Structural Conductivity of Carbon Nanotubes

In this paper

Carbon nanotubes, long, thin cylinders of carbon, about 10.000 times thinner than a human hair, were discovered in 1991 by Sumio Iijima.These are large macromolecules that are unique for their size, shape, and remarkable physical properties.From chemical point of view, the carbon nanotubes consist of SP 2 bonded carbon atoms, the same as found in graphite.The structure of graphite is very strong in one plane due to the covalent bonding between the carbon atoms, but is weak between planes because there is no covalent bonding, only Van der Waals forces acting.
The electrical conductance determination of single-wall and double-wall carbon nanotube systems was a central task, theoretical and experimental, of the scientific research at nanometer scale on advanced materials, in the past five years.Due to the importance of the chosen structure configuration in solving this difficult problem, further on we shall present few general notions on the reactivity and the structure of nanotubes.Reactivity of the fullerene molecules with respect to addition chemistries is strongly dependent on the curvature of the carbon framework.Their outer surface (exohedral) reactivity increases with increase in curvature.In comparison with fullerene molecules Single-Walled Carbon Nanotubes (SWCN's) are moderately curved.Consecutively, nanotubes are expected to be less reactive than most fullerene molecules due to their smaller curvature, but more reactive than a graphene sheet due to pyramidalization and misalignment of piorbitals.
The simplest Carbon Nanotubes (CN's) can be obtained from a hexagonal carbon monolayer rolled into a cylinder.This type is called Single-Walled Carbon Nanotube (SWCN).The first single walled nanotubes were reported in 1993 by Dr. Sumio Iijima and separately by Dr. Bethune of IBM, in simultaneous publications.
Univocally the SWNT can be characterized by the chiral vector, indicated coordinates reported to orthogonal references, figure1 and a the lattice constant of the 2D graphite.
A more intuitive, crystallographic interpretation can also be made.Therefore, the vectors and are graphene lattice vectors, in our case both of modulus , .The carbon nanotube lattice can be thought of as a wrapping (i.e. a conformal mapping) of a graphite layer into a tube.The wrapping is preformed such that the chiral vector becomes the circumferential of the (n,m) nanotube and this determines the lattice completely.Any (n,m) nanotube lattice has three symmetries.The first is a discrete translational symmetry along the tube, the second is a discrete rotational symmetry around the tube axis and the third is a helical symmetry (i.e. a screw operation).
For this type of nanotubes, the following relation must exist between the two coefficients (2) where, n, m and N are integer.If k=0 then the SWCN is metallic and otherwise, the tube is semiconductor [1][2][3].

Experimental part
The nanotube is one of these types of structures that were frequently studied, as the basic element of the future electronic nanodevices.
In the figure 1 we represented an example of construction of nanotubes wires, from the hexagonal plane structures.These constructions are studied in [1][2][3][4].Other type of nanotubes often studied is the so called Multi-Wall Carbon Nanotubes (MWCN's), and specially Double-Wall Carbon Nanotube (DWCN's).For all these CN's it is very important to know their electrical behavior.
Carbon nanotubes are metallic or semiconductor (SC), based upon delocalized electrons occupying a 1-D density of states.However, any covalent bond on SWNT sidewall causes localization of these electrons.In the vicinity of localized electrons, the SWNT can no longer be described using a band model that assumes delocalized electrons moving in a periodic potential.
Properties such as conductance, thermal conductivity (and their dependence by other factors) represent one of the most important aspects in the fields of nanotechnology.For metallic single-wall carbon nanotubes, two channels with transmission coefficient of 0.88 contribute to ballistic electronic transport at room temperature [5].
At the quantum level, the conductance of the nanotubes is at the order of e 2 /πh and its value depends on the temperature [6], the impurities of the doped carbon nanotubes [7], vibrational effects in the CN's [2][3][4][5][6], and, in some certain case, its value is quantized [4].For this reason, there are studied various cases, to clarify the ways for the utilization of CN's in the construction of nanodevices.
It is now time to make some additional assignations.The majority of known papers present a classical approach of the transport process, even though sometimes the differential equations are stochastic [9,10].On the other hand, recent results on the transport phenomena at nanometer scale require the development of new "scale" physical theories, of fractal space-time type, in which the macroscopic scale (specific to the classical quantities) coexist and it is compatible, simultaneously, with the microscopic "scale" (specific to the quantum quantities), [10,11].
Regarding the developed theory in this paper, we can say that it explicitly relies on the high structural symmetry of the carbon nanotubes.In this sense we shall emphasize the manner in which this special geometrical form of SWNT structure influences the electrical conductivity value.

The drift velocity along a lattice
Let assume that the electrons are transmitted along the single wall nanotube from one node to another with a mean velocity u.The total current density depends on the form of different path of the electron.We can consider various forms for those paths along the SWNT.In figure 2 an example for electron path in a 3D model is considered.The same path in 2D is represented in figure 3.
For a long SWNT, a fractal characteristic is supposed for different paths.The total current density, for a small electric field applied along x axis, is (3) where n s is the surface concentration of electrons, r the SWNT radius and ν drift is de drift velocity of electrons along x axis.At its turn the drift velocity ν drift depends on the mean velocity u and the path form.In this way, the total current density and the conductivity are connected with the path form in our model.To find an approximate connection between conductivity and SWNT's structures, we use an important form characterization number, the fractal dimension.

Results and discussions Fractal dimension of an ideal hexagonal fractal path
To construct the fractal path of the electron along a hexagonal lattice, we can use the following construction, ( 4a-d).
Starting with a direct path between two points (fig.4a) and dividing it in 4 equal pieces, we can construct in the first step a symmetric semihexagonal form indicated on figure 4b.We repeat the first step on each side of the semihexagonal form obtained in figure 4b and results a more complex form, as indicated in figure 4c.After the third step, we obtain the fractal structures, (fig.4d) and after more identical step we obtain a Koch curve in hexagonal form [12].To evaluate the fractal dimension of this path, in case of perfect selfsimilarity, we can use similitude dimension (4)
In realistic cases, the selfsimilarity of fractal path is statistic and the propagation of electron on bidimensional lattice is more directional along the external electric field lines.The fractal dimensions of electron path characterize the dimensionality of the space occupied by the particle.bidimensional.In the first case ν drift →u and in the second case ν drift →0 which means that the probability for the electron to arrive at the other side is null.In this case, in a first approximation, we can assume that (6) Using relations ( 3) and ( 6) (7) In this way, for a bidimensional lattice, the conductivity σ x (j x = σ x E x , where E x is the electric field along x axis) depends on the structural form, (which determines the shape of electron path) through the fractal dimension of this path (8) A similar philosophy of conductance evaluation, through simple calculus, is used by other authors, but in principle different situations.Explicit numerical estimations have been performed for nearly a hundred different double-wall nanotubes with incommensurate lattice structure [13].Because inter-tube transfer is incoherent between different segments the conductance becomes proportional to the number of such segments, i.e., σ x ∝ A/Lφ, where A is the length of the double-wall region and Lφ is the finite phase coherence length.
Using a phase coherence length of 300 nm at about 10 K obtained experimentally and the average conductance of 10 -3 e 2 / πh, we need A = 3 mm to have the conductance of the order of e 2 / πh.

Conclusions
The processes at mesoscopic level implied quantum treatment of these phenomena, but all these processes are commanded by the form and structure of nanotubes.A new way to connect the electrical properties to the structure of nanotubes was presented.An interesting path effect on the transport properties has been proved.
The fractal dimension of electron path has a major influence on the conductance estimated value.The higher the fractal dimension of electron path, the lower the microscopic electrical conductivity of Single-Walled Carbon Nanotubes.