Skip to main content

Part of the book series: Lecture Notes in Networks and Systems ((LNNS,volume 6))

Included in the following conference series:

  • 439 Accesses

Abstract

Multistability is a common phenomenon in nonlinear dynamical systems. It can also arise in nanoscale systems. This review paper presents three such systems exhibiting multistability: an electrically driven nanowire, a semiconductor superlattice, and a coupled system of ferromagnet and topological insulator. Potential applications are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. C. Grebogi, S.W. McDonald, E. Ott, J.A. Yorke, Final state sensitivity: an obstruction to predictability. Phys. Lett. A 99, 415–418 (1983)

    Article  MathSciNet  Google Scholar 

  2. S.W. McDonald, C. Grebogi, E. Ott, J.A. Yorke, Fractal basin boundaries. Physica D 17, 125–153 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  3. Y.-C. Lai, C. Grebogi, Intermingled basins and two-state on-off intermittency. Phys. Rev. E 52, R3313–R3316 (1995)

    Article  Google Scholar 

  4. Y.-C. Lai, C. Grebogi, J.A. Yorke, S. Venkataramani, Riddling bifurcation in chaotic dynamical systems. Phys. Rev. Lett. 77, 55–58 (1996)

    Article  Google Scholar 

  5. Y.-C. Lai, C. Grebogi, Noise-induced riddling in chaotic dynamical systems. Phys. Rev. Lett. 77, 5047–5050 (1996)

    Article  Google Scholar 

  6. U. Feudel, C. Grebogi, Multistability and the control of complexity. Chaos 7, 597–604 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  7. U. Feudel, C. Grebogi, Why are chaotic attractors rare in multistable systems? Phys. Rev. Lett. 91, 134102 (2003)

    Article  Google Scholar 

  8. Y.-C. Lai, T. Tél, Transient Chaos-Complex Dynamics on Finite Time Scales (Springer, New York, 2011)

    MATH  Google Scholar 

  9. X. Ni, L. Ying, Y.-C. Lai, Y. Do, C. Grebogi, Complex dynamics in nanosystems. Phys. Rev. E 87, 052911 (2013)

    Article  Google Scholar 

  10. A.N. Pisarchik, U. Feudel, Control of multistability. Phys. Rep. 540, 167–218 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. G.-L. Wang, H.-Y. Xu, Y.-C. Lai, Nonlinear dynamics induced anomalous hall effect in topological insulators. Sci. Rep. 6, 19803 (2016)

    Article  Google Scholar 

  12. F. Prengel, A. Wacker, E. Schöll, Simple model for multistability and domain formation in semiconductor superlattices. Phys. Rev. B 50, 1705–1712 (1994)

    Article  Google Scholar 

  13. N.G. Sun, G.P. Tsironis, Multistability of conductance in doped semiconductor superlattices. Phys. Rev. B 51, 11221–11224 (1995)

    Article  Google Scholar 

  14. A. Amann, A. Wacker, L.L. Bonilla, E. Schöll, Dynamic scenarios of multistable switching in semiconductor superlattices. Phys. Rev. E 63, 066207 (2001)

    Article  Google Scholar 

  15. Q. Chen, L. Huang, Y.-C. Lai, C. Grebogi, D. Dietz, Extensively chaotic motion in electrostatically driven nanowires and applications. Nano lett. 10, 406–413 (2010)

    Article  Google Scholar 

  16. Z. Kovács, K.G. Szabó, T. Tél, Controlling chaos on fractal basin boundaries, in Nonlinearity and Chaos in Engineering Dynamics, ed by J.M.T. Thompson, S.R. Bishop, (Wiley, Chichester (England), 1994) pp. 155–162

    Google Scholar 

  17. Y.-C. Lai, Driving trajectories to a desirable attractor by using small control. Phys. Lett. A 221, 375–383 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. A.N. Pisarchik, Controlling the multistability of nonlinear systems with coexisting attractors. Phys. Rev. E 64, 046203 (2001)

