Dynamics of mechanical waves in periodic graphene nanoribbon assemblies
© Scarpa et al; licensee Springer. 2011
Received: 21 September 2010
Accepted: 17 June 2011
Published: 17 June 2011
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 atomistic-continuum FE model previously developed by some of the authors, where the C-C 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 Floquet-based wave technique used to predict the pass-stop 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 band-gaps 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 nano-oscillators and novel types of mass sensors based on periodic arrangements of nanostructures.
PACS 62.23.Kn · 62.25.Fg · 62.25.Jk
Graphene nanoribbons (GNRs)  have attracted a significant interest in the nanoelectronics community as possible replacements to silicon semiconductors, quasi-THz oscillators and quantum dots . The electronic state of GNRs depend significantly on the edge structure. The zigzag layout provides the edge localized state with non-bonding 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 half-metallic 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. , and recently proposed for thermal phononics to control the reduction of thermal conductivity by Yosevich and Savin .
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. ). Hod and Scuseria have also observed that the presence of a central mechanical load (or uniform inposed displacements) in bridged-bridged nanoribbons induces a significant electromechanical response in bending and torsional deformations . 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 continuum-Finite Element (FE) model (also called lattice ), in which the carbon-carbon (C-C) 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 C-C 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 C-C 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 ). 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 free-free vibration conditions, and as periodic elements in a one-dimensional (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 in-plane and out-plane 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.
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. . Some of the authors have successfully used a similar MM approach to describe the mechanical vibrations of single-walled carbon nanotubes  and boron-nitride nanotubes . Other molecular mechanics approaches have been successfully used to describe the structural mechanics aspects of SWCNTs and MWCNTs (see for example Sears and Batra ).
Results and discussions
Molecular mechanics and atomistic-FE models
Wave propagation in bridged nanoribbons with different chirality
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 pass-stop 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 C-C 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 electro-mechanical approach to design multifunctional waveguide-type band filters.
The use of periodic assemblies of graphene nanoribbons seems also a design feature that could lead to potential breakthroughs in terms of mass-sensors 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.
single layer graphene sheet
single layer graphene sheets
universal force field.
The authors would like to thank the referees for their useful suggestions.
- Nakada K, Fujita M, Dresselhaus G, Dresselhaus MS: Edge state in graphene nanoribbons: nanometer size effect and edge shape dependence. Phys Rev B 1996, 54(24):17954. 10.1103/PhysRevB.54.17954View Article
- Wang ZF, Shi QW, Li Q, Wang X, Hou JC: Z-shaped graphene nanoribbon quantum dot device. Appl Phys Lett 2007, 91: 053109. 10.1063/1.2761266View Article
- Barone V, Hod O, Scuseria GV: Electronic structure and stability of semiconducting graphene nanoribbons. Nano Lett 2006, 6(12):2748. 10.1021/nl0617033View Article
