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CHIRAL-INVARIANT ELECTRONIC INTERACTIONS IN CARBON NANOTUBES
Stefano BELLUCCI
INFN – Laboratori Nazionali di Frascati, P.O. Box 13, 00044 Frascati, Italy
INFM-Dipartimento di Fisica, Universita’ di Roma Tor Vergata, Via Ricerca Scientifica 1, 00133 Roma, Italy
ABSTRACT
We consider a one-dimensional electron system, suitable for the description of the
electronic correlations in a metallic carbon nanotube. Renormalization group methods are
used to study the low-energy behavior of the unscreened Coulomb interaction between
currents of well-defined chirality In the limit of a very large number N of subbands we find a
strong renormalization of the Fermi velocity, reminiscent of a similar phenomenon in the
graphite sheet. For small N or sufficiently low energy, the Luttinger liquid behavior takes
over, with a strong wavefunction renormalization leading to a vanishing quasiparticle
weight. We apply the approach to study the crossover from two-dimensional to onedimensional behavior in carbon nanotubes of large radius. Our analysis extends the
dimensional crossover from Fermi to Luttinger liquid and the stability of the former, even in
the case of long-range interactions. Future applications of this approach to explain the
proximity-induced superconductivity in carbon nanotubes [3] are briefly discussed.
1. INTRODUCTION
Carbon nanotubes stick out in the field of nanostructures, owing to their exceptional
mechanical, capillarity, electronic transport and (recently) superconducting properties.
They are cylindrical molecules with a diameter of order 1 nm and a length of many microns
[1]. They are made of carbon atoms and can be thought of as a graphene sheet rolled
around a cylinder [2].
Nanostructured systems yield an important contribution to the fundamental
understanding of novel quantum mechanical features in solid state physics. They are at
the border overlapping between macroscopic and quantum physics (mesoscopic systems).
Their electronic properties play a special role in the interdisciplinary field, which lies in
between theoretical physics and statistical mechanics, with applications to condensed
matter. Methods developed in one area of research can be used, in order to explain
features in the other one: e.g. field theory techniques allow to try explaining the physics of
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such quasi one-dimensional (1D) systems, which are connected to new surprising
quantum properties of condensed matter.
The exotic properties of carbon nanotubes, which can range from metallic to
semiconducting ones, even with a large gap, allowing to consider them as molecular
quantum wires, depend on two factors, i.e. the nanometric diameter of the tube and the
peculiarities of the graphitic electronic structure [3]. Nanotubes are an exception to Peierls
instability, which states the essential instability of unidimensional metallic wires, owing to
the opening of a gap in the electronic energy at the Fermi energy, which turns them into
semiconductors, as in the case of conducting polymers, e.g. polyacetilene [4]. Because of
their special geometric and electronic band structure, nanotubes yield a protoype for
single-molecule quantum wires.
For large diameters one has a crossover from a purely 1D behavior to the 2D character
of a graphene sheet. While thin single-walled nanotubes (SWNT) can be considered in the
1D limit, thick multi-walled nanotubes (MWNT) might instead fall in a 2D regime.
We started out several years ago with a research line, which aims at clarifying both the
electronic transport properties of carbon nanotubes, and allowed us to obtain and publish
already interesting results. We report here an overview of our activity about nanostructured
systems.
2. THEORETICAL ASPECTS AND ELECTRON CORRELATIONS IN SWNT
There is much interest in the search of unconventional electron behavior deviating from
the Fermi liquid picture [5]. The other paradigm that is well-established on theoretical
grounds is the Luttinger liquid behavior of 1D electron systems [6]. In a 1D system,
electrons do not form a Fermi liquid, which characterizes the customary state of a metal in
the 3D world [7]. Instead, a liquid of correlated electrons is formed, which possesses
unusual and peculiar features, e.g. spin and charge separation. Tomonaga and Luttinger,
studying the effect of correlations in 1D electron liquids, found that they follow power laws.
The theory of electron-electron correlation was then developed for carbon nanotubes,
which can be represented as prototype 1D conductors [8]. In recent experiments on ropes
of nanotubes, the electron spin has been shown to play a role in the transport through the
tube, hence proving the relevance of electron correlations. A Luttinger signature can be
observed in recent works about the power-law dependence of the conductance on the
temperature and bias potential, with an increasing resistivity at low temperature [9]
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The theoretical investigation we carried out on the electron-electron correlation effects
so far, allowed us to explain various phenomena concerning transport properties and,
more recently, also superconductivity in carbon nanotubes, in some cases even .preceding
experimental observations. We showed several years ago that the Coulomb interaction in
SWNT produces new effects, which deserve experimental investigation [10,11]. One of
them concerns nanotubes with a radius large enough, to keep the same electronic
properties of a single graphite sheet, as the energy decreases (at least down to a certain
scale).
A second phenomenologically interesting effect has to do with the transverse
dimension, leading to scale-dipendent critical exponents. The resulting electron anomalous
dimension grows, as the energy scale goes to zero. A first confirmation of the prediction
comes from recent observations of the temperature dependence of the conductance at low
bias [9]. In fact, such attempts to find Luttinger signatures in SWNT packed in the form of
ropes, display deviations, in the low-temperature regime, from the power-law dependence
typical of the Luttinger liquid behavior, which indeed agree with the suppression we
predicted in the electron quasiparticles, and thus also in the electron density of states, at
low energy [11]. Our work on electron transport in nanotubes will continue, aiming at a
better understanding of the temperature dependence of the conductance, in view of
possible improvements in its measurements.
