Superconductivity in carbon nanotube ropes
J. Gonz\'alez
\widetext We investigate the conditions in which superconductivity may develop in ropes of carbon nanotubes. It is shown that the interaction among a large number of metallic nanotubes favors the appearance of a metallic phase in the ropes, intermediate between respective phases with spin-density-wave and superconducting correlations. These arise in samples with about 100 metallic nanotubes or more, where the long-range Coulomb interaction is very effectively reduced and it may be overcome by the attractive interaction from the exchange of optical phonons within each nanotube. We estimate that the probability for the tunneling of Cooper pairs between neighboring nanotubes is much higher than that for single electrons in a disordered rope. The effect of pair hopping is therefore what establishes the intertube coherence, and the tunneling amplitude of the Cooper pairs dictates the scale of the transition to the superconducting state.
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\title{Superconductivity in carbon nanotube ropes}
\author{J. Gonz\'alez   \\}
\address{
        Instituto de Estructura de la Materia.
        Consejo Superior de Investigaciones Cient{\'\i}ficas.
        Serrano 123, 28006 Madrid. Spain.}
%%        $^2$Instituto de Ciencia de Materiales.
%%        Consejo Superior de Investigaciones Cient{\'\i}ficas.
%%        Cantoblanco. 28049 Madrid. Spain.}


\date{\today}
\maketitle
\begin{abstract}
\widetext
We investigate the conditions in which superconductivity may
develop in ropes of carbon nanotubes. It is shown that the
interaction among a large number of metallic nanotubes favors
the appearance of a metallic phase in the ropes, intermediate
between respective phases with spin-density-wave and
superconducting correlations. These arise in samples with about 
100 metallic nanotubes or more, where the long-range Coulomb 
interaction is very effectively reduced and it may be overcome 
by the attractive interaction from the exchange of optical 
phonons within each nanotube. We estimate that the probability 
for the tunneling of Cooper pairs between neighboring nanotubes 
is much higher than that for single electrons in a disordered rope.
The effect of pair hopping is therefore what establishes the 
intertube coherence, and the tunneling amplitude of the Cooper 
pairs dictates the scale of the transition to the superconducting 
state.



\end{abstract}
\pacs{71.10.Pm,74.50.+r,71.20.Tx}

]



\narrowtext 
\tightenlines

%\newpage



\section{Introduction}



During the last years there has been growing evidence of the 
existence of superconducting correlations in carbon nanotubes.
In the first place, the proximity effect has been observed 
in ropes of nanotubes placed between superconducting 
contacts\cite{kas,marcus}. Quite remarkably, it has been shown 
that the nanotubes may support supercurrents below the critical 
temperature of the contacts\cite{kas}. Later on, there have been 
experiments directed to probe the superconductivity inherent to
the nanotubes\cite{koc}. In some of the ropes, a transition has
been observed at a temperature below $1 \; {\rm K}$, with a drop 
of two orders of magnitude in the resistance down to a value 
consistent with the finite number of channels in the rope. 

From a theoretical point of view, the electronic interactions 
in the nanotubes have been also the subject of much 
attention\cite{bal,berk,eg,kane}. The nanotubes may show metallic 
properties depending on the helicity with which the graphene
sheet is wrapped to form the tubule. The experimental 
results\cite{wild} have agreed on that point with the 
theoretical predictions\cite{saito}. Given that the conduction 
takes place in a one-dimensional (1D) structure, it has been 
proposed that the nanotubes should be ideal systems for the 
observation of the so-called Luttinger liquid 
behavior\cite{emery,lutt1,lutt2,sch}. Actually, there have been
measurements providing evidence of the power-law dependence of 
the tunneling conductance\cite{exp,yao}, what is a signature of 
the mentioned behavior. These experiments seem therefore to
probe a regime in which the repulsive Coulomb interaction turns
out to be dominant in the nanotubes.

On the other hand, the interaction with the elastic modes of 
the nanotube plays an important role in the development of the 
superconducting correlations\cite{prl}. The magnitude of the 
supercurrents measured in Ref. \onlinecite{kas} is in some 
instances up to 40 times larger than expected from the estimate 
within the conventional proximity effect. It has been shown that 
the temperature dependence of the supercurrents is characteristic 
of the 1D behavior of the system. Moreover, their 
values can be only explained by taking into account the 
attractive electronic interaction coming from the exchange of 
phonons, on top of the repulsive Coulomb interaction\cite{prl}.

Recently, a microscopic model has been elaborated to account for 
the observation of superconductivity intrinsic to the ropes of 
nanotubes\cite{micro}. In these systems, the Coulomb interaction
can be significantly reduced depending on the number of metallic
nanotubes in the rope. One has to incorporate therefore the 
balance between the repulsive electron interaction and
the effective attractive interaction coming from phonon
exchange. It has been shown that, in the case of samples with
about 300 nanotubes like those displayed in Ref. 
\onlinecite{koc}, the superconducting correlations prevail in
the system. The intertube coherence is established mainly through 
the tunneling of Cooper pairs, which gives rise to the 
superconductivity in the bulk under suitable 
conditions\cite{micro}.

One of the aims of the present paper is to unveil the 
different phases that arise in the competition between the 
repulsive Coulomb interaction and the attractive interaction 
from phonon exchange in the ropes of nanotubes. For this 
purpose we will map the phase diagram of these systems taking 
the strength of the phonon couplings and the number of 
metallic nanotubes as the relevant variables. We will show 
that the region where the superconductivity may develop opens 
up for relatively large contents of metallic nanotubes. 
The phase with spin-density-wave correlations characteristic
of a repulsive interaction is confined to the cases where 
there are only a few nanotubes in the rope. 

We assume in any event that the formation of spin or charge 
order in the bulk is prevented by the fact that the nanotubes 
have a random distribution of helicities in the ropes. These 
do not have a crystalline structure from the three-dimensional
point of view, and the only possible long-range order arises 
in the superconducting phase. We will see that, between the phases 
with superconducting and spin-density-wave correlations, there is 
a metallic phase with no signal of instability in any 
direction. This is the phase to be found out most likely when 
measuring ropes with a number below $\sim 100$ 
metallic nanotubes.

We will also address the question of the maximum transition
temperature that can be reached in the ropes. There have
been recent experimental measures in carbon nanotubes of 
minimum diameter inserted in a zeolite matrix, from which a
critical temperature of about $15 \; {\rm K}$ has been 
inferred\cite{chi}. The small radius and high curvature of
those nanotubes have to lead to an enhancement of the 
electron-phonon coupling. This would explain in turn the
increase in the critical scale for superconductivity. 

One has to bear in mind, however, that the correlations
in individual nanotubes provide an indirect measure of 
the superconducting state. We will show that, in the 
three-dimensional structure of the rope, the transition
temperature is dictated by the amplitude for the Cooper 
pairs to tunnel between neighboring nanotubes. Even in
samples with large number of metallic nanotubes, the 
opening of the superconducting phase depends at last on
the existence of intertube coherence. The most efficient
way to increase the transition temperature may come actually
from devising some mechanism to enhance the tunneling rate
of the Cooper pairs between the nanotubes.

The content of this paper is organized as follows. In 
Section II we analyze the origin of the different 
interactions in the individual nanotubes. Section III is 
devoted to elaborate the model that takes into account 
the interaction among the metallic nanotubes, ending up 
with the discussion of the possible phases of a rope. 
In Section IV we incorporate the effect of the tunneling
of the Cooper pairs, performing some estimates of the 
transition temperature under different conditions.
Finally, Section V is devoted to discuss some of the 
assumptions under which our general analysis applies. 







\section{Electron interactions in carbon nanotubes}
 


We begin by considering the band structure of the individual
metallic nanotubes. We focus on the so-called armchair 
nanotubes, which have the structure depicted in Fig. 
\ref{long} of the Appendix A. They have gapless subbands 
irrespective of the influence of the nanotube curvature on
the orbitals. Fig. \ref{metal10} represents for instance 
the spectrum of a (10,10) armchair nanotube, which has a
radius $R \approx 15 a/ \pi $ in terms of the carbon-carbon
distance $a$. We will concentrate on the case of undoped
nanotubes, in which the Fermi level is placed at $E = 0$
in the scale of the figure.



\begin{figure}
\begin{center}
\mbox{\epsfxsize 7cm \epsfbox{metal10.eps}}
\end{center}
\caption{Different subbands of a (10,10) armchair nanotube.
The energy $E$ is measured in units of the hopping parameter
$t$ and the momentum $k$ is in units of the inverse of the 
lattice spacing.}
\label{metal10}
\end{figure}



The most important feature for the study of the
low-energy properties is the crossing of two subbands at the
Fermi level, that we label with the indices 1 and 2 as shown
in the figure. These subbands correspond to the modes with
vanishing transverse momentum in the nanotube, which are
governed by the one-particle hamiltonian\cite{dirac}
\begin{equation}
{\cal H} =    t  \left(
\begin{array}{cc}
0  &   1 - 2 \cos (\sqrt{3} k a/2)    \\
1 - 2 \cos (\sqrt{3} k a/2)    &   0
\end{array}         \right)
\end{equation}
$t$ being the hopping amplitude between neighboring sites.


The amplitudes of the modes behave differently in the 
nanotube lattice depending on whether they belong to one or 
the other subband. In the case of the subband with bonding 
character, the amplitude is a smooth function of the position, 
while the modes in the other subband have an amplitude that 
alternates the sign between the two sublattices of the 
honeycomb lattice. This property has important consequences 
regarding the form of the electron-electron interactions as 
well as the coupling of the electrons to the elastic modes 
of the lattice.

In what follows we undertake the analysis of the low-energy
regime in which the physical properties are dominated by
the branches with approximate linear dispersion near the Fermi
level. For this purpose, we introduce an energy cutoff $E_c$ 
below the energy of the bottom of the first unoccupied
subband, together with its counterpart $-E_c$ below the Fermi 
level. The different branches are labelled by an index
$r = \pm$ denoting the right or left-moving character, and
by the Fermi point $i = \pm$ to which they are attached, as
shown in Fig. \ref{branch}.
Correspondingly, we end up with a collection of electron
fields $\Psi_{r i \sigma} (x)$, to which one more index $a$ 
will be added later on to label the nanotube in the rope.



\begin{figure}
\begin{center}
\mbox{\epsfxsize 7cm \epsfbox{branch.eps}}
\end{center}
\caption{Linear branches in the low-energy spectrum of a 
carbon nanotube.}
\label{branch}
\end{figure}



The electron interactions that we are going to consider
have in general the form
\begin{eqnarray}
   \int_{-k_c}^{k_c} dk   \int_{-k_c}^{k_c} dp
     \int_{-k_c}^{k_c} dq
              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
    \;\;\;\;\; \;\;\;\;\;\;\;\;\;\;&   &      \nonumber       \\
   \Psi_{r i \sigma}^{\dagger}(p+k) \Psi_{s j \sigma }(p)  V(k)
  \Psi_{t k \sigma '}^{\dagger}(q-k) \Psi_{u l \sigma '}(q)   &  &
\end{eqnarray}
The processes between the different branches can be
cataloged according to the notation of Ref.
\onlinecite{berk}. They fall into 12 different classes with
respective couplings denoted by $g_i^{(j)}$, the indices taking
the values $i = 1,2,4$ and $j = 1, \ldots 4$. The lower label
pays attention to the Fermi points of the respective fields,
following the convention that assigns $i = 1$ to backscattering
processes, $i = 2$ to scattering between currents at
different Fermi points, and $i = 4$ to scattering
between currents at the same Fermi point. The upper label
follows the same rule classifying the different combinations
of left-movers and right-movers, including the possibility of
having Umklapp processes ($j = 3$). 

We analyze next the contributions to the different scattering
processes from the interactions present in the nanotubes.


{\bf Electron-electron interactions.}

We deal with the situation in which the Coulomb interaction
is not screened by external gates. It is known that the 
Coulomb potential remains long-ranged in one spatial 
dimension\cite{grab}. The appropriate expression in momentum 
space is\cite{wang}
\begin{equation}
V_C (k) = (e^2 /4\pi^2) \log |(k_c + k)/ k|
\end{equation}
The repulsive interaction is therefore enhanced at 
small momentum transfer, as in the processes shown in 
Fig. \ref{geol2}. 
However, processes like those in Fig. \ref{geol}, in which the
flavor $r$ is changed by the interaction, are highly suppressed.
This is due to the fact that, as mentioned before, the modes
corresponding to one of the gapless subbands alternate the sign
in the two sublattices of the nanotube. When the distance
between the two currents that interact is much larger than the
nanotube radius, the matrix element of the interaction averages
to zero over the section of the nanotube. 



\begin{figure}
\begin{center}
\mbox{\epsfxsize 7cm \epsfbox{geol2.eps}}
\end{center}
\caption{Small momentum-transfer processes which are enhanced
by the long-range Coulomb interaction.}
\label{geol2}
\end{figure}




Thus, the long-range repulsive interaction reflects in an 
enhancement of the interbranch and intrabranch couplings 
$g_2^{(2)}$, $g_2^{(4)}$, $g_4^{(2)}$ and $g_4^{(4)}$.
The flavor-changing processes associated to $g_2^{(1)}$,
$g_2^{(3)}$, $g_4^{(1)}$ and $g_4^{(3)}$ survive in the form 
of a residual short-range interaction. Its strength, however, 
is corrected by a factor of the order of $\sim 0.1 \; a/R$ in 
comparison to the nominal strength of the long-range Coulomb 
interaction\cite{eg,kane}. This represents in general a 
relative reduction by two orders of magnitude, for the 
nanotubes that are typically found in the ropes.



\begin{figure}
\begin{center}
\mbox{\epsfxsize 7cm \epsfbox{geol.eps}}
\end{center}
\caption{Flavor-changing processes with small contribution from
the Coulomb interaction.}
\label{geol}
\end{figure}






There remain the processes with momentum transfer of the order
of $\sim 2 k_F$. These processes probe the interaction at short 
distances. It has been shown that the corresponding couplings 
are also reduced by a factor of the order of $\sim 0.1 \; a/ R$ 
relative to the strength $e^2 /4 \pi^2 $ of the 
forward-scattering processes\cite{eg,kane}.

