Delaney, Paul

Di Ventra, M.

Pantelides, S.T.

We report calculations of the transport properties of multiwalled carbon nanotubes based on a scattering-theoretic approach that takes into account scattering within each tube, between tubes, and at the metal contacts. We find that only the outer tube contributes to the conductance, as has been implied by experiments. Referring to experiments performed with liquid-metal contacts, we also explain why the measured conductance is close to an integer number of conductance quanta, when the tubes are immersed in the liquid metal for several hundreds of nanometers and is not an integer when they are immersed for only a few nanometers. Finally, we propose that the observed conductance of only one quantu

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Quantized conductance of multiwalled carbon nanotubesDelaney, P., Di Ventra, M., & Pantelides, S. T. (1999). Quantized conductance of multiwalled carbon nanotubes.Applied Physics Letters, 75(24), 3787-3789. DOI: 10.1063/1.125456Published in:Applied Physics LettersQueen's University Belfast - Research Portal:Link to publication record in Queen's University Belfast Research PortalGeneral rightsCopyright for the publications made accessible via the Queen's University Belfast Research Portal is retained by the author(s) and / or othercopyright owners and it is a condition of accessing these publications that users recognise and abide by the legal requirements associatedwith these rights.Take down policyThe Research Portal is Queen's institutional repository that provides access to Queen's research output. Every effort has been made toensure that content in the Research Portal does not infringe any person's rights, or applicable UK laws. If you discover content in theResearch Portal that you believe breaches copyright or violates any law, please contact openaccess@qub.ac.uk.Download date:15. Feb. 2017Quantized conductance of multiwalled carbon nanotubesP. Delaney,a) M. Di Ventra, and S. T. PantelidesDepartment of Physics and Astronomy, Vanderbilt University, Nashville, Tennessee 37235~Received 8 October 1999; accepted for publication 21 October 1999!We report calculations of the transport properties of multiwalled carbon nanotubes based on ascattering-theoretic approach that takes into account scattering within each tube, between tubes, andat the metal contacts. We find that only the outer tube contributes to the conductance, as has beenimplied by experiments. Referring to experiments performed with liquid-metal contacts, we alsoexplain why the measured conductance is close to an integer number of conductance quanta, whenthe tubes are immersed in the liquid metal for several hundreds of nanometers and is not an integerwhen they are immersed for only a few nanometers. Finally, we propose that the observedconductance of only one quantum ~instead of the expected two quanta! is due to intertubeinteractions. © 1999 American Institute of Physics. @S0003-6951~99!03650-5#Single and multiwalled carbon nanotubes ~MWNTs!have attracted considerable experimental and theoretical in-terest since they were first studied by Iijima in 1991.1 In thelast year, several groups have reported conductance measure-ments of individual tubes under various types of contacts.2–4In particular, the data reported by de Heer and co-workers2revealed puzzling features for which a theoretical explana-tion is lacking. The data show that electron transport in thesenearly perfect structures is ballistic. Within the Landauerformalism,5 we then expect the conductance of a nanotube tobe determined by the number of Fermi-level conductingchannels and the transmission factors Ti for each channel:explicitly,s5(i51MTi2e2h 5(i51MTiG0 , ~1!where G051/12.9 kV is the quantum of conductance, andthe sum runs over the M available quantum channels. The Tiare the transmission factors of each channel, with 0<Ti<1. For the important ~n,n! armchair nanotubes which aremetallic, there are two pz-derived bands at the Fermi level,one even and one odd under the n axial mirror symmetries ofthe tube, and so, one expects that a single armchair nanotubehas a conductance of (Te1To)G0 , where Te , To are thetransmission coefficients of these even and odd bands, re-spectively. If the transmission coefficients are unity, we thenget a conductance of 2G0 at the Fermi level. For the case ofa MWNT composed of m individual tubes, each having twochannels at the Fermi level, the maximum ballistic conduc-tance is thus 2mG0 . However, the data of de Heer andco-workers2 indicate clearly that only one tube in a MWNTcontributes to the conductance. In this experiment one end ofa MWNT was immersed in a liquid metal, with the other endin contact with a graphitic