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Spin-wave alternating periodic–chaotic dynamics
Journal of Magnetism and Magnetic Materials 226}230 (2001) 524}526
Spin-wave alternating periodic}chaotic dynamics F.M. de Aguiar*, S. Rosenblatt, S.M. Rezende, A. Azevedo Departamento de Fisica, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil
Abstract A description is given of pulsed ferromagnetic subsidiary resonance experiments in a YIG sphere at room temperature. Beyond the Suhl instability threshold, evidences were found for the elusive alternating periodic}chaotic sequence. The spin-wave self-oscillations have recently been characterized through one-dimensional return maps. Here it is shown that the observed scenario is also in good qualitative agreement with results from numerical simulations with a standard two-mode microscopic model. 2001 Elsevier Science B.V. All rights reserved. Keywords: Chaos; Spin waves; Ferromagnetic resonance
A remarkable signature of nonlinear dynamics is that many di!erent physical systems enjoy common features as a (system dependent) control parameter is varied. In particular, interacting spin waves, resonantly driven at high microwave power levels in a ferromagnet, exhibit a wide range of universal phenomena, depending on the dynamical parameters of the system [1]. In this paper, intriguing experimental and numerical results are presented on the nonlinear behavior of the excited spin waves. Particular attention is paid to a transition where the so-called mixed-mode oscillations occur, namely, the periodic}chaotic (PC) sequence [2,3]. Discovered 20 years ago in chemical reactions [4], the PC sequence is characterized by regions of chaotic behavior (C) bracketed by periodic states (P and P>) consisting of large-amplitude peaks followed by n"1, 2, 3, 2 small-amplitude undulations. Prior to chaos, each periodic state undergoes a sequence o period-doubling bifurcations. Experimentally, this scenario has been well characterized in only a few dynamical systems [2,5]. The experiments reported here have been carried out in the subsidiary resonance con"guration. A 1 mm-diam YIG sphere was held in the center of a TE rectangular microwave cavity ( f"8.9 GHz, Q"3000, microwave magnetic "eld amplitude h), coupled to a waveguide
* Corresponding author. Tel.: #55-81-271-84-50; fax: #5581-271-03-59. E-mail address: [email protected] (F.M. de Aguiar).
through a circulator. A 10-W TWT ampli"er fed by a synthesized sweeper provided the power. The interaction with the sample results in changes in the amplitude of the radiation re#ected by the cavity. These changes are detected by a recti"er diode at the output port of the circulator and stored in a digital oscilloscope. In order to avoid heating e!ects, 2-ms-long pulses were used from a p}i}n diode. Abrupt changes in the pulse shape are "rstly observed at the Suhl instability threshold h , when ! a certain spin-wave pair is driven out of equilibrium, depending on the value of the applied static magnetic "eld H . At higher power levels, the system may exhibit, at subsequent thresholds, higher-order bifurcations such as self-oscillations and chaos [1]. Fig. 1 shows the measured time series with the static magnetic "eld H aligned parallel to the [1 1 0] crystal axis, exhibiting evidences (ordered mixed-mode self-oscillations) for the PC sequence. The microwave magnetic "eld was "xed at h"0.8 Oe (h/h &2) and H was varied in the interval ! from 1970 to 2090 Oe, corresponding to a region in the #at bottom of the &butter#y curve' h (H ), where mag! netostatic modes are primarily excited [1,6]. In Fig. 1, from top to bottom, one observes the regimes P, P, 2P, C, P, and C, respectively. Notice that the chaotic return maps exhibit a surprisingly L-shaped one-dimensional behavior [6]. A "t to the data with a nonlinear function F(x), resulted in intriguing bifurcation diagrams that display more complete sequences, i.e., periodic states with n'50, prior to a "nal tangency to a "xed point. Here were we add to those "ndings
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 9 9 7 - 5
F.M. de Aguiar et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 524}526
Fig. 1. Measured periodic}chaotic sequence in subsidiary resonance experiments in a YIG sphere for h/h &2, and varying ! static "eld, as described in the text. From top to bottom, H "1976, 2025, 2040, 2060, 2067, and 2085 Oe, respectively.
numerical results from a standard two-mode model that are in good qualitative agreement with the experiments. It is well known that, beyond the Suhl instability threshold, an entire degenerate spin-wave manifold is eligible for the parametric process [1]. We circumvent this complexity by considering the excitation of only two spin wave modes [7]. Introduced by Nakamura and coworkers in the early 1980s, this so-called two-mode model (TMM) gives rise to four coupled "rst-order nonlinear rate equations for the real and imaginary parts of the two spin variables c (i"1, 2), with e!ectively nine G phenomenological parameters (see caption of Fig. 2). In principle, by changing the static magnetic "eld one should expect to vary all parameters. Given that the experimental situation corresponds to a #at region of the &butter#y curve', this variation must not be dramatic. To mimic the experiments, we then drive the system beyond the Suhl threshold at "xed power and vary only the detuning of &mode 1', keeping the other parameters constant. An interesting scenario, in good qualitative agreement with the measured one, is shown in Fig. 2 by the square of the real part of the spin variable of &mode 2'.
