We consider a simple model of carbon nanotubes (CNTs) subject to external electric field E(t). Using a tight-binding approximation for the description of energy bands of CNTs, together with the standard Boltzmann transport equation and constant relaxation time, we predict the effect of self-induced transparency and absolute negative conductivity. The predicted effects may be useful in diagnostics of carbon nanotubes as well as in the amplification and efficiency conversion of electromagnetic signals.
The electrical transport properties of carbon nanotubes (CNs) have been the subject of much research ever since the discovery by Iijima [
In this paper we study theoretically two phenomena in CNs, which are self-induced transparency and absolute negative conductance for the following cases respectively:
1) When the CNs is exposed to an a. c. electric field i.e.
.
2) When the CNs is exposed to an a. c. and d.c electric field i.e.
.
Following references [3,4] and using the approach similar to reference [
We use the semiclassical approximation in which π- electrons are considered as classical particles with dispersion law extracted from the quantum theory in the tight-binding approximation [
Considering the hexagonal crystalline structure of zigzag and armchair CNTs within tight binding approximation, the dispersion relation is given as [
here γ0 ~3.0 eV is the overlapping integral, pz is the axial component of quasimomentum, 
is transverse quasimomentum level spacing and s is an integer. The expression for 
in Equation (2) is given as 
with the C-C bond length 
and 
is the Plank’s constant, we shall assume 
. The − and + signs correspond to the valence and conduction bands, respectively. Due to the transverse quantization of the quasi-momentum, its transverse component can take n discrete values,
. Unlike transverse quasimomentum 
, the axial quasimomentum 
is assumed to vary continuously within the range 
, which corresponds to the model of infinitely long CNT
. This model is applicable to the case under consideration because of the restriction to the temperatures and/or voltages well above the level spacing [
, where 
is Boltzmann constant, T is the temperature, 
is the charging energy. The energy level spacing 
is given by 
=
, where
is the Fermi velocity and 
is the carbon nanotube length [
Proceeding as in 4, we employ Boltzmann equation with a constant relaxation time
where 
is the electron charge, 
is the equilibrium distribution function, 
is the distribution function, and 
is the relaxation time. The electric field 
is applied along CNTs axis. In this problem the relaxation term 
is assumed to be constant. The justification for being constant can be found in [
.
Expanding the distribution functions of interest in Fourier series as;
and
here the coefficient, 
is the Dirac delta function, 
is the coefficient of the Fourier series and 
is the factor by which the Fourier transform of the nonequilibrium distribution function differs from its equilibrium distribution counterpart. The equilibrium distribution function 
can be expanded in the analogous series with coefficients as follows
Substituting Equations (5) and (6) into Equation (4), and solving with Equation (1) we obtain
where
, 
is the Bessel function of the kth order and 
for zigzag CNTs and 
for armchair CNTs.
We determine the surface current density as
or
and the integration is carried over the first Brillouin zone. We consider the relation
, and represent 
in Fourier series with the coefficients defined similar to Equation (7).
Substituting Equations (6) and (8) into (9) we find the current density for the CNs after averaging over a period of time t, as
From Equation (10) we obtain
If 
and
, Equation (11) takes the form
where
, 
is the Bessel function of the zeroth order.
From Equation (12) it is seen that near the zeros of 
and
, the CNTs exhibit absolute negative resistance in the transparency regions. That is, the low frequency oscillations are amplified in these regions. This phenomenon is called absolute negative conductivity and was first observed by Kryuchkov et al. [
In the limiting case of
, E0 = 0, we find that the current density for the CNTs is
From Equation (13) at each zero of the J0(rβ) function the current density vanishes and the CNTs become transparent. See also Ignatov et al. [
Equations (12) and (13) describe the theory of electron transport in the presence of an external electric field E(t). We assume that the equations do not take into account both interband transitions and quantum-mechanical corrections to the intraband motion or resonant transitions inherent in the CN lattice. To estimate the restriction on the fundamental frequency necessary for resonant transitions to be negligible, we apply a general approximate relation for the electron state density in a CNT [3,8] that establishes the low-frequency edge 
of the optical transition band in metallic CNTs; 
where 
is the CNTs radius. This condition is decisive for theapplicability of the developed theory in CNTs. As the CNTs radius increases, the upper limit for the fundamental frequency is shifted to the red. Thus, CNs with not too large radii are of the most interest. Another limitation is related to the strength of the driving field where interband tunneling is negligible: the Stark frequency corresponding to this field must be smaller than the conduction zone bandwidth 
[
an analytical expression for the current density was obtained in Equation (13). Figures 1 and 2 illustrate the graphs of 
on 
for (12, 0) metallic zigzag and (10, 10)
metallic armchair CNTs calculated by Equation (13). From the figure it is seen that the amplitude of the current density oscillates with increasing amplitude of the high frequency and in the zeros of the 
function or when 
is equal to the roots of the zeroth order Bessel function (2.4, 4.8, 8.4, 11.8, 14.8, 18.0) the total current is zero (the self-transparency effect). This effect is due to the non-parabolicity of the electron energy band which is stronger in the CNTs. However, when the CNTs is exposed to an a.c. and d.c electric fields, i.e. 
for 
and
, we obtained expression (12) which is an indication for absolute negative conductivity i.e. current flows against the applied d.c field. See 
on 
for (12, 0) metallic zigzag and (10, 10) metallic armchair CNTs obtained via the calculation of Equation (12). In the low limit of (rβ), we observed that as rβ increases, the normalized current 
decreases and at a certain value e.g., rβ = rβmin the normalized current reaches a minimum value
. Thuswe predict the region with absolute negative conductivity (ANC). Also we note that such state will be unstable and will lead to CNTs segmentation into domains. The region of absolute negative conductivity decreases when external electric field increases. We noted that due to the high density of states of conduction electrons in carbon nanotubes and the specific dispersion law inherent in hexagonal crystalline structure result in a uniquely higher current densities for the metallic zigzag CNTs than the metallic armchair CNTs.
In conclusion, using the solution to the Boltzmanns transport equation with constant relaxation time, we report on a theoretical analysis of a simple model of carbon nanotubes (CNTs) subject to an external electric field. We predict the effect self-induced transparency and absolute negative conductivity. The predicted effects may be useful in diagnostics of carbon nanotubes as well as in the amplification and efficiency conversion of electromagnetic signals. We noted that the metallic zigzag CNT has higher current density than the metallic armchair CNT. This is attributed to the fact that the metallic zigzag CNT has higher density of states of conduction electrons than the metallic armchair CNT. Another reason for the metallic ziagzag CNT to exhibit higher current density than the armchair is due to its dispersion law which is derived from the hexagonal structure.