VOLUME84, NUMBER 19 PHYSICAL REVIEW LETTERS 8MAY2000Electrical Transport in Rings of Single-Wall Nanotubes: One-Dimensional LocalizationH.R. Shea, R. Martel, and Ph. Avouris*IBM T.J. Watson Research Center, Yorktown Heights, New York 10598(Received 14 April 1999)We report low-temperature magnetoresistance (MR) measurements on rings of single-wall carbon nano-tubes. Negative MR characteristic of weak one-dimensional localization is clearly observed from 3.0 to60 K, and the coherence length Lwis obtained as a function of temperature. The dominant dephasingmechanism is identified as electron-electron scattering. Below 1 K, we observe a transition from weakto strong localization, and below 0.7 K a weak antilocalization is induced by spin-orbit scattering.PACS numbers: 73.61.Wp, 61.48.+c, 72.15.Rn, 73.50.–hCarbon nanotubes (NTs) provide ideal model sys-tems to test theories describing transport phenomenain low-dimensional systems. Nanotubes come in twoforms: large diameter (typically 10–30 nm) multiwallnanotubes (MWNTs) and smaller diameter (typically1–2 nm) single-wall nanotubes (SWNTs). The power-ful technique of magnetoresistance (MR) has alreadybeen used to investigate the transport mechanism inMWNTs [1–3]. Weak localization [4] was observedwhich allowed the coherence length Lwin MWNTsto be determined. SWNTs are attracting even moreinterest, their being closer to ideal one-dimensional (1D)systems. It has been suggested that backscattering isineffective in SWNTs, but it can be switched on byan external magnetic field leading to a large positivemagnetoresistance [5], and that transport is ballistic[6–9]. However, recent electrical measurements onsemiconducting SWNTs were found to be consistentwith diffusive transport [9,10]. The nature of transportin metallic SWNTs is still a matter of debate. MRmeasurements on metallic SWNTs could in principledetermine the transport mechanism and reveal the natureof the inelastic scattering (dephasing) processes involved.Attempts to measure MR in SWNT bundles have notbeen successful so far [11]; however, negative MR atlow fields has been reported from entangled SWNT mats[12]. Recently, we have been able to fabricate ringsfrom SWNTs [13], and we observed MR from someof these rings at low temperatures. Furthermore, unlikepast low temperature studies (for example, see Refs. [6]and [14]), our SWNT rings do not exhibit Coulombblockade even at the lowest temperatures (0.3 K). Thesestudies enable us to determine the transport mechanism,the dominant electron dephasing mechanism, to observethe transition from a weakly to a strongly localizedstate, and spin-related scattering phenomena.Figure 1 is an atomic force microscope image ofa SWNT ring deposited over two 25 nm thick Ti兾Auelectrodes. The electrodes are patterned by e-beam lithog-raphy on 100 nm of SiO2grown on degenerately dopedSi. Typical ring resistances range from 20 to 50 kV at300 K.The magnetoresistance of a 0.82 mm diam and 20 nmthick ring is shown in Fig. 2a for temperatures between3.0 and 6.0 K. The MR is negative; i.e., the resistancedecreases with increasing magnetic field. Negative MR ischaracteristic of materials in a state of weak localization[15], in our case one-dimensional weak localization(1D-WL). WL results from the constructive interferencebetween conjugate electron waves counterpropagatingaround self-intersecting electron trajectories inside thematerial [4,16]. The closed ring geometry in principlemay provide an additional path for interference. Theenhanced backscattering produced by the constructiveinterference leads to an increased nanotube resistance,the magnitude of which depends on the number and thestrength of the dephasing collisions that the electronsexperience inside the nanotube. By applying a magneticfield perpendicular to the ring, the conjugate electronwaves acquire opposite phases and the constructive inter-ference is destroyed. From the effect of the field on theresistance, the coherence length Lwcan thus be obtained.The change of the conductance DG共H兲 of a unit lengthalong the circumference of a metallic ring of radius R andof wall thickness w smaller than Lwcan be written as [17]FIG. 1. Atomic force microscope image of a nanotube ringspanning two gold electrodes.0031-9007兾00兾84(19)兾4441(4)$15.00 © 2000 The American Physical Society 4441VOLUME84, NUMBER 19 PHYSICAL REVIEW LETTERS 8MAY20001101009080dV/dI (kΩ)-4 -2 0 2 4Magnetic Field (T)6.0 KLϕ = 435 nmLϕ = 306 nm3.0 K500450400350300250Lϕ (nm)6.05.55.04.54.03.53.0Temperature (K)(a)(b)FIG. 2. (a) Differential resistance dV 兾dI of a nanotube ringas a function of the magnetic field perpendicular to the planeof the ring. The probe current was 10 pA. The four data sets(in gray), from top to bottom, were taken at 3.00, 4.00, 5.00,and 6.00 K and are not offset. The solid black lines are fitsto the 1D weak localization theory using w 苷 1.4 nm. Similarfits were performed using different wall thicknesses w between1.4 and 20 nm. For example, the obtained values of Lwat3.00 K are 342 nm for w 苷 2 nm and 216 nm for w 苷 3 nm.(b) Coherence length from Eqs. (1) and (2) vs temperature (w 苷1.4 nm). The line is a fit to Lw~ T21兾3.DG共H兲 苷 22e2Lw共H兲hsinh共2pRLw共H兲兲cosh共2pRLw共H兲兲 2 cos关2p2FF0兴,(1)where H is the magnetic field perpendicular to the ringthrough which a flux F passes, F0苷 h兾e is the flux quan-tum, and Lw共H兲 is the magnetic field dependent coherencelength [16]:1L2w共H兲苷1L2w113µ2pwHF0∂2. (2)Lwcan be determined by fitting the measured MR toEqs. (1) and (2), with DG scaled by a factor A to accountfor the transmission of the gold electrode–NT barriers.We determined A from fits to the 4 K data to be A ⬃ 0.2,which corresponds to a contact resistance of 22 kV.Weused the same A to fit the data at other temperatures.The width w entering Eq. (2) needs careful considera-tion. It can be argued that the most appropriate value is1.4 nm, i.e., the diameter of the most abundant nanotubein the SWNT sample. A very good fit of the negative MRdata to Eqs. (1) and (2) of 1D-WL theory is obtained asshown by the solid dark lines in Fig. 2(a). The obtainedLware plotted in Fig. 2(b) vs temperature, and range from306 nm at 6 K to 435 nm at 3 K. A possible explanationfor the fact that MR was not observed in straight bundlesof SWNTs and in some of our ring samples may be thatMR requires either a stronger metallic tube-tube coupling(which may be a function of sample