Domain Suppression in the Negative Differential Conductivity Region of Carbon Nanotubes by Applied AC Electric Field

We study theoretically the electron transport properties in achiral carbon nanotubes under the influence of an external electric field E(t) using Boltzmann’s transport equation to derive the current-density. A negative differential conductivity (NDC) is predicted in quasi-static approximation i.e., ωτ << 1, similar to that observed in superlattice. However, a strong enhancement in the current density intensity is observed in NDC of the achiral carbon nanotubes. This is observed at where the constant electric field E0 is equal to the amplitude of the AC electric field E1. The peak of the NDC intensity occurs at very weaker fields than that of superlattice under the same conditions. The peak intensity decreases and shifts to right with the increase in the amplitude of the ac field. This mechanism suppresses the domain formation and therefore could be used in terahertz frequency generation.


Introduction
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 B c , T      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 1


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.  0 1 cos where the dc bias 0 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 a-SWCNT are expressed respectively, as [2] and where 0 ~3.0 eV  is the overlapping integral, z p is the axial component of quasimomentum, p   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 qua-simomentum  , the axial quasimomentum z p is assumed to vary continuously within the range 0 2π , 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  is the charging energy.The energy level spacing   is given by where F  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,   0 f p is the equilibrium distribution function,   , f p t 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 Expanding the distribution functions of interest in Fourier series as; and where the coefficient,   Substituting Equations ( 6) and ( 7) into Equation (5), and solving with Equation (1) we obtain 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

Results and Discussion
The current density expression for z-SWCNT subjected to dc bias field 0 and quasi static-field   is obtained by using the solution to the Boltzmann equation with constant relaxation time The normalized current density of the nanotubes structures (z-SWCNT, a-SWCNT) exhibits linear dependence on c Z at weak applied external static electric field strengths (i.e. the region of ohmic conductivity).As Z c increases, the current density z o j j 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 z o j j against c Z for 0.1, 0.2, 0.5 and 0.9 We observed that when 0.1    , the normalized current density z o j j for the nanotubes structures (z-SWCNT, a-SWCNT) linearly depends on c Z at weak applied external static electric field strengths.On the other hand for 0.5   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].
[9] F. G. Bass and A. A. Bulgakov, "Kinetic and Electrodynamic Phenomena in Classical and Quantum Semiconductor Superlattices," Nova Science, New York, 1997.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 1


(quasi-static case).We suggest that this phenomenon can also be used for the generation of terahertz radiation without electric current instability.
) 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 I V  characteristics have shown that scattering by twistons increases region.The effect is stronger for smaller ms.Quantitative changes of the I V  curves turn out to be insignificant in comparison with the case of  = constant[7,8].
which the Fourier transform of the nonequilibrium distribution function differs from its equilibrium distribution counterpart.

E 1 
. The behavior of the normalized current density z o j j as a function of the dimensionless parameter c Z    (Equation 14) for given  and  values was analyzed using a Matlabversion 7.6 2008a.