Carbon nanotubes (CNTs) were first discovered by Iijima in 1991 [1], and since then there has been exponential growth in the interest of the quasi-one-dimensional monomolecular structure due to their unique and outstanding electrical, mechanical, and chemical properties. Nonlinear effects on CNTs are of great interest for potential applications in nanoelctronics and negative differential conductivity (NDC) has been predicted at room temperature under the condition, when, in a certain range of electric field strength [2]. The NDC is believed to provide current instability in CNTs [2] which is destructive for the formation of terahertz (THz) radiation as in semiconducting superlattices (SLs). Simultaneous application of both dc-and ac-fields to the CNTs will result in nonlinear phase of the instability [2] as observed in semiconducting superlattices (SL). Mensah [3] studied the NDC effect in a semiconductor SL in the presence of an external electric field and showed that the NDC occurred in the current density characteristics when i.e. near where the constant electric field is equal to the amplitude of the ac electric field and the peak decreases with increasing the amplitude of the ac field. The theory agrees fairly well with an experiment [4] that indicated “right shift” of the current density versus static electric field characteristics, which is typical for a SL without domain formation. Klappenberger and coworkers [4] demonstrated ultrafast creation and annihilation of space-charge domains in a semiconductor superlattice using terahertz fields. Up to now, NDC has been observed only in a d.c electric field in both doped and undoped CNTs [2]. However, there is no report on simultaneous application of dc and ac fields to CNT to date. In this paper, we report a systematic theoretical investigation of effect of simultaneous application of dc and ac field to armchair CNT (a-SWCNT) and zigzag CNT (z-SWCNT) using the Boltzmann’s transport equation to derive the current density and analyze the behavior of the normalized current density as a function of dc electric field.

We consider a response of electrons in an undoped achiral single-wall carbon nanotubes (a-SWCNT and z-SWCNT) (a-SWCNT), to the action of a strong pumping electric field.

where the dc bias is small and the ac field is quasi static The investigation is done within the semiclassical approximation in which the motion of π-electrons is considered as classical motion of free quasi-particles in the field of the crystalline lattice with dispersion law extracted from quantum theory. Taking into account the hexagonal crystalline structure of a rolled graphene in a form of SWCNT and using the tight binding approximation, the energy dispersion for z-SWCNT and aSWCNT are expressed respectively, as [2]

and

where is the overlapping integral, is the axial component of quasimomentum, is transverse quasimomentum level spacing and s is an integer. The expression for in Equations (2) and (3) is given as, is the C-C bond length. The – and + signs correspond to the valence and the 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 SWCNT. This model is applicable to the case under consideration because of the restriction to the temperatures and /or voltages well above the level spacing [5], i.e. 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 L is the carbon nanotube length [6].

Employing Boltzmann equation with a single relaxation time approximation

where e is the electron charge, is the equilibrium distribution function, is the distribution function, and is the relaxation time. The electric field E is applied along the SWCNT axis. In this problem the relaxation term is assumed to be constant. The justification for being constant can be found in ref [7]. The relaxation term of Equation (5) describes the effects of the dominant type of scattering (e.g. electron-phonon and electron-twistons) [8]. For the electron scattering by twistons (thermally activated twist deformations of the tube lattice), is proportional to m and the characteristics have shown that scattering by twistons increases and decreases in the NDC region. The effect is stronger for smaller ms. Quantitative changes of the curves turn out to be insignificant in comparison with the case of = constant [7,8].

Expanding the distribution functions of interest in Fourier series as;

and

where 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.

Substituting Equations (6) and (7) into Equation (5), and solving with Equation (1) we obtain

where, and is the Bessel function of the k^th order and.

Similarly, expanding in Fourier series with coefficients

where

and expressing the velocity as

we determine the surface current density as

or

where the integration is taken over the first Brillouin zone. Substituting Equations (7), (9) and (12) into (13) we find the current density for the z-SWCNT after averaging over a period of time t, as

which is similar to expression (8) in ref. [3]. See Figure 1.

Where for z-SWCNT and for a-SWCNT.

The current density expression for z-SWCNT subjected to dc bias field and quasi static-field is obtained by using the solution to the Boltzmann equation with constant relaxation time. The behavior of the normalized current density as a function of the dimensionless parameter (Equation 14) for given and values was analyzed using a Matlabversion 7.6 2008a.

The normalized current density of the nanotubes structures (z-SWCNT, a-SWCNT) exhibits linear dependence on at weak applied external static electric field strengths (i.e. the region of ohmic conductivity). As Z_c increases, the current density increases and researches a maximum then drops off, and hence experiencing negative differential conductivity (NDC) i.e.

. We observed that as the amplitude increases the normalized current density maximum decreases and shifts towards large electric field values. This “right shift” of the current density maximum is due to a nonlinearity of the Esaki-Tsu characteristics (i.e. nonlin

earity associated with the non-parabolicity of the electron energy band) which is very strong in SWCNT because of the high stark component (summation over r). We noted that the nonlinearity in SWCNT is much higher than in SL [9,10]. See Figure 1.

The current density-electric field characteristics curves for the nanotubes structures are qualitatively similar to those of SL. However, the NDC effect on the SL appeared at larger electric field strengths in comparison with the nanotubes. See Figures 1(a)-(c).

We sketched also the graph of against for when. See Figure 2. We observed that when, the normalized current density for the nanotubes structures (z-SWCNT, a-SWCNT) linearly depends on at weak applied external static electric field strengths. On the other hand for the normalized current density oscillates at weak applied external static electric field strengths.

The estimations of the restrictions to the theoretical approach can be found in [11].

We have studied theoretically the current-density electric field characteristic in the presence of ac-dc driven field and negative differential conductivity was observed. The current-density electric field characteristic shows a negative differential conductivity when (quasi-static case). We suggest that this phenomenon can also be used for the generation of terahertz radiation without electric current instability.