By means of {\it ab initio} simulations, we investigate the phonon band
structure and electron-phonon coupling in small 4-\AA diameter nanotubes. We
show that both the C(5,0) and C(3,3) tubes undergo above room temperature a
Peierls transition mediated by an acoustical long-wavelength and an optical
$q=2k_F$ phonons respectively. In the armchair geometry, we verify that the
electron-phonon coupling parameter $\lambda$ originates mainly from phonons at
$q=2k_F$ and is strongly enhanced when the diameter decreases. These results
question the origin of superconductivity in small diameter nanotubes.Comment: submitted 21oct2004 accepted 6jan2005 (Phys.Rev.Lett.

D. Connétable

G.-M. Rignanese

J.-C. Charlier

K. Tanigaki

X. Blase

English

arXiv

submitted 21oct2004 accepted 6jan2005 (Phys.Rev.Lett.)By means of {\it ab initio} simulations, we investigate the phonon band structure and electron-phonon coupling in small 4-Å\ diameter nanotubes. We show that both the C(5,0) and C(3,3) tubes undergo above room temperature a Peierls transition mediated by an acoustical long-wavelength and an optical $q=2k_F$ phonons respectively. In the armchair geometry, we verify that the electron-phonon coupling parameter $\lambda$ originates mainly from phonons at $q=2k_F$ and is strongly enhanced when the diameter decreases. These results question the origin of superconductivity in small diameter nanotubes

Connétable, D.

Rignanese, G.-M.

Charlier, J.-C.

Blase, Xavier

HAL Descartes

Room temperature Peierls distortion in small radius nanotubes

By means of ab initio simulations, we investigate the phonon band structure and electron-phonon coupling in small 4-Å diameter nanotubes. We show that both the C(5,0) and C(3,3) tubes undergo above room temperature a Peierls transition mediated by an acoustical long wavelength and an optical q = 2kF phonon, respectively. In the armchair geometry, we verify that the electron-phonon coupling parameter lambda originates mainly from phonons at q = 2kF and is strongly enhanced when the diameter decreases. These results question the origin of superconductivity in small diameter nanotubes

Blase, X.

Room Temperature Peierls Distortion in Small Diameter Nanotubes

By means of ab initio simulations, we investigate the phonon band structure and electron-phonon coupling in small 4-Angstrom diameter nanotubes. We show that both the C(5,0) and C(3,3) tubes undergo above room temperature a Peierls transition mediated by an acoustical long wavelength and an optical q=2k(F) phonon, respectively. In the armchair geometry, we verify that the electron-phonon coupling parameter lambda originates mainly from phonons at q=2k(F) and is strongly enhanced when the diameter decreases. These results question the origin of superconductivity in small diameter nanotubes

