Modulational instability of gap solitons in single-walled carbon nanotube lattices

doi:10.1016/j.wavemoti.2020.102511

Wave Motion

Volume 94, April 2020, 102511

https://doi.org/10.1016/j.wavemoti.2020.102511 Get rights and content

Highlights

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Nonlinear localized excitations are addressed in a single-walled carbon nanotube lattice.
•
Two frequency modes are studied through the quasi-discrete approximation.
•
The modulational instability and breather solutions are studied in the upper lower frequency cutoff modes.

Abstract

Modulational instability and nonlinear localized excitations are addressed, in the framework of a one-dimensional diatomic carbon nanotube (CNT) model, using the quasi-discrete approximation. Gap soliton solutions, based on the modulational instability criterion, are studied, where one considers the solutions arising in the upper and lower optical frequency cutoff regimes, and in the upper acoustic frequency cutoff mode. Solutions are found as breathers and double breathers, and their response to interatomic interaction parameters is discussed. Vibrations of the CNTs from the two modes are compared based on their capability of carrying the amount of energy required for specific purposes, either in Microelectronics or in Nano-devices.

Introduction

Over the last two decades, carbon nanotubes (CNTs) have drawn considerable attention, and significant amount of investigations has revealed their extraordinary, and sometimes unexpected mechanical, optical, chemical and structural properties [1], [2], [3]. One of the main research directions devoted to mechanical behaviors has also revealed that CNT properties fundamentally depend on structure, so that any uniform [4], [5] or nonuniform [6], [7] deformation of a CNT can considerably affect its properties. Attention has been paid extensively to nonlinear excitations in graphene nanoribbons and their role in transporting and localizing energy has been reported [8]. In that respect, CNTs, that are made by rolling up graphene sheet, may support nonlinear excitations and solitons. The existence of spatially localized nonlinear modes in the form of discrete breathers was predicted in CNTs with chirality $(m, 0)$ and $(m, m)$ [9], [10]. Principally, three types of discrete breathers (DBs) were discussed, namely the longitudinal, radial, and twisting DBs. The structure and stability of stationary DBs in zigzag and armchair CNTs were studied by Doi and Nakatani [11], along with the involved structural physical processes and dynamical implications. By considering a large radius of the CNT, a one-dimensional diatomic chain model was proposed by Savin and Kivshar [9], [10], and numerical studies were performed, using molecular dynamics simulations with realistic interatomic potentials, to confirm the intrinsic character of DBs in CNTs. The present work focuses on such a model.

Diatomic lattices are generalized versions of monoatomic chains, and their dispersion curve contains two frequency branches known as the acoustic branch and the optical branch. Between the optical and the acoustic frequencies, there exist a gap where wave propagation of certain wavelengths is forbidden. More clearly, linear approximation is valid for small amplitude vibrations described as combinations of the linear modes within the phonon band. However, when vibration with large displacement takes place, nonlinearity of the interaction between atoms becomes dominant, leading to the so-called self-induced nonlinearity, and there appears some new effects in the system, mainly the formation of gap solitons in the forbidden zone of the linear wave spectrum. Introduced by Chen and Mills and applied to nonlinear optical response of superlattices [12], the concept of gap solitons has gained interest in other physical contexts such as solid-state lattice vibrations [13], [14], [15], nonlinear optics [16], [17], biophysics [18], [19], electrical transmission lines [20], [21], [22], and so on. The effect of cubic and quartic nonlinearities on gap solitons, in one-dimensional diatomic lattices, was addressed by Huang and Hu [13], [15] who further confirmed that such excitations can be described by two coupled nonlinear Schrödinger (NLS) equations, each corresponding to a specific frequency mode. The nonlinear localized excitations corresponding to each of the modes have been discussed in different physical situations [15], [18], [20], with emphasis on the different dynamical features of gap solitons. The model we consider in the present study, to the best of our knowledge, has been exploited only for numerical calculations until now, probably due to the complexity of the interatomic potentials.

One of the natural mechanisms leading to the formation of solitonic structures is modulational instability (MI). MI, which can be defined as an exponential growth of some modulation sidebands of a plane wave due to the balance between nonlinear and dispersive effects, appears in different physical settings, including nonlinear optics [23], [24], [25], plasma physics [26], [27], [28], [29], biological physics [30], [31], [32], [33], [34], [35], [36], [37], Bose–Einstein condensation [38], [39], [40], just to name a few. NLS breather solutions appear in general in regions of MI, i.e., where the product between the dispersion and nonlinearity coefficients is positive, which mainly depends on the choice of right model parameters that constitute the nonlinear interatomic potentials. In that direction, an attempt was recently made to analyze the stability/instability of nonlinear waves in single walled CNTs under the effect of higher order interaction terms from the Brenner’s potential [41], which was found to strongly influence energy localization in the discrete context, through a discrete NLS equation. Moreover, intense energy exchange between different parts of CNT and weak energy localization in the excited part of CNT were reported in the framework of the continuum shell theory, based on the NLS equations resulting from the multi-scale expansion procedure [42]. All the above clearly supports the idea that in one-dimensional lattices, weekly nonlinear excitations with a large spatial extension, or envelope solitons, are governed by the NLS equation which is straightforwardly related to short wave wavepacket excitations [43].

In the rest of this paper, we first present the CNT model, via its Hamiltonian, in Section 2. In Section 3, the quasi-discrete approximation (QDA) is utilized to show that each of the modes, optical and acoustic, is described by a NLS equation. Gap solitons are discussed in Section 4, and attention is given to breather solitons that are related to MI. The paper ends with some concluding remarks in Section 5.

Section snippets

Model

The model adopted in the present paper was introduced in Refs. [9], [10], where the CNT is considered to have a radius $R$ . Considering a chirality $(m, 0)$ , the Hamiltonian for the vibration of the lattice of carbon atoms is written as $H = \sum_{n} \sum_{l} [\frac{1}{2} M ({\dot{u}}_{n, l, 0}^{2} + {\dot{u}}_{n, l, 1}^{2}) + P_{n, l}],$ where the three indexes $(n, l, k)$ define the structure of the CNT, with $(n, l)$ defining an elementary cell $(n = 0, \pm 1, \dots, l = 1, 2, \dots, m)$ , and $k = 0, 1$ being the atom index in the cell. $l$ is unlimited for a planar sheet of carbon atoms. $M = 12 \times 1.6603 \cdot 1$

Amplitude equation

The Hamiltonian (6) contains complicated anharmonic interatomic potentials which may prevent direct exact solutions to be found. This further explains why most of the works devoted to the studied model have been essentially numerical. In order to proceed with the QDA, we expand the potentials $V_{1}$ and $V_{2}$ of Eq. (7) in Taylor’s series up to the fourth order, i.e., $V_{1} (w) ≃ \frac{1}{2} I_{2} w^{2} + \frac{1}{3} I_{3} w^{3} + \frac{1}{4} I_{4} w^{4} and V_{2} (w) ≃ \frac{1}{2} J_{2} w^{2} + \frac{1}{3} J_{3} w^{3} + \frac{1}{4} J_{4} w^{4},$ where $I_{2} = \frac{2 D α^{2}}{M}$ , $I_{3} = - \frac{6 D α^{3}}{M}$ , $I_{4} = \frac{14 D α^{4}}{M}$ , $J_{2} = \frac{1}{M} (\frac{27 ϵ_{v}}{2 ρ_{0}^{2}} + D α^{2})$ , $J_{3} = \frac{3 D α^{2} (3 - α ρ_{0})}{2 M ρ_{0}} - 81 ϵ_{v}$

Nonlinear localized modes and gap solitons in CNTs

In general, solutions whose amplitudes are governed by Eq. (28) depend on the sign of the product $P_{\pm} \times Q_{\pm}$ . To this effect, let us assume plane solutions for Eq. (28) to be of the form $F_{\pm} (t, x_{n}^{\pm}) = F_{0} (x_{n}^{\pm}) e^{i Q_{\pm} F_{0}^{2} t}$ , and subjected to a small perturbation $δ F_{\pm} (t, x_{n}^{\pm}) = (U_{0} + i V_{0}) e^{i (K x_{n}^{\pm} - Ω_{\pm} t)}$ , with $K$ and $Ω_{\pm}$ being, respectively, the wavenumber and frequency of the perturbation. The use of the standard calculations of MI leads to the nonlinear dispersion relation $Ω_{\pm}^{2} = {(K^{2} P_{\pm})}^{2} (1 - \frac{K_{c r}^{2}}{K^{2}}),$ with the critical wavenumber

Conclusion

The present paper was devoted to the investigation of gap envelope solitons, as consequence of MI, in single-walled CNTs. As proposed by Savin and Kivshar [9], [10], when the radius of the nanotube is large, the dynamics of the system becomes similar to that of a one-dimensional diatomic lattice. This case was considered here and, under the linear approximation, two frequency modes have been detected: the acoustical and the optical frequency modes. In order to specifically study each of the

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The work by CBT is supported by the Botswana International University of Science and Technology under the grant DVC/RDI/2/1/16I (25). CBT thanks the Kavli Institute for Theoretical Physics (KITP), University of California Santa Barbara (USA), for invitation.

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View full text

Modulational instability of gap solitons in single-walled carbon nanotube lattices

Highlights

Abstract

Introduction

Section snippets

Model

Amplitude equation

Nonlinear localized modes and gap solitons in CNTs

Conclusion

Declaration of Competing Interest

Acknowledgments

Chaos Solitons Fractals

Commun. Nonlinear Sci. Numer. Simul.

Physica A

Commun. Nonlinear Sci. Numer. Simul.

Chaos Solitons Fractals

Wave Motion

Physica D

Proced. Eng.

Physica D

Physica A

Science

Nat. Nanotechnol.

Carbon Nanotubes: Synthesis, Structure, Properties and Applications

Phys. Rev. B

Phys. Rev. B

Phys. Rev. B

Phys. Rev. B

Europhys. Lett.

Europhys. Lett.

Fiz. Nizk. Temp.

Lett. Mater.

Phys. Rev. Lett.

Int. J. Bifurc. Chaos

Phys. Rev. B