fiber. This experiment has alsoshown that, unlike the expected conductance of 2G0 ~assum-ing only the outermost tube conducts!, the conductance ofthe MWNT is G0 when the tube is immersed in the liquidmetal for several hundred nanometers and a fraction of G0when only the tip of the MWNT ~around 30 nm! is in theliquid.In this letter, we consider the puzzling results of theabove experiments and set up a model of a multiwallednanotube composed of armchair tubes to show that only theoutermost tube gives rise to conduction, as is indeed ob-served. We then consider the issue of the numerical value ofthe transmission coefficient for any open channel: this is notnecessarily unity and should, in principle, depend sensitivelyon the details of the contacts. We suggest, however, thatsaturation and decoherence effects associated with the liquid-metal contacts give rise to a transmission of unity for anyopen channel. This result applies also to many other contactgeometries used in transport experiments6 and is thus of gen-eral validity.We describe the scattering properties of a finite-lengthMWNT by placing at each C site a potential with two prop-erties: it acts as a splitter7 for the propagating electron wavefunction with three channels for in-tube scattering ~along thesp2 bonds! and two channels for intertube scattering ~oneabove and one below the tube!, and it contains a singlebound state that captures the physics of the pz-derived p andp* Fermi-level bands. The C – C distance, which governsthe decay in the wave function as it propagates along thebonds, has been fixed to the experimental value of 1.42 Å.The single-splitter bound-state energy is fixed in such a waythat the band structure of a single infinite nanotube is cor-rectly reproduced in the vicinity of the Fermi level, which isthe energy region of interest in our problem. The MWNT isformed by putting m ~n,n! nanotubes inside each other andconnecting the vertical connections of the splitters in theappropriate way. This is the simplest possible type ofMWNT where all the tubes have the same indices and arealigned exactly with each other. A schematic picture of aMWNT composed of two tubes is presented in Fig. 1. Weterminate the MWNT at each end by connecting each of the2n outward-pointing scattering connections on the outermostnanotube to a potential step, which models the presence ofthe contact on the outermost tube. We follow the methodoutlined in Ref. 7 to generate the scattering matrix of thea!Electronic mail: delaney@dynix.phy.vanderbilt.eduAPPLIED PHYSICS LETTERS VOLUME 75, NUMBER 24 13 DECEMBER 199937870003-6951/99/75(24)/3787/3/$15.00 © 1999 American Institute of PhysicsDownloaded 01 Feb 2005 to 169.229.15.142. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspsingle splitter and then of the entire network. Using this 4n34n matrix, we calculate the transmission coefficients of thenetwork at any given energy. The conductance is then calcu-lated according to Eq. ~1!. Other details of the calculationswill be given elsewhere.8We first note that with this model we reproduce someknown results for single-tube transport: for the simplest caseof a perfect nanotube we find, as expected, two channels atthe Fermi level, with a maximum conductance of 2G0 .9With a single vacancy present one channel is lost,9 and withan axial magnetic field present an Aharonov–Bohm effect isseen where the resistance varies periodically with the numberof flux quanta through the tube.10 The model is thus robustenough to investigate the effects.Only the outermost tube conducts. The experiments ofRefs. 2 and 11 strongly suggest that only a few, or possiblyone, of the nanotubes in a MWNT contribute to the conduc-tance. Indeed, with our model we confirm that the presenceof the other tubes in the MWNT does not affect the overallconductance significantly. The current is injected into theoutermost tube of the MWNT, as in the experiments, buteven with m tubes present we found only two open channelsat the Fermi level, yielding a maximum conductance of 2G0 .We note that, due to the intertube interaction, the states ofone tube mix with those of another to form the eigenstates ofthe composite system. In this way, the states of the innertubes, and therefore their conductance channels, should bepresent on the outermost nanotube, and so leak to the con-tacts, suggesting an increase of conductance. However, onlythe local density of states ~LDOS! at the position where theelectrons are injected is important for determining the con-ductance, and the LDOS on the outermost tube is notchanged even if the electrons delocalize from tube to tube.12Furthermore, the single-nanotube wave functions at theFermi level decay very quickly away from the tube surfaces.The work