Effective low-energy theory of superconductivity in carbon nanotube ropes

We derive and analyze the low-energy theory of superconductivity in carbon nanotube ropes. A rope is modelled as an array of metallic nanotubes, taking into account phonon-mediated as well as Coulomb interactions, and arbitrary Cooper pair hopping amplitudes (Josephson couplings) between different tubes. We use a systematic cumulant expansion to construct the Ginzburg-Landau action including quantum fluctuations. The regime of validity is carefully established, and the effect of phase slips is assessed. Quantum phase slips are shown to cause a depression of the critical temperature $T_c$ below the mean-field value, and a temperature-dependent resistance below $T_c$. We compare our theoretical results to recent experimental data of Kasumov et al. [Phys. Rev. B 68, 214521 (2003)] for the sub-$T_c$ resistance, and find good agreement with only one free fit parameter. Ropes of nanotubes therefore represent superconductors in the one-dimensional few-channel limit.

Figure 1

Temperature dependence of ${\Delta }_0∕2\pi T$ versus $T∕T_c^0$ for $N=31$ (open circles) and $N=253$ (filled circles).

Figure 2

Temperature-dependent resistance $R(T<T_c)$ predicted by Eq. (5.4) for $\nu =1$ and different $N$. The smaller is $N$, the broader is the transition. From the leftmost to the rightmost curve, $N=4$, 7, 19, 37, 61, 91, 127, 169, 217.

Figure 3

Temperature dependence of the resistance below $T_c$ for superconducting rope $R2$ experimentally studied in Ref. 4. Open squares denote experimental data (with subtracted residual resistance), the curve is the theoretical result.

Figure 4

Same as Fig. 3, but for sample $R4$ experimentally studied in Ref. 4.