    Article  Google Scholar 

  19. M. Dequesnes, S.V. Rotkin, N.R. Aluru, Calculation of pull-in voltages for carbon-nanotube-based nanoelectromechanical switches. Nanotechnology 13, 120–131 (2002)

    Article  Google Scholar 

  20. S.N. Cha et al., Fabrication of a nanoelectromechanical switch using a suspended carbon nanotube. Appl. Phys. Lett. 86, 083105 (2005)

    Article  Google Scholar 

  21. J.B. Ma, L. Jiang, S.F. Asokanthan, Influence of surface effects on the pull-in instability of nems electrostatic switches. Nanotechnology 21, 505708 (2010)

    Article  Google Scholar 

  22. Y.T. Yang, C. Callegari, X.L. Feng, M.L. Roukes, Surface adsorbate fluctuations and noise in nanoelectromechanical systems. Nano Lett. 11, 1753–1759 (2011)

    Article  Google Scholar 

  23. W. Conley, A. Raman, C. Krousgrill, S. Mohammadi, Nonlinear and nonplanar dynamics of suspended nanotube and nanowire resonators. Nano Lett. 8, 1590–1595 (2008)

    Article  Google Scholar 

  24. J.R. Dormand, P.J. Prince, A family of embedded runge-kutta formula. J. Comput. Appl. Math. 6, 19–26 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  25. L. Esaki, R. Tsu, Superlattice and negative differential conductivity in semiconductors. IBM J. Res. Dev. 14, 61–65 (1970)

    Article  Google Scholar 

  26. H.T. Grahn, Semiconductor Supperlattices, Growth and Electronic Properties (World Scientific, Singapore, 1995)

    Book  Google Scholar 

  27. Y. Zhang, J. Kastrup, R. Klann, K.H. Ploog, H.T. Grahn, Synchronization and chaos induced by resonant tunneling in gaas/alas superlattices. Phys. Rev. Lett. 77, 3001–3004 (1996)

    Article  Google Scholar 

  28. E. Schöll, Nonlinear Spatiotemporal Dynamics and Chaos in Semiconductors (Cambridge University Press, Cambridge, UK, 2001)

    Book  Google Scholar 

  29. A. Wacker, Semiconductor superlattices: a model system for nonlinear transport. Phys. Rep. 357, 1–111 (2002)

    Article  MATH  Google Scholar 

  30. X.L. Lei, C.S. Ting, Theory of nonlinear electron transport for solids in a strong electric field. Phys. Rev. B 30, 4809–4812 (1984)

    Article  Google Scholar 

  31. X.L. Lei, C.S. Ting, Green’s-function approach to nonlinear electronic transport for an electron-impurity-phonon system in a strong electric field. Phys. Rev. B 32, 1112–1132 (1985)

    Article  Google Scholar 

  32. X.L. Lei, N.J.M. Horing, H.L. Cui, Theory of negative differential conductivity in a superlattice miniband. Phys. Rev. Lett. 66, 3277–3280 (1991)

    Article  Google Scholar 

  33. X.L. Lei, High-frequency differential mobility in vertical transport of a confined superlattice. J. Phys. Cond. Matt. 6, 10043 (1994). http://stacks.iop.org/0953-8984/6/i=46/a=021

  34. X.L. Lei, Balance equations for electron transport in a general energy band. J. Phys. Cond. Matt. 6, 9189 (1994)

    Article  Google Scholar 

  35. A.A. Ignatov, E.P. Dodin, V.I. Shashkin, Transient response theory for semiconductor superlattices: connection with bloch oscillations. Mod. Phys. Lett. B 5, 1087 (1991)

    Article  Google Scholar 

  36. R.R. Gerhardts, Effect of elastic scattering on miniband transport in semiconductor superlattices. Phys. Rev. B 48, 9178–9181 (1993)

