Nonlinear Dynamics of Silicon Nanowire Resonator Considering Nonlocal Effect
© The Author(s). 2017
Received: 19 January 2017
Accepted: 24 April 2017
Published: 4 May 2017
In this work, nonlinear dynamics of silicon nanowire resonator considering nonlocal effect has been investigated. For the first time, dynamical parameters (e.g., resonant frequency, Duffing coefficient, and the damping ratio) that directly influence the nonlinear dynamics of the nanostructure have been derived. Subsequently, by calculating their response with the varied nonlocal coefficient, it is unveiled that the nonlocal effect makes more obvious impacts at the starting range (from zero to a small value), while the impact of nonlocal effect becomes weaker when the nonlocal term reaches to a certain threshold value. Furthermore, to characterize the role played by nonlocal effect in exerting influence on nonlinear behaviors such as bifurcation and chaos (typical phenomena in nonlinear dynamics of nanoscale devices), we have calculated the Lyapunov exponents and bifurcation diagram with and without nonlocal effect, and results shows the nonlocal effect causes the most significant effect as the device is at resonance. This work advances the development of nanowire resonators that are working beyond linear regime.
KeywordsSilicon nanowire resonator Chaotic vibration Nonlocal effect
Nanoscale resonators working at certain parameters exhibit rich nonlinear dynamics such as chaos and bifurcation [1–4]. To thoroughly investigate the nonlinear dynamics in such nanostructures considering various effects that are brought by reducing the size of the device and/or choosing different fabrication materials is crucial for developing real applications [5–9]. Nonlocal effect, essentially originated from device’s size-reducing, is usually taken into account when studying nanoscale structures in which lattice node interaction is not only affected by its surrounding nodes, but also from the nodes neighboring to the surrounding nodes [10, 11]. This effect has been proved to be playing an important role in nanoscale structures with respect to dynamical response, taking pulling-in as an example [12, 13]. To be more specific, in previous work, nonlocal effects on the elastic behavior of statically bent nanowires have been investigated in . The influences of nonlocal effect on the thermo-electro-mechanical vibration characteristics of piezoelectric nanoplates have been discussed by Chen Liu et al. . Based on a refined nonlocal theory, dynamical behavior of core-shell nanowires with weak interfaces has been analyzed in . Numerically, linear optical response of conducting nanostructures was proved to be altered dramatically by nonlocality . Vibration characteristic of piezoelectric nanobeam under the influence of nonlocal effect has been reported in . However, most of the works so far have been confined in analyzing the nonlocal effect based on linear regime. Even though F. Najar et al. recently investigated the nonlinear static and dynamical response in a nanoactuator taking nonlocal effect into account , in which they only studied the puling-in and bulking. How the nonlocal effect exerts its impact on nonlinear dynamics, in particular, the bifurcation and chaos, of nanostructures deserves further investigation.
Here, in this work, we employ silicon nanowire resonator as paradigm and try to systematically derive the expressions linking the nonlocal term with the dynamical parameters such as the resonant frequency, Duffing coefficient, and damping ratio that are directly influence the nonlinear dynamics of the device. Nonlinear dynamics of the resonator is then investigated through the key analysis such as Lyapunov exponent and bifurcation calculation by considering varied nonlocal parameters. Interesting remarks are drawn from the analysis, which in a fundamental way provides a significant result for future device design and modeling and gives useful guidance for the development and design of novel applications based on the nonlinear dynamics of nanowires, e.g., applications such as random generators and secure communications .
The report is presented as follows: Model Construction section presents the mathematic derivation of the dynamic equations taking the nonlocal effect into consideration. Numeric response al simulation results are described in Numerical Analysis section. Finally, the key conclusion remarks are summarized in the Conclusions section.
where W(x, t) is the dynamical displacement of the resonator along the x-axis, with the dot and prime are denoting the differentiation with respect to the t and x, respectively. T 0 and T are the initial and induced mechanical tension in the nanowire, respectively.
Results and discussion
Analysis of the nonlinear behavior of the double clamped silicon nanowire resonator with the consideration of the nonlocal effect has been made based on the Duffing motion equation and the Galerkin’s method. Relations between the nonlocal coefficient and dynamic parameters such as resonant frequency, Duffing coefficient, and the damping ratio have been derived. Calculations on the indicator (Lyapunov exponents) of the chaotic vibrations have been conducted, and it is concluded that as the nonlocal term is taken into account, the structure effectively gets hardened and the nonlinear performance has also changed. Importantly, from the bifurcation analysis, the nonlocal effect causes the most significant impact when the driving frequency is at the resonating frequency of the structure. The work provides useful guidance in designing future nanowire resonator-related applications.
