Abstract
Silicon crystal puller (SCP) is key equipment in silicon wafer manufacture, which is, in turn, the base material for the most currently used integrated circuit chips. With the development of the techniques, the demand for longer mono-silicon crystal rod with larger diameter is continuously increasing in order to reduce the manufacture time and the price of the wafer. This demand calls for larger SCP with an increasing height, though it causes serious swing phenomenon of the crystal seed. The strong swing of the seed increases the possibility of defects in the mono-silicon rod and the risk of mono-silicon growth failure. The main aim of this paper is to analyze the nonlinear dynamics in flexible shaft rotating–lifting (FSRL) system of the SCP. A mathematical model for the swing motion of the FSRL system is derived. The influence of relevant parameters, such as system damping, excitation amplitude, and rotation speed, on the stability and the responses of the system is analyzed. The stability of the equilibrium, bifurcation, and chaotic motion is demonstrated, which have been observed in practical situations. Melnikov method is used to derive the possible parameter region which leads to chaotic motion. Three routes to chaos are identified in the FSRL system, including period doubling, symmetry-breaking bifurcation, and crisis. The work in this paper analyzes and explains the complex dynamics in FSRL system of the SCP, which will be helpful for the designers in the designing process in order to avoid the swing phenomenon in the SCP.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Lan, C.W.: Recent progress of crystal growth modeling and growth control. Chem. Eng. Sci. 59, 1437–1457 (2004)
Stelian, C., Nehari, A., Lasloudji, I., Lebbou, K., Dumortier, M., Cabane, H., Duffar, T.: Modeling the effect of crystal and crucible rotation on the interface shape in Czochralski growth of piezoelectric langatate crystals. J. Cryst. Growth 475, 368–377 (2017)
Zhang, J., Ren, J.C., Liu, D.: Effect of crucible rotation and crystal rotation on the oxygen distribution at the solid–liquid interface during the growth of Czochralski monocrystalline silicon under superconducting horizontal magnetic field. Results Phys. 13, 102127 (2019)
Li, Y.R., Zhang, L., Zhang, L., Wu, C.M.: Experimental study on complex flow of a binary mixture in Czochralski configurations with different aspect ratios and rotation rates. Int. J. Heat Mass Transf. 117, 835–845 (2018)
Yuan, D.N., Ma, J.P., Yang, R., Fu, W.P., Liu, H.Z.: Dynamic simulation of oscillation phenomenon of single-crystal growth furnace lifting system based on double pendulum model. J. Xi’an Univ. Technol. 24, 177–181 (2008)
Yuan, D.N., Shi, J.W.: Vibration model and simulation for pulling system of single-crystal growth furnace. J. Synth. Cryst. 39, 545–551 (2010)
Depetri, G.I., Pereira, F., Marin, B., Baptista, M.S., Sartorelli, J.C.: Dynamics of a parametrically excited simple pendulum. Chaos 28, 033103 (2018)
Zhen, B., Xu, J., Song, Z.: Influence of nonlinearity on transition curves in a parametric pendulum system. Commun. Nonlinear Sci. Numer. Simul. 42, 275–284 (2017)
Andreeva, T., Alevras, P., Naess, A., Yurchenko, D.: Dynamics of a parametric rotating pendulum under a realistic wave profile. Int. J. Dyn. Control 4(2), 233–238 (2016)
Han, N., Cao, Q.: A parametrically excited pendulum with irrational nonlinearity. Int. J. Non-Linear Mech. 88, 122–134 (2017)
Alevras, P., Brown, I., Yurchenko, D.: Experimental investigation of a rotating parametric pendulum. Nonlinear Dyn. 81, 201–213 (2015)
Belyakov, A.O., Seyranian, A.P.: Homoclinic, subharmonic, and superharmonic bifurcations for a pendulum with periodically varying length. Nonlinear Dyn. 77, 1617–1627 (2014)
Soto, I., Campa, R.: Modelling and control of a spherical inverted pendulum on a five-bar mechanism. Int. J. Adv. Robot. Syst. 12, 1–16 (2015)
Reddy, B.S., Ghosal, A.: Chaotic motion in a flexible rotating beam and synchronization. J. Comput. Nonlinear Dyn. 12(4), 044505 (2017)
Yuan, G., Hunt, B., Grebogi, C., Ott, E., Yorke, J.A., Kostelich, E.: Design and control of shipboard cranes. In: Proceedings of the ASME Design Engineering Technical Conference VIB. ASEM, p. 4095 (1997)
Abdel-Rahman, E.M., Nayfeh, A.H.: Pendulation reduction in boom cranes using cable length manipulation. Nonlinear Dyn. 27, 255–269 (2002)
Pan, J.N., Qin, W.Y., Deng, W.Z., Zhou, H.L.: Harvesting base vibration energy by a piezoelectric inverted beam with pendulum. Chin. Phys. B 28(1), 017701 (2019)
Wiggins, S.S.: Chaotic dynamics of a whirling pendulum. Phys. D Nonlinear Phenom. 31, 190–211 (1988)
Chen, L.J., Li, J.B.: Chaotic behavior and subharmonic bifurcations for a rotating pendulum equation. Int. J. Bifurc. Chaos 14, 3477–3488 (2004)
Melnikov, V.K.: On the stability of the center for time periodic perturbations. Trans. Mosc. Math. Soc. 12, 3–52 (1963)
Bishop, S.R., Sofroniou, A., Shi, P.: Symmetry-breaking in the response of the parametrically excited pendulum model. Chaos Solitons Fractals 25, 257–264 (2005)
Grebogi, C., Ott, E., Yorke, J.A.: Crises, sudden changes in chaotic attractors, and transient chaos. Phys. D Nonlinear Phenom. 1983(7), 181–200 (1983)
Acknowledgements
The project is supported by the Key Program of National Natural Science Foundation of China (Grant No. 61533014).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ren, HP., Zhou, ZX. & Grebogi, C. Nonlinear dynamics in the flexible shaft rotating–lifting system of silicon crystal puller using Czochralski method. Nonlinear Dyn 102, 771–784 (2020). https://doi.org/10.1007/s11071-020-05592-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-05592-9