Microscopic model of superconductivity in carbon nanotubes
J. Gonz\'alez
\widetext We propose the model of a manifold of one-dimensional interacting electron systems to account for the superconductivity observed in ropes of nanotubes. We rely on the strong suppression of single-particle hopping between neighboring nanotubes in a disordered rope and conclude that the tunnelling takes place in pairs of electrons, which are formed within each nanotube due to the existence of large superconducting correlations. Our estimate of the transition temperature is consistent with the values that have been measured experimentally in ropes with about 100 metallic nanotubes.
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\title{Microscopic model of superconductivity 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 propose the model of a manifold of one-dimensional interacting 
electron systems to account for the superconductivity observed 
in ropes of nanotubes. We rely on the strong suppression of 
single-particle hopping between neighboring nanotubes in 
a disordered rope and conclude that the tunnelling takes place 
in pairs of electrons, which are formed within each nanotube 
due to the existence of large superconducting correlations. 
Our estimate of the transition temperature is consistent 
with the values that have been measured experimentally in 
ropes with about 100 metallic nanotubes.




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

]



\narrowtext 
\tightenlines

%\newpage



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.











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.







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does not play an essential role in the
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\end{references}






\end{document}








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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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\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.



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