Extreme Magneto-transport of Bulk Carbon Nanotubes in Sorted Electronic Concentrations and Aligned High Performance Fiber

We explored high-field (60 T) magneto-resistance (MR) with two carbon nanotube (CNT) material classes: (1) unaligned single-wall CNTs (SWCNT) films with controlled metallic SWCNT concentrations and doping degree and (2) CNT fiber with aligned, long-length microstructure. All unaligned SWCNT films showed localized hopping transport where high-field MR saturation definitively supports spin polarization instead of a more prevalent wave function shrinking mechanism. Nitric acid exposure induced an insulator to metal transition and reduced the positive MR component. Aligned CNT fiber, already on the metal side of the insulator to metal transition, had positive MR without saturation and was assigned to classical MR involving electronic mobility. Subtracting high-field fits from the aligned fiber’s MR yielded an unconfounded negative MR, which was assigned to weak localization. It is concluded that fluctuation induced tunnelling, an extrinsic transport model accounting for most of the aligned fiber’s room temperature resistance, appears to lack MR field dependence.


S1--Fitting Hopping Conduction.
Qualitatively for the as-is unaligned SWCNT films, the asymptotic increase of resistance R as temperature T approaches absolute zero is a strong indicator of the variable range hopping (VRH) model. Following are the model fits to variable range hopping for the as-is unaligned SWCNT films. Although they have not been deliberately chemically treated, they have doping from atmosphere exposure. Here is the temperature dependent variable range hopping equation where R0T is the high temperature limit of the resistance, and q equals ½ for ES variable range hopping, 1/3 for two dimensional variable range hopping, and ¼ for three dimensional variable range hopping. The fits show that the as-is 98% semi-conducting film fits best to ES-variable range hopping, the unsorted film fits best to two dimensional variable range hopping, and the 95% metallic film fits bests to three dimensional variable range hopping. These results are largely consistent with literature [1] [2].
Figure S1-1| Two dimensional variable range hopping gives the best fit for the as-is unsorted unaligned SWCNT film. Black is the data and red is the fit.
Figure S1-2| Three dimensional variable range hopping gives the best fit for the as-is metallic unaligned SWCNT film. Black is the data and red is the fit.
Figure S1-3| ES variable range hopping gives the best fit for the as-is 98% semi-conducting unaligned SWCNT film. Black is the data and red is the fit. Figure S2-1 plots MRSAT against temperature T in a way that linearizes equation (3) in the main text, the MR saturation equation. The fitted exponents are -0.39 (for the metallic SWCNT film) and -0.82 (for the unsorted SWCNT film), whereas the spin saturation model expects exponents of -1/4 [3] and -1/3 [4] respectively. As explained in the main text, this disagreement is from the offset of another negative MR mechanisms.