    Article  Google Scholar 

  37. M. Büttiker, H. Thomas, Current instability and domain propagation due to bragg scattering. Phys. Rev. Lett. 38, 78–80 (1977)

    Article  Google Scholar 

  38. M. Büttiker, H. Thomas, Hydrodynamic modes, soft modes and fluctuation spectra near the threshold of a current instability. Z. Phys. B 33, 275–287 (1979)

    Article  MathSciNet  Google Scholar 

  39. M. Büttiker, H. Thomas, Bifurcation and stability of dynamical structures at a current instability. Z. Phys. B 34, 301–311 (1979)

    Article  Google Scholar 

  40. X.L. Lei, Balance equations for hot electron transport in an arbitrary energy band. Phys. Status Solidi B 170, 519–529 (1992)

    Article  Google Scholar 

  41. X.L. Lei, Investigation of the buttiker-thomas momentum balance equation from the heisenberg equation of motion for bloch electrons. J. Phys. Condens. Matt. 7, L429 (1995)

    Article  Google Scholar 

  42. K.N. Alekseev, E.H. Cannon, J.C. McKinney, F.V. Kusmartsev, D.K. Campbell, Spontaneous dc current generation in a resistively shunted semiconductor superlattice driven by a terahertz field. Phys. Rev. Lett. 80, 2669–2672 (1998)

    Article  Google Scholar 

  43. K.N. Alekseev, G.P. Berman, D.K. Campbell, E.H. Cannon, M.C. Cargo, Dissipative chaos in semiconductor superlattices. Phys. Rev. B 54, 10625–10636 (1996)

    Article  Google Scholar 

  44. O.M. Bulashenko, M.J. García, L.L. Bonilla, Chaotic dynamics of electric-field domains in periodically driven superlattices. Phys. Rev. B 53, 10008–10018 (1996)

    Article  Google Scholar 

  45. M. Patra, G. Schwarz, E. Schöll, Bifurcation analysis of stationary and oscillating domains in semiconductor superlattices with doping fluctuations. Phys. Rev. B 57, 1824–1833 (1998)

    Article  Google Scholar 

  46. K.J. Luo, H.T. Grahn, K.H. Ploog, L.L. Bonilla, Explosive bifurcation to chaos in weakly coupled semiconductor superlattices. Phys. Rev. Lett. 81, 1290–1293 (1998)

    Article  Google Scholar 

  47. K.N. Alekseev, E.H. Cannon, J.C. McKinney, F.V. Kusmartsev, D.K. Campbell, Symmetry-breaking and chaos in electron transport in semiconductor superlattices. Phys. D 113, 129–133 (1998)

    Article  MATH  Google Scholar 

  48. J.C. Cao, X.L. Lei, Synchronization and chaos in miniband semiconductor superlattices. Phys. Rev. B 60, 1871–1878 (1999)

    Article  Google Scholar 

  49. J.C. Cao, H.C. Liu, X.L. Lei, Chaotic dynamics in quantum-dot miniband superlattices. Phys. Rev. B 61, 5546–5555 (2000)

    Article  Google Scholar 

  50. D. Sánchez, G. Platero, L.L. Bonilla, Quasiperiodic current and strange attractors in ac-driven superlattices. Phys. Rev. B 63, 201306 (2001)

    Article  Google Scholar 

  51. K.N. Alekseev, F.V. Kusmartsev, Pendulum limit, chaos and phase-locking in the dynamics of ac-driven semiconductor superlattices. Phys. Lett. A 305, 281–288 (2002)

    Article  MATH  Google Scholar 

  52. K.N. Alekseev, G.P. Bermana, D.K. Campbell, Dynamical instabilities and deterministic chaos in ballistic electron motion in semiconductor superlattices. Phys. Lett. A 193, 54–60 (1994)

    Article  Google Scholar 

  53. A. Amann, J. Schlesner, A. Wacker, E. Schöll, Chaotic front dynamics in semiconductor superlattices. Phys. Rev. B 65, 193313 (2002)

