Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Chaotic electron diffusion through stochastic webs enhances current flow in superlattices

Nature volume 428, pages 726–730 (2004)Cite this article

Abstract

Understanding how complex systems respond to change is of fundamental importance in the natural sciences. There is particular interest in systems whose classical newtonian motion becomes chaotic1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22 as an applied perturbation grows. The transition to chaos usually occurs by the gradual destruction of stable orbits in parameter space, in accordance with the Kolmogorov–Arnold–Moser (KAM) theorem1,2,3,6,7,8,9—a cornerstone of nonlinear dynamics that explains, for example, gaps in the asteroid belt2. By contrast, ‘non-KAM’ chaos switches on and off abruptly at critical values of the perturbation frequency6,7,8,9. This type of dynamics has wide-ranging implications in the theory of plasma physics10, tokamak fusion11, turbulence6,7,12, ion traps13, and quasicrystals6,8. Here we realize non-KAM chaos experimentally by exploiting the quantum properties of electrons in the periodic potential of a semiconductor superlattice22,23,24,25,26,27 with an applied voltage and magnetic field. The onset of chaos at discrete voltages is observed as a large increase in the current flow due to the creation of unbound electron orbits, which propagate through intricate web patterns6,7,8,9,10,12,13,14,15,16 in phase space. Non-KAM chaos therefore provides a mechanism for controlling the electrical conductivity of a condensed matter device: its extreme sensitivity could find applications in quantum electronics and photonics.

This is a preview of subscription content, access via your institution

Relevant articles

Open Access articles citing this article.

Access options

Rent or buy this article

Get just this article for as long as you need it

$39.95

Learn more

Prices may be subject to local taxes which are calculated during checkout

Figure 1: Properties of the SL.
Figure 2: Resonant enhancement of current.
Figure 3: Electron trajectories and wavefunctions.
Figure 4: Classical and quantum phase space structure.

References

  1. Lichtenberg, A. J. & Lieberman, M. A. Regular and Chaotic Dynamics (Springer, New York, 1992)

    Book  Google Scholar 

  2. Gutzwiller, M. C. Chaos in Classical and Quantum Mechanics (Springer, New York, 1990)

    Book  Google Scholar 

  3. Stöckmann, H.-J. Quantum Chaos: an Introduction (Cambridge Univ. Press, Cambridge, 1999)

    Book  Google Scholar 

  4. Berry, M. V. Quantum chaology. Proc. R. Soc. Lond. A 413, 183–198 (1987)

    Article  ADS  Google Scholar 

  5. Heller, E. J. Bound-state eigenfunctions of classically chaotic Hamiltonian systems—scars of periodic orbits. Phys. Rev. Lett. 53, 1515–1518 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  6. Chernikov, A. A., Sagdeev, R. Z., Usikov, D. A., Yu Zakharov, M. & Zaslavsky, G. M. Minimal chaos and stochastic webs. Nature 326, 559–563 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  7. Shlesinger, M. F., Zaslavsky, G. M. & Klafter, J. Strange kinetics. Nature 363, 31–37 (1993)

    Article  ADS  CAS  Google Scholar 

  8. Zaslavsky, G. M., Sagdeev, R. Z., Usikov, D. A. & Chernikov, A. A. Weak Chaos and Quasi-Regular Patterns (Cambridge Univ. Press, Cambridge, 1991)

    Book  Google Scholar 

  9. Kamenev, D. I. & Berman, G. P. Quantum Chaos: a Harmonic Oscillator in Monochromatic Wave (Rinton, Princeton, New Jersey, 2000)

    MATH  Google Scholar 

  10. Chia, P.-K., Schmitz, L. & Conn, R. W. Stochastic ion behaviour in subharmonic and superharmonic electrostatic waves. Phys. Plasmas 3, 1545–1568 (1996)

    Article  ADS  CAS  Google Scholar 

  11. Karney, C. F. F. & Bers, A. Stochastic ion heating by a perpendicularly propagating electrostatic wave. Phys. Rev. Lett. 39, 550–554 (1977)

    Article  ADS  Google Scholar 

  12. Beloshapkin, V. V. et al. Chaotic streamlines in pre-turbulent states. Nature 337, 133–137 (1989)

    Article  ADS  Google Scholar 

  13. Gardiner, S. A., Cirac, J. I. & Zoller, P. Quantum chaos in an ion trap: the delta-kicked harmonic oscillator. Phys. Rev. Lett. 79, 4790–4793 (1997)

    Article  ADS  CAS  Google Scholar 

  14. Demikhovskii, V. Ya., Kamenev, D. I. & Luna-Acosta, G. A. Quantum weak chaos in a degenerate system. Phys. Rev. E 59, 294–302 (1999)

    Article  ADS  CAS  Google Scholar 

  15. Fromhold, T. M. et al. Effects of stochastic webs on chaotic electron transport in semiconductor superlattices. Phys. Rev. Lett. 87, 046803 (2001)

