table I, we report the parameter λ(ν,q ⊂ {Γ}) de- bands (correspondingto ω0 ) arerepresentedby dashed q,ν fined as the q-sum of all the λ (see Eq. 1) for q in the lines in Fig. 3(a). The related frequencies and λ q,ν 2kF,ν neighborhoodofΓ ona givenν-band[30]. This coupling values are underlined in Table I. This treatment is ap- strengthvariesfromonebandtoanotherandisenhanced pliedtobothstatespresentingthelargestKohnanomaly, when the tube diameter decreases. However, the varia- butitseffectisclearlymorepronouncedforthelow-lying tion with the diameter differs from one mode to another ν =1 band. and it is difficult to extract a simple scaling law. Using the ω0 frequencies leads to providing a 2kF,ν We now discuss the e-ph coupling at q = 2k . lower bound for the corresponding λ e-ph param- F 2kF,ν The strong variation of the lowest frequency ω eter. Even in that limit, we observe that the coupling 2kF,ν=1 with temperature raises the question of the meaning of with modes at q = 2k is significantly larger than with F λ calculatedwiththeg andω valuesobtained long-wavelength phonons. As at zone-center, few modes 2kF,ν=1 qν qν from ab initio calculations. The difficulties have been with q = 2k couple to the electrons and the strength F clearly exposed in Ref. 12. If one is interested in study- of the coupling is significantly enhanced with decreasing ing the SC transition, one faces the difficulty that at the diameter. In the C(3,3) case, we obtain in the undis- experimental T , the self-consistent g and ω torted phase a density of states N(E ) = 0.4 states/eV- SC 2kF,ν 2kF,ν F values are those of the CDW phase. As T is larger cell-spin, yielding an e-ph potential Vep = λ/N(E ) = CDW F than any realistic SC transition temperature, we will 440 meV. This is larger than that in the fullerenes (60- not attempt to estimate T with e-ph vertices and ph- 70 meV), but it is not sufficient to lead to any signifi- SC propagator properly “undressed” from the CDW insta- cantT value. The much largerdensity ofstates in the SC bility [12]. We limit ourselves to identifying the relevant “D ” C(5,0) tube (N(E ) = 1.8 states/eV-cell-spin) 10h F 4 may lead in principle to a large TSC value. However, as (2002). shown above, the prevailing structure around the exper- [12] R. Barnett, E. Demler, E. Kaxiras, cond-mat/0407644; imental T temperature is the insulating D one. ibid.,cond-mat/0305006 SC 2h [13] K.-P.Bohnenet al.,Phys.Rev.Lett.93,245501 (2004); In conclusion, we have shown that the “metallic” Phys. Rev. Lett. 93, 259901(E) (2004). Paper published C(5,0)andC(3,3)nanotubesundergoaPeierlstransition after submission of our manuscript. with a critical temperature larger than 300 K. Our re- [14] N. Troullier and J.L. Martins, Phys. Rev. B 43, 1993 sults are derived within Mermin’s generalizationof DFT (1991). to finite temperature systems and do not rely on models [15] S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Gi- for the e-ph interactions nor the electronic susceptibil- annozzi, Rev. Mod. Phys. 73 (2), 515 (2001); S. Ba- ity. Additionalphysicalingredients(interactionwiththe roni, A. Dal Corso, S. de Gironcoli, and P. Giannozzi, zeolite network, possible defective structure, Luttinger http://www.pwscf.org. [16] M. Methfessel and A. Paxton, Phys. Rev. B 40, 3136 liquid character, etc.) might reconcile these results with (1989) theexperimentallyobservedSCtransitionatT =15K. SC [17] N. Mermin, Phys. Rev. 137, A1441 (1965). Acknowledgements: Calculations have been per- [18] This deformation occurs for small enough a FD energy formed at the French CNRS national computer center broadening. This explains why it has not been reported at IDRIS (Orsay). GMR and JCC acknowledge the Na- so far. The present results have been obtained indepen- tional Fund for Scientific Research [FNRS] of Belgium dently with the ABINIT (http://www.abinit.org) and for financial support. Parts of this work are also con- PWSCF (http://www.pwscf.org) codes. [19] X. Blase, L.X. Benedict, E.L. Shirley, S.G. Louie, Phys. nected to the Belgian Program on Interuniversity At- Rev. Lett.72, 1878 (1994). traction Poles (PAI5/1/1) on Quantum Size Effects in [20] M. Machón, S. Reich, C. Thomsen, D. Sánchez-Portal, Nanostructured Materials, to the Action de Recherche and P. Ordejón, Phys. Rev. B 66, 155410 (2002); H.J. Concertée “Interaction électron-phonon dans les nanos- Liu, and C.T. Chan, Phys. 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