    Article  Google Scholar 

  54. L.L. Bonilla, H.T. Grahn, Non-linear dynamics of semiconductor superlattices. Rep. Prog. Phys. 68, 577–683 (2005)

    Article  Google Scholar 

  55. J. Galán, L.L. Bonilla, M. Moscoso, Bifurcation behavior of a superlattice model. SIAM J. Appl. Math. 60, 2029–2057 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  56. M.T. Greenaway, A.G. Balanov, E. Schöll, T.M. Fromhold, Controlling and enhancing terahertz collective electron dynamics in superlattices by chaos-assisted miniband transport. Phys. Rev. B 80, 205318 (2009)

    Article  Google Scholar 

  57. S.P. Stapleton et al., Use of stochastic web patterns to control electron transport in semiconductor superlattices. Phys. D 199, 166–172 (2004)

    Article  MATH  Google Scholar 

  58. C. Wang, J.-C. Cao, Current oscillation and chaotic dynamics in superlattices driven by crossed electric and magnetic fields. Chaos 15, 013111 (2005)

    Article  Google Scholar 

  59. A.G. Balanov, D. Fowler, A. Patanè, L. Eaves, T.M. Fromhold, Bifurcations and chaos in semiconductor superlattices with a tilted magnetic field. Phys. Rev. E 77, 026209 (2008)

    Article  Google Scholar 

  60. K.J. Luo, H.T. Grahn, S.W. Teitsworth, K.H. Ploog, Influence of higher harmonics on poincaré maps derived from current self-oscillations in a semiconductor superlattice. Phys. Rev. B 58, 12613–12616 (1998)

    Article  Google Scholar 

  61. Y.-H. Zhang, R. Klann, H.T. Grahn, K.H. Ploog, Transition between synchronization and chaos in doped gaas/alas superlattices. Superlatt. Microstruc. 21, 565–568 (1997)

    Article  Google Scholar 

  62. T.M. Fromhold et al., Chaotic electron diffusion through stochastic webs enhances current flow in superlattices. Nature (London) 428, 726–730 (2004)

    Article  Google Scholar 

  63. A.E. Hramov et al., Subterahertz chaos generation by coupling a superlattice to a linear resonator. Phys. Rev. Lett. 112, 116603 (2014)

    Article  Google Scholar 

  64. T. Hyart, A.V. Shorokhov, K.N. Alekseev, Theory of parametric amplification in superlattices. Phys. Rev. Lett. 98, 220404 (2007)

    Article  Google Scholar 

  65. T. Hyart, K.N. Alekseev, E.V. Thuneberg, Bloch gain in dc-ac-driven semiconductor superlattices in the absence of electric domains. Phys. Rev. B 77, 165330 (2008)

    Article  Google Scholar 

  66. T. Hyart, N.V. Alexeeva, J. Mattas, K.N. Alekseev, Terahertz bloch oscillator with a modulated bias. Phys. Rev. Lett. 102, 140405 (2009)

    Article  Google Scholar 

  67. T. Hyart, J. Mattas, K.N. Alekseev, Model of the influence of an external magnetic field on the gain of terahertz radiation from semiconductor superlattices. Phys. Rev. Lett. 103, 117401 (2009)

    Article  Google Scholar 

  68. T. Hyart, Tunable Superlattice Amplifiers Based on Dynamics of Miniband Electrons in Electric and Magnetic Fields (Ph.D. Dissertation) (University of Oulu, Finland, 2009)

    Google Scholar 

  69. P.H. Siegel, Terahertz technology. IEEE Trans. Microwave Theory Tech. 50, 910–928 (2002)

    Article  Google Scholar 

  70. B. Ferguson, X.-C. Zhang, Materials for terahertz science and technology. Nature Mater. 1, 26–33 (2002)

    Article  Google Scholar 

  71. T.W. Crowe, W.L. Bishop, D.W. Porterfield, J.L. Hesler, R.M. Weikle, Opening the terahertz window with integrated diode circuits. IEEE J. Solid-State Cir. 40, 2104–2110 (2005)