525
Fig. 2. Calculated periodic}chaotic sequence with the twomode model [6] for r"4.8, / "0.7, / "5.0, (S #2¹ )/ "!1.0, (S #2¹ )/ "0.5, S / " S / "2.5, ¹ / "¹ / "!1.2, "0. From top to bottom, / "!1.5, 2.8, 4.0, 5.96, 5.97, and 6.9, respectively. The parameter r is proportional to h/h , is the ! G relaxation rate of mode i (i"1, 2), Ss and ¹s vertexes of spin-wave interactions, and is the detuning of mode i with G respect to the frequency of the primary mode, excited at the Suhl threshold h !
One can see that the frequency depends more strongly on the control parameter in the model than in the experiments. Besides, the calculated frequency and pumping "eld h are larger than the measured ones by a factor between 2 and 3, probably because we ignore the variation in the other parameters and the "nite size of the sample [1]. Work to check these ideas is in progress. Here we would like to stress that multibranched structures similar to the ones observed in suitably constructed one-dimensional return maps in both experiments and mapping [6] are also present in the TMM. These results are examples of the richness of the longstanding subject of microwave driven nonlinear spin-waves, whose number and nature are hitherto unsolved problems. This work has been supported by CNPq CAPES, FINEP, PADCT, and FACEPE (Brazilian agencies).
526
F.M. de Aguiar et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 524}526
References [1] P.E. Wigen (Ed.), Nonlinear Phenomena and Chaos in Magnetic Materials, World Scienti"c, Singapore, 1994, and references therein. [2] T. Braun, J.A. Lisboa, J.A.C. Gallas, Phys. Rev. Lett. 68 (1992) 2770. [3] T. Hayashi, Phys. Rev. Lett. 84 (2000) 3334.
[4] J.L. Hudson, M. Hart, D. Marinko, J. Chem. Phys. 71 (1979) 1601. [5] M. Lefranc, D. Hannequin, D. Dangoisse, J. Opt. Soc. Am. B 8 (1991) 239. [6] S. Rosenblatt, F.M. de Aguiar, S.M. Rezende, A. Azevedo, J. Appl. Phys. 87 (2000) 6917. [7] S.M. Rezende, F.M. de Aguiar, Proc. IEEE 78 (1990) 893.
Spin-wave alternating periodic}chaotic dynamics F.M. de Aguiar*, S. Rosenblatt, S.M. Rezende, A. Azevedo Departamento de Fisica, Universidade Federal de Pernambuco, Recife, PE 50670-901, Brazil
Abstract A description is given of pulsed ferromagnetic subsidiary resonance experiments in a YIG sphere at room temperature. Beyond the Suhl instability threshold, evidences were found for the elusive alternating periodic}chaotic sequence. The spin-wave self-oscillations have recently been characterized through one-dimensional return maps. Here it is shown that the observed scenario is also in good qualitative agreement with results from numerical simulations with a standard two-mode microscopic model. 2001 Elsevier Science B.V. All rights reserved. Keywords: Chaos; Spin waves; Ferromagnetic resonance
A remarkable signature of nonlinear dynamics is that many di!erent physical systems enjoy common features as a (system dependent) control parameter is varied. In particular, interacting spin waves, resonantly driven at high microwave power levels in a ferromagnet, exhibit a wide range of universal phenomena, depending on the dynamical parameters of the system [1]. In this paper, intriguing experimental and numerical results are presented on the nonlinear behavior of the excited spin waves. Particular attention is paid to a transition where the so-called mixed-mode oscillations occur, namely, the periodic}chaotic (PC) sequence [2,3]. Discovered 20 years ago in chemical reactions [4], the PC sequence is characterized by regions of chaotic behavior (C) bracketed by periodic states (P and P>) consisting of large-amplitude peaks followed by n"1, 2, 3, 2 small-amplitude undulations. Prior to chaos, each periodic state undergoes a sequence o period-doubling bifurcations. Experimentally, this scenario has been well characterized in only a few dynamical systems [2,5]. The experiments reported here have been carried out in the subsidiary resonance con"guration. A 1 mm-diam YIG sphere was held in the center of a TE rectangular microwave cavity ( f"8.9 GHz, Q"3000, microwave magnetic "eld amplitude h), coupled to a waveguide
* Corresponding author. Tel.: #55-81-271-84-50; fax: #5581-271-03-59. E-mail address: [email protected] (F.M. de Aguiar).