We recall from mesoscopic physics that one has single electron tunneling, when the
capacitance of a conducting nanostructure is so small, that even adding a single electron
requires an electrostatic charging energy larger than the thermal energy. Transport at low
bias is then blocked, yielding a situation described as the so-called Coulomb blockade
[12]. The anomalous increase we discovered in the intensity of the repulsive interaction at
large distances, yields a quantitative evaluation of the effect of the Coulomb blockade
mechanism in carbon nanotubes [11]. It will be necessary to measure the tunneling density
of states and the conductance at even lower energy, in order to establish a quantitative
relationship between the observed deviations from the scaling, and the long-ranged nature
of the Coulomb interaction.
3. CRITICAL EXPONENT AND TUNNELING CONDUCTANCE IN MWNT
This year we extended our model to MWNT, whose properties we investigated in
connection with the physical observables of the tunneling conductance and critical
exponents [13]. In connection with superconductivity in Cu-prates at high Tc, Anderson
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suggested that the Luttinger model could be extended to two-dimensional systems, in the
hope that it may explain some of the features of the copper-oxide materials [14]. However,
the analytic continuation in the number D of dimensions has shown that the Luttinger liquid
behavior is lost as soon as one departs from D = 1 [15].
Our investigation concerns the possibility that the long-range Coulomb interaction,
owing to the singular long-ranged character, may lead to the breakdown of the Fermi liquid
behavior at any dimension between D = 1 and D=2. The issue is significant for the purpose
of comparing with a recent experimental observations of a power-law behavior of the
tunneling conductance in MWNT [16]. These are systems whose description lies between
that of a pure 1D system and the 2D graphite layer. It turns out, for instance, that the
critical exponent measured for tunneling into the bulk of the multi-walled nanotubes is
approximately 0.3. This value is close to the exponent found for the single-walled
nanotubes in [9]. However, it is much larger than expected by taking into account the
reduction due to screening (which goes like 1/N) in a wire with a large number N of subbands, what points towards sizeable effects of the long-range Coulomb interaction in the
system.
By developing an analytic continuation in the number of dimensions, and having in mind
the low-energy modes of metallic nanotubes, we get a vanishing density of states at the
Fermi level above D = 1. This ensures that the Coulomb interaction remains unscreened in
the analytic continuation. At D = 2 we recover the low-energy description of the electronic
properties of a graphite layer, dominated by the presence of isolated Fermi points. At this
point we can solve the model in a self-consistent way, assuming basically the persistence
a complete and unbroken chiral symmetry, which forbids any mixing through interactions
between the two non-equivalent Fermi points common to the low-energy spectra of
graphite layers and metallic nanotubes. On the other hand, it appears that nanophases
with broken symmetry can only be realized, by doping the system about half-filling [17] or
in a strong coupling regime [18].
Our solution determines the equivalence class that the model belongs to. In D=2 we
are bound to recover the low-energy behavior of the model of Dirac fermions with the
Coulomb interaction, whereas the 1D model exhibits a Luttinger liquid behavior, as it
happens in the case of a short-range interaction. Hence we recover the anomalous
dimension we found earlier in the solution of the Luttinger model [11]. We have therefore a
model that interpolates between the marginal Fermi liquid behavior, that is known to
characterize the 2D model, and a non-Fermi liquid behavior at D = 1. We conclude then
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that, even in a model that keeps the Coulomb interaction unscreened, the breakdown of
the Fermi liquid behavior only takes place formally at D = 1.
In the vicinity of D = 1, a crossover takes place to a behavior with a sharp reduction of
the electron quasiparticle weight. For values of D above 1.2 (approximately), we have a
clear signature of quasiparticles at low energies. For lower values of D, the picture cannot
be distinguished from that of a vanishing quasiparticle weight for all practical purposes.
This picture allows us to make contact with the experiments carried out in MWNT. In the
proximity of the D = 1 fixed-point, the density of states displays an effective power-law
behavior, with an increasingly large exponent. Moving to the other side of the crossover,
the density of states approaches the well-known behavior of the graphite layer. We
observe that the exponents of the density of states at different dimensions are always
larger than a lower bound of approximately 0.26, which is in agreement with the value
measured experimentally. Our analysis stresses the need of an appropriate description of
the dimensional crossover between one and two dimensions, showing that the picture of a
thick nanotube as an aggregate of 1D channels does not allow to obtain the correct values
of the critical exponents.
4. FUTURE DEVELOPMENTS: SUPERCONDUCTIVITY OF SWNT
It is possible to improve the conductance of carbon nanotubes using doping, even at
room temperature, through a mechanism whose origin is still unclear. One of the main
motivations to introduce doping in such systems is the search for their possible
superconductivity. Recent experimental observations by the groups of Kasumov and
Bouchiat on superconductivity in molecular wires show the existence of supercurrents
induced
by
proximity
effects
in
undoped
carbon
nanotubes
mounted
across
superconducting electrodes [19]. The dependence of the critical current on the
temperature and magnetic field displays some unusual features that cannot be fully
explained in terms of the customary BCS mechanism, but have to do rather with the strong
1D character of the samples.
Our model appears to yield a natural explaination of such features, in terms of the
effective reduction of interactions due to the long-range nature of the Coulomb interaction.
Work is currently in progress and could be possibly extended to include Kasumov’s
measurements on DNA molecules deposited across superconducting contacts (below 1
K). The DNA molecules can be metallic, down to a few mK, with a phase coherence
maintained over a range a few hundreds of nanometers [20].
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ACKNOWLEDGEMENTS
I wish to thank warmly the Organizers of the Workshop for a very stimulating
atmosphere, and especially Giovanni Cuniberti for the kind invitation.
PUBLICATIONS
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