The couplings for large momentum transfer, as well as 
$g_2^{(1)}$, $g_2^{(3)}$, $g_4^{(1)}$ and $g_4^{(3)}$, are 
marginally relevant in the renormalization group sense. 
This means that they have greater strength as the theory is
scaled to smaller energies, but the rate of increase starts
being quadratic in the own couplings\cite{berk,eg}. 
Thus, one has to proceed
to extremely low energies, many orders of magnitude below
$E_c$, to have a significant increase of the couplings with
respect to their nominal values\cite{eg}.


{\bf Electron-phonon interactions.}

The electron-phonon interactions give rise to a retarded
interaction between the electrons, which may become attractive
at frequencies below the typical energy of the phonons. 
The potential for the effective interaction has the form
\begin{equation}
V (k,\omega ) = - g_{p,p'}(k) g_{q,q'}(k) 
  \frac{\omega_k }{- \omega^2 + \omega_k^2 }
\end{equation}
where $\omega_k $ is the phonon energy and the $g_{p,p'}(k)$ 
are appropriate electron-phonon couplings described in the
Appendix A. The form of the interaction is 
dictated by the latter, whose dependence on the momentum 
transfer varies according to the kind of phonons exchanged.

{\em Acoustic phonons.}--- 
In this case,  $\omega_k $ is proportional to
$|k|$ at small momentum transfer. This means that, regarding the 
interbranch and intrabranch processes, the range of energies in
which the effective interaction becomes attractive is greatly
reduced. This instance has been studied with detail in Ref.
\onlinecite{loss}, looking for the regime in which the 
retarded interaction may drive the transition to a state with
superconducting correlations. Taking the parameters
appropriate for the nanotubes, it can be
seen that the critical exponents estimated in this way are
very small. Therefore, the acoustic phonons at small momentum 
transfer do not play a role in the superconductivity
observed in the nanotubes.
 
Moving now to the processes with large momentum transfer 
$k \sim 2 k_F$, the phonon energy $\omega_{2 k_F}$ becomes 
of the order of the Debye frequency $\omega_D$. This provides
then a significant range in which the effective interaction 
becomes attractive. The precise couplings to the acoustic
phonons in the nanotubes can be found in Ref. \onlinecite{jdd}.
It has been shown there that some restrictions arise in the 
exchange of phonons depending on the bonding or antibonding
character of the subbands involved. In the case of longitudinal 
acoustic phonons, the processes cannot take one electron from a 
given subband to the other. Moreover, the couplings have 
opposite signs when the scattered electron is in one or the 
other subband. The exchange of phonons leads then to the 
effective couplings below the Debye frequency
\begin{equation}
g_{{\rm eff},1}^{(1)} = - g_{{\rm eff},1}^{(3)} < 0
\label{klarge}
\end{equation}

In the case of the transverse acoustic phonons, the 
phonon-exchange processes in which the electrons remain in the 
same subband are forbidden. Processes associated to 
$g_{1}^{(2)}$ and $g_{1}^{(4)}$ in which 
the electron shifts from one subband to the other could play 
in principle a role, but there is a kinematic factor that 
vanishes when the incoming and outgoing modes have opposite
momenta. There are therefore no more contributions from the
acoustic phonons when the system is precisely at half-filling.

{\em Optical phonons.}--- The Debye frequency is of the 
same order that the energy of the acoustic phonons at $2 k_F$, 
so that we can expect the effective couplings to be comparable 
to those in Eq. (\ref{klarge}). 
The different symmetry of the modes with bonding and  
antibonding character imposes again restrictions in the way
the optical phonons can be exchanged, as shown in the 
Appendix A. In the case of the transverse optical phonons,
the electron modes have to remain within the same subband,
while the longitudinal phonons force the electron to shift
from one of the subbands to the other. The effective
couplings that we get have the following signs
\begin{eqnarray}
g_{{\rm eff},2}^{(1)}  & = &  - g_{{\rm eff},2}^{(3)} < 0    \\
g_{{\rm eff},2}^{(2)}  & = &  - g_{{\rm eff},2}^{(4)} < 0    \\
g_{{\rm eff},4}^{(1)}  & = &  - g_{{\rm eff},4}^{(3)} < 0    \\
g_{{\rm eff},4}^{(4)}  & = &  - g_{{\rm eff},4}^{(2)} < 0   
\end{eqnarray}


We finally remark that the Debye frequency of the phonons 
in the nanotubes can be estimated to be between 0.1 and 
$0.2 \; {\rm eV}$. This is of the same order of magnitude that 
the cutoff $E_c$, that marks the energy below which
the nanotubes can be viewed as simple 1D systems. In that
regime, the effective interactions coming from the exchange of
phonons can be matched against the interaction of purely
electronic origin. We will adopt henceforth this procedure, 
as long as the electronic properties in which we are going to 
be interested refer to energies much lower than $E_c$.






\section{Electronic interactions in ropes of nanotubes}




We turn now to the processes that take place between the 
nanotubes in a rope. A property to be taken into 
account is that the rope is made of a disordered mixture of 
nanotubes with all kind of diameters and  
helicities. Given their random distribution, the lattices of 
neighboring nanotubes are not aligned in general. The 
consequences of this fact have been studied in Ref.
\onlinecite{mkm}. As long as there is a mismatch in the position
of the respective Fermi points of neighboring nanotubes, the 
conservation of the longitudinal momentum cannot be precisely
enforced in the tunneling processes. At the one-particle level,
the electronic states appear to be localized along the individual
nanotubes, and the intertube coherence is strongly
suppressed.

The authors of Ref. \onlinecite{mkm} have computed the
expression of the tunneling amplitude between nanotubes in
the so-called {\em compositionally disordered} ropes.
In the case of nanotubes with the same helicity and aligned 
lattices, the estimate for the tunneling amplitude is
$t_T \sim 0.01 \; {\rm eV}$. When there is a mismatch $\delta k$
between the Fermi points of neighboring nanotubes, it has been 
found that the amplitude for an electron hopping between them
is suppressed by a relative factor $\sim \exp [-R a_0
(\delta k)^2 /4]$, where $a_0$ is a parameter of the order of 
$\sim 0.5$ \AA. Taking a    
typical nanotube radius $R = 7$ \AA, the average value of that
factor is of the order of $\sim 0.005$.

The presence of compositional disordered has also important 
consequences regarding the kind of electronic instabilities that
may develop in the rope. Depending on the point of the phase   
diagram, the individual nanotubes may support $2 k_F$
charge-density-wave and superconducting correlations, or 
$2 k_F$ spin-density-wave correlations. It is clear, however, 
that the absence of a three-dimensional crystalline structure 
precludes the
development of charge or spin order in the bulk of the rope.
The transition to a superconducting state is therefore favored
when the correlations are strong enough to force the symmetry
breaking in that direction.

The possibility of having a superconducting transition in the 
rope of nanotubes arises from the balance of the effective 
attractive interactions due to phonon exchange and the  
repulsive interactions of purely electronic character. At the
diagrammatic level, for instance, the correlations with $s$-wave 
order parameter are driven by the couplings $g_2^{(2)}$, 
$g_1^{(1)}$, $g_2^{(1)}$ and $g_1^{(2)}$, as shown in the 
Appendix B \cite{caron}. The obstruction to
having superconducting correlations in the nanotubes
comes, in principle, from the large contribution of the Coulomb
potential to the coupling $g_2^{(2)}$.
In this respect, the processes associated to $g_1^{(1)}$ and
$g_2^{(1)}$ contribute to enhance the superconducting
correlations when they are able to develop, but they play a
secondary role in the determination of the phase diagram of the 
ropes.

It is therefore necessary to build a model that takes into
account the competition between attractive and repulsive
interactions in the interbranch and intrabranch processes
$g_2^{(2)}$, $g_2^{(4)}$, $g_4^{(2)}$ and $g_4^{(4)}$.
This model can be solved by means of bosonization   
techniques. With this method, one boson field is introduced for
each of the linear branches below $E_c$ in each metallic
nanotube. At this point we need
to introduce a new index $a$ labelling the metallic nanotubes
in the rope, so that the fermion fields read now
$\Psi^{(a)}_{r i \sigma }(x)$. The correspondence between each of
these and the respective boson operator is given by 
\begin{equation}
: \Psi^{(a) \dagger}_{r i \sigma}(x) \Psi^{(a)}_{r i \sigma }(x) : =
   \rho^{(a)}_{r i \sigma }(x)
\end{equation}
where the colons represent normal ordering and
$\rho^{(a)}_{r i \sigma }(x)$ stands for the density operator
associated to the given branch.

Neglecting for the time being the tunneling between the
nanotubes, the hamiltonian for the model, including a collection
$a = 1, \ldots n$ of metallic nanotubes, takes the form   
\begin{eqnarray}
H_0  & = &    \frac{1}{2} v_F \int_{-k_c}^{k_c} dk
 \sum_{a r i \sigma }  : \rho^{(a)}_{r i \sigma} (k)
             \rho^{(a)}_{r i \sigma} (-k)  :   \nonumber      \\   
  &  &    + \frac{1}{2}  \int_{-k_c}^{k_c} dk \;
      \sum_{a r i \sigma } \rho^{(a)}_{r i \sigma} (k) \;  
  \sum_{b s j \sigma'  }  V^{(ab)}_{r i, s j} (k)  \;
                      \rho^{(b)}_{s j \sigma'} (-k) 
\label{ham}
\end{eqnarray}
where $k_c$ is related to $E_c$ through the Fermi velocity
$v_F$, $k_c = E_c/v_F$.
   
The interaction term in (\ref{ham}) can be better organized
by introducing the charge density operators
\begin{equation}
\rho^{(a)}_{r i \rho }(x)  = \frac{1}{\sqrt{2}} 
 \left(  \rho^{(a)}_{r i \uparrow }(x) +
       \rho^{(a)}_{r i \downarrow }(x)   \right)
\end{equation}
and their spin density counterparts. We define
further the linear combinations
\begin{eqnarray}
\tilde{\rho}^{(a)}_{1 \rho }(x) & = &
                  \rho^{(a)}_{+ + \rho }(x) +
              \rho^{(a)}_{- - \rho }(x)                \\
\tilde{\rho}^{(a)}_{2 \rho }(x) & = &
                   \rho^{(a)}_{+ - \rho }(x) +
              \rho^{(a)}_{- + \rho }(x)
\end{eqnarray}
Obviously, $\tilde{\rho}^{(a)}_{1 \rho }(x)$ is made of the
modes with bonding character, while
$\tilde{\rho}^{(a)}_{2 \rho }(x)$ contains those
with antibonding character.

