We reiterate that we did not observe any Hall-coefficient sign anomaly in any of our films, although such a phenomenon is often observed in hopping-conduction-dominated systems17-20. The observed sign consistency indicates the existence of delocalized carriers that respond to the Lorentz force, as opposed to the hopping carriers whose existence is clear from the temperature dependence of resistance (see Supplementary Fig. 10)17,20,24. The existence of weakly localized carriers was confirmed by the appearance of negative magnetoresistance in any conditions (Supplementary Fig. 11), as described as a signature of weak localization23,24. Therefore, we assume that such weakly localized carriers contributed the Hall voltage to some degree and ensured the sign consistency in the Hall effect in our SWCNT films, although the hopping conduction mechanism reduced a.
Finally, we discuss the possible origins of the very small a values we observed. First, the Hall voltage can be compensated by the hopping carriers moving in the direction opposite to the Lorentz force3. In organic semiconductors, this contribution is usually related to the reduction of carrier mobility. Table 1 shows the FET carrier mobility values for the SWCNT film devices, calculated using the formula
where L is the channel length, W is the channel width, and σ is the sheet conductance. We note that our SWCNT films exhibited relatively high carrier mobilities (> 52 cm2 V-1 s-1), comparable to those reported for similar gated SWCNT films (~ 59 cm2 V-1 s-1)29,30. However, these mobilities are much smaller than that expected for a single SWCNT rope (~79,000 cm2 V-1 s-1)31, indicating the contribution of hopping conduction that can reduce a. The effect of grain boundaries, reported in organic composite systems, can be also the origin to reduce the Hall voltage7. SWCNT films have many interfaces, both tube-tube and bundle-bundle junctions (see AFM images in Fig. 1 and Fig. 3), which can be thought of as grain boundaries in the context of carrier localization. However, the above two contributions are generally used to explain a of at most 0.1, and this extremely small a we obtained for the SWCNT films suggests that there may be other factors in addition to the above two. We assume the following effect of dimensionality. If the conduction path (e.g., a SWCNT bundle) of the delocalized carriersis closer to one-dimensional, their contribution to the two-dimensional Hall effect should be smaller, resulting in smaller a.
Finally, we highlight two trends of a values observed in sample dependence: (i) a (Metal) > a (Semi) > a (Mix), and (ii) a (Mix) > a (L-Mix). These trends suggest that the aforementioned factors that affect the Hall voltage, such as weak localization, hopping conduction, grain boundaries, depend on the sample. The Metal sample exhibited the largest a, which is reasonable because it shows weak localization even without carrier doping23,24. The Mix samples containing metallic SWCNTs showing a smaller a than the Semi sample indicates that a in the high-density region is affected not only by the electronic structure but also by sample inhomogeneity. Regarding (ii), it is noteworthy that there is no significant difference in a between the Mix and L-Mix samples, despite the significant difference in film morphology (see Fig. 3). This suggests that the morphology of films does not considerably affect the Hall effect.
In conclusion, we evaluated the validity of the Hall effect in SWCNT films with various electronic types and morphologies. We observed clear Hall voltages in all samples in high carrier density regions and confirmed that the carrier types were consistent between the Hall effect and FET measurements. However, the Hall carrier densities were significantly overestimated, compared to the FET densities, resulting in extremely small values of a (< 10-3). The values of a for SWCNT films are the smallest among the values reported for all the materials, indicating that thin films of one-dimensional SWCNTs are quite different from conventional hopping transport systems. This study suggests that, even though Hall measurements are a general method to evaluate carrier types and carrier densities, a careful evaluation is crucial for correct estimation of the carrier density when interpreting Hall-effect results for SWCNT films.