Connetable, D

Rignanese, Gian-Marco

Charlier, Jean-Christophe

Blase, X

DIAL UCLouvain

Room temperature Peierls distortion in small diameter nanotubes

Connétable, Damien

Open Archive Toulouse Archive Ouverte

PRL 94, 015503 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending14 JANUARY 2005Room Temperature Peierls Distortion in Small Diameter NanotubesD. Conne´table,1 G.-M. Rignanese,2,3 J.-C. Charlier,2,3 and X. Blase11Laboratoire de Physique de la Matie`re Condense´e et des Nanostructures (LPMCN), CNRS and Universite´ Claude Bernard Lyon I,Baˆtiment Brillouin, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France2Unite´ de Physico-Chimie et de Physique des Mate´riaux (PCPM), Universite´ Catholique de Louvain,1 Place Croix du Sud, B-1348 Louvain-la-Neuve, Belgium3Research Center on Microscopic and Nanoscopic Electronic Devices and Materials (CERMIN), Universite´ Catholique de Louvain,B-1348 Louvain-la-Neuve, Belgium(Received 21 October 2004; published 14 January 2005)0031-9007=By means of ab initio simulations, we investigate the phonon band structure and electron-phononcoupling in small 4- A diameter nanotubes. We show that both the C(5,0) and C(3,3) tubes undergo aboveroom temperature a Peierls transition mediated by an acoustical long wavelength and an optical q  2kFphonon, respectively. In the armchair geometry, we verify that the electron-phonon coupling parameter originates mainly from phonons at q  2kF and is strongly enhanced when the diameter decreases. Theseresults question the origin of superconductivity in small diameter nanotubes.DOI: 10.1103/PhysRevLett.94.015503 PACS numbers: 61.46.+w, 63.20.Kr, 63.22.+m, 68.35.RhSuperconductivity (SC) has been recently discovered inthe 4- A diameter carbon nanotubes (CNTs) embedded in azeolite matrix [1] with a transition temperature TSC 15 K, much larger than that observed in the bundles oflarger diameter tubes [2]. This has stimulated a significantamount of work at the theoretical level to understand theorigin of the SC transition [3–6]. While both reducedscreening and quantum or thermal fluctuations disfavorsuperconductivity in 1D systems, early work on fullerenes[7] and nanotubes [8] suggested that the curvature signifi-cantly enhances the strength of the electron-phonon (e-ph)coupling. This is consistent with the softening of theRaman modes under tube diameter reduction [9].In 1D systems, the enhancement of the e-ph deformationpotential should also favor the occurrence of Peierls, orcharge density wave (CDW), transitions driving the systeminto an insulating phase. This possibility has been exploredin early works on the basis of model Hamiltonians [10].The strength of the e-ph coupling was either extrapolatedfrom its value in graphite or taken as an adjustable parame-ter. The Peierls instability transition temperatures TCDWwere estimated to be much smaller than room temperature:for instance, Huang et al. [10] find TCDW  9:1 K for aC(5,5) tube. Further, long-wavelength optical [9] or acous-tical [11] ‘‘opening the gap’’ phonon distortions have alsobeen explored and the transition temperature was againpredicted to be a few degrees Kelvin.The Peierls transition in 4- A diameter CNTs has notbeen addressed explicitly, except for a recent tight-bindingexploration of the C(5,0) tube [12] and an ab initio study ofthe C(3,3) one [13]. While the inclusion of electronicscreening in the non-self-consistent approach [12] for theC(5,0) case yields a negligible TCDW, the ab initio simula-tions predict for the C(3,3) tube a TCDW temperature of240 K. These results suggest that the C(5,0) tube is the05=94(1)=015503(4)$23.00 01550only candidate for a SC transition at 15 K. However, such alarge discrepancy between the two results calls for a uni-fied treatment of both nanotubes within a parameter-freeapproach.In the present Letter, we study by means of ab initiosimulations the phonon band structure and e-ph couplingin 4- A diameter CNTs. We show that the C(5,0) and C(3,3)tubes both undergo a Peierls distortion with a criticaltemperature TCDW larger than room temperature. Whilethe distortion is associated with a long-wavelength acous-tical mode in the C(5,0) case, the e-ph coupling is domi-nated by phonons at q  2kF in the case of C(n; n)nanotubes, with a coupling parameter  which is stronglyenhanced