Laser induced resistivity of chiral single wall carbon nanotubes

Using the Boltzmann transport equation within the semi-classical approximation with constant relaxation time, we theoretically studied the dynamics of electrons in chiral single wall nanotubes (SWNTs) subjected to a temperature gradient (∇T) in the presence of a combined direct current and high frequency alternating fields. We obtained an expression for the resistivity (ρc) of the SWNTs which varies with temperature and depends among others on material’s chiral angle (θh), dc field strength (Eo) and ac field amplitude (Es). Our results show that chiral SWNTs exhibit metallic behavior with resistivity increasing approximately linearly with temperature over a wide temperature range well above 100 K. Based on the low chiral resistivity obtained for the SWNTs at room temperatures, we propose these materials as good candidates for possible optoelectronic applications.


Introduction
Carbon nanotubes (CNTs) are a class of nanomaterials that consist of two-dimensional hexagonal lattice of carbon atoms, bent and joined seamlessly in one direction so as to form a hollow cylinder [1,2]. For the past few decades, interest in the study of CNTs has increased due to their unique electrical, electronic and mechanical properties caused by their small diameters and lattice orientation [3]; they can either be single-walled or multiwalled [3]. Single wall nanotubes (SWNTs) are produced in the outflow of a carbon arc and in much higher yield by laser vaporization of a graphite rod in an oven at 1200°C [4]. SWNTs can either be semiconducting or metallic, depending on their diameter and chirality [5]. Authors in [6] studied the temperature dependent resistivity of single wall carbon nanotubes and compared their results with the predictions of the twiston theory [7]. The outcome of their study predicted that intrinsic resistivity of twistons is proportional to the absolute temperature. In this study, we investigate theoretically temperature dependent electrical resistivity of chiral SWNTs along the base helix in the presence of a laser using the Boltzmann's transport equation (BTE) with constant relaxation time. To the best of our knowledge, no such study has been reported.

Theory
We consider a chiral SWNT under a temperature gradient ∇T placed in an electric field applied along the nanotube axis. The carrier current density and resistivity in the SWNT are studied using the BTE following the approach of [8] together with the phenomenological model of a SWNT developed in references [9,10]. The BTE which is based on the principle of conservation of charge, is a semi-classical formulation of transport and is capable of including self-consistently the transport of both electrons and phonons, as well as externally applied electric fields, electron-phonon interaction, anharmonic phonon decay, and many other types of scattering to a desired level of accuracy and detail [11]. Within the classical domain of the BTE, each particle is assumed to occupy a spatial and momentum coordinate and is described by a distribution function f r p t , , ( )which counts the number of particles occupying each set of coordinate in space and momentum. To express the conservation of particles in both space and time, the total rate of change in time is equated to the total rate of change in the distribution function due to various scattering mechanisms. In the semi-classical formulation of the BTE, Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. position and momentum are both independent and functions of time only and can be expressed as [12]: where f (r, p, t) is the distribution function, f 0 (p) is the equilibrium distribution function, v(p) is the electron velocity, r is the electron position, p is the electron dynamical momentum, t is the elapsed time , τ is the electron relaxation time which is assumed to be constant and e is the electronic charge. The applied dc-ac field, where E o is the constant electric field. E s and w are the amplitude and frequency of the ac field, respectively. Using the perturbation approach, equation (1) is solved in which the second term on the left-hand side is treated as a weak perturbation. In the linear approximation of ∇T and quasi-fermi level ∇μ, the solution to equation (1) is expressed as where μ is the electrochemical potential which ensures conservation of electrons and ε(p) is the tight-binding energy of the electron.
The current density is defined as Using the phenomenological model [9,10], a SWNT is considered as an infinitely long periodic chain of carbon atoms wrapped along a base helix. Based on this model, the circumferential and axial current densities can be expressed respectively in the form where Z′ and S′ are respectively the components of the current density along the nanotube axis and the base helix, and θ h is the chiral angle. Neglecting the interference between the axial and helical paths connecting a pair of atoms, transverse motion quantization is ignored [9,10]. This approximation best describes doped chiral carbon nanotubes and was experimentally confirmed in [13]. In our study, the analysis is restricted to the properties of the current density along the circumferential direction. Therefore, we resolve the current density along the base helix and obtain in equation (8) where d z and d s are the inter-atomic distance along the nanotube axis and the base helix respectively, the flux along the base helix can further be expressed as The circumferential component of the electrical conductivity is the coefficient of the electric field E zn * and is given as The resistivity of the SWNT along its circumferential direction is defined as

