Strong Localization

The increasing coherence length of nanotubes upon cooling can lead to the formation of a strongly localized state. This will occur when the coherence lifetime t$ becomes so long that the width of the coherent state h/rv becomes smaller than the energy separation between coherent states [75]. Localization in the context of nanotubes, but ignoring interference effects, was studied theoretically by Kostyrko et al. and Roche and Saito [76,77]. These theories are expected to be valid in cases where disorder is dominant over electron interactions. Experimentally strong localization has been observed in SWNT mats [78] and SWNT rings [57]. As Fig. 10b shows, continued cooling of the ring below 3 K leads at about 1 K to a sharp transition to a state of high

Fig. 10. (a) Temperature dependence of the zero-field SWNT ring resistance from 6-200K (gray curve), and fit to one-dimensional weak localization theory (dashed line). (b) Ring resistance from 0.3 to 6K [57]

12 3 4 Temperature (K)

12 3 4 Temperature (K)

resistance [57]. Below this point transport becomes thermally activated with the resistance following a R x exp(To/T) dependence with T0 « 0.8 K. This behavior is characteristic of a conductor in a strongly localized state.

This conclusion is further supported by corresponding changes in magneto-transport [57]. The effect of the magnetic field is now stronger, and the MR curves cannot be fit by weak localization theory. Side bands to the central MR peak appear, and based on the fact that their detailed structure is altered by mild thermal annealing and re-cooling [79], these bands are ascribed to conductance fluctuations, a low temperature phenomenon involving defect sites.

From knowing the localization temperature, the strong localization length L^ can be estimated as the coherence length L$ at the localization temperature (about 1K). For this ring a localization length of L^=750 nm is obtained [57]. This localization length implies a mean free path in the range of 250-350 nm [76,75] in good agreement with mean free path of about 300 nm deduced in studies of individual SWNTs [10].

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