with decreasing diameter.The electronic and vibrational properties are studiedwithin a pseudopotential [14] plane wave approach todensity functional theory (DFT). We adopt the local den-sity approximation (LDA) and a 50 Ry cutoff to expand theelectronic eigenstates. The phonon modes are calculated atarbitrary q vectors using the perturbative DFT approach[15] (in what follows q and k vectors are labeled phononand electron momenta, respectively). Special care is takenin sampling the electronic states around the Fermi levelusing an adaptative k-grid technique. The energy levels arepopulated using either a Gaussian broadening [16] or aFermi-Dirac (FD) distribution within Mermin’s general-ization [17] of DFT to the canonical ensemble. The lattertechnique explicitly accounts for the effect of temperature.We use a periodic cell allowing for 10 A vacuum betweenthe nanotubes.We start by studying the nanotube band structure aftercareful structural relaxation with a 25 meV FD distributionbroadening (T  300 K). The first important outcome ofthis calculation is the spontaneous zone-center deforma-tion of the C(5,0) tube [18]. From the D10h symmetry, the3-1  2005 The American Physical SocietyFIG. 1 (color online). Symbolic representations of the out-of-plane acoustical and optical modes at  for the phonon bandsdriving the Peierls instability in the (a) C(5,0) and (b) C(3,3)tubes, respectively.PRL 94, 015503 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending14 JANUARY 2005structure relaxes to form an elliptic tube (D2h symmetry),as indicated by the arrows in Fig. 1(a), with principal axesof 3:83 A and 4:13 A.The corresponding band structure is provided inFig. 2(a). As a result of the reduction of symmetry, therepulsion of the hybridized - bands [19] at the Fermilevel opens a 0:2 eV band gap (LDA value). This impliesthat the C(5,0) nanotube undergoes a metallic-semiconducting transition with TCDW larger than roomtemperature. Following Ref. [9], such a transformationcan be referred to as a ‘‘pseudo-Peierls’’ transition sinceit opens the band gap through a (long-wavelength) struc-tural deformation.In the C(3,3) case, we do not observe any zone-centerdeformation leading the band structure away from the well-documented [20] zone-folding picture of two linear bandscrossing at the Fermi level EF. However, compared to thezone-folding analysis, the Fermi wave vector kF is nolonger at 2=3X but at 0:58X so that 2=2kF is nolonger commensurate with the unperturbed lattice (or com-mensurate with a very large supercell). This is a well-documented effect of the curvature.We further present in Fig. 3(a) the phonon band structureof the C(3,3) tube [21]. The most salient feature is the giantFIG. 2. Band structures of (a) the ‘‘distorted’’ D2h C(5,0)nanotube and (b) the C(3,3) tube. The zero of energy has beenset to the top of the valence bands and at the Fermi level,respectively.01550Kohn anomaly [22] at q  2kF leading to a dramatic soft-ening of a few phonon modes [23]. In particular, the opticalband starting at 620 cm1 at  becomes the lowest vibra-tional state at q  2kF [thick/dotted low-lying band inFig. 3(a)]. The associated atomic displacements, indicatedin Fig. 1(b), correspond to an out-of-plane optical mode inagreement with the results of Ref. [12]. This is consistentwith the analysis provided by Mahan [24]: the coupling ofelectrons with out-of-plane optical modes should be en-hanced away from zone center and with decreasing diame-ter. The predominance of such a mode for e-ph coupling isa strong manifestation of the importance of curvature insmall diameter tubes.The value of the associated frequency at q  2kFstrongly depends on the energy distribution broadeningused in the phonon calculation, as clearly illustrated inFig. 3(a) by the difference between the thick and dottedlines. Using a 25 meV FD distribution (T  300 K), wefind a negative phonon mode !2kF;1 of 200 cm1.This means that the C(3,3) tube undergoes a Peierlsdistortion with TCDW larger than room temperature [25].Consequently, at the mean-field SC transition temperaturemeasured for the 4- A diameter CNTs (TSC  15 K), theC(5,0) and C(3,3) tubes should both be semiconducting,seriously questioning the origin of superconductivity insmall diameter tubes.Our results are in good agreement with those of Ref. [13]concerning the C(3,3) tube, but contrast significantly withthose based on a model for the interaction Hamiltoniansand the electronic susceptibility [12]. We emphasize thatour self-consistent treatment automatically includes therenormalization of both the e-ph and e-e interactions byelectronic screening at the DFT level.FIG. 3. Phonon band structures for the C(n; n) nanotubes withn  3, 4, and 5 from left to right obtained with a Gaussianbroadening of 136 meV (10 mRy). The dotted line for the C(3,3)nanotube corresponds to the additional explicit calculation of thephonon modes around q  2kF with a reduced 68 meV (5 mRy)broadening. Important bands appear as thicker lines. The un-dressed!0q bands (see text) are indicated with thick dashed lines.3-2TABLE I. Contributions () to the coupling constant for allthe q vectors in the neighborhood of  q  fg and 2kF q f2kFg for the C(n; n) tubes with n  3, 4, and 5. The corre-sponding frequencies (!) at  and 2kF are indicated in cm1.For the C(3,3) tube, the values underlined for q  f2kFg corre-spond to the undressed frequencies (see text). The global cou-pling constant () is indicated at the last line.C(3,3) C(4,4) C(5,5)!  !  ! q  fgRBM 536 0.009 416 0.005 338 0.001ZO 692 0.009 799 0.004 848 0.001A1(L) 1354 0.007 1468 0.009 1538 0.002A1(T) 1431 0.012 1505 0.009 1570 0.002q  f2kFg520 0:074 414 0.022 497 0.010499 0.026 509 0.010 533 0.0041320 0:018 1013 0.010 1154 0.0091144 0.001 1180 0.014 1237 0.0110.156 0.083 0.040PRL 94, 015503 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending14 JANUARY 2005To study the influence of tube diameter on the softeningof the modes at q  2kF, we represent in Fig. 3(b) and 3(c)the phonon band structure of the C(n; n) tubes with n  4and 5 [21]. Again, we observe that the same optical band(thick low-lying band) is softened at 2kF, but clearly theeffect is strongly reduced with increasing diameter. Thisconfirms that TCDW will quickly lower with increasingdiameter. The large amount of sp3 character in ultrasmalltubes invites to draw a comparison with the SC transition indoped clathrates [26] and diamond [27].Surprisingly, the softening of another mode at higherenergy, starting as an optical mode around 1450 cm1,(also indicated by a thick line) seems to be less sensitiveto the diameter. This softening is the signature of a largee-ph coupling which might thus play an important role inthe transport properties of larger CNTs [28].To further quantify the strength of the e-ph interaction,we can calculate the e-ph coupling matrix elementsgqknn0 and the coupling constant  Xq;q;  2NEfXq;hjgqj2i= h!q (1)hjgqj2i Z dkLBZXn;n0jgqknn0j2kq;n0 knNEF2(2)gqknn0 h2M!q1=2h 0nkj^q Vu^qj 0n0kqi (3)Because of conservation of energy and momentum, onlyvibrational states around q   and 2kF contribute to ,as the nesting factor [29] nq quickly decays away fromthese points. At , only four modes contribute to  due tosymmetry considerations. They are the radial breathingmode (RBM), the optical out-of-plane (ZO) mode, andthe in-plane optical longitudinal A1(L) and transverseA1(T) (G-band) modes (see Ref. [9]) located, respectively,at 536 cm1, 692 cm1, 1354 cm1, and 1431 cm1 inthe C(3,3) tube.In Table I, we report the parameter ; q  fg definedas the q sum of all the q; [see Eq. (1)] for q in theneighborhood of  on a given  band [30]. This couplingstrength varies from one band to another and is enhancedwhen the tube diameter decreases. However, the variationwith the diameter differs from one mode to another and it isdifficult to extract a simple scaling law.We now discuss the e-ph coupling at q  2kF. Thestrong variation of the lowest frequency !2kF;1 withtemperature raises the question of the meaning of2kF;1 calculated with the gq and !q values obtainedfrom ab initio calculations. The difficulties have beenclearly exposed in Ref. [12]. If one is interested in studyingthe SC transition, one faces the difficulty that at the ex-perimental TSC, the self-consistent g2kF; and !2kF; valuesare those of the CDW phase. As TCDW is larger than any01550realistic SC transition temperature, we will not attemptto estimate TSC with e-ph vertices and ph-propagatorsproperly ‘‘undressed’’ from the CDW instability [12]. Welimit ourselves to identifying the relevant modes and pro-viding the values for 2kF;1 obtained above TCDW, whichare those that matter for transport measurements in thenormal state.While the g2kF parameters are rather stable with tem-perature above TCDW, the phonon frequencies still varysignificantly. Following Ref. [12], we provide an upperbound for !2kF; by removing the Kohn’s anomaly.Namely, we assume that around q  2kF,!2q;  !0q;2  A lnjq 2kF=q 2kFj (4)and fit the !2q; bands by a cubic polynomial for !0q;2plus the logarithmic term that we subtract to obtain the barephonon frequency. The important undressed bands (corre-sponding to !0q;) are represented by dashed lines inFig. 