- Han MY, Ozyilmaz B, Zhang Y, Kim P: Energy band-gap engineering of graphene nanoribbons. Phys Rev Lett 2007, 98(20):206805–1-206805–4.View Article
- Hod O, Scuseria GE: Electromechanical properties of suspended graphene nanoribbons. Nano Lett 2009, 9(7):2619–2622. 10.1021/nl900913cView Article
- Law M, Sirbuly DJ, Johnson JC, Goldberger J, Saykally RJ, Yang P: Nanoribbon waveguides for subwavelength photonics integration. Science 2004, 305(5688):269.View Article
- Yosevich YA, Savin AV: Reduction of phonon thermal conductivity in nanowires and nanoribbons with dynamically rough surfaces and edges. Eur Phys Lett 2009, 88: 14002. 10.1209/0295-5075/88/14002View Article
- Ruzzene M, Baz A: Attenuation and localization of wave propagation in periodic rods using shape memory inserts. Smart Mater Struct 2000, 9: 805. 10.1088/0964-1726/9/6/310View Article
- Gonella S, Ruzzene M: Homogenization of vibrating periodic lattice structures. Appl Math Model 2008, 32(4):459. 10.1016/j.apm.2006.12.014View Article
- Tee KF, Spadoni A, Scarpa F, Ruzzene M: Wave propagation in auxetic tetrachiral honeycombs. ASME J Vibr Acoust 2010, 132(3):031007. 10.1115/1.4000785View Article
- Kempa K, Kimball B, Rybczynski J, Huang ZP, Wu PF, Steeves D, Sennett M, Giersig M, Rao DVGLN, Carnahan DL, Wang DZ, Lao JY, Li WZ, Ren ZF: Photonic crystals based on periodic arrays of aligned carbon nanotubes. Nano Lett 2003, 3(1):13. 10.1021/nl0258271View Article
- Scarpa F, Adhikari S, Gil AJ, Remillat C: The bending of single layer graphene sheets: lattice versus continuum approach. Nanotechnology 2010, 21(12):125702. 10.1088/0957-4484/21/12/125702View Article
- Scarpa F, Adhikari S, Phani AS: Effective elastic mechanical properties of single layer graphene sheets. Nanotechnology 2009, 20: 065709. 10.1088/0957-4484/20/6/065709View Article
- Scarpa F, Adhikari S: A mechanical equivalence for Poisson's ratio and thickness of C-C bonds in single wall carbon nanotubes. J Phys D 2008, 41(8):085306. 10.1088/0022-3727/41/8/085306View Article
- Goldstein H, Poole CP, Safko JL: Classical Mechanics. Cambridge, MA: Addison-Wesley; 1950.
- Meirovitch L: Analytical Methods in Vibrations. 1st edition. Englewood Cliffs: Prentice-Hall; 1997.
- Reddy CD, Rajendran S, Liew KM: Equilibrium configuration and elastic properties of finite graphene. Nanotechnology 2006, 17: 864. 10.1088/0957-4484/17/3/042View Article
- Reddy CD, Ramasubramaniam A, Shenoy VB, Zhang YW: Edge elastic properties of defect-free single-layer graphene sheets. Appl Phys Lett 2009, 94(10):101904. 10.1063/1.3094878View Article
- Sun CQ, Sun Yi, Nie YG, Wang Y, Pan JS, Ouyang G, Pan LK, Sun Z: Coordinationresolved C-C bond length and the C1s binding energy of carbon allotropes and the effective atomic coordination of the few-layer graphene. J Phys Chem C 2009, 113(37):16464. 10.1021/jp905336jView Article
- Sakhaee-Pour A, Ahmadian MT, Vafai A: Potential application of single-layered graphene sheet as strainsensor. Solid State Commun 2008, 147(7–8):336–340. 10.1016/j.ssc.2008.04.016View Article
- Sakhaee-Pour A, Ahmadian MT, Naghdabadi R: Vibrational analysis of singlelayered graphene sheets. Nanotechnology 2008, 19: 085702. 10.1088/0957-4484/19/8/085702View Article
- Sakhaee-Pour A: Elastic properties of single-layered graphene sheet. Solid State Commun 2009, 149(1–2):91. 10.1016/j.ssc.2008.09.050View Article
- Tserpes KI, Papanikos P: Finite Element modelling of single-walled carbon nanotubes. Composites B 2005, 36: 468. 10.1016/j.compositesb.2004.10.003View Article
- Timoshenko S: Theory of Plates and Shells. London: McGraw-Hill, Inc; 1940.
- Huang Y, Wu J, Hwang KC: Thickness of graphene and single wall carbon nanotubes. Phys Rev B 2006, 74: 245413.View Article
- Kaneko T: On Timoshenko's correction for shear in vibrating beams. J Phys D 1974, 8: 1927.View Article
- Przemienicki JS: Theory of Matrix Structural Analysis. New York: McGraw-Hill; 1968.
