There have been several experiments revealing the existence of superconducting correlations in the carbon nanotubes. These observations have taken the form of a drastic drop in the resistance of the nanotube samples below certain temperature. In one of the most remarkable experiments, reported in , it has been shown that a rope of carbon nanotubes is able to carry an electric current with zero voltage drop, when embedded between superconducting contacts. The measurement of that so-called supercurrent implies therefore a vanishing resistance of the conductor. The experiment provides a realization of the proximity effect, by which the electronic properties of a normal metal change drastically when placed in contact with a superconductor [77, 78]. In the latter, there is no sign of electron-like particles at low energies and, instead, a condensate formed by pairs of bound electrons is found . These so-called Cooper pairs may extend their propagation to the nearby normal metal, giving rise to the electric current without dissipation of energy.
The influence of the superconducting electrodes in the electronic properties of single-walled nanotubes has been investigated in the experiments reported in  and . One of the main differences between these experiments is that, in the latter, the transport properties have been measured in a set of individual single-walled nanotubes. In the former experiment, the supercurrents have been observed in a massive rope made of about 200 nanotubes, and in a thin rope leading to a single nanotube at one of its ends. On the other hand, a common feature in both experiments is the low resistance attained for the samples that have shown the proximity effect. Typically the values measured at room temperature have been consistent with a resistance of the individual metallic nanotubes of the order or below the inverse of the conductance quantum, h/e2 ~ 25.8 kH [76, 80]. Such values are comparable to the resistance (2G0)-1 = h/4e2 corresponding to the ballistic transport in individual nanotubes, which gives a measure of the high transparency of the contacts produced in the experiments.
In the experiments reported in , the ability to produce highly transparent junctions has been the result of using a remarkable technique allowing to suspend the nanotube ropes between the contacts. In the transport measurements, a drop to a vanishing resistance has been observed in the two nanotube samples mentioned earlier, below the temperature Tc of the transition of the electrodes to the superconducting state. The contacts were made of bilayer electrodes with respective temperatures Tc ~ 1.1 K for the Re/Au bilayer in the case of the thick rope, and Tc ~ 0.4 K for the Ta/Au bilayer in the case of the thin rope. By applying a magnetic field perpendicular to the nanotube axes, it has been possible to reduce the value of Tc as measured in the ropes, up to a point in which the transition disappears for a suitably large field . This effect of the magnetic field is one of the genuine features of superconductivity, and it serves to corroborate the nature of the phenomenon observed in the experiment.
When increasing the current that flows along the rope, it can be supported without developing any resistance up to a maximum value, that is called the critical current. The behavior of the critical currents for the ropes studied in  has shown unconventional features, regarding their magnitude as well as their dependence with the temperature. The critical current should vanish, for instance, at the transition temperature of the contacts, but in the thick rope of  the behavior is very smooth instead near Tc. In the conventional picture of the proximity effect, the magnitude of the critical current should correspond to the expression Ic = (w/2)A/eRN, Rn being the normal state resistance and A the binding energy of the Cooper pairs in the superconducting condensate. As it has been pointed out in , the value of Ic estimated in that way is, however, 40 times smaller than what is actually measured in the thick rope. The thin rope shows a better agreement in the magnitude of the critical current, but this also displays a very unusual temperature dependence, with a flat behavior until the neighborhood of Tc is reached .
An explanation of the unconventional behavior of the critical currents in ropes has been presented in . That work has shown the relevance of taking into account appropriately the interaction among the large number of metallic nanotubes that may be present in a rope. The Coulomb potential is not screened in an individual nanotube, but the interaction between the charges in different metallic nanotubes leads to a significant reduction of the effective interaction strength . This can be understood by thinking that, instead of Eq. (9), the Hamiltonian appropriate for a rope with n metallic nanotubes is
the indices a, b labelling the electronic densities in the different metallic nanotubes. The Coulomb interaction is long-ranged and takes place therefore between all of them, so that it contributes equally to all the V^ab)(k) terms. The Hamiltonian (13) can be diagonalized by passing to the total densities pra(k) = J2a pi"J(k). It becomes clear that the Coulomb interaction is only felt in the channel of the total charge, while there are still 4n — 1 noninteracting partial channels .
The preceding argument explains that the repulsive electron-electron interaction becomes less relevant as the number n of metallic nanotubes increases in the rope. The proximity effect for a Luttinger liquid in contact with a macroscopic superconductor has been studied in  and , showing that the Cooper pairs propagate along the one-dimensional metal but giving rise to a supercurrent Ic that decays with the length L as
where / is the Luttinger liquid parameter quoted above Eq. (10). Recalling that / < 1 in the case of a repulsive interaction, that kind of behavior can only account for the large critical current measured in the rope of  after the appropriate reduction in the strength of the Coulomb interaction is considered. It has been also shown in  that the temperature dependence of the critical currents can be reproduced by considering the one-dimensional propagation of the Cooper pairs, which gives further support to the picture of the single-walled nanotubes as genuine one-dimensional conductors.
In the experiment presented in , the resistance of individual single-walled nanotubes placed between Nb electrodes has been measured. The nanotubes have been capac-itively coupled to the Si substrate, and changing the gate voltage Vg has allowed to increase the already high transparency of the contacts. Below the transition temperature of the Nb electrodes (Tc ~ 9.2 K) and for some interval of Vg, a dip has been observed in the broad peak of the resistance centered at zero bias voltage. That structure has disappeared by increasing the temperature above Tc, which shows its relation to the superconducting character of the electrodes . Although the room-temperature resistances of the samples were comparable to those in the experiment of , no supercurrents have been found in this case. This can be attributed to the large strength of the repulsive electron-electron interaction in the individual nanotubes, supporting the point of view that the superconducting correlations are more likely to develop in ropes of nanotubes.
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