    Article  ADS  CAS  Google Scholar 

  16. Demikhovskii, V. Ya., Izrailev, F. M. & Malyshev, A. I. Manifestation of Arnol'd diffusion in quantum systems. Phys. Rev. Lett. 88, 154101 (2002)

    Article  ADS  Google Scholar 

  17. Jensen, R. V. Quantum chaos. Nature 355, 311–318 (1992)

    Article  ADS  CAS  Google Scholar 

  18. Wilkinson, P. B. et al. Observation of ‘scarred’ wavefunctions in a quantum well with chaotic electron dynamics. Nature 380, 608–610 (1996)

    Article  ADS  CAS  Google Scholar 

  19. Fromhold, T. M. et al. Tunneling spectroscopy of mixed stable-chaotic electron dynamics in a quantum well. Phys. Rev. B 65, 155312 (2002)

    Article  ADS  Google Scholar 

  20. Hensinger, W. K. et al. Dynamical tunnelling of ultra-cold atoms. Nature 412, 52–55 (2001)

    Article  ADS  CAS  Google Scholar 

  21. Steck, D. A., Oskay, W. H. & Raizen, M. G. Observation of chaos-assisted tunneling between islands of stability. Science 293, 274–278 (2001)

    Article  ADS  CAS  Google Scholar 

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

    Article  ADS  Google Scholar 

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

    Article  CAS  Google Scholar 

  24. Tsu, R. & Döhler, G. Hopping conduction in a “superlattice”. Phys. Rev. B 12, 680–686 (1975)

    Article  ADS  Google Scholar 

  25. Patanè, A. et al. Tailoring the electronic properties of GaAs/AlAs superlattices by InAs layer insertions. Appl. Phys. Lett. 81, 661–663 (2002)

    Article  ADS  Google Scholar 

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

    Article  ADS  CAS  Google Scholar 

  27. Canali, L., Lazzarino, M., Sorba, L. & Beltram, F. Stark-cyclotron resonance in a semiconductor superlattice. Phys. Rev. Lett. 76, 3618–3621 (1996)

    Article  ADS  CAS  Google Scholar 

  28. Penrose, R. The role of aesthetics in pure and applied mathematical research. Bull. Inst. Math. Appl. 10, 266–271 (1974)

    Google Scholar 

  29. Scott, R. G., Bujkiewicz, S., Fromhold, T. M., Wilkinson, P. B. & Sheard, F. W. Effects of chaotic energy-band transport on the quantized states of ultracold sodium atoms in an optical lattice with a tilted harmonic trap. Phys. Rev. A. 66, 023407 (2002)

    Article  ADS  Google Scholar 

  30. Wilkinson, P. B. & Fromhold, T. M. Chaotic ray dynamics in slowly varying two-dimensional photonic crystals. Opt. Lett. 28, 1034–1036 (2003)

    Article  ADS  Google Scholar 

Download references

Acknowledgements

This work was supported by the EPSRC, The Royal Society, CONACyT, and INSA. S.B. is funded by an EU Marie Curie Fellowship and is grateful for support from the Institute of Physics, Wrocław University of Technology, Poland. We are grateful to R. Airey for processing our samples.

Author information

Author notes

  1. A. A. Krokhin

    Present address: Center for Nonlinear Science, University of North Texas, PO Box 311427, Denton, Texas, 76203, USA

  2. N. S. Sankeshwar

    Present address: Department of Physics, Karnatak University, Dharwad, 580 003, India

Authors and Affiliations

  1. School of Physics and Astronomy, University of Nottingham, NG7 2RD, Nottingham, UK

    T. M. Fromhold, A. Patanè, S. Bujkiewicz, P. B. Wilkinson, D. Fowler, D. Sherwood, S. P. Stapleton, L. Eaves, M. Henini & F. W. Sheard

  2. Instituto de Fisica, Universidad Autonoma de Puebla, Puebla, 72570, Mexico

    A. A. Krokhin

Authors
  1. T. M. Fromhold
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. Patanè
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Bujkiewicz
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. P. B. Wilkinson
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. D. Fowler
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. D. Sherwood
    View author publications

    You can also search for this author in PubMed Google Scholar

  7. S. P. Stapleton
    View author publications

    You can also search for this author in PubMed Google Scholar

  8. A. A. Krokhin
    View author publications

    You can also search for this author in PubMed Google Scholar

  9. L. Eaves
    View author publications

    You can also search for this author in PubMed Google Scholar

  10. M. Henini
    View author publications

    You can also search for this author in PubMed Google Scholar

  11. N. S. Sankeshwar
    View author publications

    You can also search for this author in PubMed Google Scholar

  12. F. W. Sheard
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. M. Fromhold.

Ethics declarations

Competing interests

The authors declare that they have no competing financial interests.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Fromhold, T., Patanè, A., Bujkiewicz, S. et al. Chaotic electron diffusion through stochastic webs enhances current flow in superlattices. Nature 428, 726–730 (2004). https://doi.org/10.1038/nature02445

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nature02445

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing