We investigate theoretically the high frequency complex conductivity in carbon nanotubes that are stimulated axially by a strong inhomogeneous electric field of the form E( t)= E 0+ E 1cos( ωt). 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.
Carbon nanotube (CNT) is an allotrope of carbon with nanometers size diameter and an aspect ratio as high as ~107. In 1952, images of 50 nanometer diameter tubes of carbon were reported by Radushkevich and Lukyanovich [
The electron transport properties of the CNTs continue to be the subject of intense research. CNTs have been shown to exhibit ballistic transport [
In this work, we report on a high frequency (hf) complex conductivity
Using the simple model of the tight-binding approximation, we describe the energy spectrum of the CNTs as [
where the indices s and z correspond to the circumferential and axial directions, 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 [
The distribution function
and
where no 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,
In addition to Equation (2), we employ the Poisson equation that will allow for continuous current, i.e.,
where
neous perturbations with frequency w and wave-vector k of the following form:
tion (2)), we obtained [
where
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:
Defining the axial
and
Using
and
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
circumferential [
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.
Sulemana S.Abukari,Samuel Y.Mensah,MusahRabiu,Kofi W.Adu,11,Natalia G.Mensah,AnthonyTwum,AlfredOwusu,Kwadwo A.Dompreh,PatrickMensah-Amoah,MatthewAmekpewu, (2015) High-Frequency Electric Field Induced Nonlinear Electron Transport in Chiral Carbon Nanotubes. World Journal of Condensed Matter Physics,05,294-300. doi: 10.4236/wjcmp.2015.54030