Figure S2-1|
The saturated MR value, MRSAT, as a function of temperature T, displayed in a way that linearizes equation (3) from the main text. Note that a separate, negative MR contribution makes the measured value lower than that predicted by the spin saturation model. SWCNT film) and 2.04 (for the unsorted unaligned SWCNT film), where the shrinking wave function model expects 3/4 (for three dimensional hopping [5]) and 1 (for two dimensional hopping [6]). The quality of these linear fits are not as good and the fitted parameters are far off the theoretical values, at least compared to the fit quality of the rivalling spin saturation model.
Thus, the fit of the spin saturation model from higher field data is better than the fit of the wave function shrinking model from lower field data. this slope, collected for a range of temperatures, is plotted against temperature in a log-log plot.
S4-Fitting the field saturation value, HSAT, for the insulator to metal transition. As discussed in the text, chemical treatment brought the metallic and unsorted SWCNT films close to the insulator to metal transition. The spin saturation model is for variable range hopping and it seems that the theory has not been extended to the insulator to metal transition yet. Still, we plot HSAT against T a way to reveal power laws. Figure S4-1| a, the field saturation values, HSAT, plotted against temperature T on a log-log plot for the unaligned SWCNT films when chemically treated.
S5-Weak Localization. We next discuss the low-field, negative MR that remains after subtraction of the high-field MR component. We stress that correct, unconfounded analysis of the negative MR at low-field would not have been possible if the high-field data had not been collected and subtracted out. For the purposes of discussion, assume from now on that this positive MR component has been subtracted. Figure   Weak localization is a quantum mechanical phenomenon that explains the small resistance increase in disordered metallic thin films as they are cooled to cryogenic temperatures [7] [10].
Later, the mechanism was used to explain the unusual electronic transport of turbostatic/ Applying a magnetic field also de-phases the backscatter, leading to a negative MR. Lφ is the coherence length, the average distance a charge carrier wave function looses coherence due to inelastic collisions. It is related to temperature and field via the following equation [8] [10] [17] [18]: where ℏ is Plank's constant divided by 2π, e is the electric charge, Hφ is the dephasing field that destroys the backscatter coherence. Diff is the diffusion constant and τ is the relaxation time between inelastic collisions. p is an exponent dependent on the exact inelastic scattering mechanism and the dimensionality of the transport. Because mobility μ is proportional to τ, it is expected that In addition to its temperature dependence, the field dependence of the remaining negative MR component also supports a weak localization mechanism. Weak localization is a scaling theory meaning that all weak localization curves will match some universal curve f such that = C f(H/Hφ), where C is a scaling parameter and the second scaling parameter is the dephasing field [15]. Figure 5-S2 a shows the negative MR component curves at different temperatures aligning to a universal curve, here assigned as the one at 10 K. MR curves with temperatures less than 10 K are not considered here because of the expected confounding influence of EEI interaction. Using this scaling approach, the temperature dependent scaling parameter, Hφ, is plotted against temperature relative to Hφ at 10 K ( Figure S5-2 c). The dephasing field Hφ follows the power law T p where the power law exponent is p= 0.52 and then departs from this power law behavior at higher temperatures. From equation (S3), this means μ∝T -0.52 and this matches the temperature dependent mobility component calculated independently from the high-field MR data.
Moving beyond the generalized scaling approach, for thoroughness, we fit the subtracted negative MR component to specific field dependent expressions for weak localization, given in [19] [20] for two dimensional weak localization and [10] [14] [21] for three dimensional weak localization. Figure S5-2b shows an example of a two dimensional weak localization fit to the low-field component at 10 K, which is representative across the temperatures measured 10 K and above. Figure S5-2d shows the fitted dephasing field parameter Hφ as a function of temperature, which is similar to the scaling approach except it is now in absolute terms and is not normalized. Note that results between the two different analytical approaches, fitting and scaling, nearly agree in terms of the power law dependence and the departure from the power law at higher temperatures.
Fitting to three dimensional weak localization expressions across the temperature range was not as effective. It is noteworthy that weak localization, an effect based in quantum mechanics, is present approaching room temperature. While weak localization was originally applied to thin film metals for conductivity corrections below 25 K [7], it has been shown to apply to pre-graphitic carbon materials approaching room temperature as well [12]. The presence of weak localization implies that elastic collisions between defects are more probable than phonon interactions and indicates the extent of crystal imperfection on the CNT structure.
Weak localization in two dimensions means that only two spatial coordinates are required to specify localization of the wave function. In CNT materials, two dimensional weak localization has been attributed to confinement of the charge carrier wave function on the surface of the CNT bundle [1] [17] , the natural aggregation of individual CNTs held together by van der Waals forces.
In this picture, bundle bundle interfaces account for most of the overall dissipation; transport between CNTs within a bundle has little influence on the transport. Two dimensional weak localization applies provided the coherence length Lφ is larger than the bundle diameter, otherwise it is three dimensional [17]. Figure S5-2 e shows Lφ as a function of temperature, calculated from Hφ and equation (S2). At 10 K, Lφ = 60 nm and decreases with temperature until Lφ appears to level-off approaching 30 nm, which is a typical approximate value for bundle diameters in the fiber ( Figure 1 from the main text). Because the trajectory of Lφ changes when the coherence length reaches the approximate bundle diameter, it is possible this is a cross-over from two dimensional to three dimensional weak localization (where a change in power law is expected). Similar crossovers have been seen before in CNT systems [17], although in our case more analysis and model fitting on higher temperature data will be required to confirm the cross-over and a possibly temperature dependent dimensionality. Conclusive determination of dimensionality at a given temperature could be made by changing the field direction relative to the current flow. Two dimensional weak localization is anisotropic and is strongest when the field is perpendicular to the current flow [7]; three dimensional weak localization is isotropic and field direction does not matter [10]. Figure S5-2| Weak localization fitting of the low-field MR data, after the positive MR component is subtracted. a, Examples at various temperatures of the scaling approach-the low-field MR component collapsing to a universal curve by adjusting the scaling parameters, C and Hφ. b, Example of a direct fitting of a low-field MR component to a specific expression of two dimensional weak localization. c, From the scaling approach, relative dephasing field as a function of temperature. d, From specific fitting of two dimensional weak localization, the dephasing field in absolute terms as a function of temperature. e, The coherence length, Lφ, calculated from the dephasing field.

S6-Fluctuation Induced
Tunnelling. This next section shows various fits for the fluctuation induced tunneling model on resistance R versus temperature T data of the aligned CNT fiber, zero field. Different fitting situations are considered, such as exclusion of temperatures equal to and less than 4 K. This was attempted because of concern about EEI interaction becoming significant and confounding the data.
Another situation considered was the contribution from weak localization subtracted from the data. Two different metallic terms were considered, the quasi 1D metallic term and the standard metallic term. From these fits, the extrinsic contribution accounts for approximately 80% of the room temperature resistance.
This was accomplished according to the equation and procedures given in [22] [23] [24]. Figure S6-1| Fitting R versus T to the fluctuation model for the data as is, then excluding data less than 4 K (because of possible EEI interactions), and then excluding the data less than 4 K and excluding the contribution from weak localization. a, covers the fitting with an anisotropic, quasi 1D conduction term and b is with the standard metallic conduction term. R0 is the resistance at room temperature.