- Li C, Chou TW: Single-walled carbon nanotubes as ultrahigh frequency nanomechanical resonators. Phys Rev B 2003, 68: 073405.View Article
- Friswell MI, Mottershead JE: Finite Element Updating in Structural Dynamics. Dordrecht: Kluwer Academic Publishing; 1995.View Article
- Rajendran S, Reddy CD: Determination of elastic properties of graphene and carbon-nanotubesusing brenner potential: the maximum attainable numerical precision. J Comput Theor Nanosci 2006, 3: 1.
- Aberg M, Gudmundson P: The usage of standard finite element codes for computation of dispersion relations in materials with periodic microstructure. J Acoust Soc Am 1997, 102(4):2007. 10.1121/1.419652View Article
- Brillouin L: Wave Propagation in Periodic Structures. Dover Phoenix edition. New York: Dover; 1953.
- Frisch MJ, Trucks GW, Schlegel HB, Scuseria GE, Robb MA, Cheeseman JR, Scalmani G, Barone V, Mennucci B, Petersson GA, Nakatsuji H, Caricato M, Li X, Hratchian HP, Izmaylov AF, Bloino J, Zheng G, Sonnenberg JL, Hada M, Ehara M, Toyota K, Fukuda R, Hasegawa J, Ishida M, Nakajima T, Honda Y, Kitao O, Nakai H, Vreven T, Montgomery JA Jr, Peralta JE, Ogliaro F, Bearpark M, Heyd JJ, Brothers E, Kudin KN, Staroverov VN, Kobayashi R, Normand J, Raghavachari K, Rendell A, Burant JC, Iyengar SS, Tomasi J, Cossi M, Rega N, Millam JM, Klene M, Knox JE, Cross JB, Bakken V, Adamo C, Jaramillo J, Gomperts R, Stratmann RE, Yazyev O, Austin AJ, Cammi R, Pomelli C, Ochterski JW, Martin RL, Morokuma K, Zakrzewski VG, Voth GA, Salvador P, Dannenberg JJ, Dapprich S, Daniels AD, Farkas O, Foresman JB, Ortiz JV, Cioslowski J, Fox DJ: Gaussian 09 Revision A.1.
- Rappe AK, Casewit CJ, Colwell KS, Goddard WA, Skiff WM: UFF, a full periodictable force-field for molecular mechanics and molecular dynamics simulations. J Am Chem Soc 1992, 114(25):10024. 10.1021/ja00051a040View Article
- Chowdhury R, Adhikari S, Wang C, Scarpa F: A molecular mechanics approach for the vibration of single-walled carbon nanotubes. Comput Mater Sci 2010, 48(4):730–735. 10.1016/j.commatsci.2010.03.020View Article
- Chowdhury R, Wang CY, Adhikari S, Scarpa F: Vibration and symmetry-breaking of boron nitride nanotubes. Nanotechnology 2010, 21(36):365702. 10.1088/0957-4484/21/36/365702View Article
- Sears A, Batra RC: Macroscopic properties of carbon nanotubes from molecularmechanics simulations. Phys Rev B 2004, 69(23):235406.View Article
- Kudin KN, Scuseria GE, Yakobson BI: C2F, BN and C nanoshell elasticity from ab initio computations. Phys Rev B 2001, 64: 235406.View Article
- Gupta SS, Batra RC: Elastic properties and frequencies of single-layer graphene sheets. J Comput Theor Nanosci 2010, 7: 1–14. 10.1166/jctn.2010.1332View Article
- Mead DJ: Free wave propagation in periodically supported, infinite beams. J Sound Vibr 1970, 11(2):181. 10.1016/S0022-460X(70)80062-1View Article
- Reddy CD, Rajendran S, Liew KM: Equivalent continuum modeling of graphene sheets. Int J Nanosci 2005, 4(4):631. 10.1142/S0219581X05003528View Article
- Horgan CO: Recent developments concerning Saint Venant's principle: an update. Appl Mech Rev 1989, 42: 295. 10.1115/1.3152414View Article
- Mead DJ: Wave propagation in continuous periodic structures: research contributions from Southampton 1964–1995. J Sound Vibr 1996, 190(3):495. 10.1006/jsvi.1996.0076View Article
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