3(a). The related frequencies and 2kF; values areunderlined in Table I. This treatment is applied to bothstates presenting the largest Kohn anomaly, but its effect isclearly more pronounced for the low-lying   1 band.Using the !02kF; frequencies leads to providing a lowerbound for the corresponding 2kF; e-ph parameter. Evenin that limit, we observe that the coupling with modes atq  2kF is significantly larger than with long-wavelengthphonons. As at zone center, few modes with q  2kFcouple to the electrons and the strength of the coupling issignificantly enhanced with decreasing diameter. In theC(3,3) case, we obtain in the undistorted phase a densityof states NEF  0:4 states=eV=cell=spin, yielding ane-ph potential Vep  =NEF  440 meV. This is largerthan that in the fullerenes (60–70 meV), but it is not3-3PRL 94, 015503 (2005) P H Y S I C A L R E V I E W L E T T E R S week ending14 JANUARY 2005sufficient to lead to any significant TSC value. The muchlarger density of states in the ‘‘D10h’’ C(5,0) tube(NEF  1:8 states=eV=cell=spin) may lead in principleto a large TSC value. However, as shown above, the pre-vailing structure around the experimental TSC temperatureis the insulating D2h one.In conclusion, we have shown that the ‘‘metallic’’ C(5,0)and C(3,3) nanotubes undergo a Peierls transition with acritical temperature larger than 300 K. Our results arederived within Mermin’s generalization of DFT to finitetemperature systems and do not rely on models for the e-phinteractions nor the electronic susceptibility. Additionalphysical ingredients (interaction with the zeolite network,possible defective structure, Luttinger liquid character,etc.) might reconcile these results with the experimentallyobserved SC transition at TSC  15 K.Calculations have been performed at the French CNRSnational computer center at IDRIS (Orsay). G. M. R. andJ. C. C. acknowledge the National Fund for ScientificResearch [FNRS] of Belgium for financial support. Partsof this work are also connected to the Belgian Program onInteruniversity Attraction Poles (PAI5/1/1) on QuantumSize Effects in Nanostructured Materials, to the Actionde Recherche Concerte´e ‘‘Interaction e´lectron-phonondans les nanostructures’’ sponsored by the Communaute´Franc¸aise de Belgique, and to the NanoQuanta and FAMEEuropean networks of excellence.[1] Z. K. Tang et al., Science 292, 2462 (2001).[2] M. Kociak et al., Phys. Rev. Lett. 86, 2416 (2001).[3] J. Gonza´lez, Phys. Rev. Lett. 87, 136401 (2001); Phys.Rev. Lett. 88, 076403 (2002); J. V. 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Reich, C. Thomsen, D. Sa´nchez-Portal,and P. Ordejo´n, Phys. Rev. B 66, 155410 (2002); H. J. Liu,and C. T. Chan, Phys. Rev. B 66, 115416 (2002).[21] The phonon band structures are obtained by a specialinterpolation technique [15] based on the explicit calcu-lation of the dynamical matrices on a 20 q-point grid.[22] W. Kohn, Phys. Rev. Lett. 2, 393 (1959).[23] As expected, this softening does not appear in classicallattice dynamical models. See, Z. M. Li et al., Appl. Phys.Lett. 84, 4101 (2004). Our results do not exclude theexistence at very low temperature of an acoustic-phonon-mediated instability [6,11], but such an instabilitywill be clearly dominated by the one at q  2kF.[24] G. D. Mahan, Phys. Rev. B 68, 125409 (2003).[25] Using various values of FD broadening, TCDW can beestimated to lie with 300 K and 400 K.[26] H. Kawaji et al., Phys. Rev. Lett. 74, 1427 (1995);K. Tanigaki et al., Nat. Mater. 2, 653 (2003);D. Conne´table et al., Phys. Rev. Lett. 91, 247001 (2003).[27] E. A. 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Room temperature Peierls distortion in small radius nanotubes

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