function for graphite is about 4.7 eV,13 which cor-responds to a wave-function decay constant of 0.9 Å. If weassume exponential behavior to be valid 1 Å outside thesurface of the tube, within a tube–tube distance of 3.5 Å thewave function has decayed to 25% of its initial value. Oncewe include the three-dimensional nature of the Schro¨dingerequation, however, a much larger decay rate is predicted.14The decay constant is amplified by the kinetic energy of theelectron along the surface of the tube. For graphite, at theFermi level, the surface oscillations of the wave functioncorrespond to an energy of 11 eV. This effective increase inthe work function generates much greater decay, and follow-ing the above analysis we now find that between tubes thewave function should have decayed to 5% of its initial value.Thus, the transmission coefficient for propagation from thesurface into the inner tubes will be very small on average,even for the tube closest to the outermost tube.The outermost transmission coefficients are unity. In ad-dition to the scattering inside the tube, our study also in-cludes the reflections that occur when the electron enters andleaves the nanotube via the contacts. We found that the re-sulting transmission factors depend sensitively on the poten-tial structure in the interface region between the nanotubeand the contacts, and so, should be some arbitrary real num-bers that would need to be calculated from an ab initioapproach.15 On the other hand, de Heer and co-workers2have shown that the transmission coefficients turn out to beclose to integers. It is hard to reconcile this result with elastictransmission factors as in the Landauer formula ~1!. There-fore, the question arises as to why the measured conduc-tances are of the form iG0 , where i is an integer, and not ofthe expected form iT¯ G0 , where T¯ is the average transmis-sion of the available channels which may be much less than1. The answer lies in the nature of the contacts made to thenanotube, and here we consider both the liquid–tube and thetube–fiber interface.First, for the contact with the liquid metal, we note thatunlike the configuration pictured in the Landauer approach5in which the reservoirs are linked to the sample by horn-shaped ballistic conductors, here the nanotube is directly in-serted into the reservoir. Two effects tend to enhance theeffective transmission and cause it to saturate at one. As weare really in a time-dependent inelastic regime, the transmis-sion coefficient at a given energy for hopping to a particulararea of the tube will oscillate with time, due to thermal ef-fects. Also, the more surface area of the nanotube that makescontact with the liquid metal, the more transitions happen perunit time, thus saturating any sufficiently open channel byfilling it with electrons. One could think of the liquid asbeing composed of a series of layers of thickness 10 nm orso, with each layer acting as a separate reservoir, and eachmaking contact with the same nanotube. Each time an elec-tron tunnels into the nanotube the individual probability oftunneling is not just of the form of a transmission as inLandauer’s formula ~1!, but must also include the nonequi-librium distribution function reflecting the occupation of thenanotube mode into which the electron is tunneling. Givensufficiently many ~incoherent! opportunities for tunnelinginto this mode, the latter will eventually be completely filled,and the measured current will be that appropriate to a com-pletely filled mode. This yields an entire quantum of conduc-tance, and so, working backwards from Eq. ~1!, we deducean effective transmission factor of 1 for this mode. We callthis transmission factor effective because it does not comefrom a calculation for any one particular atomic geometry,but from the cumulative effects of many different geom-etries, at many different sites on the MWNT surface. Weassume here that the exponential decay between the tubes issufficiently strong to render the LDOS of the inner tubesinaccessible to this saturation mechanism.As for the other contact, the nanotube is embedded in agraphitic fiber composed of nanotubes and graphitic par-FIG. 1. Schematic view of a multiwalled nanotube formed from two tubes.3788 Appl. Phys. Lett., Vol. 75, No. 24, 13 December 1999 Delaney, Di Ventra, and PantelidesDownloaded 01 Feb 2005 to 169.229.15.142. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspticles, connected to a gold wire with silver paint.2 Thus, thereshould be many contacts to the nanotube generating manyconducting paths leading up to the gold wire. Therefore, inthe elastic scattering approach, the effective transmissionfrom the tube to the fiber should be high. The rest of the fiberbeyond this immediate contact region has a small resistance:taking the nominal figures given in Ref. 2 and a resistivity ofroughly 500 mV cm, we calculate a total resistance of 200 Vfor the fiber.16 This small series resistance might explain whythe conductance plateaus observed in Ref. 2 seem to beslightly below integer values.Now, if there is an elastic scatterer in between the twocontacts, the conductance will be reduced according to theLandauer formula ~1!. The cap–shaft interface between thenanotube tip and the bulk acts as such a scattering center,and the conductance would lower to noninteger values whenonly the tip is immersed into the liquid metal, as the authorsproposed in Ref. 2 to explain their experimental observa-tions. Once this junction is pushed below the surface, itseffect on electron transmission is removed and the conduc-tance rises up to the full integer value: the electrons can nowtunnel from the liquid metal directly into the bulk of thenanotube.We conclude the letter by analyzing the last puzzlingaspect of the liquid-metal experiment, i.e., the discovery thatonly one quantum of conductance is measured for theMWNT, and not two. From our arguments above, it is un-likely that this reduction arises from the physics of theliquid–tube interface, as the disorder and high contact areashould guarantee that all available conductance channels arefilled. However, intertube interactions in ropes can substan-tially alter the physics of ropes precisely at the Fermi level.13These interactions perturb the band structure of the singletubes significantly and cause the existing even and odd bandsto repel each other, creating a depletion of states at the Fermilevel. This effect is similar to that observed in AB graphite,where instead of four bands being present at the Fermi level,only two are found.13 In graphite, this interlayer interactionreduces the LDOS at the Fermi level by a factor of 1/2 ex-actly, as the other states that would have completed theLDOS of the graphene sheets have been pushed away inenergy from the Fermi level. As long as this repulsion isgreater than the small bias voltages used in the experiment,these other states will not be available for transport. A clearpicture of this effect can be seen in scanning tunneling mi-croscopy ~STM! images of graphite, where exactly one halfof the surface atoms are visible at low biases.17 We thuspropose that the intertube interaction, either between thetubes in the MWNT or at the contacts between tubes in thefiber can cause similar band repulsion, reducing the LDOSpresent precisely at the Fermi level by a factor of 1/2 andthus reducing the number of channels available at this energyfrom 2 to 1.This work was supported in part by Grant No. MDA972-98-1-0007 from DARPA.1 S. Iijima, Nature ~London! 354, 56 ~1991!.2 S. Frank, P. Poncharal, Z. L. Wang, and W. A. de Heer, Science 280, 1744~1998!.3 J. W. G. Wildoer, L. C. Venema, A. G. Rinzler, R. E. Smalley, and C.Dekker, Nature ~London! 391, 59 ~1998!.4 R. Martel, T. Schmidt, H. R. Shea, T. Hertel, and Ph. Avouris, Appl. Phys.Lett. 73, 2447 ~1998!.5 R. Landauer, J. Phys.: Condens. Matter 1, 8099 ~1989!.6 B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Williamson, L.P. Kouwenhouven, D. van der Marel, and C. T. Foxon, Phys. Rev. Lett.60, 848 ~1988!.7 F. Gagel and K. Maschke, Phys. Rev. B 49, 17170 ~1994!; M. Di Ventra,F. Gagel, and K. Maschke, ibid. 55, 1353 ~1997!.8 P. Delaney, M. Di Ventra, and S. T. Pantelides ~unpublished!.9 L. Chico, L. X. Benedict, S. G. Louie, and M. L. Cohen, Phys. Rev. B 54,2600 ~1996!.10 H. Ajiki and T. Ando, J. Phys. Soc. Jpn. 62, 1255 ~1993!.11 A. Bachtold, C. Strunk, J.-P. Salvetat, L. Forro, T. Nussbaumer, and C.Scho¨nenberger, Nature ~London! 397, 673 ~1999!.12 Similar considerations apply to transport in the STM. See, e.g., J. Tersoffand D. R. Hamann, Phys. Rev. B 31, 805 ~1985!.13 P. Delaney, H. J. Choi, J. Ihm, S. G. Louie, and M. L. Cohen, Nature~London! 391, 466 ~1998!; Phys. Rev. B 60, 7899 ~1999!.14 J. Tersoff, Appl. Phys. Lett. 74, 2122 ~1999!. See, also, P. Delaney and M.Di Ventra, Appl. Phys. Lett. ~to be published!.15 M. Di Ventra and S. T. Pantelides, Phys. Rev. B 59, R5320 ~1999!; M. DiVentra, S. T. Pantelides, and N. D. Lang ~unpublished!.16 See, e.g., V. Natarajan and J. A. Woollam, Mater. Res. Soc. Symp. Proc.20, 45 ~1983!.17 D. Tomanek, S. G. Louie, H. J. Mamin, D. W. Abraham, R. E. Thomson,E. Gainz, and J. Clarke, Phys. Rev. B 35, 7790 ~1987!.3789Appl. Phys. Lett., Vol. 75, No. 24, 13 December 1999 Delaney, Di Ventra, and PantelidesDownloaded 01 Feb 2005 to 169.229.15.142. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

Quantized conductance of multiwalled carbon nanotubes

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