    Article  Google Scholar 

  72. M. Tonouchi, Cutting-edge terahertz technology. Nat. Photonics 1, 97–105 (2009)

    Article  Google Scholar 

  73. L. Kocarev, Chaos-based cryptography: a brief overview. IEEE Cir. Sys. Magaz. 1, 6–21 (2001)

    Article  Google Scholar 

  74. T. Stojanovski, L. Kocarev, Chaos-based random number generators-part i: analysis [cryptography]. IEEE Trans. Cir. Sys. I. Funda. Theo. App. 48, 281–288 (2001)

    Google Scholar 

  75. T. Stojanovski, J. Pihl, L. Kocarev, Chaos-based random number generators. part ii: practical realization. IEEE Trans. Cir. Sys. I. Funda. Theo. App. 48, 382–385 (2001)

    Google Scholar 

  76. M. Drutarovský, P. Galajda, Chaos-based true random number generator embedded in a mixed-signal reconfigurable hardware. J. Elec. Eng. 57, 218–225 (2006)

    Google Scholar 

  77. T. Lin, L.O. Chua, A new class of pseudo-random number generator based on chaos in digital filters. Int. J. Cir. Theo. App. 21, 473–480 (2006)

    Article  MATH  Google Scholar 

  78. A. Uchida1 et al., Fast physical random bit generation with chaotic semiconductor lasers. Nat. Photonics 2, 728–732 (2008)

    Google Scholar 

  79. I. Reidler, Y. Aviad, M. Rosenbluh, I. Kanter, Ultrahigh-speed random number generation based on a chaotic semiconductor laser. Phys. Rev. Lett. 103, 024102 (2009)

    Article  Google Scholar 

  80. J.-Z. Zhang et al., A robust random number generator based on differential comparison of chaotic laser signals. Opt. Expr. 20, 7496–7506 (2012)

    Article  Google Scholar 

  81. L. Ying, D.-H. Huang, Y.-C. Lai, Multistability, chaos, and random signal generation in semiconductor superlattices. Phys. Rev. E 93, 062204 (2016)

    Article  Google Scholar 

  82. D. Huang, P.M. Alsing, T. Apostolova, D.A. Cardimona, Coupled energy-drift and force-balance equations for high-field hot-carrier transport. Phys. Rev. B 71, 195205 (2005)

    Article  Google Scholar 

  83. D. Huang, P.M. Alsing, Many-body effects on optical carrier cooling in intrinsic semiconductors at low lattice temperatures. Phys. Rev. B 78, 035206 (2008)

    Article  Google Scholar 

  84. D. Huang, D.A. Cardimona, Nonadiabatic effects in a self-consistent hartree model for electrons under an ac electric field in multiple quantum wells. Phys. Rev. B 67, 245306 (2003)

    Article  Google Scholar 

  85. D. Huang, S.K. Lyo, G. Gumbs, Bloch oscillation, dynamical localization, and optical probing of electron gases in quantum-dot superlattices in high electric fields. Phys. Rev. B 79, 155308 (2009)

    Article  Google Scholar 

  86. C. Grebogi, E. Ott, J.A. Yorke, Crises, sudden changes in chaotic attractors and chaotic transients. Phys. D 7, 181–200 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  87. J.C. Slonczewski, Current-driven excitation of magnetic multilayers. J. Mag. Mag. Mate. 159, L1–L7 (1996)

    Article  Google Scholar 

  88. C. Chappert, A. Fert, F.N. Van Dau, The emergence of spin electronics in data storage. Nat. Mater. 6, 813–823 (2007)

    Article  Google Scholar 

  89. L. Liu, T. Moriyama, D.C. Ralph, R.A. Buhrman, Spin-torque ferromagnetic resonance induced by the spin hall effect. Phys. Rev. Lett. 106, 036601 (2011)