We recall that, in the case of the attractive interaction from
the exchange of phonons, the sign of the coupling depends on the
bonding or antibonding character of the currents involved. Thus,
a great simplification is achieved by introducing the operators
\begin{eqnarray}
\tilde{\rho}^{(a)}_{+ \rho }(x) & = &   \frac{1}{\sqrt{2}}
   \left(   \tilde{\rho}^{(a)}_{1 \rho }(x) +
              \tilde{\rho}^{(a)}_{2 \rho }(x)   \right)     \\  
\tilde{\rho}^{(a)}_{- \rho }(x) & = &  \frac{1}{\sqrt{2}}
    \left(    \tilde{\rho}^{(a)}_{1 \rho }(x) -
              \tilde{\rho }^{(a)}_{2 \rho }(x)   \right)
\end{eqnarray}
The Coulomb interaction acts on the symmetric combinations
$\tilde{\rho}^{(a)}_{+ \rho }(x)$, while the effective attractive
interaction only couples the antisymmetric combinations
$\tilde{\rho}^{(a)}_{- \rho }(x)$. The hamiltonian (\ref{ham})
can be rewritten in the form
\begin{eqnarray}
H_0  & = &    \frac{1}{2} v_F \int_{-k_c}^{k_c} dk
 \sum_{a r i \sigma }  : \rho^{(a)}_{r i \sigma} (k)
             \rho^{(a)}_{r i \sigma} (-k)  :   \nonumber    \\   
  &  &    + \frac{1}{2}  \int_{-k_c}^{k_c} dk \;
    \left(  \sum_a \tilde{\rho}^{(a)}_{+ \rho} (k) \;
         V_C (k)   \sum_b     \;
   \tilde{\rho}^{(b)}_{+\rho} (-k)   \right.     \nonumber     \\
  &  &  \left.    \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
         +   g  \sum_a \tilde{\rho}^{(a)}_{- \rho} (k) \;
                \tilde{\rho}^{(a)}_{- \rho} (-k)    \right)
\label{ham2}
\end{eqnarray}  
where we have defined
\begin{equation}
g = g_2^{(2)} = g_4^{(4)} = - g_2^{(4)} = - g_4^{(2)}  
\end{equation}


After the decoupling of the Coulomb and the effective
attractive interaction, we observe that the latter affects to 
a number $n$ of the $4n$ density operators, while the Coulomb 
interaction acts only on the channel of the total charge density. 
That is, the hamiltonian (\ref{ham2}) can be completely 
diagonalized by changing variables to the symmetric combination 
of the charge density operators of all the metallic nanotubes. 
This is a consequence of the long-range character of the
Coulomb interaction, that allows us to assume that, in the 
interbranch and intrabranch processes,
the different nanotubes interact between them with the
same strength. 

The balance between the long-range repulsive interaction and
the effective attractive interaction is possible as the 
former turns out to be greatly reduced for large values of $n$.
The attraction from phonon exchange is
otherwise an intratube effect that does not depend on the number
of nanotubes. It is then interesting to draw the phase diagram
of the ropes in terms of the number of metallic nanotubes and
the strength of the attractive interaction.
At large values of $n$ and sufficiently large values of $|g|$,
there must exist a phase where the attractive interaction
prevails, with the development of superconducting correlations
in the individual nanotubes. For low values of $n$ and small $|g|$,
the electronic properties have to be dictated by the Coulomb
interaction, with the appearance of the spin-density-wave
correlations characteristic of the Luttinger liquid behavior.

The different phases can be identified by looking at the 
correlators of the model.
The bosonization techniques allow to compute
them by using the correspondence between the fermion fields
and the respective boson operators\cite{emery,lutt1}
\begin{eqnarray}
 \Psi^{(a)}_{r + \sigma }(x)  & = &
   \exp \left( i \Phi^{(a)}_{r + \sigma } (x) \right)      \\
 \Psi^{(a)}_{r - \sigma }(x)  & = &
   \exp \left( - i \Phi^{(a)}_{r - \sigma } (x) \right)   
\end{eqnarray}
where $\partial_x \Phi^{(a)}_{r i \sigma } (x) = 
 2\pi  \rho^{(a)}_{r i \sigma } (x)$ .

We deal first with the propagator $D_{\rm sc}^{(0)} (x,t)$
of the Cooper
pairs in the individual nanotubes. This object factorizes into 
the different channels that appear after diagonalizing the 
hamiltonian
\begin{eqnarray}
 D_{\rm sc}^{(0)}(x,t) & \equiv &
          \langle \Psi^{(a) \dagger}_{+ + \uparrow} (x,t)
  \Psi^{(a) \dagger}_{- - \downarrow} (x,t)
      \Psi^{(a)}_{- - \downarrow} (0,0)  
          \Psi^{(a)}_{+ + \uparrow} (0,0)  \rangle  \nonumber  \\
      &  =  &   C_{\rm sc}(x,t) \prod_{1}^{n} N_{\rm sc}(x,t)
               \prod_{1}^{3n-1} F (x,t)
\label{coop}
\end{eqnarray}
The first factor corresponds to the channel of the total charge 
density of the metallic nanotubes, while $N_{{\rm sc}}$ stands for 
the contribution of the antisymmetric combination of the charge in 
the two subbands. The rest of the factors corresponds to the 
channels with no interaction.

The different factors in Eq. (\ref{coop}) can be computed in 
terms of the respective boson propagators, what leads to 
expressions of the form
\begin{eqnarray}
X_{\rm sc}(x,t) & = &  \exp \left( -\frac{1}{2n} \int_{0}^{k_c}
 dk \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;   \right.   \nonumber    \\
   &  &  \;\;\;\;\;\;\;\;\;  \left.  \frac{1}  {\mu (k)\: k}
   \left(1 - \cos (kx) \: \cos (\tilde{v}_F kt) \right) \right)
\label{prop}
\end{eqnarray}
In the case $X_{\rm sc}(x,t) = C_{\rm sc}(x,t)$, we have 
$\mu (k) = 1/\sqrt{1 + 8n V_{C}(k)/v_F}$ and $\tilde{v}_F (k) = 
v_F/\mu (k)$. For $X_{\rm sc}(x,t) = N_{\rm sc}(x,t)$, we have 
the constant parameters $\mu  = 1/\sqrt{1 - 4|g|/\pi v_F}$ and 
$\tilde{v}_F  = v_F/\mu $.

The same technique can be applied to the computation of the 
propagator $D_{\rm sdw}^{(0)}(x,t)$ of the spin-density-waves
along the nanotubes. The correlator is given by
\begin{eqnarray}
 D_{\rm sdw}^{(0)}(x,t) & \equiv &
          \langle \Psi^{(a) \dagger}_{+ + \uparrow} (x,t)
      \Psi^{(a)}_{- - \downarrow} (x,t)
         \Psi^{(a) \dagger}_{- - \downarrow} (0,0)
   \Psi^{(a)}_{+ + \uparrow} (0,0)  \rangle  \nonumber  \\
      &  =  &   C_{\rm sdw}(x,t) \prod_{1}^{n} N_{\rm sdw}(x,t)
               \prod_{1}^{3n-1} F (x,t)
\label{sdw}
\end{eqnarray}
Each of the factors in Eq. (\ref{sdw}) has now the 
representation
\begin{eqnarray}
X_{\rm sdw}(x,t) & = &  \exp \left( -\frac{1}{2n} \int_{0}^{k_c}
 dk \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;   \right.   \nonumber    \\
   &  &  \;\;\;\;\;\;\;\;\;  \left.  \frac{ \mu (k) }  {k}
   \left(1 - \cos (kx) \: \cos (\tilde{v}_F kt) \right) \right)
\label{prop2}
\end{eqnarray}
with the same correspondence between $\mu (k)$ and $C$, $N$ and
$F$ as for the propagator of the Cooper pairs.

The correlators in the individual nanotubes cannot display an 
instability for any finite value of the frequency\cite{sch}.
The way to
discern the tendency to the formation of long-range order is
to determine whether the correlations are enhanced at large 
distances over the values in the absence of interaction. When
this happens, the Fourier transform of the corresponding
propagator at zero temperature diverges at zero frequency
and momentum. At nonvanishing temperature, the propagators
remain finite but the intertube coupling may give rise to 
the breakdown of the symmetry if the correlations in the 
nanotubes are sufficiently enhanced. We will address this 
question in the next section. The computation of the propagators 
(\ref{coop}) and (\ref{sdw}) can be extended to the case of 
temperature $T \neq 0$, just by inserting the factor
$1 + 2/[\exp (\tilde{v}_F |k|/T) - 1]$ in the integrand of
expressions such as (\ref{prop}) and (\ref{prop2}) \cite{emery}.

At this point, we stick to the model at zero temperature and
map the regions with enhanced superconducting or  
spin-density-wave correlations. The propagators (\ref{coop}) 
and (\ref{sdw}) do not show a perfect scaling behavior at large 
distances, due to the momentum dependence of the Coulomb 
potential. We may take however an effective value $k_0$ of the 
momentum in the infrared to approximate the behavior of each
propagator by a power-law with a constant exponent.

In the case of $D_{\rm sc}^{(0)}(x,t)$, the decay at large
distances takes the form
\begin{equation}
D_{\rm sc}^{(0)}(x,0) \sim 1/x^{2+\gamma}
\end{equation}
where the anomalous scaling dimension is given by 
\begin{equation}
\gamma = \frac{1}{2n \mu (k_0)} + \frac{1}{2 \mu } 
            - \frac{1}{2} - \frac{1}{2n}
\label{asd}
\end{equation}
Similarly, $D_{\rm sdw}^{(0)}(x,t)$ has a large-distance 
behavior
\begin{equation}
D_{\rm sdw}^{(0)}(x,0 ) \sim 1/x^{2+\delta}
\end{equation}
with an anomalous exponent
\begin{equation}
\delta = \frac{\mu (k_0)}{2n} + \frac{\mu }{2}     
            - \frac{1}{2} - \frac{1}{2n}          
\end{equation}

We have represented in Fig. \ref{scsdw} the different
phases that arise by varying the number $n$ of metallic 
nanotubes and the strength $|g|$ of the intratube attractive 
interaction. For the Coulomb interaction, we have taken the
values $2 e^2/\pi^2 v_F = 1.0$ and $k_0 = 10^{-3} k_c $,
which are appropriate for typical experimental 
samples\cite{prl}. At sufficiently large values of $n$, 
a phase with superconducting
correlations opens up, characterized by negative values of 
the anomalous dimension $\gamma $. This phase is placed 
above the upper full line in the diagram. Another phase with 
spin-density-wave correlations shows up, below the lower 
full line in the diagram. This region corresponds to the
points characterized by negative values of $\delta $.




\begin{figure}
\begin{center}
\mbox{\epsfxsize 8.5cm \epsfbox{scsdw.eps}}
\end{center}
\caption{Phase diagram showing the regions where the 
superconducting (SC) correlations and the spin-density-wave
(SDW) correlations prevail, in terms of the strength $|g|$ of
the effective attractive interaction and the number $n$ of
metallic nanotubes.}
\label{scsdw}
\end{figure}





We observe also the existence of an intermediate region in 
which neither $\gamma $ nor $\delta $ are negative. This 
represents a phase with intrinsic metallic properties, where 
no correlations are enhanced and no order would develop 
irrespective of the value of the intertube coupling in the 
rope\cite{foot}. In our model, this phase arises from the 
balance between the Coulomb and the effective attractive 
interaction. The former remains long-ranged, but the 
interaction among a large number of metallic nanotubes is 
what favors the appearance of the metallic phase, 
as a bridge to the development of the superconducting 
correlations.






\section{Superconducting transition in ropes of nanotubes}



In the preceding section we have neglected the effect of 
tunneling between metallic nanotubes, relying on the smallness 
of the amplitude for that process. We have seen however that, 
in the absence of intertube hopping, the model cannot develop 
any instability at finite temperature. 
In this respect, the upper full line in Fig. \ref{scsdw} has to 
be considered as a first approximation to the boundary where the
superconducting phase opens up in real samples.
The tunneling amplitude 
between neighboring nanotubes in a rope is actually what 
dictates the transition temperature to the superconducting 
state, as we study in what follows.

As mentioned before, the single-particle hopping between 
neighboring nanotubes is strongly suppressed in a disordered 
rope. The tunneling amplitude between nanotubes with different 
helicities can be estimated to be of the order of $\sim 0.5 
\times 10^{-4} \; {\rm eV}$. Its smallness stems from the 
obstruction to having precise momentum conservation in the 
hopping between the misaligned lattices of the 
nanotubes\cite{mkm}. In these conditions, the tunneling of 
Cooper pairs turns out to be a much more important effect.

The probabilities of the single-particle hopping and the pair 
hopping can be compared by measuring the respective amplitudes 
with respect to the energy cutoff $E_c$. We can take this 
quantity of the order of $\sim 0.1 \; {\rm eV}$, which is the 
scale below which the  nanotubes in a rope are seen as 1D 
objects. Thus, we find that the single-particle hopping has a 
relative weight of the order of $\sim 0.5 \times 10^{-3}$. The 
amplitude for the tunneling of Cooper pairs can be computed 
roughly as the square of the amplitude $t_T \sim 0.01 \; 
{\rm eV}$, so that the relative weight $\lambda_2$ of that 
process becomes of the order of $\sim 10^{-2}$.

The model of the preceding section has to be corrected then
by adding to the hamiltonian (\ref{ham2}) the term describing
the tunneling of the Cooper pairs
\begin{eqnarray}
 H_2 =  \sum_{\langle a,b \rangle}  ( \lambda_2 )_{ab}
  \int_{-k_c}^{k_c} dk   \int_{-k_c}^{k_c} dp
     \int_{-k_c}^{k_c} dp'
              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
                  \;\;\;\;\;&   &      \nonumber       \\
   \Psi^{(a) \dagger}_{r i \uparrow}(k+p)
          \Psi^{(a) \dagger}_{-r -i \downarrow}(-p)
        \Psi^{(b)}_{-s -j \downarrow}(-p')
           \Psi^{(b)}_{s j \uparrow}(k+p')  &  &
\label{ham3}
\end{eqnarray}
where the sum runs over all pairs $\langle a,b \rangle$ of
nearest-neighbor metallic nanotubes.

Given that the model without pair hopping is exactly solvable,
we can compute the scaling dimension of the operator in $H_2$.
We can check in this way whether this is a relevant perturbation
or otherwise it fades away at low energies. The na\"{\i}ve 
dimension of the four-fermion operator is 2, and the deviation
from that value can be obtained from the large-distance behavior 
of the correlator in Eq. (\ref{coop}). The anomalous scaling 
dimension coincides with the exponent given in Eq. (\ref{asd}). 
Therefore, we observe that it is negative in the phase with 
superconducting correlations, what points at the enhancement of 
the pair-hopping perturbation at low energies. 

The values that can be obtained for $\gamma $ are rather small
anyhow. Taking again an effective value of $k_0 = 10^{-3} k_c$ , 
we find $\gamma \approx -0.02$ for $n = 60$ and 
$4|g|/\pi v_F = 0.6$, and
$\gamma \approx -0.16$ for $n = 120$ and $4|g|/\pi v_F = 0.8$.
In the latter case, for instance, the anomalous scaling leads to
an enhancement by a factor that, four orders of magnitude below
$E_c$, can be at most of the order of $(10^{-4})^{\gamma } \sim 4$. 
Such a scaling does not affect our estimate of the order of 
magnitude for the relative weight $\lambda_2 $.
  
As long as we have the small parameter $\lambda_2 $ in the 
system, we may include the effects of pair hopping by summing
up all the processes in which a Cooper pair, propagating along 
a nanotube with the amplitude (\ref{coop}), tunnels to a 
neighboring metallic nanotube. Let us denote by $l_a$ the 
position of nanotube $a$ in the transverse section of the rope.
We will assume that the metallic nanotubes are dense in the 
collection of nanotubes of the rope\cite{priv}. 
Then, we may write the propagator of the Cooper pairs between 
the metallic nanotube $a$ and the metallic 
nanotube $b$ as a function $D (l_a,l_b;x,t)$. This object is
related to $D_{\rm sc}^{(0)}(x,t)$ through the self-consistent
equation represented in Fig. \ref{dyson}, in the approach that
takes $H_2$ as a perturbation to the hamiltonian 
(\ref{ham2}).


\begin{figure}
\begin{center}
\mbox{\epsfxsize 8cm \epsfbox{dyson.eps}}
\end{center}
\caption{Self-consistent diagrammatic equation for the
propagator $D$ of Cooper pairs along the rope.}
\label{dyson}
\end{figure}




For the sake of simplifying the calculation, we will suppose
that the positions $l_a$ form a periodic arrangement in the
transverse section of the rope. Then we can take the Fourier
transform of these variables as well as of the distance $x$
along the nanotube. The equation for the Fourier transformed
propagator $\widetilde{D} (q;k,\omega_k)$ reads
\begin{equation}
\widetilde{D} (q;k,\omega_k) =
         \widetilde{D}_{\rm sc}^{(0)} (k,\omega_k) +
    \widetilde{D}_{\rm sc}^{(0)} (k,\omega_k) \lambda_2 (q)
                       \widetilde{D} (q;k,\omega_k)
\label{fourier}
\end{equation}

The measure of the condensation of Cooper pairs in the rope
is given by the propagator at zero frequency and momentum
$\widetilde{D} (0;0,0)$. This accounts for the propagation of
a Cooper pair from a metallic nanotube to the rest in the rope. 
From Eq. (\ref{fourier}) we obtain the expression
\begin{equation}
\widetilde{D} (0;0,0) = 
     \frac{ \widetilde{D}_{\rm sc}^{(0)} (0,0) }
  {  1 - \lambda_2 (0) \widetilde{D}_{\rm sc}^{(0)} (0,0)  }
\end{equation}

According to the conventional interpretation, the transition
to the superconducting state is given by the point at which
$\widetilde{D} (0;0,0)$ develops a pole. This may happen only
when $\widetilde{D}_{\rm sc}^{(0)} (0,0)$ becomes large enough 
at low temperatures. In the superconducting region
of the phase diagram, the correlations grow large in the limit
of vanishing temperature, as shown in Fig. \ref{corre}. The
only limitation to develop a real divergence is placed by the
finite length of the system, as we discuss later on.




\begin{figure}
\begin{center}
\mbox{\epsfxsize 8cm \epsfbox{corre.eps}}
\end{center}
\caption{Plot of the propagator
$\widetilde{D}_{\rm sc}^{(0)}$ 
at zero frequency and momentum versus $T/E_c$, for
$2 e^2/(\pi^2 v_F) = 1.0$ . The dashed line
corresponds to the case $n = 1$ and $g = 0$, and the
solid lines to $n = 100$ and the respective values  
(from top to bottom) $4|g|/(\pi v_F) = 0.75, 0.5, 0$ .}
\label{corre}
\end{figure}





The parameter that plays the major role in
setting the values of the transition temperature is the
weight $\lambda_2 (0)$ for pair hopping at zero
transverse momentum. One can look, for instance, for the
points of the phase diagram with transition temperature
$T_c = 10^{-3} E_c$ . This value corresponds to a temperature 
of $\approx 1 \; {\rm K}$ for $E_c = 0.1 \; {\rm eV}$. 
Taking the intermediate value $\lambda_2 (0) = 0.025$, the 
points form the boundary represented by the dashed line
in Fig. \ref{scsdw}. The curve has the same shape that the 
boundary of the superconducting phase determined from the 
expression (\ref{asd}) of the anomalous dimension. However, 
we see that the region with $T_c > 10^{-3} E_c$ above the 
dashed line is sensibly smaller than that of the whole
superconducting phase.

We have represented in Fig. \ref{lambdan} the contour lines
for different critical temperatures in the space of the pair 
hopping parameter $\lambda_2 (0)$ and the number $n$ of metallic 
nanotubes, fixing the coupling of the attractive interaction
at $4|g|/\pi v_F = 0.75$. We observe that a slight change in the
value of the tunneling amplitude leads to a significant
increase in the transition temperature. Considering ropes with
larger content of metallic nanotubes may also help to enhance
$T_c$, although the figure shows that very high values of $n$
have to be reached to find a sensible variation.


\begin{figure}
\begin{center}
\mbox{\epsfxsize 8cm \epsfbox{lambdan.eps}}
\end{center}
\caption{Contour lines for the critical temperature $T_c$ 
depending on the pair-hopping parameter $\lambda_2 (0)$ and 
the number $n$ of metallic nanotubes.}                                            
\label{lambdan}                                  
\end{figure}


Finally, we remark that the finite length of the nanotubes 
imposes a limit on the strength of the correlations. 
The existence of a length scale $L$ spoils the scaling of
the model and, therefore, the approximate power-law behavior 
of the correlators. Thus, the divergence of the propagators at
zero frequency and momentum is cut off in practice at a 
temperature scale which is between $v_F /L$ and one order of
magnitude below that value. This effect is illustrated in 
Fig. \ref{nlarge}, which displays the behavior of 
$\widetilde{D}^{(0)}_{{\rm sc}} (0,0)$ for ropes with different 
number of metallic nanotubes $n = 40, 100, 400, 1000$
and finite length $L = 1000/k_c$.



\begin{figure}
\begin{center}
\mbox{\epsfxsize 8cm \epsfbox{nlarge.eps}}
\end{center}
\caption{Logarithmic plot of $\widetilde{D}^{(0)}_{{\rm sc}}$ 
at zero frequency and momentum versus $T/E_c$, for 
$2 e^2/(\pi^2 v_F) = 1.0$ and $4|g|/(\pi v_F) = 0.75$. From 
top to bottom, the curves correspond to different numbers
of metallic nanotubes $n = 1000, 400, 100$ and 40 .}
\label{nlarge}
\end{figure}




The constraint on the superconducting correlations due to the
finite-size scaling may have been observed in the experiments
reported in Ref. \onlinecite{koc}. It is shown there that
the large drop in the resistance is present in two samples
with respective lengths of $1.6 \; {\rm \mu m}$ and 
$1 \; {\rm \mu m}$. The effect is absent in a third sample 
which is $0.3 \; {\rm \mu m}$ long. 
Assuming that the typical transition temperature for these
samples is around $T_c \sim 0.5 \; {\rm K}$, the relative small 
length of the third sample would explain the absence of 
superconductivity. The minimum transition temperature that
could be supported in that case is of the order of
$T_c \sim 0.1 \; v_F/L \sim 10^{-4} \; {\rm eV }\sim 1 \; 
{\rm K}$. The same argument provides a definite check
of the model elaborated in this paper since, according to it, 
a transition with critical temperature $T_c = 0.1 \; {\rm K}$, 
for instance, should not be present in the samples below 
$1 \; {\rm \mu m}$. In general, it has to be true that 
any sample with length $L$ cannot have a transition at temperatures
lower than $0.1 v_F/L$, as it is observed in the samples 
considered in Ref. \onlinecite{koc}.






\section{Discussion}





In this paper we have shown that superconductivity is
a plausible effect in the ropes of nanotubes. The ropes with
greater number of metallic nanotubes have in general larger
superconducting correlations. We have seen that the long-range
Coulomb interaction is reduced very effectively in 
ropes with 100 metallic nanotubes or more. On the other hand,
the coupling to the elastic modes within each nanotube provides
the attractive interaction leading to the electronic pairing.
We have shown that the low-momentum optical phonons have suitable
properties to balance the effect of the repulsive Coulomb
interaction.

We have dealt with a model providing an exact description of
the competition between the Coulomb interaction and the effective 
attractive interaction in interbranch and intrabranch processes.
Proceeding in this way, we have disregarded the effect of
other processes coming from phonon exchange. These contribute to 
the backscattering and Umklapp couplings $g_1^{(1)}$,
$g_1^{(3)}$, $g_2^{(1)}$ and $g_2^{(3)}$. Only the first and the
third kind of processes play a role in the development of
the superconducting correlations, as analyzed in the Appendix
B. These coupling are marginally relevant in the renormalization
group sense but, as we have already remarked in the paper, the
effects derived from renormalization group scaling turn out to 
be very soft down to the energies where to look for a 
superconducting transition.

The significance of the backscattering processes is found in
that they determine the symmetry of the order parameter
whenever the system becomes superconducting. According to the
analysis in Section II, both $g_1^{(1)}$ and $g_2^{(1)}$
correspond to an effective attractive interaction. Then, from
inspection of the different order parameters considered in
the Appendix B, we conclude that singlet pairing is enhanced
by the backscattering interactions. We observe also
that the $s$-wave symmetry with positive amplitude in all the 
branches is favored over more exotic possibilities like the 
$d$-wave symmetry of the order parameter.

In our description of the mechanism of superconductivity, we 
have taken into account the fact that the ropes are made of 
nanotubes with a random distribution of helicities. This 
prevents the development of any ordered charge or spin structure 
from the coupling of the nanotubes. Moreover, this kind of 
disorder implies a strong suppression of the single-particle 
hopping and that the relevant tunneling process is given by 
the hopping of Cooper pairs between neighboring nanotubes. 
The amplitude for that process is in general very small, 
which explains the relatively low transition temperatures 
measured experimentally.

In any event, the effect of pair hopping has the most direct 
influence on the transition to the superconducting state.
The disorder present in the samples used in the experiments 
has to be very high, and its effect should be studied on more
quantitative grounds to have an estimate of the maximum
transition temperature reachable in the ropes. In this sense,
good prospects should exist to increase the transition 
temperatures, either by suitable refinement of the experimental
samples or by intercalation or modification of the internal
structure of the ropes.





Fruitful discussions with F. Guinea and A. Kasumov are
gratefully acknowledged. This work has been partly
supported by CICyT (Spain) and CAM (Madrid, Spain)
through grants PB96/0875 and 07N/0045/98.






\appendix

\section{Selection rules for the coupling to optical phonons}



The different symmetry of the modes with bonding and  
antibonding character leads to definite relations between the
various electron-phonon couplings. This can be observed in the 
results of Ref. \onlinecite{jdd}, where the full expressions of 
the couplings to the acoustic phonons in armchair nanotubes
have been obtained. In
this appendix we exploit the mentioned symmetry to get the 
corresponding relations in the case of the optical phonons.

The use of the tight-binding approximation is appropriate for
the carbon nanotubes\cite{pietro}. The electron-phonon coupling 
can be written in terms of the polarization vector 
$\mbox{\boldmath $\epsilon$}_s (k)$ depending on site $s$ and 
the amplitudes of the incoming and outgoing modes 
$u_s^{(p)} (k)$ and $u_{s'}^{(p')} (k')$ in 
the respective subbands $p$ and $p'$, taking the form
\begin{eqnarray}
g_{p,p'} (k,k')  & = &  \frac{1}{( \mu \; \omega_{k-k'} )^{1/2} }
  \sum_{\langle s , s' \rangle } 
    u_s^{(p) *} (k)   u_{s'}^{(p')} (k')   \nonumber   \\ 
   &   &     ( \mbox{\boldmath $\epsilon$}_s (k-k') - 
        \mbox{\boldmath $\epsilon$}_{s'} (k-k') ) 
      \mbox{\boldmath $\cdot \nabla$}   J (s,s')
\label{tight}
\end{eqnarray}
In the above expression, the sum is restricted to
nearest-neighbors of the atoms in the unit cell of the nanotube,
$J (s,s')$ is the matrix element of the atomic potential 
connecting orbitals at $s$ and $s'$, $\omega_k$ is the phonon
frequency and $\mu $ is the mass per unit length.

Let us consider first the case of optical phonons with 
longitudinal polarization. The polarization vector depends only
on the distance $z$ measured along the nanotube, but it has
opposite amplitudes in the two sublattices shown in 
Fig. \ref{long} so that
\begin{eqnarray}
\mbox{\boldmath $\epsilon$}_s (k)  & =  & \hat{z}
 \exp (ikz_s)  \;\;\;\;\;\;\;\;\; {\rm for \;\; black \;\; points}  \\
\mbox{\boldmath $\epsilon$}_s (k)  & =  &  - \hat{z}
 \exp (ikz_s)  \;\;\;\;\;\;\;\;\; {\rm for \;\; white \;\; points}
\end{eqnarray}
$\hat{z}$ being the unit vector in the axis direction. 



\begin{figure}
\begin{center}
\mbox{\epsfxsize 7cm \epsfbox{long.eps}}
\end{center}
\caption{Scheme of the unit cell of an armchair nanotube, 
represented between the vertical dashed lines. The arrows
correspond to the displacements for a longitudinal optical
phonon.}
\label{long}
\end{figure}





It is not difficult to see that, if the modes belong to the same 
subband $p = p'$, the different terms in the sum of the 
expression (\ref{tight}) cancel among themselves. Then, we have
that in the case of longitudinal optical phonons
\begin{equation}
g_{1,1} (k,k') = g_{2,2} (k,k') = 0
\label{long1}
\end{equation}
When the incoming and outgoing electron modes are in different 
subbands, the electron-phonon coupling does not vanish. Using the 
fact that the amplitude of the modes with antibonding character 
changes its sign under the exchange of the two sublattices, 
we obtain the result 
\begin{equation}
g_{1,2} (k,k') = - g_{2,1} (k,k')
\label{long2}
\end{equation}

Moving now to the case of transverse optical phonons, we have
a picture like that shown in Fig. \ref{trans}. The polarization 
vector is
\begin{eqnarray}
\mbox{\boldmath $\epsilon$}_s (k)  &  =  &  \hat{\theta} 
 \exp (ikz_s) \;\;\;\;\;\;\;\;\;   {\rm for \;\; black \;\; points}  \\   
\mbox{\boldmath $\epsilon$}_s (k)  &  =  &  - \hat{\theta} 
 \exp (ikz_s) \;\;\;\;\;\;\;\;\;   {\rm for \;\; white \;\; points}
\end{eqnarray}
$\hat{\theta} $ being a unit vector tangential to the nanotube. 
When the incoming and outgoing electron modes belong to the
same subband, we have now 
\begin{equation}
g_{1,1} (k,k') = - g_{2,2} (k,k')
\label{trans1}
\end{equation}
If the modes are in different subbands, the terms in the 
expression (\ref{tight}) cancel out in pairs and we have
\begin{equation}
g_{1,2} (k,k') =  g_{2,1} (k,k') = 0
\label{trans2}
\end{equation}



\begin{figure}
\begin{center}
\mbox{\epsfxsize 7cm \epsfbox{trans.eps}}
\end{center}
\caption{Same scheme as in Fig. \ref{long}, but representing
the displacements for a transverse optical phonon.}
\label{trans}
\end{figure}




We observe that the situation is reversed with respect to the 
case of the acoustic phonons, in which the relations 
(\ref{trans1}) and (\ref{trans2}) apply to the longitudinal branch
while (\ref{long1}) and (\ref{long2}) hold for the transverse
branch.



\section{Pairing symmetry of the superconducting correlations}



There are several response functions that give a measure 
of the superconducting correlations in the nanotubes, each 
of them corresponding to a different symmetry of the pair
wavefunction. They have the general form
\begin{equation}
R(x,t) = \langle {\cal O} (x,t) 
      {\cal O}^{\dagger} (0,0) \rangle
\end{equation}
In the case of singlet and triplet pairing, the operator
${\cal O} (x,t)$ corresponds respectively to the upper $-$
sign and the lower $+$ of the expression
\begin{eqnarray}
{\cal O} (x,t)  & = & \Psi_{++\uparrow}  
       \Psi_{--\downarrow}  \mp  \Psi_{++\downarrow}  
         \Psi_{--\uparrow}       \nonumber \\   
  &  & + \Psi_{-+\uparrow}   \Psi_{+-\downarrow} 
       \mp  \Psi_{-+\downarrow} \Psi_{+-\uparrow} 
\end{eqnarray}
There are also more exotic possibilities like the $d$-wave 
symmetry of the Cooper pairs, corresponding to
\begin{eqnarray}
{\cal O} (x,t)  & = & \Psi_{++\uparrow} \Psi_{--\downarrow}
   -  \Psi_{++\downarrow}  \Psi_{--\uparrow}  \nonumber \\
  &  & - ( \Psi_{-+\uparrow} \Psi_{+-\downarrow}
       -  \Psi_{-+\downarrow}  \Psi_{+-\uparrow} )
\end{eqnarray}

The symmetry of the superconducting correlations 
can be obtained by using a diagrammatic
analysis of the different response functions. In the case of
singlet and triplet pairing, the contributions to the 
correlator have the structure depicted in Fig. \ref{pairs},
with respective upper and lower signs. Then, the enhancement
of the singlet pairing response function is driven by the
combination of renormalized couplings $g_2^{(2)} + g_1^{(1)}
+ g_2^{(1)} + g_1^{(2)}$, while in the case of triplet 
pairing the combination is $g_2^{(2)} - g_1^{(1)} + g_2^{(1)} 
- g_1^{(2)}$.



\begin{figure}
\begin{center}
\mbox{\epsfxsize 7cm \epsfbox{pairs.eps}}
\end{center}
\caption{Contributions to the response functions for singlet and 
triplet pairing, where the filled circles represent the different
renormalized vertices.}
\label{pairs}
\end{figure}



For the response function with $d$-wave symmetry of the 
Cooper pairs, the structure of the contributions is represented
in Fig. \ref{paird}. The enhancement is controlled now by the 
combination of couplings $g_2^{(2)} + g_1^{(1)} - g_2^{(1)}
- g_1^{(2)}$.



\begin{figure}
\begin{center}
\mbox{\epsfxsize 7cm \epsfbox{paird.eps}}
\end{center}
\caption{Contributions to the response function for $d$-wave
pairing, with the same representation as in Fig. \ref{pairs}.}
\label{paird}
\end{figure}







\begin{references}


\bibitem{kas}
A. Yu. Kasumov {\em et al.}, Science {\bf 284}, 1508 (1999). 

\bibitem{marcus}
A. F. Morpurgo {\em et al.}, Science {\bf 286}, 263 (1999).

\bibitem{koc}
M. Kociak {\em et al.}, Phys. Rev. Lett. {\bf 86}, 2416 (2001).

\bibitem{bal}
L. Balents and M. P. A. Fisher, Phys. Rev. B {\bf 55}, R11973
(1997).

\bibitem{berk}
Yu. A. Krotov, D.-H. Lee and S. G. Louie, Phys. Rev. Lett.
{\bf 78}, 4245 (1997).

\bibitem{eg}
R. Egger and A. O. Gogolin, Phys. Rev. Lett. {\bf 79}, 5082
(1997); Eur. Phys. J. B {\bf 3}, 281 (1998).

\bibitem{kane}
C. Kane, L. Balents and M. P. A. Fisher, Phys. Rev. Lett. {\bf
79}, 5086 (1997).

\bibitem{wild} 
J. W. G. Wild\"{o}er {\em et al.}, Nature {\bf 391}, 59 (1998).
T. W. Odom {\em et al.}, Nature {\bf 391}, 62 (1998).

\bibitem{saito}
J. W. Mintmire, B. I. Dunlap and C. T. White, Phys. Rev. Lett.
{\bf 68}, 631 (1992).
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\bibitem{emery}
V. J. Emery,
in {\em Highly Conducting One-Dimensional Solids},
ed. J. T. Devreese, R. P. Evrard and V. E. Van Doren
(Plenum, New York, 1979).

\bibitem{lutt1}
J. S\'olyom, Adv. Phys. {\bf 28}, 201 (1979).

\bibitem{lutt2}
F. D. M. Haldane, J. Phys. C {\bf 14}, 2585 (1981).

\bibitem{sch}  
H. J. Schulz, in {\em Correlated Electron Systems}, Vol. 9,
ed. V. J. Emery (World Scientific, Singapore, 1993).

\bibitem{exp}
M. Bockrath {\em et al.}, Nature {\bf 397}, 598 (1999).  

\bibitem{yao}
Z. Yao {\em et al.}, Nature {\bf 402}, 273 (1999).

\bibitem{prl}
J. Gonz\'alez, Phys. Rev. Lett. {\bf 87}, 136401 (2001).

\bibitem{micro}
J. Gonz\'alez, Phys. Rev. Lett. {\bf 88}, 076403 (2002).

\bibitem{chi}
Z. K. Tang {\em et al.}, Science {\bf 292}, 2462 (2001).

\bibitem{dirac}
J. Gonz\'alez, F. Guinea and M. A. H. Vozmediano, Nucl. Phys. B
{\bf 406}, 771 (1993).

\bibitem{grab}
R. Egger and H. Grabert, Phys. Rev. Lett. {\bf 79}, 3463 (1997).
S. Bellucci and J. Gonz\'alez, Eur. Phys. J. B {\bf 18}, 3
(2000).

\bibitem{wang}
D. W. Wang, A. J. Millis and S. Das Sarma, Phys. Rev. B
{\bf 64}, 193 307 (2001).

\bibitem{loss}
D. Loss and T. Martin, Phys. Rev. B {\bf 50}, 12160 (1994).

\bibitem{jdd}
R. A. Jishi, M. S. Dresselhaus and G. Dresselhaus, 
Phys. Rev. B {\bf 48}, 11385 (1993).

\bibitem{mkm}
A. A. Maarouf, C. L. Kane and E. J. Mele, Phys. Rev. B {\bf 61},
11156 (2000).

\bibitem{caron}
See also A. S\'ed\'eki, L. G. Caron and C. Bourbonnais, report 
cond-mat/0111296.

\bibitem{foot}
A phase of this kind has been also found in Ref. 
\onlinecite{loss} in the study of the 1D electron system 
coupled to acoustic phonons.

\bibitem{priv}
This is the case of the superconducting ropes
of Ref. \onlinecite{koc}, although to meet such
experimental condition one has to find the appropriate
sample out of a large number of them.

\bibitem{pietro}
L. Pietronero, S. Str\"assler, H. R. Zeller and M. J. Rice,
Phys. Rev. B {\bf 22}, 904 (1980).






\end{references}






\end{document}





Carbon nanotubes have nowadays a great potential for 
technological applications in devices at the nanometer
scale. This makes very important the precise knowledge
of their electronic properties, taken as individual
structures as well as when packed in the form 
of ropes. The first step of discerning their metallic 
or insulating behavior has been completed, as the 
results obtained from theoretical 
considerations\cite{saito} 
have been confirmed experimentally\cite{wild}. 
It turns out that the low-energy spectrum of a carbon 
nanotube may be gapless or not
depending on the particular wrapping of the graphene
sheet to form the tubular structure. 


The electron interactions are also known to modify
significantly the transport properties of the 
nanotubes\cite{exp,yao}. 
In the case of metallic single-walled nanotubes, the
one-dimensional (1D) character of the system leads to a strong
correlation among the electrons, making the so-called 
Luttinger liquid\cite{emery,lutt1,lutt2,sch} 
the most appropriate paradigm 
to describe the electronic properties\cite{bal,eg,kane}. 
Evidence of Luttinger liquid
behavior has been found in measurements of the tunnelling
conductance in ropes\cite{exp} and in individual 
single-walled nanotubes\cite{yao}. 

Experiments have been also carried out to probe 
superconducting correlations in the carbon 
nanotubes\cite{kas,marcus}. By
suspending them between superconducting contacts, large 
supercurrents have been measured in samples almost made of
a single-walled nanotube and in ropes\cite{kas}. 
More recently, superconducting properties have been 
measured in ropes suspended between nonsuperconducting,
good metallic contacts\cite{sup}. Evidence of 
superconducting fluctuations has been also obtained 
in individual single-walled nanotubes\cite{chi}. 
In the experiments presented in Ref. \onlinecite{sup}, 
a drop by two orders of magnitude in the resistance
has been found, down to the minimum value consistent 
with the number of conducting channels in the rope. 
This feature, together with its suppression under a 
suitably high magnetic field, shows the existence of
superconductivity inherent to the ropes of 
nanotubes\cite{sup}. The value of the 
transition temperature varies from one sample to other, 
being below 1 K in the two where the resistance drop 
has been measured. 

In this Letter we develop a model to account for the 
superconductivity intrinsic to the ropes of nanotubes. 
The problem that faces any model trying to describe 
such effect is twofold. In the first place, 
it is known that the single-particle hopping between
neighboring nanotubes in a rope is strongly 
suppressed\cite{mkm}. This is because, in general, 
the different helical structure of the
nanotubes leads to the misalignement of their lattices
and, therefore, to the difficulty of conserving 
momentum for an electron hopping from one nanotube
to the other. On the other hand, the effect of 
superconductivity cannot rely exclusively on the 
properties of the individual nanotubes, since any
correlation in a 1D system can only
develop a divergence at zero temperature\cite{sch}. 

The experiments on the proximity effect
of Ref. \onlinecite{kas} show anyhow 
the existence of sensible superconducting fluctuations
in carbon nanotubes. The precise value of the 
critical supercurrents found there can be only 
explained by the presence of a short-range attractive 
interaction coming from the coupling to the elastic 
modes of the nanotube\cite{prl}. The point is that,
as long as Cooper pairs are formed locally with zero
total momentum, they may overcome the special
difficulty that single electrons find to tunnel
between neighboring nanotubes.

According to the work of Ref. \onlinecite{mkm}, the
amplitude $t_T$ for tunnelling between nanotubes with 
the same chirality and orientation is of the order
$t_T \sim 0.01$ eV. In a disordered rope, the
misalignement of the lattices of neighboring nanotubes
introduces in general an additional suppression of the
hopping amplitude by a factor $ \sim
\exp (- R a_0 (\delta k)^2/4) $, where $R$ is the radius
of the tube, $\delta k$ is related to the mismatch of
the Fermi points, and $a_0$ is a parameter of the order 
$\sim 0.5$ \AA \cite{mkm}.
For a typical nanotube radius $R = 7$ \AA, this factor
is $ \sim 0.005 $. 

The tunnelling amplitudes have to be compared with 
the energy scale at which the metallic nanotubes behave 
as 1D objects. 
This is the scale $E_c$ below which the gapless
linear subbands dominate the physical properties, 
and it can be estimated as $E_c \sim 0.1$ eV. 
The probability $\lambda_2$ of tunnelling of a Cooper 
pair, given by the square of $t_T$ in units of $E_c$,
has a relative weight of the order $\sim 0.01$. 
On the other hand, the probability of
tunnelling of a single electron has a relative weight
of the order $\sim 0.0005$. This means that pair hopping
is the dominant process for tunnelling between neighboring
nanotubes in a disordered rope.

Prior to considering pair hopping, the rope can be described
at low energies by a model in which each nanotube is treated 
as a 1D system, whose charge can interact
with the charge in the other nanotubes of the rope. 
The hamiltonian for this model, including a collection of
metallic nanotubes $a = 1, \ldots n$ with linear
branches of different chirality $i = +,-$, can be written
in terms of the respective density operators
$\rho_{a i \sigma }$ \cite{rem}:
\begin{eqnarray}
H_1  & = &    \frac{1}{2} v_F \int_{-k_c}^{k_c} dk 
 \sum_{a i \sigma }  : \rho_{a i \sigma} (k)
             \rho_{a i \sigma} (-k)  :   \nonumber      \\
  &  &    + \frac{1}{2}  \int_{-k_c}^{k_c} dk \; 
      \sum_{a i \sigma } \rho_{a i \sigma} (k) \; 
  \sum_{b j \sigma'  }  V_{ab}(k)  \;
                      \rho_{b j \sigma'} (-k)
\label{ham}
\end{eqnarray}
where 
%$k_c$ is related to $E_c$ through the Fermi velocity $v_F$, 
$k_c = E_c/v_F$.


For the potential between different nanotubes 
$V_{ab}$ we take the Coulomb interaction     
$V_C (k) = e^2 /(4\pi^2)
      \log |(k_c + k)/ k|$ \cite{wang}, which remains 
long-ranged in one spatial dimension\cite{grab}. 
According to the results of Ref. \onlinecite{prl}, the 
interaction potential $V_{aa}$ within each nanotube includes
moreover the effective short-range attraction coming from
the coupling to the elastic modes, so that
$V_{aa}(k) = e^2 /(4\pi^2) \log |(k_c + k)/ k| - g/(2\pi )$.
The strength $g$ of the attractive interaction
is inversely proportional to the mass of the atoms and
directly proportional to $(t'/v_s)^2$, where $t'$ is the
modulation of the lattice hopping and $v_s$ is the speed
of sound\cite{loss}. A rough estimate for a carbon 
nanotube gives $g/v_F \sim 0.2$ .


Terms which couple the spin densities have been neglected
in writing the hamiltonian (\ref{ham}).
These are backscattering (BS) interactions, which arise as 
a remnant of the Coulomb interaction at short distances.
Being local in the nanotube lattice, their couplings are 
reduced by a relative factor of the order $\sim 0.1 a/R$, 
where $a$ is the nanotube lattice spacing\cite{bal,eg,kane}. 
These terms are marginally relevant in the renormalization 
group sense. This means that they have greater strength as 
the model is scaled to smaller energies, but the rate of
increase starts being quadratic in the own couplings. 
Thus, the theory has to be scaled to extremely low 
energies, many orders of magnitude below $E_c$, to have the 
BS couplings comparable to the couplings in 
(\ref{ham}) \cite{eg}.


The bundle of 1D electron systems coupled only
by charge interactions ressembles the system proposed in
Ref. \onlinecite{slide} for the description of the sliding
Luttinger liquid. Here the Coulomb interaction couples each 
of the nanotubes to all the others in the bundle.
We recall that, for the samples of the experiments reported
in Ref. \onlinecite{sup},
the number of metallic nanotubes is very
large, of the order $n \sim 100$. Then, the coupling
of the charge in the different nanotubes leads to a
significant reduction of the effective repulsive
interaction. On the other hand, the effect of the intra-tube
attractive interaction does not depend on the number of
nanotubes and, in certain regime, it may dominate over
the Coulomb interaction.



The above point can be checked by looking at the
correlators of the model governed by $H_1$. These can be
computed exactly by means of bosonization techniques.
For instance, the propagator $D^{(0)}(x,t)$ for the Cooper 
pairs within each nanotube factorizes into the different 
interaction channels, taking the form 
\begin{eqnarray}
 D^{(0)}(x,t) & \equiv &   
          \langle \Psi^{+}_{a + \uparrow} (x,t)
   \Psi^{+}_{a - \downarrow} (x,t) \Psi_{a + \uparrow} (0,0)   
     \Psi_{a - \downarrow} (0,0)  \rangle  \nonumber  \\ 
      &  =  &   C(x,t) \prod_{1}^{n-1} N(x,t)
               \prod_{1}^{3n} F(x,t)
\label{coop}
\end{eqnarray}

The first factor in Eq. (\ref{coop}) corresponds to the 
contribution of the total charge density, which is given 
at zero temperature by the expression
\begin{equation}
C(x,t) = \exp \left( -\frac{1}{2n} \int_{0}^{k_c}
     dk \frac{1}  {\mu (k)\: k}
      \left(1 - \cos (kx) \: \cos (\tilde{v}_F kt) \right)
            \right)
\label{prop}
\end{equation}
where $\mu (k) = 1/\sqrt{1 + 8n V_{C}(k)/v_F - 4g/(\pi v_F)}$ 
and $\tilde{v}_F (k) = v_F/\mu (k)$.
The factors $N(x,t)$ correspond to the rest of the charge
channels, and they have a form similar to (\ref{prop}) but
with $\mu  = 1/\sqrt{1 - 4g/(\pi v_F)}$ instead of 
$\mu (k)$ and $\tilde{v}_F  = v_F/\mu $ instead of
$\tilde{v}_F (k)$. The factors $F(x,t)$ correspond to the
noninteracting channels with $\mu = 1$.
The computation can be extended to the case of 
temperature $T \neq 0$, just by inserting the factor 
$1 + 2/[\exp (\tilde{v}_F |k|/T) - 1]$ in the integrand of 
expressions such as (\ref{prop}) \cite{emery}.


The couplings that produce the most accurate fit of the
critical supercurrents reported in Ref. \onlinecite{kas} 
are $2 e^2/(\pi^2 v_F) \approx 1.0$ and $4g/(\pi v_F) 
\approx 0.6$ \cite{prl}. For $n \sim 100$, the strong
reduction of the Coulomb interaction implies that the 
effect of the intra-tube attractive interaction prevails
in the system. In the $g$-ology 
description\cite{lutt1}, the model is in the regime with 
short-range attractive coupling $g_2 < 0$, where the singlet
superconductor response is enhanced over the 
charge-density-wave response. 
The temperature dependence of the Fourier
transform $\widetilde{D}^{(0)}$ of the propagator at zero 
frequency and momentum is represented in Fig. \ref{one}.
The enhancement of the propagator is the signal that
large superconducting correlations exist in the individual
nanotubes at low temperatures.



The above considerations are pertinent
to the system at half-filling, in which the linear subbands
cross at the Fermi level. When the nanotubes are slightly doped,  
the shift of the Fermi level gives rise to four Fermi points. 
The Cooper pairs have then the possibility to resonate between 
the outer and the inner gapless subbands. 
Anyhow, as long as the repulsive interactions
mediating these processes are reduced by a relative 
factor as small as that of
the BS interactions\cite{bal,eg}, the $s$-wave pairing is 
favored over other channels with different symmetry.










%For these
%parameters, the model is in a regime where 
%the propagator $D^{(0)}$ is greatly enhanced at low 
%temperatures. This is the signal that
%large superconducting correlations exist in the individual 
%nanotubes. We have represented in
%Fig. \ref{one} the temperature dependence of the Fourier
%transform $\widetilde{D}^{(0)}$ of the propagator at zero
%frequency and momentum. The divergence of the correlator is
%cut off at zero temperature, as a consequence of computing
%$\widetilde{D}^{(0)}$ for a system with finite size to 
%establish a proper correspondence with the experimental 
%conditions (see below). 
%We observe in Figs. \ref{one} and \ref{three} that the
%peak at $T = 0$ is more enhanced for larger values
%of $n$, what follows from the strong renormalization of
%the Coulomb interaction for a large number of nanotubes in
%the rope.


\begin{figure}[H]

\epsfig{file=micro1.eps, height=7cm, width=7cm,
         bbllx=0, bblly=0, bburx=600, bbury=600,
         angle=270}

\caption{Plot of the propagator $\widetilde{D}^{(0)}$ at
zero frequency and momentum versus $T/E_c$, for 
$2 e^2/(\pi^2 v_F) = 1.0$ . The dashed line 
corresponds to the case $n = 1$ and $g = 0$, and the 
solid lines to $n = 100$ and the respective values 
(from top to bottom) $4g/(\pi v_F) = 0.75, 0.5, 0$ .}
\label{one}
\end{figure}




The tunnelling of Cooper pairs between the nanotubes can 
be taken into account by modifying the hamiltonian 
(\ref{ham}) with the additional term
\begin{eqnarray}
 H_2 =  \sum_{\langle a,b \rangle}   ( \lambda_2 )_{ab} 
  \int_{-k_c}^{k_c} dk   \int_{-k_c}^{k_c} dp
     \int_{-k_c}^{k_c} dp'
              \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
                  \;\;\;\;\;&   &      \nonumber       \\
  \left(  \Psi^{+}_{a i \uparrow}(k+p)
          \Psi^{+}_{a -i \downarrow}(-p)
           \Psi_{b j \uparrow}(k+p')
           \Psi_{b -j \downarrow}(-p')
              + {\rm h. c.} \right) &  &
\label{ham2}
\end{eqnarray}
where $\Psi^{+}_{a i \sigma}$ is the electron
operator and
the sum runs over all pairs $\langle a,b \rangle$ of
nearest-neighbor metallic nanotubes.


The term (\ref{ham2}) is a relevant perturbation
from the renormalization group point of view.
However, the anomalous scaling dimensions of the relevant
perturbations turn out to be in general rather small. 
They can be computed in the boson representation and, 
in the case of the term (\ref{ham2}), the result is
$\gamma_2 = 2[1/(4n\mu (k_0)) + (n-1)/(4n\mu ) - 1/4]$.
$k_0$ represents some effective value, which does 
not affect significantly the estimate. For the above mentioned 
couplings, we obtain $\gamma_2 \approx - 0.07$. Thus, even 
four orders of magnitude below $E_c$, we observe that 
$\lambda_2$ is not enhanced by more than a factor
$(10^{-4})^{\gamma_2} \sim 2$.



The proposed model has a
superconducting instability at some finite temperature,
provided that the Cooper pairs are able to percolate   
in the transverse directions of the rope.
This is the case of the superconducting ropes
of Ref. \onlinecite{sup}, although to meet such
experimental condition one has to find the appropriate      
sample out of a large number of them\cite{com}.
Assuming the coherence in the hopping of pairs in the 
transverse directions,
we may write the propagator of the Cooper pair 
from a metallic nanotube $a$ to
another $b$ as a function of the distance $x$ along the
rope and the positions $l_a$ and $l_b$ of the nanotubes
in the transverse section of the rope, 
$D (l_a,l_b;x,t)$. This object can be related to the
propagator $D^{(0)}$ along each nanotube through the
self-consistent diagrammatic equation in Fig. \ref{two},
which takes into account the dominant terms in powers of
$\lambda_2$.


\begin{figure}
\begin{center}
\mbox{\epsfxsize 8.5cm \epsfbox{micro2.eps}}
\end{center}
\caption{Self-consistent diagrammatic equation for the 
propagator $D$ of Cooper pairs along the rope.}
\label{two}
\end{figure}



By introducing the Fourier transform with respect to 
$(x,t)$ as well as in the $l_a$ variables\cite{rem2}, 
the equation in Fig. \ref{two} can be written for the 
Fourier transformed propagator $\widetilde{D}$ in the 
form
\begin{equation}
\widetilde{D} (q;k,\omega_k) =
         \widetilde{D}^{(0)} (k,\omega_k) +
    \widetilde{D}^{(0)} (k,\omega_k) \lambda_2 (q) 
                       \widetilde{D} (q;k,\omega_k)
\label{fourier}
\end{equation}

We are interested in the propagation of the Cooper pairs
from a metallic nanotube to the rest in the rope, which
is given by $\widetilde{D} (0;0,0) =
    \sum_{l_b} \int dx \int dt \;  D(l_a,l_b;x,t) $.
By solving Eq. (\ref{fourier}), we get the result
\begin{equation}
\widetilde{D} (0;0,0) = \frac{ \widetilde{D}^{(0)} (0,0) }
        {  1 - \lambda_2 (0) \widetilde{D}^{(0)} (0,0)  }
\end{equation}

The relevant dependence of $\widetilde{D} (0;0,0)$, as well
as of $\widetilde{D}^{(0)} (0,0)$, is on the temperature.
It becomes clear that, if the superconducting correlations
are such that $\widetilde{D}^{(0)} (0,0)$ has a divergence
at $T = 0$, then the effect of pair hopping gives rise to
the appearance of a pole at a finite value of $T$ in the
propagator of the Cooper pairs in the rope. According to
the conventional interpretation, 
this is the signal of the
condensation of Cooper pairs and the onset of the
superconducting transition in the system.

In practice, when dealing with a rope of finite length $L$,
the divergence of $\widetilde{D}^{(0)} (0,0)$ is cut off at 
a temperature scale about one order of magnitude below 
$ v_F /L $. The curves shown in Figs. \ref{one} and 
\ref{three}, for instance, have been obtained for a 
system with $L = 1000/k_c $, which corresponds to $L \sim 1$ 
${\rm \mu m}$ with an appropriate choice of the length scale.


Taking our estimate of the pair-hopping amplitude
$\lambda_2 \sim 0.01$, we observe that the length $L =
1000/k_c $ may be in some samples at the limit below 
which a superconducting instability cannot arise in 
the system. This depends on the spatial distribution 
of metallic nanotubes, which should determine more precisely 
the effective value of $\lambda_2$ to be used. Quite 
remarkably, a superconducting transition has
been found in samples whose length is 1 ${\rm \mu m}$ or
greater, while a sample with low resistance and 
0.3 ${\rm \mu m}$ long has shown no 
transition at all\cite{sup}.


\begin{figure}[H]

\epsfig{file=micro3.eps, height=7cm, width=7cm,
         bbllx=0, bblly=0, bburx=600, bbury=600,
         angle=270}

\caption{Logarithmic plot of $\widetilde{D}^{(0)}$ at zero
frequency and momentum versus $T/E_c$. The solid lines are
the transposition of the solid curves in Fig. \ref{one} for
$n = 100$ and the respective values
$4g/(\pi v_F) = 0.75, 0.5, 0$ . The dashed line corresponds
to the case $n = 400$ and $4g/(\pi v_F) = 0.75$ .}
\label{three}
\end{figure}



Relying also on our estimate for $\lambda_2$, we observe 
from Fig. \ref{three} that the temperature of the transition 
to the superconducting phase in a disordered rope with about
100 metallic nanotubes can be in the range between
$10^{-4} E_c$ and $10^{-3} E_c$. As the natural energy
scale in our model is $E_c \sim 0.1$ eV, this sets
the scale for typical transition temperatures in the range
between 0.1 K and 1 K . These values are consistent with 
the transition temperatures $T_c \approx $ 0.1 K and 
$T_c \approx $ 0.4 K measured experimentally in the two 
samples of Ref. \onlinecite{sup}.


The results of our analysis show that the superconductivity
in the ropes of nanotubes is close in nature to that of 
the alkali-doped fullerenes\cite{gun}. 
We have seen that the 
tunnelling of electrons between neighboring nanotubes in
a rope takes place in pairs,
which are formed within each nanotube due to the large
superconducting correlations which develop at low
temperatures.

The low values of $T_c$ compared to those of the alkali-doped 
fullerenes can be understood on phenomenological grounds 
by the fact that the electron-phonon coupling is smaller
in the nanotubes.
%\cite{bene}. 
This is consistent with a higher 
estimate of $T_c$ obtained in Ref. \onlinecite{chi}
from measurements on small-diameter single-walled nanotubes,
which have a larger electron-phonon coupling than 
the nanotubes in the ropes. As seen in Fig. \ref{three},
another way to obtain a higher $T_c$ in the ropes
may be to increase the number of nanotubes of the 
samples. Finally, the feasibility of controlling the 
rate of tunnelling between the nanotubes should be 
studied since, as observed from Figs. \ref{one} and 
\ref{three}, a slight change in that parameter may
result in an increase of $T_c$ by more than one order
of magnitude.























\tolerance = 10000
\documentstyle[prl,aps,epsfig,twocolumn]{revtex}
\begin{document}
\draft
\flushbottom
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\title{Consistency of superconducting correlations with one-dimensional
electron interactions in carbon nanotubes}
\author{J. Gonz\'alez   \\}
\address{
        Instituto de Estructura de la Materia.
        Consejo Superior de Investigaciones Cient{\'\i}ficas.
        Serrano 123, 28006 Madrid. Spain.}
%%        $^2$Instituto de Ciencia de Materiales.
%%        Consejo Superior de Investigaciones Cient{\'\i}ficas.
%%        Cantoblanco. 28049 Madrid. Spain.}


\date{\today}
\maketitle
\begin{abstract}
\widetext
We show that a model of interacting electrons in one dimension
is able to explain the order of magnitude as well as the
temperature dependence of the critical supercurrents recently
measured in nanotube samples placed between superconducting
contacts. We use bosonization methods to deal with the long-range Coulomb
interaction, ending up with a picture in which the critical current
does not follow the temperature dependence of the gap
in the contacts. 
Our results also reveal the presence of a
short-range attractive interaction in the nanotubes,
which accounts for a significant enhancement of the critical
supercurrents.


\end{abstract}
\pacs{71.10.Pm,74.50.+r,71.20.Tx}

]



\narrowtext 
\tightenlines

%\newpage





Since the discovery of carbon nanotubes, these have offered a
great potential for novel electronic properties and 
technological applications. 
It has been checked experimentally\cite{wild}
the theoretical prediction that 
there should be semiconducting as well as metallic 
nanotubes\cite{ham}. 
It has been also remarkable the experimental observation of 
unconventional transport properties\cite{exp}, that seem to be 
compatible with the expected Luttinger liquid behavior of 
one-dimensional electron systems\cite{lutt}. 
Different approaches have predicted the 
appearance of phases with broken symmetry in the carbon nanotubes 
at very low energies\cite{bal,louie,eg,odint}. Anyhow, the estimates 
are in general that there should be enough margin to observe the 
characteristic scaling behavior of the Luttinger liquid over a wide
range of temperatures.

 
A different class of experiments has been aimed to test the
superconducting properties of the carbon nanotubes\cite{kas,marcus}. 
One of the most striking results has been the observation of 
supercurrents along carbon nanotubes placed between superconducting 
contacts\cite{kas}. In a sample made of a single-walled nanotube, 
for instance, critical supercurrents have been measured that are 
about 40 times higher than expected from the value of the gap in the
contacts\cite{kas}. They also show a very flat
dependence with temperature, until the critical value of the
superconducting contacts is approached. In that respect, there is a
marked difference from the behavior of another sample made of a rope  
of nanotubes, where the critical supercurrent seems to follow the
BCS gap in a certain range of low temperature\cite{kas}.

A model of the electron interaction in the carbon
nanotubes should give a quantitative account of all these
different observations of superconducting correlations. We show
in this letter that the mentioned features of the critical
supercurrent can be understood in the framework of a
one-dimensional theory of interacting electrons. We will see
that the experimental data are consistent with a definite
form of the one-dimensional interaction, as it is actually
nontrivial to reproduce both the shape and the order of
magnitude of the supercurrents in the single-walled nanotube and
in the rope of nanotubes. In particular, the experimental values 
of the supercurrent point at a sensible renormalization of the 
strength of the long-range Coulomb interaction, specially 
in the sample made of a rope of nanotubes, in agreement with 
earlier theoretical predictions\cite{eur,graph}.


A metallic single-walled nanotube has several one-dimensional
subbands, with two pairs of linear branches crossing at
Fermi points $k_F$ and $-k_F$.
We deal in this letter with an effective description of the
nanotubes for energies below the scale $E_c$ at which all the
gapped subbands decouple in the computation of low-energy
properties, so that the relevant modes left belong to the
linear branches close to the Fermi level. We
can estimate this energy $E_c$ as a few 
tenths of ${\rm eV}$, for a typical single-walled nanotube 
with about 10 subbands. 

The low-energy excitations can be encoded into four boson fields,
each boson corresponding to a linear branch in the same fashion
as in the Luttinger model\cite{lutt}. The hamiltonian of the 
effective theory
can be written in terms of the respective density operators 
$\rho_{i a \sigma}$, labelled by the Fermi point $a = 1,2$ and
by the chirality $i = L,R$,
\begin{eqnarray}
H  & = &      \frac{1}{2} v_F \int_{-k_c}^{k_c} dk 
 \sum_{i a \sigma }  : \rho_{i a \sigma} (k)
             \rho_{i a \sigma} (-k)  :   \nonumber      \\
  &  &    + \frac{1}{2}  \int_{-k_c}^{k_c} dk \; 
      \sum_{i a \sigma } \rho_{i a \sigma} (k) \; V(k)
      \sum_{j b \sigma'  } \rho_{j b \sigma'} (-k)
\label{ham}
\end{eqnarray}
In the above expression, $k_c$ is related to $E_c$ through the 
Fermi velocity $v_F$, $k_c = E_c/v_F$. 

Our assumption regarding the
interaction will be the presence of the long-range 
Coulomb interaction 
$V(k) \approx e^2 /(4\pi^2) \log |(k_c + k)/ k|$ \cite{wang}, 
which remains unscreened in one spatial dimension\cite{grab}, 
plus an additional short-range effective attraction
coming from the coupling to the elastic modes of the nanotube. 
In this framework, we are neglecting backscattering and Umklapp
processes that mix different chiralities and Fermi points, relying on
the fact that those interactions have smaller relative strength
($\sim 0.1/n$, in terms of the number $n$ of subbands\cite{eg,kane})
and they remain small down to extremely low energies\cite{eg}.

The correlators in the model governed by (\ref{ham}) can be
computed by changing variables to the total charge density
operators $ \rho_i (k) = (1 / \sqrt{N})  \sum_{a \sigma } 
      \rho_{i a \sigma} (k) , i = L, R  $ ,
where $N$ stands in general for the number of channels 
$\{ (a,\sigma) \}$, so that $N = 4$ in the case of a
single-walled nanotube.

