Self-sustained oscillations and chaos in space charge limited currents

Yu. N. Gartstein; P. S. Ramesh

doi:10.1103/PhysRevE.60.1069

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Self-sustained oscillations and chaos in space charge limited currents

Yu. N. Gartstein and P. S. Ramesh

Xerox Corporation, Wilson Center for Research and Technology, 147-59B, 800 Phillips Road, Webster, New York 14580

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Phys. Rev. E 60, 1069 – Published 1 July, 1999

DOI: https://doi.org/10.1103/PhysRevE.60.1069

Abstract

In kinetic simulations of a flow of charged particles between two parallel plate electrodes, it is found that chaotic responses in space charge limited currents can be induced by a periodically varying applied voltage.

References (20)

I. Langmuir and K. T. Compton, Rev. Mod. Phys. 3, 191 (1931).
J. Millman, Vacuum-tube and Semiconductor Electronics (McGraw-Hill, New York, 1958).
See, e.g., C. E. Fay, A. L. Samuel, and W. Shockley, Bell Syst. Tech. J. 17, 49 (1938), and references therein, and an extensive bibliography in Ref. [7].
E. W. B. Gill, Philos. Mag. 49, 993 (1925).
L. Tonks, Phys. Rev. 30, 501 (1927).
C. K. Birdsall and W. B. Bridges, J. Appl. Phys. 32, 2611 (1961); ibid.W. B. Bridges and C. K. Birdsall, 34, 29461 (1963).
C. K. Birdsall and W. B. Bridges, Electron Dynamics of Diode Regions (Academic, New York, 1966).
Yu. N. Gartstein and P. S. Ramesh, J. Appl. Phys. 83, 2958 (1998); ibid.84, 1158 (1998).
If $φ_{0}$ is used as an appropriate voltage scale, then the magnitude of space charge effects is related to the parameter ${P = j L}^{2} {(q m / 2)}^{1 / 2} / ε φ_{0}^{3 / 2}$ where L is the distance between the electrodes, m the particle mass, and ε the permeability of the vacuum (or another medium between electrodes). In simulations for Fig. 11, $P = 5.98.$ From the Langmuir-Child law, the voltage for the onset of SLC on curve (1) $V_{m} / φ_{0} {= (9 P / 4)}^{2 / 3} = 5.67,$ as indeed is found in the simulations.
E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981).
L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon, Oxford, 1980).
See, e.g., S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wiley, Reading, MA, 1995).
C. Hayashi, Nonlinear Oscillations in Physical Systems (Princeton University Press, Princeton, NJ, 1985).
As was found in our simulations [8], the oscillations of the charged cloud in the gap persist even at such negative voltages where no net current flows through the system. Only at some negative enough voltage, the oscillations cease to exist and the dynamic simulation picture would correspond to the one expected from the steady-state classic analysis. This transition voltage point signifies a Hopf limit cycle bifurcation [12].
The finite thickness of limit cycles in Fig. 22 is due to finite time steps in our simulations. A better resolution is achievable when the time steps are decreased.
F. C. Hoppensteadt, Analysis and Simulation of Chaotic Systems (Springer, New York, 1993).
J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983).
A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer, New York, 1992).
G. M. Zaslavskii and R. Z. Sagdeev, Introduction to Nonlinear Physics (Nauka, Moscow, 1988).
P. Bergé, Y. Pomeau, and C. Vidal, Order Within Chaos (Wiley, New York, 1984).

[c1] I. Langmuir and K. T. Compton, Rev. Mod. Phys. 3, 191 (1931).

[c2] J. Millman, Vacuum-tube and Semiconductor Electronics (McGraw-Hill, New York, 1958).

[c3] See, e.g., C. E. Fay, A. L. Samuel, and W. Shockley, Bell Syst. Tech. J. 17, 49 (1938), and references therein, and an extensive bibliography in Ref. [7].

[c4] E. W. B. Gill, Philos. Mag. 49, 993 (1925).

[c5] L. Tonks, Phys. Rev. 30, 501 (1927).

[c6] C. K. Birdsall and W. B. Bridges, J. Appl. Phys. 32, 2611 (1961); ibid.W. B. Bridges and C. K. Birdsall, 34, 29461 (1963).

[c7] C. K. Birdsall and W. B. Bridges, Electron Dynamics of Diode Regions (Academic, New York, 1966).

[c8] Yu. N. Gartstein and P. S. Ramesh, J. Appl. Phys. 83, 2958 (1998); ibid.84, 1158 (1998).

[c9] If $φ_{0}$ is used as an appropriate voltage scale, then the magnitude of space charge effects is related to the parameter ${P = j L}^{2} {(q m / 2)}^{1 / 2} / ε φ_{0}^{3 / 2}$ where L is the distance between the electrodes, m the particle mass, and ε the permeability of the vacuum (or another medium between electrodes). In simulations for Fig. 11, $P = 5.98.$ From the Langmuir-Child law, the voltage for the onset of SLC on curve (1) $V_{m} / φ_{0} {= (9 P / 4)}^{2 / 3} = 5.67,$ as indeed is found in the simulations.

[c10] E. M. Lifshitz and L. P. Pitaevskii, Physical Kinetics (Pergamon, Oxford, 1981).

[c11] L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1 (Pergamon, Oxford, 1980).

[c12] See, e.g., S. H. Strogatz, Nonlinear Dynamics and Chaos (Addison-Wiley, Reading, MA, 1995).

[c13] C. Hayashi, Nonlinear Oscillations in Physical Systems (Princeton University Press, Princeton, NJ, 1985).

[c14] As was found in our simulations [8], the oscillations of the charged cloud in the gap persist even at such negative voltages where no net current flows through the system. Only at some negative enough voltage, the oscillations cease to exist and the dynamic simulation picture would correspond to the one expected from the steady-state classic analysis. This transition voltage point signifies a Hopf limit cycle bifurcation [12].

[c15] The finite thickness of limit cycles in Fig. 22 is due to finite time steps in our simulations. A better resolution is achievable when the time steps are decreased.

[c16] F. C. Hoppensteadt, Analysis and Simulation of Chaotic Systems (Springer, New York, 1993).

[c17] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Springer, New York, 1983).

[c18] A. J. Lichtenberg and M. A. Lieberman, Regular and Chaotic Dynamics (Springer, New York, 1992).

[c19] G. M. Zaslavskii and R. Z. Sagdeev, Introduction to Nonlinear Physics (Nauka, Moscow, 1988).

[c20] P. Bergé, Y. Pomeau, and C. Vidal, Order Within Chaos (Wiley, New York, 1984).

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Self-sustained oscillations and chaos in space charge limited currents

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