Metallic single-wall carbon nanotubes (SWCNTs) assembled into crystalline ropes and mats show a resistivity that rises approximately linearly with temperature from the crossover temperature up to 580 K, with a low-temperature upturn below 10-100 K. Kane, Mele, Lee, Fischer and coauthors (EPL 41, 683 (1998)) attributed the linear regime to backscattering of electrons by thermally excited twistons, the long-wavelength torsional acoustic modes of a tube, and predicted a rope slope of 0.005 μΩ cm/K0.005\,\mu\Omega\,\mathrm{cm/K}, a factor of 8-20 below the measured slopes. They posed four linked open issues: the origin of the low-temperature upturn (collective charge- or spin-density-wave state versus disorder and inter-tube coupling effects), the extrinsic-versus-intrinsic nature of the slope discrepancy (non-metallic chiral tube fraction, orientation-dependent inter-tube tunneling, uncertain parameters), the absence of decisive high-temperature transport measurements on single isolated tubes with which to test the twiston mechanism and its diameter dependence, and the lack of quantitative determinations of the inter-tube tunneling bandwidth WW and the librational gap ℏω0\hbar\omega_0 that control the one- to three-dimensional crossovers. The quantitative determinations of the inter-tube tunneling bandwidth WW and the librational gap ℏω0\hbar\omega_0 in realistic mixed-chirality ropes, with the resulting one- to three-dimensional crossover quantification, and a direct wide-temperature-range, diameter-resolved test of the linear-in-TT twiston resistivity on single isolated tubes, remain open.

Background

A single-wall carbon nanotube (SWCNT) is a graphene sheet rolled into a cylinder, its wrapping fixed by two integers; the so-called armchair tubes, labeled [N,N][N,N], are metallic by band theory. The double-laser-ablation synthesis of Thess et al. (Science 273, 483 (1996)) produced high-purity material dominated by [10,10][10,10] tubes that self-assemble into ropes (crystalline bundles on a triangular lattice) and mats of entangled ropes. Four-probe and microwave measurements showed metallic behavior with a resistivity increasing approximately linearly with temperature, first up to 470 K (J. E. Fischer et al., Phys. Rev. B 55, R4921 (1997)) and then up to 580 K on bulk, rope and single-rope samples in the Kane et al. EPL 41, 683 (1998) study.

C. L. Kane and E. J. Mele (Phys. Rev. Lett. 78, 1932 (1997)) had shown that among the long-wavelength shape fluctuations of a tube only the twist couples right-moving and left-moving conduction electrons; the EPL 1998 paper developed this into a transport theory of twistons, the acoustic torsional modes. Because the twistons relevant at temperature TT carry energy of order 2kBT vt/vF2k_BT\,v_t/v_F (with vt≪vFv_t\ll v_F the twiston and Fermi velocities), they are always heavily thermally populated, so backscattering is not frozen out at low temperature as it would be in the usual Bloch–Grüneisen picture of phonon scattering in a metal; the resulting one-dimensional resistivity is proportional to the absolute temperature,

ρ1D=932π2he2β2N ct kBT,\rho_{1D}=\frac{9}{32\pi^2}\frac{h}{e^2}\frac{\beta^2}{N\,c_t}\,k_B T,

where β\beta is the logarithmic derivative of the hopping with bond length, ctc_t is the (diameter-independent) twist modulus per unit of NN, and 1/N1/N gives the tube-diameter dependence. For a rope of close-packed [10,10][10,10] tubes this predicts dρ3D/dT≈0.005 μΩcm/Kd\rho_{3D}/dT\approx0.005\,\mu\Omega\mathrm{cm/K}, whereas the measured slopes were 0.040.04-0.1 μΩcm/K0.1\,\mu\Omega\mathrm{cm/K}, i.e. 8-20 times larger. An alternative intrinsic linear-in-TT mechanism, Umklapp scattering of repulsively interacting electrons (the two-channel ladder mapping of L. Balents and M. P. A. Fisher, Phys. Rev. B 55, R11973 (1997), where Umklapp means momentum-non-conserving electron-electron scattering onto a reciprocal lattice vector), scales instead as 1/N21/N^2 and was estimated to be of comparable magnitude at N=10N=10.

Rope-level complications were central to the 1998 paper. Electron diffraction on similar material suggested that more than half of the tubes in a rope are chiral and hence non-metallic, so that a fraction of the rope cross-section is electrically dead. Inter-tube tunneling of electrons, characterized by a bandwidth WW that depends on the relative orientational registry of neighboring tubes, both frustrates the nesting of the one-dimensional Fermi surface and, because the tunneling matrix elements vary with orientation, can provide an additional backscattering channel. Elastic coupling between twistons on neighboring tubes opens a librational gap ℏω0\hbar\omega_0 in the twiston spectrum as q→0q\to0 (estimated near 4 meV from the corresponding graphite zone-boundary phonon and the libron energies of crystalline C60). The one-dimensional regime holds for kBT>max⁡(ℏω0,(vt/vF)W)k_BT>\max(\hbar\omega_0,(v_t/v_F)W), so WW and ℏω0\hbar\omega_0 set the crossovers between one- and three-dimensional transport. Finally, all samples showed an upturn of the resistivity at low temperature with an onset between about 10 and 100 K that tracked sample morphology and quality; it had been suggested that this signals condensation into a collective charge- or spin-density-wave ground state (a periodic modulation of the electronic charge or spin density driven by interactions), but the morphology dependence pointed instead to disorder or other three-dimensional effects. At the time, transport through a single tube had only been measured at very low temperature, where Coulomb blockade (single-electron charging of the tube) dominates the conductance.

Later studies revisited these four issues. The low-temperature upturn has been attributed to extrinsic effects — disorder-induced localization together with inter-tube coupling — rather than to a collective charge- or spin-density-wave ground state, and clean ropes instead show superconductivity below about 0.55 K. Room-temperature transport through single isolated tubes has been measured and found consistent with acoustic-phonon scattering, and the theory of inter-tube tunneling in mixed-chirality ropes has been refined, with momentum conservation along the tube strongly reducing the effective transverse coherence. Quantitative determinations of the inter-tube tunneling bandwidth WW and the librational gap ℏω0\hbar\omega_0 in realistic ropes, and a direct diameter-resolved test of the linear-in-TT twiston resistivity on isolated tubes, have not been established.

Problem Statement

Establish, quantitatively and mechanism-specifically, the origin of the temperature-dependent resistivity of metallic single-wall carbon nanotube materials, as posed by Kane et al. (EPL 41, 683 (1998)) at the close of their twiston theory. Among the linked issues that paper raised, this task addresses the surviving open core: (1) determine the inter-tube tunneling bandwidth WW and the librational gap ℏω0\hbar\omega_0 of realistic mixed-chirality ropes (the [10,10][10,10]-dominated material characterized in the Background), reporting WW as the orientational average over the thermally accessible inter-tube registries together with the spread across registries and reporting ℏω0\hbar\omega_0 as a value with its uncertainty, and thereby quantify the crossovers between one-dimensional and three-dimensional transport regimes by giving the resulting crossover temperatures; and (2) test the twiston mechanism and its predicted 1/N1/N diameter dependence against high-temperature transport in single isolated tubes, free of rope averaging and inter-tube effects, over a wide temperature range.

References

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