through a circulator. A 10-W TWT ampli"er fed by a synthesized sweeper provided the power. The interaction with the sample results in changes in the amplitude of the radiation re#ected by the cavity. These changes are detected by a recti"er diode at the output port of the circulator and stored in a digital oscilloscope. In order to avoid heating e!ects, 2-ms-long pulses were used from a p}i}n diode. Abrupt changes in the pulse shape are "rstly observed at the Suhl instability threshold h , when ! a certain spin-wave pair is driven out of equilibrium, depending on the value of the applied static magnetic "eld H . At higher power levels, the system may exhibit, at subsequent thresholds, higher-order bifurcations such as self-oscillations and chaos [1]. Fig. 1 shows the measured time series with the static magnetic "eld H aligned parallel to the [1 1 0] crystal axis, exhibiting evidences (ordered mixed-mode self-oscillations) for the PC sequence. The microwave magnetic "eld was "xed at h"0.8 Oe (h/h &2) and H was varied in the interval ! from 1970 to 2090 Oe, corresponding to a region in the #at bottom of the &butter#y curve' h (H ), where mag! netostatic modes are primarily excited [1,6]. In Fig. 1, from top to bottom, one observes the regimes P, P, 2P, C, P, and C, respectively. Notice that the chaotic return maps exhibit a surprisingly L-shaped one-dimensional behavior [6]. A "t to the data with a nonlinear function F(x), resulted in intriguing bifurcation diagrams that display more complete sequences, i.e., periodic states with n'50, prior to a "nal tangency to a "xed point. Here were we add to those "ndings
0304-8853/01/$ - see front matter 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 0 ) 0 0 9 9 7 - 5
F.M. de Aguiar et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 524}526
Fig. 1. Measured periodic}chaotic sequence in subsidiary resonance experiments in a YIG sphere for h/h &2, and varying ! static "eld, as described in the text. From top to bottom, H "1976, 2025, 2040, 2060, 2067, and 2085 Oe, respectively.
numerical results from a standard two-mode model that are in good qualitative agreement with the experiments. It is well known that, beyond the Suhl instability threshold, an entire degenerate spin-wave manifold is eligible for the parametric process [1]. We circumvent this complexity by considering the excitation of only two spin wave modes [7]. Introduced by Nakamura and coworkers in the early 1980s, this so-called two-mode model (TMM) gives rise to four coupled "rst-order nonlinear rate equations for the real and imaginary parts of the two spin variables c (i"1, 2), with e!ectively nine G phenomenological parameters (see caption of Fig. 2). In principle, by changing the static magnetic "eld one should expect to vary all parameters. Given that the experimental situation corresponds to a #at region of the &butter#y curve', this variation must not be dramatic. To mimic the experiments, we then drive the system beyond the Suhl threshold at "xed power and vary only the detuning of &mode 1', keeping the other parameters constant. An interesting scenario, in good qualitative agreement with the measured one, is shown in Fig. 2 by the square of the real part of the spin variable of &mode 2'.
525
Fig. 2. Calculated periodic}chaotic sequence with the twomode model [6] for r"4.8, / "0.7, / "5.0, (S #2¹ )/ "!1.0, (S #2¹ )/ "0.5, S / " S / "2.5, ¹ / "¹ / "!1.2, "0. From top to bottom, / "!1.5, 2.8, 4.0, 5.96, 5.97, and 6.9, respectively. The parameter r is proportional to h/h , is the ! G relaxation rate of mode i (i"1, 2), Ss and ¹s vertexes of spin-wave interactions, and is the detuning of mode i with G respect to the frequency of the primary mode, excited at the Suhl threshold h !
One can see that the frequency depends more strongly on the control parameter in the model than in the experiments. Besides, the calculated frequency and pumping "eld h are larger than the measured ones by a factor between 2 and 3, probably because we ignore the variation in the other parameters and the "nite size of the sample [1]. Work to check these ideas is in progress. Here we would like to stress that multibranched structures similar to the ones observed in suitably constructed one-dimensional return maps in both experiments and mapping [6] are also present in the TMM. These results are examples of the richness of the longstanding subject of microwave driven nonlinear spin-waves, whose number and nature are hitherto unsolved problems. This work has been supported by CNPq CAPES, FINEP, PADCT, and FACEPE (Brazilian agencies).
526
F.M. de Aguiar et al. / Journal of Magnetism and Magnetic Materials 226}230 (2001) 524}526
References [1] P.E. Wigen (Ed.), Nonlinear Phenomena and Chaos in Magnetic Materials, World Scienti"c, Singapore, 1994, and references therein. [2] T. Braun, J.A. Lisboa, J.A.C. Gallas, Phys. Rev. Lett. 68 (1992) 2770. [3] T. Hayashi, Phys. Rev. Lett. 84 (2000) 3334.
[4] J.L. Hudson, M. Hart, D. Marinko, J. Chem. Phys. 71 (1979) 1601. [5] M. Lefranc, D. Hannequin, D. Dangoisse, J. Opt. Soc. Am. B 8 (1991) 239. [6] S. Rosenblatt, F.M. de Aguiar, S.M. Rezende, A. Azevedo, J. Appl. Phys. 87 (2000) 6917. [7] S.M. Rezende, F.M. de Aguiar, Proc. IEEE 78 (1990) 893.