A typical propagator of Cooper pairs, for instance, becomes
\begin{eqnarray}
 G(x,t) & \equiv &    \langle \Psi_{L 1 \uparrow} (x,t) 
   \Psi_{R 2 \downarrow} (x,t) \Psi^{+}_{L 1 \uparrow} (0,0)   
     \Psi^{+}_{R 2 \downarrow} (0,0)  \rangle  \nonumber  \\ 
      &  =  &   C(x,t) F(x,t)
\end{eqnarray}
where $F$ is the part that does not depend on the interaction and
$C$ corresponds to the propagation of the total charge. At zero
temperature, for instance, we have 
\begin{equation}
C(x,t) = \exp \left( -\frac{2}{N} \int_{0}^{k_c} dk \frac{1}
   {\mu (k)\: k} \left(1 - \cos (kx) \: \cos (\tilde{v}_F kt) \right)
            \right)
\label{prop}
\end{equation}
where $\mu (k) = 1/\sqrt{1 + 2N V(k)/v_F}$ and $\tilde{v}_F = 
v_F/\mu (k)$. The other factor has the simple dependence
\begin{equation}
F(x,t) = 1/ \left|k_c^2 (x - v_F t) (x + v_F t)\right|^{\frac{N-1}{N}}
\end{equation}

The critical supercurrent ${\cal I}$ can be estimated under
the assumption that (i) the normal-superconductor junctions are 
perfectly transmitting\cite{maslov}, or 
(ii) the single-particle scattering is
relevant at the interfaces\cite{fazio}. The latter is
more realistic for the experiments that we are considering. The
distance $L$ between the superconducting contacts is large
enough that ${\cal I}$ can be expressed as a function of $L$ and 
the temperature $T$ as 
\begin{equation}
{\cal I}_L (T)  = e v_F k_c \int_{0}^{1/T} d \tau \;  G(L, -i\tau )
\label{curr}
\end{equation}
In the above equation, $G$ stands
for the appropriate expression at finite temperature. The analytic
continuation to imaginary time, however, cannot be taken directly
in expressions like (\ref{prop}), and for computational purposes
it is more convenient to introduce the temperature dependence through 
the Matsubara formalism
\begin{eqnarray}
C(x,-i\tau) & = &  \exp \left( -\frac{2}{N} \int_{0}^{k_c} dk \;
   \frac{2T}{v_F}   \right.     \nonumber            \\  
    &  &  \left.        \sum_{m = -\infty}^{m = +\infty}
    \frac{1 - \cos (kx) \: \cos (2\pi m T \tau )}
   {(2\pi m T/\tilde{v}_F)^2 + k^2}  \right)
\end{eqnarray}

We can use Eq. (\ref{curr}) to test whether the behavior of the 
critical currents measured in Ref. \onlinecite{kas} can be 
reproduced within the present framework. The comparison should be
fairly direct for the sample that is made at one end of a single
nanotube (called $ST_1$ in Ref. \onlinecite{kas}). 
According to the above discussion, we consider a 
momentum-dependent parameter
$\mu (k) = 1/\sqrt{ 1 + N \left( e^2/(2\pi^2 v_F) \log |(k_c + k)/k|
 - g/(\pi v_F) \right) }$, taking in this case
a number of channels $N = 4$ in the above equations. 



We have checked first that the critical current ${\cal I}
\approx 0.1 {\rm \mu A}$ of the $ST_1$ sample at $T \approx
0 {\rm K}$, found anomalously high in the BCS framework, 
can be explained with the present 
model. The results represented in Fig. \ref{two} show
the magnitude of ${\cal I}_L (0)$ for different values of the
interaction, the distance being measured
in units of $k_c^{-1}$. The actual values of the 
supercurrent are obtained by multiplying the magnitudes in Fig.
\ref{two} by the prefactor in Eq. (\ref{curr}).
The Fermi velocity can be obtained from the hopping amplitude
$t \approx 2.1 {\rm eV}$ and the nearest-neighbor distance
$a \approx 1.4 {\rm \AA} $, by using the expression $v_F = 3ta/2$.
A reasonable estimate of the cutoff $k_c$ for the single-walled
nanotube is $k_c \approx 0.5 {\rm nm}^{-1}$, which gives
$e v_F k_c \approx 30 {\rm \mu A}$. 


\begin{figure}[H]

\epsfig{file=mst0.eps, height=7cm, width=7cm,
         bbllx=0, bblly=0, bburx=600, bbury=600,
         angle=270}

\caption{Plots of the critical current (in units of $e v_F k_c$)
versus distance, at $T = 0$, for different strengths of the Coulomb
interaction. From top to bottom, the solid curves correspond to
$2 e^2/(\pi^2 v_F) = 1.0, 2.0, 4.0, 8.0$. The dotted lines
correspond in each case to the correction by effect of the additional
short-range interaction, with $4g/(\pi v_F) = 0.8$.}
\label{two}
\end{figure}


We observe that the coupling corresponding to the bare parameters 
of a graphite layer, $2 e^2/(\pi^2 v_F) \approx 8.0$, does not lead 
to sensible results. The total length of the sample $ST_1$ is
$\approx 300 {\rm nm}$, and we should expect to get the correct
order of magnitude of the critical current at $L \sim 50/k_c$.
The most appropriate value for the Coulomb interaction seems to be 
$2 e^2/(\pi^2 v_F) \approx 1.0$. The order of magnitude ${\cal I} 
\approx 0.1 {\rm \mu A}$ is then reached, most precisely if one  
takes into account a coupling for the short-range attractive 
interaction $g/(\pi v_F) \sim 0.2$. The sensible reduction in 
the value of $2 e^2/(\pi^2 v_F)$ can be understood by the presence
of nearby charges and the renormalization of the 
interaction in the narrow rope into which the
nanotube merges, as we discuss afterwards. 


Moreover, the dependence of the critical current on $T$ for the
mentioned interaction strength reproduces the shape that has been
observed in the measurements of the sample $ST_1$. In the 
theoretical model, the temperature $T$ is given in units of the 
unique energy scale $E_c$. This means that the critical temperature
$T_c \approx 0.4 {\rm K}$ of the contacts for the sample $ST_1$ 
corresponds to the 
dimensionless value $T_c/E_c \approx 2 \times 10^{-4}$.
We have represented in Fig. \ref{three} the results for the 
critical current ${\cal I}_L (T)$ at $L = 50/k_c$, with and
without the effect of the short-range attractive interaction. It is
remarkable the smooth behavior of the critical current in the
low-temperature regime below $T_c$. The slight increase observed
near zero temperature is related to the renormalization of the
transmission at the interfaces\cite{kf}. Near $T_c$, the suppression
of superconductivity in the contacts should be incorporated to
produce a sharp decrease, leading then to the full agreement with
the experimental results of Ref. \onlinecite{kas}.


\begin{figure}[H]

\epsfig{file=ms501.eps, height=7cm, width=7cm,
         bbllx=0, bblly=0, bburx=600, bbury=600,
         angle=270}

\caption{Plots of the critical current (in units of $e v_F k_c$)
versus $T/E_c$, for $2 e^2/(\pi^2 v_F) = 1.0$ and a coupling of   
the short-range attractive interaction $4g/(\pi v_F) = 0$ (lower
curve) and 0.8 (upper curve).}
\label{three}
\end{figure}



Moving now to the sample made of a rope of nanotubes (called 
$R0_3$ in Ref. \onlinecite{kas}), we have to take into account
two main differences with respect to the preceding discussion.
First, the energy scale $E_c$ up to which the rope can be seen as a 
purely one-dimensional system is smaller, compared to the cutoff
introduced for the sample $ST_1$. Given that the diameter
of $R0_3$ can be approximately 15 times larger than that of 
the single-walled nanotube of $ST_1$, we may assume that the 
energy cutoffs in the two samples differ in the inverse proportion 
by a factor of 15. This means that a given temperature of
the sample $R0_3$, when measured in units of the corresponding 
$E_c$, looks comparatively higher than the same temperature in
the sample $ST_1$. Thus, the critical temperature $T_c \approx 
1.1 {\rm K}$ of the contacts used for the sample $R0_3$ gives 
a dimensionless value $T_c/E_c \approx 9 \times 10^{-3}$, 
which is more than one order of
magnitude higher than the ratio for the sample $ST_1$.



The second important difference between the rope of nanotubes 
$R0_3$ and the sample $ST_1$ is the interaction among the large 
number of nanotubes ($\approx 200$) in the former\cite{mele},
which leads to  
a significant renormalization of the strength of the Coulomb 
interaction. This can be understood
in the bosonization approach developed above, if we consider that
each metallic nanotube in the rope contributes with four units to
the number $N$, while there is only one channel for the interaction
of the total charge density\cite{egger}.
The picture is more involved considering the whole number of
nanotubes in the rope, but the overall
physical effect can be taken into account by assuming a
convenient renormalization of the bare coupling.
It turns out, for instance, that the correlations in an aggregate
of 100 metallic nanotubes with a bare coupling 
$2 e^2/(\pi^2 v_F) = 1.0 $ can be reproduced in a system of 
decoupled nanotubes in which the interaction has been renormalized 
down to 0.2, as observed in Fig. \ref{four}. 
On the other hand,
the attractive coupling $g$ is not affected by this kind of
renormalization, as it refers to an interaction with the elastic
modes that takes place within each nanotube.


Given that the total length of the sample $R0_3$ is 
$\approx 1.7 {\rm \mu m}$, we have 
estimated the supercurrent by the decay of ${\cal I}_L$ through a 
distance $L = 50/k_c \approx 1.5 {\rm \mu m}$. The evaluation of the 
prefactor in front of Eq. (\ref{curr}) is now more delicate, compared
to that for the sample $ST_1$. On the one hand, we have to 
bear in mind that the value of $k_c$ decreases according
to the increase in the diameter of the sample. On the other hand, 
there are more metallic nanotubes in the sample $R0_3$, in a number 
that may be estimated as $1/3$ of the 
total number, which gives $\approx 60$ metallic nanotubes. Balancing 
both points, it is appropriate to take now a prefactor in Eq. (\ref{curr}) 
that is four times the value for the single-walled nanotube. 

We show in Fig. \ref{four} the plots of ${\cal I}_L (T)$,
within the above picture of renormalization of the Coulomb 
interaction and including the short-range attractive
interaction in each nanotube. The
critical current in the sample $R0_3$
at $T \approx 0 {\rm K}$ is ${\cal I} \approx 2.5 {\rm \mu A}$.
We observe that the correct order of magnitude can be obtained
from our results by considering a renormalization
of the coupling $2 e^2/(\pi^2 v_F)$ down to a value $\approx 0.2$,
together with the effect of a weak short-range attractive
interaction with coupling $g/(\pi v_F) \sim 0.15$. 

The value of the attractive coupling $g$
is similar to that required for the sample $ST_1$, and it actually
matches what is expected from the coupling to the elastic modes   
of the nanotube. The short-range effective attraction can be
estimated on theoretical grounds 
from the modulation of the hopping $t' = \partial t/
\partial a \approx 4.2 {\rm eV} {\rm \AA}^{-1} $, the speed of sound
$v_s \approx 2.1 \times 10^4 {\rm m} {\rm s}^{-1}$, and the mass
$M$ of the atoms. This gives $g/v_F \sim t'^2 a^3 /(M v_s^2 v_F)
\sim 0.2$, which is of the same order of magnitude needed in our
fits.


As a final check of the consistency
of our approach, we observe that the curves in Fig. \ref{four}
reproduce the dependence on temperature measured experimentally
in the sample $R0_3$, with the characteristic inflection point 
and the very slow decay around the critical temperature
at $T \sim 10^{-2} E_c$\cite{kas}. 

\begin{figure}[H]

\epsfig{file=ms502.eps, height=7cm, width=7cm,
         bbllx=0, bblly=0, bburx=600, bbury=600,
         angle=270}

\caption{Plots of the critical current (in units of $e v_F k_c$)
versus $T/E_c$ 
for a renormalized interaction $2 e^2/(\pi^2 v_F) = 0.2$ 
corrected within each tube by the additional short-range
attraction with $4g/(\pi v_F) = 1.0, 0.75, 0.5$ and 0 (solid lines
from top to bottom), and
for an aggregate of
100 metallic nanotubes interacting with strength
$2 e^2/(\pi^2 v_F) = 1.0$ (dotted line).}
\label{four}
\end{figure}



To summarize, we have seen that a model of interacting electrons
in one dimension is able to explain the order of magnitude as
well as the temperature dependence of the critical currents in
both the $ST_1$ and the $R0_3$ samples of Ref. \onlinecite{kas}.
Our description is free of the shortcomings arising from the
conventional picture of the proximity effect, which relates the
value of the critical supercurrent to the gap $\Delta $ and the
normal resistance $R$ through the expression ${\cal I} = \pi
\Delta / (e R)$. Our approach focuses on the strong correlations
in the one-dimensional electron system, explaining in this way
why the experimental data of the supercurrent do not follow in
general the temperature dependence of the gap in the
superconducting contacts.

Our discussion also stresses the relevance of the coupling to
the elastic modes of the nanotube, which reveals itself through
the presence of a short-range attractive electron interaction.
This is also supported by recent experiments on the intrinsic
superconductivity of ropes of nanotubes\cite{sup}. In a sample
like $R0_3$, it can be already observed that the supercurrent
measured experimentally does not vanish near $T_c$, which is at
odds with the conventional picture of the proximity effect but
in accordance with the results of our model.
This enhancement of the superconducting correlations should
deserve further study, in order to understand the experimental
conditions under which the effect of the short-range attraction 
may dominate over the Coulomb repulsion.

Fruitful discussions with S. Bellucci and F. Guinea are gratefully
acknowledged. This work has been partly supported by CICyT (Spain)
and CAM (Madrid, Spain) through grants PB96/0875 and
07N/0045/98.






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\end{document}











\begin{figure}
%%\epsfxsize=12cm
%%\centerline{\epsfbox{nt50.eps}}

\epsfig{file=nt10.eps, height=8cm, width=8cm,
         bbllx=-100, bblly=550, bburx=188, bbury=838,
         angle=0}

\caption{Low-energy band structure of a carbon nanotube with
$n = 10$ subbands. The energy is measured in units of the
hopping amplitude ($t \approx 2.1 {\rm eV}$) and the momentum
is in units of the inverse of the nearest-neighbor distance.}
\label{one}
\end{figure}

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