Dynamics of mechanical waves in periodic graphene nanoribbon assemblies
 Fabrizio Scarpa^{1}Email author,
 Rajib Chowdhury^{2},
 Kenneth Kam^{1},
 Sondipon Adhikari^{2} and
 Massimo Ruzzene^{3}
DOI: 10.1186/1556276X6430
© Scarpa et al; licensee Springer. 2011
Received: 21 September 2010
Accepted: 17 June 2011
Published: 17 June 2011
Abstract
We simulate the natural frequencies and the acoustic wave propagation characteristics of graphene nanoribbons (GNRs) of the type (8,0) and (0,8) using an equivalent atomisticcontinuum FE model previously developed by some of the authors, where the CC bonds thickness and average equilibrium lengths during the dynamic loading are identified through the minimisation of the system Hamiltonian. A molecular mechanics model based on the UFF potential is used to benchmark the hybrid FE models developed. The acoustic wave dispersion characteristics of the GNRs are simulated using a Floquetbased wave technique used to predict the passstop bands of periodic mechanical structures. We show that the thickness and equilibrium lengths do depend on the specific vibration and dispersion mode considered, and that they are in general different from the classical constant values used in open literature (0.34 nm for thickness and 0.142 nm for equilibrium length). We also show the dependence of the wave dispersion characteristics versus the aspect ratio and edge configurations of the nanoribbons, with widening bandgaps that depend on the chirality of the configurations. The thickness, average equilibrium length and edge type have to be taken into account when nanoribbons are used to design nanooscillators and novel types of mass sensors based on periodic arrangements of nanostructures.
PACS 62.23.Kn · 62.25.Fg · 62.25.Jk
Introduction
Graphene nanoribbons (GNRs) [1] have attracted a significant interest in the nanoelectronics community as possible replacements to silicon semiconductors, quasiTHz oscillators and quantum dots [2]. The electronic state of GNRs depend significantly on the edge structure. The zigzag layout provides the edge localized state with nonbonding molecular orbitals near the Fermi energy, with induced large changes in optical and electronic properties from quantization. DFT calculations and experimental measurements have shown that zigzag edge GNRs can show metallic or halfmetallic behaviour (depending on the spin polarization in DFT simulations), while armchair nanoribbons are semiconducting with an energy gap decreasing with the increase of the GNR width [3–5]. GNRs have also been prototyped as photonics waveguides by Law et al. [6], and recently proposed for thermal phononics to control the reduction of thermal conductivity by Yosevich and Savin [7].
In this study, we describe the mechanical vibration natural frequencies and acoustic wave dispersion characteristics of graphene nanoribbons considered as periodic structures. In structural dynamics design, the wave propagation characteristics of periodic systems (both 1D and 2D) have been extensively used to tune the acoustic and vibrational signature of structures, materials and sensors [8–10], while at nanoscale level the periodicity of nanotubes array has also been used to develop nanophotonics crystals (see for example the study of Kempa and et al. [11]). Hod and Scuseria have also observed that the presence of a central mechanical load (or uniform inposed displacements) in bridgedbridged nanoribbons induces a significant electromechanical response in bending and torsional deformations [5]. We focus in this article on nanoribbon architectures of the type (8,0) and (0,8). While the results present in this manuscript are related to these specific nanoribbon topologies, the general algorith that we proposed can be readily extended to analyse more general graphene architectures. The nanoribbon models are developed using a hybrid atomistic continuumFinite Element (FE) model (also called lattice [12]), in which the carboncarbon (CC) covalent bonds are represented by Timoshenko structural beams with equivalent mechanical properties (Young's modulus and Poisson's ratio) derived by the minimisation of the Hamiltonian of the structural system, or total potential energy for the static case [12–14]. It is worth to notice that the concept of the Hamiltonian of a system is not limited to problems associated to quantum mechanics, but it is also used in a large variety of variational problems related to the dynamics and stability of engineering and mechanical structures [15, 16]. The equivalent mechanical properties for the sp ^{2} CC bond are expressed in terms of the thickness of the bond itself. It is useful to reiterate that there is neither a physical thickness per se for the covalent bonds, nor for the carbon atoms involved in the bond. Nonetheless, when subjected to a mechanical static loading, the nanostructure tends to reach its equilibrium state corresponding to the minimum potential energy. The geometric and material configuration of the equivalent continuum mechanics structures used to represent the graphene (plates and/or shells) will be therefore be defined by the energy equilibrium conditions of the nanostructure, and cannot be ascribed as fixed. The length of the covalent bonds merits also some considerations. In finite size rectangular single layer graphene sheets (SLGS), the lengths of the CC bonds at equilibrium after mechanical loading are unequal, ranging between 0.136 and 0.144 nm, and depend on the type of loading, size and boundary conditions [17, 18], as well as the location on the SLGS itself (i.e. the edges [19]). This fact contrasts with the classical use of the fixed value of 0.142 nm at equilibrium considered in most mechanical simulations [20–23]. The variation of the thickness and the distributions of lengths at equilibrium is important factors to consider when computing the homogenised mechanical properties of the graphene, i.e. the equivalent mechanical performance of the graphene seen as a continuum. In this study, we will show that the thickness and the equilibrium length distributions assume some specific values in GNRs also when undergoing a mechanical resonant behaviour, both as a single nanostructure in freefree vibration conditions, and as periodic elements in a onedimensional (1D) acoustic wave propagation case. However, the thickness and equilibrium lengths for the mechanical vibration case will be determined minimimsing the Hamiltonian of the system, rather that the total potential energy of the static loading case. Similar to the static inplane and outplane loading cases [12, 13], those values can be different from the ones usually adopted in open literature. We will also show that the chirality of the GNRs (and their edge effects in nanoribbons with short widths) provides different acoustic wave dispersion properties, which should be taken into account when GNRs are considered for potential nanoelectromechanical systems (NEMS) applications.
Modeling
AtomisticFE model
The real and imaginary parts of the domain in the FE representation are produced creating two superimposed meshes, linked by the boundary conditions [10, 31] (12). For a given wave propagation constant k _{ x } , the resultant eigenvalue problem provides the frequency associated to the acoustic wave dispersion curve. Similar to the undamped eigenvalue problem, the minimisation of the Hamiltonian (10) is also carried out for the wave propagation case to identify the set of thickness and average bond length required for the eigenvalue solution.
Molecular mechanics approach
Here k _{1}, k _{2}, k _{3} and k _{4} are force constants, θ _{0} is the natural bond angle, D is the van der Waals well depth, r* is the van der Waals length, q _{ i } is the net charge of an atom, ε is the dielectric constant and r _{ ij } is the distance between two atoms. In nanotubes, the atoms have no net charge, so the E _{el} term is always zero. The torsion term, E _{ ϕ } , turns out to be of great importance. Detailed values of these parameters in Equation 14 can be found in Ref. [34]. Some of the authors have successfully used a similar MM approach to describe the mechanical vibrations of singlewalled carbon nanotubes [35] and boronnitride nanotubes [36]. Other molecular mechanics approaches have been successfully used to describe the structural mechanics aspects of SWCNTs and MWCNTs (see for example Sears and Batra [37]).
Results and discussions
Molecular mechanics and atomisticFE models
Wave propagation in bridged nanoribbons with different chirality
Conclusions
In this study, we have presented a new methodology to derive the mechanical structural dynamics characteristics and acoustic wave dispersion relations for graphene nanoribbons using an hybrid Finite Element approach. The technique, benchmarked against a Molecular Mechanics model, allows to identify the mechanical natural frequencies and associated modes shapes, as well as the passstop band acoustic characteristics of periodic arrays of GNRs.
The numerical results from the minimisation of the Hamiltonian in the hybrid FE method show that the commonly used value in nanomechanical simulations for the thickness (0.34 nm) is not adequate to represent the effective structural dynamics of the system. Thickness values identified through the minimisation of the Hamiltonian vary in a restricted range around 0.07 nm for the AMBER force model used in this study. We also observe a distribution of the CC bond lengths corresponding to average values between 0.142 nm and 0.145 nm, after the minimisation for specific modes. However, the minimised thickness does not show any particular dependence over the type of mode shape considered. Only for pure torsional modes a small percentage variation from the baseline d = 0.074 nm value is observed.
We also show that graphene manoribbons exhibit a significant dependence of the acoustic wave propagation properties over the type of edge and aspect ratio, quite similarly to what observed for their electronic state. This feature suggests a possible combined electromechanical approach to design multifunctional waveguidetype band filters.
The use of periodic assemblies of graphene nanoribbons seems also a design feature that could lead to potential breakthroughs in terms of masssensors concepts, with enhanced selectivity provided by the periodic distribution of constraints and supports. The model proposed in this study allows to design and simulate these novel devices.
Abbreviations
 AR:

aspect ratio
 GNRs:

graphene nanoribbons
 MM:

molecular mechanics
 NEMS:

nanoelectromechanical systems
 1D:

onedimensional
 SS:

simply supported
 SLGS:

single layer graphene sheet
 SLGS:

single layer graphene sheets
 UFF:

universal force field.
Declarations
Acknowledgements
The authors would like to thank the referees for their useful suggestions.
Authors’ Affiliations
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