Direct Current Generation in Carbon Nanotubes by Terahertz Field

Sulemana S. Abukari^{1}^{*},
Frederick Sam^{1},
Samuel Y. Mensah^{1},
Natalia G. Mensah^{2},
Rabiu Musah^{3},
Anthony Twum^{1},
Patrick M. Amoah^{1},
Alfred Owusu^{1}

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Received 12 September 2015; accepted 26 February 2016; published 29 February 2016

1. Introduction

Investigation into the electronic properties of carbon nanotubes (CNTs) has attracted a great deal of interests ever since the discovery of these quasi-one-dimensional monomolecular structures by Iijima [1] . This may be due to their intriguing properties. The oscillatory response of the CNTs to moderate electric field strength makes CNTs inherently nonlinear, and as such, can exhibit plethora of transport phenomena. Under different conditions of an external electric field, an electron in CNT is predicted to reveal a variety of physical effects such as Bloch oscillations, self-induced transparency, negative differential conductivity, absolute negative conductance [2] - [9] , and many more. Furthermore, CNTs have been shown to exhibit ballistic transport [10] - [12] , Coulomb-blockade [13] [14] , Luttinger Liquid [15] [16] and superconductivity [17] [18] . The unique architecture and physicochemical properties have undoubtedly led to CNT which has been identified as a promising candidate for the fundamental building block of new generation of nanoelectronics [19] - [22] , sensors [23] [24] , electrochemical capacitors [25] -[27] , Li-ion batteries [28] - [30] and terahertz (THz) generation and amplification [31] - [35] , just to mention a few. There are several reports on CNTs for THz application [31] - [35] . Most of these reports focus on room temperature generation of THz radiation. Different proposals of CNTs for THz applications have been made. These ranges from multipliers [31] [32] , amplifies [33] , switches [34] to antennas [35] (see also [36] - [41] ).

However, there are very limited reports on the effect of THz-fields on the transport properties of CNTs.

In this report, we present a theoretical investigation of the influence of THz-fields on the conductivity of CNTs that are stimulated axially, by considering the kinetic approach based on the semiclassical Boltzmann’s transport equation with constant relaxation time approximation, together with the energy spectrum of electron in the tight-binding approximation, and predict the generation of DC current in both axial and circumferential directions of the CNTs.

2. Theory

We start with the Boltzmann equation and proceed as in refs [42] -[44] ,

(1)

In accordance with ref. [4] - [6] [11] , we find that the distribution function is periodic in the quasimomentum and can be written in Fourier series as;

(2)

where e is electronic charge, the index ; and are components of the electron dynamical momentum along the axial and tubular axes respectively; is the distribution function, and is the equilibrium distribution function; while is the electron relaxation time and assumed to be

constant. The electric field is related to the vector potential A as. is the modified Bessel function

of the order m and is Planck’s constant, finally and are the distances between along the axis and

helix respectively.

The solution of Equation (1) by the method of characteristics is obtained in [5] [6] [11] as;

(3)

Interesting to us is the situation that and Equation (2) reduces to the form

(4)

We proceed as in [5] [6] by considering an infinitely long chain of carbon atoms wrapped along a base helix as a model of a SWNT. This phenomenological model gives analytical tractability, which easily provides physically interpretable results. Also, the model gives a correct qualitative description of various electronic processes which are corroborated by the first principle numerical simulations. Thus, using the simple model of the tight- binding approximation, we describe the energy spectrum of the CNTs as ref [5] [6] .

(5)

where is the energy of an outer-shell electron in an isolated carbon atom, and are the real overlapping integrals for jumps along the respective coordinates.

The electron quasi classical velocity components for and are obtained as in refs [5] [13] ;

(6a)

(6b)

and the electron fluxes along the tubular axis and the base helix are given by after making the transformation as ref [5] [6] ;

(7a)

(7b)

where the integration is done over the first Brilloiun zone. The expressions for the axial and the circumferential components are as in [5] ;

(8a)

(8b)

where is the chiral angle. We consider the CNTs stimulated by a uniform electric field with a uniform sinusoidal THz radiation field of frequency and amplitude i.e.,. Substituting Equations (4) and (6) into Equation (7), we obtained the following direct current density expression from Equation (8) for circumferential and axial directions, respectively.

(9a)

(9b)

where, and

If we re-write expressions (9) in the form of ref [44] and using the simplest case for [5] [6] that, , and, we obtain the static current density as;

(10a)

and

(10b)

respectively.

where and

Equation (10) show the ac field can open up new transport channel for DC current as a result of multiphoton absorption or emission with probability of emission or absorption of photon given by (see [44] ).

For quasi static case i.e., expression (9) reduces to;

(11a)

(11b)

here and

From Equation (11) we obtain the normalized differential conductivity as;

(12)

where

3. Results and Discussion

We present the results of the kinetic equation approach of a CNT subjected to inhomogeneous THz field of the form. The Boltzmann’s equation is solved in the framework of constant relaxation time approximation. The expressions for the direct current densities along the axial and circumferential directions of the chiral CNT are given in Equation (9). The nonlinearity is analyzed using the dependence of the normalized direct current density as a function of the ac amplitude for, 0.5, 0.9, 1 and 2.

In Figure 1, we show the dependence of the current density on the ac amplitude β_{i}, Equation (9), i.e., when the CNTs are stimulated axially with a combination of a uniform electric field E_{0} and a uniform sinusoidal THz radiation field of frequency and amplitude. The behavior is similar in the axial and circumferential directions. In the region of strong scattering () the DC conductivity remains positive, but decreases with increasing the strength of the ac amplitude. However, for (), the current density decreases strongly with increasing β_{i} and then oscillates around with intervals of negative conductivity. The arrows in Figure 1 indicate the regions of negative conductivity. We observed that the width of the intervals of the negative conductivity is largest at and decreases with decreasing scattering. The results indicate that negative conductance can be generated in chiral CNTs at room temperature by axial stimulation of the CNTs with THz frequencies (→10^{12} Hz). On the other hand, the stable state with THz induced DC current is the region of dynamic localization for and is indicated by black boxes (see Figure 2). This suggests that stimulation of chiral CNTs with THz fields at room temperature can lead to switching from zero DC current to a finite DC current state, a process of quasi-ballistic electron transport where electron scattering is a critical component for the switching.

(a)(b)

Figure 1. A plot of a normalized DC current density of CNTs obtained from expression (9) as a function of ac current amplitude for (a) axial component (b) circumferential component when, 0.5, and 2.

Figure 2. A plot of a DC conductivity of CNTs obtained from expression (12) as a function of ac current amplitude. The arrows indicate regions of negative conductivity and the black shades show regions of dynamical location which is the stable state with THz induced DC current.

4. Conclusion

In conclusion, we considered the nonlinear electronic properties in CNTs stimulated by a high frequency THz field, i.e., using the Boltzmann’s kinetic equation in the constant relaxation time. The results indicate the creation of DC current densities along the axial and circumferential directions of the chiral CNTs.

NOTES

^{*}Corresponding author.

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