High-Frequency Electric Field Induced Nonlinear Electron Transport in Chiral Carbon Nanotubes

We investigate theoretically the high frequency complex conductivity in carbon nanotubes that are stimulated axially by a strong inhomogeneous electric field of the form ( ) ( ) o E t E E t 1cos ω = + . Using the kinetic approach based on Boltzmann’s transport equation with constant relaxation time approximation and the energy spectrum of the electron in the tight-binding approximation, together with Bhatnagar-Gross-Krook collision integral, we predict high-frequency nonlinear effects along the axial and the circumferential directions of the carbon nanotubes that may be useful for the generation of high frequency radiation in the carbon nanotubes.


Introduction
Carbon nanotube (CNT) is an allotrope of carbon with nanometers size diameter and an aspect ratio as high as ~10 7 .
In 1952, images of 50 nanometer diameter tubes of carbon were reported by Radushkevich and Lukyanovich [1].
Using the vapor-growth technique, Orberlin and coworkers [2] reported observation of hollow carbon fibers of nanometer size diameters.Other reports also published similar observation of tubular carbon nanostructures [3].However, the credit for the discovery of CNTs goes to S. Iijima [4].This one-atom thick sheet of graphene rolled up into a seamless cylinder has since attracted a great deal of interest, mainly due to their unique thermal, chemical and physical properties [5] [6].These properties depend on the fundamental indices (n,m) of the CNTs.The indices (n,m) determine the diameter and the chiral angle of the CNTs.As n and m vary, the conduction ranges from metallic to semiconducting [7], with an inverse diameter dependent band gap of ≤1 eV [7].
In this work, we report on a high frequency (hf) complex conductivity ( ) σ ω in CNTs using the standard li- near electrodynamical approach [15]- [17].We use the kinetic approach based on Boltzmann's transport equation with constant relaxation time approximation and the energy spectrum of the electron in the tight-binding approximation, together with Bhatnagar-Gross-Krook (BGK) collision integral.This approach permits adequate allowance for the particle-number conservation law for scattering inhomogeneous field as well as the influence on the electron spectrum by both resonance effects due to Bloch oscillations of the electrons and by the effects connected with the carrier drift and diffusion under conditions of strong spatial dispersions.

Theory
Using the simple model of the tight-binding approximation, we describe the energy spectrum of the CNTs as [39] ( ) where the indices s and z correspond to the circumferential and axial directions, respectively.o ξ is the energy of an outer-shell electron in an isolated carbon atom, ∆ z and ∆ s are the real overlapping integrals for jumps along the respective coordinates, s p and z p are the components of momentum tangential to the base helix and along the nanotube axis, respectively. is Planck's constant, and d z and d s are the distances along the CNT axis and helix, respectively.
We consider the Boltzmann transport equation with constant relaxation time together with Bhatnagar-Gross-Krook (BGK), which account for the spatial effects due to the conservation of the number of scattered particles [41] [42], The distribution function ( ) , , f x p t and the equilibrium distribution function ( ) o f p are periodic and can be written in Fourier series as in Equations ( 3) and (4), respectively.

( ) ( )
, , e ( ) where n o is the equilibrium carrier density, n(x) is the electron density at position x, v(p) is the electron velocity, p is the electron dynamical momentum, t is elapsed time, τ is the electron relaxation time, e is the electron charge, E(t) is the external electric field, B κ is the Boltzmann's constant and T is the temperature.The collision integral is taken in the relaxation time τ approximation and further assumed to be constant.
( ) is the factor by which the Fourier transform of the nonequilibrium distribution function differs from its equilibrium distribution counterpart.I m (I n ) is the modified Bessel function of the order m(n) and I o is the modified Bessel function of the zeroth order.
In addition to Equation (2), we employ the Poisson equation that will allow for continuous current, i.e., ( ) ( )  , where is the lattice dielectric constant.For a constant homogeneous electric field i E , the electron distribution for both the tubular axis z and the base helix s is expressed as: ) is the scattering frequency.We use the conditions of inhomogeneous perturbations with frequency ω and wave-vector k of the following form: ( ) where  is the normalized wave number of the perturbations, and is the dimensionless quasimomentum.The electron flux along the tubular axis (Φ z ) and the base helix (Φ s ) can be expressed as [39] [40], where the integration is done over the first Brillouin zone.Using Equation (7) together with the solution of Equation (6) we obtain the following expressions: , , 1 .
Defining the axial z j and the circumferential s j components of the current density as sin , respectively [39] [40], where θ h is the chiral angle of the CNTs, the current densities are expressed as [43] [44], and linearizing the result with ( ) o J β = for 0 β → , the expressions the complex conductivities along the axial and the circumferential directions are expressed, respectively, as: )

Results and Discussions
Using the solution to the Boltzmann's transport equation with constant relaxation time together with Bhatnagar-Gross-Krook collision integral, we obtained the expressions for the complex conductivities along the axial and circumferential directions of the CNTs.We analyzed the effects of the high frequency electric field by considering the dependence of the complex conductivity ( ) o σ ω σ on the dimensionless frequency ω Ω (Equation The red and the blue curves represent the real and the imaginary parts of the conductivity, respectively.The complex conductivity along both directions exhibits similar characteristics. The dimensionless complex conductivity ( ) o σ ω σ depends strongly and nonlinearly on the normalized fre- quency ω Ω .It can be seen from the plots that the real part of the complex conductivity becomes more negative with increasing the frequency until a resonance minimum occurs just before the Bloch frequency Ω.This negative-conductivity resonance near the Bloch frequency makes the CNT a potential active medium for Bloch oscillations without domain instabilities induced by negative dc conductivity.

Conclusion
In summary, we have obtained the nonlinear high frequency conductivity in CNTs that are stimulated axial with a strong inhomogeneous electric field by using the Boltzmann's equation with constant relaxation time approximation together with the Bhatnagar-Gross-Krook collision integral.We predict this high-frequency nonlinear effect along the axial and the circumferential directions of the carbon nanotubes may be useful for the generation of a high frequency radiation in the carbon nanotubes.

( 10 )Figure 1 .
Figure 1.The real (red solid curve) and the imaginary (blue solid curve) parts of the normalized complex conductivity σ/σ o as a function of normalized frequency ω/Ω along the circumferential (top) and axial (bottom) directions of CNT.