Results and discussion
In this paper, we analytically studied the electrical resistivity of a chiral SWNT using the Boltzmann transport equation. An expression was derived for the circumferential component of the electrical conductivity in the presence of applied field E and is shown in equation (22). To analyze this expression numerically, a chiral SWNT having the following parameters was chosen: d s =1Å, d z =2Å, τ=0.3×10 -12 s and θ h =4.0°.
The ac source has a frequency, w=10 12 s −1 and amplitude E s =5×10 7 V m −1 . The d. c. electric field, E o =6.9063×10 7 V m −1 , chosen such that Ωτ=1, where Ωτ=ed z E o /. Figure 1(a) shows the dependence of the circumferential electrical resistivity, ρ c on temperature, T, for various fixed values of the dc field E o .
It is observed that ρ c changes slowly with temperature at low temperatures up to about 200 K and then increases linearly with increasing temperature. This trend is attributed to electron-phonon interactions which cause scattering of charge carriers along the circumferential direction of the chiral SWNT as temperature increases. The low values of resistivity observed clearly suggest that chiral SWNTs do exhibit metallic properties. Also noted is a significant increase in the circumferential resistivity as the electric field strength E o increases. As E o is increased, the electrons in the SWNT become more energetic, leading to increased collision with carbon atoms within the walls of the SWNT which sets these carbon atoms into large amplitude oscillations which enhance the scattering of the electrons. Experimentally, it has been observed that the resistivity of carbon nanotubes varies with temperature [14][15][16]. The resistance generally decreases with decreasing temperature (as shown theoretically in our study) up to a threshold temperature which is determined by the type and purity of the carbon nanotube. Based on available experimental results, interesting electrical models have been developed as a function of temperature range and carbon nanotube fabrication technique [17,18]. For chiral SWNTs however, our results are the first of such low temperature resitivity dependence study; to the best of knowledge. On the other hand, figure 1(b) shows that the resistivity decreases markedly with increasing Δ s .
The dependence of the circumferential resistivity on temperature for given values of the chiral angle θ h is depicted in figure 2(a). The figure shows that increasing θ h results in a decrease in the resistivity ρ c of the chiral SWNT.
Interestingly, figure 2(b) indicates that keeping Δ s constant and altering Δ z reduces ρ c by an order of magnitude.
When the ac source is switched off, E s =0, a=0, w=0 and J a n 2 ( ) becomes unity and equation (17) reduces to   Figure 3 shows the circumferential current density dependence on temperature in the presence of a laser (laser on) and absence of a laser (laser off).
It is noted that when the laser source is off, ρ c reduces by a factor of two which indicates that within the temeperature range under consideration, the laser field modulates the dc field and enhances the momentum and kinetic energy of those electrons which are deficient in energy.

Conclusion
The circumferential resistivity, ρ c of a chiral SWNT induced with a laser field has been investigated using the semi-classical approach. Our results indicate that the nanotube parameters (Δ s, Δ z , θ h) and the d.c. field E o and laser source strength E s have significant influence on the resistivity with ρ c increasing with increasing E o . The results reveal that increases in both Δ s and θ h decrease ρ c . The observed low resistivity exhibited by the chiral SWNT at increasing temperatures indicate that this unique material is a good conductor of electricity and could serve as a good candidate for optoelectronic applications.