This work was supported by the National Natural Science Foundation of China (61604078), Natural Science Youth Foundation of Jiangsu Province (BK20160905), Natural Science Foundation of the Jiangsu Higher Education Institutions of China (16KJB510029), and NUPTSF (NY216010).
LJ and LL both conceived the idea. LJ conducted the numerical simulation. LJ and LL both analyzed the results. Both authors read and approved the final manuscript.
The authors declare that they have no competing interests.
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- Ni X, Ying L, Lai Y-C, Do Y, Grebogi C (2013) Complex dynamics in nanosystems. Phys Rev E 87:052911View ArticleGoogle Scholar
- Kenfack A (2003) Bifurcation structure of two coupled periodically driven double-well Duffing oscillators. Chaos Solitons Fractals 15:205–218View ArticleGoogle Scholar
- Jin L, Mei J, Li L (2014) Chaos control of parametric driven Duffing oscillators. Appl Phys Lett 104:134101View ArticleGoogle Scholar
- Conley WG, Raman A, Krousgrill CM, Mohammadi S (2008) Nonlinear and nonplanar dynamics of suspended nanotube and nanowire resonators. Nano Lett 8:1590–1595View ArticleGoogle Scholar
- Rafii-Tabar H, Ghavanloo E, Fazelzadeh SA (2016) Nonlocal continuum-based modeling of mechanical characteristics of nanoscopic structures. Phys Rep 638:1–97View ArticleGoogle Scholar
- Dai MD, Kim C-W, Eom K (2011) Finite size effect on nanomechanical mass detection: the role of surface elasticity. Nanotechnology 22:265502View ArticleGoogle Scholar
- Lee H-L, Hsu J-C, Chang W-J (2010) Frequency Shift of Carbon-Nanotube-Based Mass Sensor Using Nonlocal Elasticity Theory. Nanoscale Res Lett 5:1774–1778View ArticleGoogle Scholar
- Zhang J, Wang C (2013) Size-dependent pyroelectric properties of gallium nitride nanowires. J Appl Phys 106:167–174Google Scholar
- Zhang J, Wang C, Adhikari S (2013) Fracture and buckling of piezoelectric nanowires subject to an electric field. Opt Lett 114:235303Google Scholar
- Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16View ArticleGoogle Scholar
- Reddy JN (2007) Nonlocal theories for bending, buckling and vibration of beams. Int J Eng Sci 45:288–307View ArticleGoogle Scholar
- Wang CM, Zhang YY, He XQ (2007) Vibration of nonlocal Timoshenko beams. Nanotechnology 18(9):105401View ArticleGoogle Scholar
- Soltani P, Kassaei A, Taherian MM, Farshidianfar A (2012) Vibration of wavy single-walled carbon nanotubes based on nonlocal Euler Bernoulli and Timoshenko models. Int J Adv Struct Eng 4:3Google Scholar
- Wu Q, Volinsky AA, Qiao L (2011) Surface effects on static bending of nanowires based on non-local elasticity theory. Prog Nat Sci-Mater Int 11:1002–1008Google Scholar
- Liu C, Ke L-L, Wang Y-S (2006) Thermo-electro-mechanical vibration of piezoelectric nanoplates based on the nonlocal theory. Compos Struct. 203:410–427Google Scholar
- Fu Y, Zhong J (2015) Nonlinear free vibration of core-shell nanowires with weak interfaces based on a refined nonlocal theory. ACTA Mechanica 226:1369View ArticleGoogle Scholar
- Kiani K (2015) Axial buckling analysis of a slender current-carrying nanowire acted upon by a magnetic field using the surface energy approach. J Phys D Appl Phys 48:245302View ArticleGoogle Scholar
- Ghorbanpour-Arani AH, Rastgoo A, Sharafi MM, Kolahchi R, Arani AG (2016) Nonlocal viscoelasticity based vibration of double viscoelastic piezoelectric nanobeam systems. Meccanica 51:25–40View ArticleGoogle Scholar
- Najar F, El-Borgi S, Reddy JN, Mrabet K (2015) Nonlinear nonlocal analysis of electrostatic nanoactuators. Compos Struct 120:117–128View ArticleGoogle Scholar
- Chen Q, Huang L, Lai Y-C, Grebogi C, Dietz D (2010) Extensively chaotic motion in electrostatically driven nanowires and applications. Nano Lett 10:406–413View ArticleGoogle Scholar
- Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710View ArticleGoogle Scholar
- Jin L, Mei J, Li L (2015) Nonlinear dynamics of doubly clamped carbon nanotube resonator. RSC Adv 5:7215–7221View ArticleGoogle Scholar