    Article  Google Scholar 

  90. L. Liu, O.J. Lee, T.J. Gudmundsen, D.C. Ralph, R.A. Buhrman, Current-induced switching of perpendicularly magnetized magnetic layers using spin torque from the spin hall effect. Phys. Rev. Lett. 109, 096602 (2012)

    Article  Google Scholar 

  91. L. Liu et al., Spin-torque switching with the giant spin hall effect of tantalum. Science 336, 555–558 (2012)

    Article  Google Scholar 

  92. I.M. Miron et al., Current-driven spin torque induced by the rashba effect in a ferromagnetic metal layer. Nat. Mater. 9, 230–234 (2010)

    Google Scholar 

  93. I.M. Miron et al., Perpendicular switching of a single ferromagnetic layer induced by in-plane current injection. Nature 476, 189–193 (2011)

    Article  Google Scholar 

  94. U.H. Pi et al., Tilting of the spin orientation induced by rashba effect in ferromagnetic metal layer. Appl. Phys. Lett. 97, 162507 (2010)

    Article  Google Scholar 

  95. X. Wang, A. Manchon, Diffusive spin dynamics in ferromagnetic thin films with a rashba interaction. Phys. Rev. Lett. 108, 117201 (2012)

    Article  Google Scholar 

  96. B.A. Bernevig, T.L. Hughes, S.-C. Zhang, Quantum spin hall effect and topological phase transition in hgte quantum wells. Science 314, 1757–1761 (2006)

    Article  Google Scholar 

  97. M.Z. Hasan, C.L. Kane, Colloquium topol. insul. Rev. Mod. Phys. 82, 3045–3067 (2010)

    Article  Google Scholar 

  98. X.-L. Qi, S.-C. Zhang, Topological insulators and superconductors. Rev. Mod. Phys. 83, 1057–1110 (2011)

    Article  Google Scholar 

  99. I. Garate, M. Franz, Inverse spin-galvanic effect in the interface between a topological insulator and a ferromagnet. Phys. Rev. Lett. 104, 146802 (2010)

    Article  Google Scholar 

  100. T. Yokoyama, Current-induced magnetization reversal on the surface of a topological insulator. Phys. Rev. B 84, 113407 (2011)

    Article  Google Scholar 

  101. T. Yokoyama, Y. Tanaka, N. Nagaosa, Anomalous magnetoresistance of a two-dimensional ferromagnet/ferromagnet junction on the surface of a topological insulator. Phys. Rev. B 81, 121401 (2010)

    Article  Google Scholar 

  102. Y.G. Semenov, X. Duan, K.W. Kim, Voltage-driven magnetic bifurcations in nanomagnet-topological insulator heterostructures. Phys. Rev. B 89, 201405 (2014)

    Article  Google Scholar 

Download references

Acknowledgements

This Review is based on Refs. [9, 11, 15, 81]. I thank my former and current students Dr. Q.-F. Chen, Dr. L Huang, Dr. X. Ni, Dr. L. Ying, Mr. G.-L. Wang, and Mr. H.-Y. Xu who contributed to different aspects of the topics discussed here. The efforts were supported by ONR under Grant No. N00014-15-1-2405 and by AFOSR under Grant No. FA9550-15-1-0151.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ying-Cheng Lai .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Lai, YC. (2017). Multistability in Nanosystems. In: In, V., Longhini, P., Palacios, A. (eds) Proceedings of the 4th International Conference on Applications in Nonlinear Dynamics (ICAND 2016). ICAND 2016. Lecture Notes in Networks and Systems, vol 6. Springer, Cham. https://doi.org/10.1007/978-3-319-52621-8_5

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-52621-8_5

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-52620-1

  • Online ISBN: 978-3-319-52621-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics