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Exemplary locking sequence during self-generated quasiperiodicity of extrinsic germanium
Volume 124, number 6,7
EXEMPLARY LOCKING SEQUENCE OF EXTRINSIC GERMANIUM
PHYSICS LETTERS A
DURING
SELF-GENERATED
5 October 1987
QUASIPERIODICITY
U. RAU, J. PEINKE, J. PARISI, R.P. HUEBENER Physikalisches Institut, Lehrstuhl Experimentalphysik II, Universitiit Tiibingen, D-7400 Tiibingen, FRG
and E. SCHOLL Institut fir Theoretische Physik, Rheinisch- Westfdlische Technische Hochschule, D-5100 Aachen, FRG Received 1 June 1987; accepted for publication 6 August 1987 Communicated by A.P. Fordy
We report on experimental investigations of self-organized quasiperiodic and mode-locked behavior during low-temperature avalanche breakdown of p-doped germanium. Under variation of the applied longitudinal magnetic field, undriven spontaneous current oscillations are demonstrated to undergo an exemplary sequence of distinct locking states via an apparent self-similar emergence of high-order quasiperiodic mixing frequencies.
The observed quasiperiodic behavior of many different nonlinear dissipative systems supports the universal predictions of the circle-map theory [ 11, particularly in the case where one fundamental frequency corresponds to a periodic oscillation driven externally. Quasiperiodicity at two incommensurate frequencies is characterized by the motion of the phase space trajectory on an invariant two-torus, the Poincare cross section of which approximates the form of an unfolded circle. Hence, the deterministic process underlying the continuous dynamics of the toroidal system flow can be modelled by simple discrete 1-D mappings of the circle onto itself. The most convenient “sine” circle map is defined as the periodic function 0 n+ 1=.I&~,> =8, +Q-
(Kl27c) sin 2x8, ,
(1)
where 8 means the iterated angular variable on the circle (modulo 1). The parameters 52 and K denote the frequency ratio and the coupling strength of the two competing oscillatory modes, respectively. Increasing strength of the nonlinear coupling between
the oscillators develops an increasing tendency to lock into commensurate motion where the ratio of the oscillatory frequencies is a rational number. Following the hierarchical Farey-tree ordering, the resonant frequency-locked states form hornlike structures within the (4 K) parameter plane known as “Arnold tongues”. The circle map exhibits universal scaling behavior due to the characteristic self-similar staircase structure of the mode-locking intervals at the critical boundary (K= l), where the map ceases to be an invertible diffeomorphism. So far, a number of these predicted features have been tested in different experimental systems #I. Best conformity with numerical results of the circle map has been obtained in experiments where an external periodic forcing is applied to a single internal oscillatory mode of the system. As an advantage, the amplitude and frequency of the driving force represent adequate control parameters which are simply related to the parameters of the circle map. In the ‘I For experiments on hydrodynamic systems see ref. [ 21; for experiments on chemical systems see ref. [ 31; for experiments on solid-state systems see ref. [4].
0375-9601/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
335
Volume 124, number 6,7
PHYSICS LETTERS A
I
/II d
@
I(t)
Tr--?T--
m
mm2
Fig. I. Scheme of the experimental set-up. The hatched surface area on the Ge sample indicates the plate-capacitor arrangement of the ohmic Al contacts.
following, we address the question whether the above circle-map formalism is directly applicable also to the more general experimental situation where the two competing frequencies arise dynamically within the undriven system. This paper reports on experimental investigations of self-organized quasiperiodic and mode-locked behavior during impact-ionization-induced avalanche breakdown of extrinsic germanium at low temperatures. In our semiconductor system two or more natural intrinsic oscillatory modes are simultaneously present as a consequence of the spatiotemporal nonlinear current transport in the breakdown regime [ 51. Therefore, our experiments deal with an interesting study object for nonlinear dynamics. Depending sensitively upon the applied longitudinal magnetic field which acts as the relevant control parameter, the undriven spontaneous current oscillations are demonstrated to undergo an exemplary sequence of distinct locking states via an apparent self-similar emergence of high-order quasiperiodic mixing frequencies. The resulting scenario is discussed in terms of the simple circle-map formalism. Our experimental system consists of a homogeneously p-doped germanium single crystal (acceptor concentration of about 1OL4 cm-3) having the dimensions of about 0.6x 1.7x2.8 mm3 and carrying ohmic aluminum contacts evaporated on the two smallest opposite surfaces. The sample geometry and the electronic measuring configuration are sketched schematically in fig. 1. To provide the ohmic contacts with an electric field, a dc bias voltage I’,, was 336
5 October 1987
applied to the series combination of the sample and the 100 Q load resistor. A dc magnetic field B parallel to the broad sample surfaces could also be applied by a superconducting solenoid surrounding the semiconductor sample. The resulting electric current Zwas found from the voltage drop at the load resistor. The voltage V was measured along the sample. All experiments were performed in a metal cryostat at 4.2 K temperature, the sample always being kept in direct contact with the liquid-helium bath and carefully protected against external electromagnetic irradiation (visible, far infrared). In extrinsic germanium cooled to liquid-helium temperatures most of the charge carriers are frozen out at the impurities and the material becomes an electric insulator. Applying an electric field of sufficiently high strength (in the range of a few V/cm), impact ionization of the shallow impurities takes place in the bulk of the homogeneously doped semiconductor. The resulting avalanche breakdown persists until1 al impurities are ionized. The underlying nonequilibrium phase transition from a low conducting state to a high conducting state is directly reflected in strongly nonlinear regions of negative differential resistivity [ 61. The corresponding nonlinear curvature of the measured current-voltage characteristic is associated with self-generated current oscillations (with a relative amplitude of about 10e3 in the frequency range 0.1-2 kHz) superimposed upon the steady dc current (of typically a few mA). Fig. 2 shows an example of a current-voltage curve obtained at B = 67 G longitudinal magnetic field. Note the multistable and hysteretic form of the characteristic which can be attributed to the formation of spatially inhomogeneous current density structures developing in the homogeneous bulk semiconductor during avalanche breakdown (cf. refs. [ 5,7,8]). Accordingly, we observed a variety of complex temporal dissipative structures along the descending branch of the current-voltage characteristic - preferably in the regions of negative differential resistance. For instance, we found in regions I and II of fig. 2 various types of two-torus quasiperiodic oscillations and period-doubling cascades (including the onset of chaos as well as period-three and period-five windows), respectively. It is emphasized that slight variations of the control parameters (bias voltage and/or longitudinal
PHYSICS LETTERS A
Volume 124, number 6,7
5 October 1987 Locking states II2 -
f/f0 Ill1
J4
f,
112
LJ/ 15
1.3
1.4
1.5
v/v
Fig. 2. Current-voltage characteristic obtained at the applied longitudinal magnetic field 8=67 G. Multistable oscillatory states were found in the regions I and II along the descending branch of the characteristic.
magnetic field) lead to a hysteretic jumping or an intermittent-like switching between different attractors. Regardless of the prevailing attractor, however, the resulting system flow generally displays similar nonlinear behavior. In the following, we concentrate on an exemplary scenario of quasiperiodic and mode-locked behavior associated with a characteristic self-similar emergence of high-order mixing frequencies. For a typical scan through control-parameter space, the longitudinal magnetic field was increased from B= 72 G to B=82 G, while the bias voltage was always kept at the constant value V,= 1.788 V. The resulting overall system flow consists of two competing intrinsic oscillatory modes, the originally incommensurate frequencies (say, fo andf, ) of which are both changing fromf0=1.15 kHz andf,=0.30 kHz (at 8~72 G) to& 1.05 kHz andf; =0.90 kHz (at B=82 G), respectively. The frequency bifurcation diagram of fig. 3 gives a schematic synopsis of the examined transition as a function of the applied magnetic control field. Quasiperiodicity was found to be interrupted by distinct stable resonances locked at the rational frequency ratiosf,lfo= l/2, 213, 314, 415, 516, and 617 over the finite field intervals ABzO.90 G, 0.45 G, 0.15 G, 0.20 G, about 0.05 G, and less than 0.05 G, respectively. Locking states filf< II2 could
a b c BIG
def
80
Fig. 3. Dependence of the two fundamental oscillatory modes (frequencies j& f; ) and their mixing components (frequencies f= ufo+ bf; ) upon the applied longitudinal magnetic field (bias voltage VO=1.788 V). All frequencies of the spectral bifurcation diagram are normalized to&. The letters a-f refer to the corresponding power spectra shown in fig. 4.
not be resolved. At magnetic fields below B=72 G quasiperiodicity becomes unstable (rising noise). Most strikingly, an increasing number of high-order mixing components (a, b) defined as the linear combinations afo + bfi (a, b integers) of the two fundamental frequencies f.and fi gradually arises with increasing magnetic field at distinct lockings of oddnumbered denominators (i.e., 213, 415, 617). It is emphasized that due to the continuous presence of the low-order mixing frequency fo-fithe second fundamental frequency fi is not uniquely defined, since fi and f.-fimay be interchanged. The apparent self-similar formation of stable highorder mixing frequencies beyond the 213 locking state becomes clearly manifest in the different current power spectra of fig. 4 obtained for distinct control parameter values (indicated by the letters a-f in fig 3). Fig. 4a gives the power spectrum of the 213 locking state. Increasing the magnetic field from B= 77.0 G to B=77.3 G leads to a bifurcation of the two spectral lines (0, 1): f;=fol3 and (1, -1): fo-fi =2f,l3 into the four components (0, l), (2, -2) and (1, -l), (- 1, 2), respectively (fig. 4b). Note the emergence of the second-order mixing frequencies 2fo - 2fi and -f. + 2fi that are preserved without further bifurcation up to the 415 locking state at B=78.7 G (fig. 4e). The intervening 314 locking state at B= 77.8 G (fig. 4c) is a result of their cross-over and is not followed by a bifurcation of new mixing 337
Volume 124, number 6,7
PHYSICS LETTERS A a)
IQ 01
d)
PldB 50 -
1
5 October 1987
al
f/kHz b)
0
n
0.S
00 1
1.0
b) f)
kdh Ok---T--
50-
I-23113r31
“J
0.6 Fig. 4. Power spectra of the current I at different magnetic control fields (a) B=77.0 G, (b) B=77.3 G, (c) 8=77.8 G, (d) B=78.3 G, (e) B=78.7 G and (f) B=79.0 G as indicated in fig. 3 (biasvoltage I’,,= 1.788 V).
components. For clarity, see fig. 4d obtained at slightly increased magnetic field B=78.3 G. However, the system flow again bifurcates beyond the 415 locking state via introducing other two third-order mixing frequencies 3fo - 3fi and - 2f,+ 3fi in an analogous self-similar way, as indicated in the power spectrum of fig. 4f obtained at B= 79.0 G. The analogous bifurcation procedure recurs at the higher 5/6 and 6/7 locking states (cf. fig. 3). The pattern of the frequency bifurcation diagram together with the power spectra clearly show an increasing number and strength of high-order mixing frequencies with increasing longitudinal magnetic field. A simple interpretation is possible, assuming that the degree of nonlinear coupling between the two fundamental oscillatory modes of the quasiperiodic state increases with increasing magnetic field. Comparing the experimental locking sequence with the resonance structure of the (9 IQ parameter space of the circle map (1 ), one might presume that our experimental system upon variation of the single control parameter B follows a distinct path through the two-parameter mapping space as indicated schematically in fig. Sa. Thus, variation of the longitudinal magnetic field B simultaneously changes both 338
n
08
1.0
Fig. 5. Mode-locking states obtained for the “sine” circle map. (a) Phase diagram in (Q, K) parameter space. The solid line K= Q indicates an exemplary path followed by the experimental system. The intersection intervals with the Arnold tongues correspond to the experimentally observed locking sequence. (b) Winding number W= (Iln)( 0, - 0,) calculated for n = 500 iterations versus frequency ratio B assuming K=Q.
the frequency ratio 52and the coupling strength K #*. The circles in fig. 5a point out the intersection intervals of the system path (K=Q) with the Arnold tongues. An approximate reproduction of the experimentally observed locking sequence is also demonstrated in the diagram of fig. 5b. Here the “effective” frequency ratio W (termed as winding number) determining the average twist of the phase space trajectory as a consequence of the nonlinear coupling is plotted for the circle map (1) versus the “bare” frequency ratio s2 under the assumption K= 52.
To conclude, we have shown that self-generated quasi-periodic and mode-locked behavior of our semiconductor system can well be modelled by distinct features of the circle-map formalism. The authors thank B. Rohricht, K.M. Mayer, M. x2Note that analogous behavior can be derived from a simple mechanical oscillatory system of two weakly coupled, undamped pendula by increasing, for example, the mass of one pendulum as the relevant control parameter.
Volume 124, number 6,7
PHYSICS LETTERS A
Weise, and O.E. RGssler for discussions. Partial support of this work from the Stiftung Volkswagenwerk is gratefully acknowledged.
References [ 1] M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. A 30 (1984) 1960; T. Bohr, P. Bak and M.H. Jensen, Phys. Rev. A 30 (1984) 1970; P. Bak, T. Bohr and M.H. Jensen, Phys. Ser. T 9 (1985) 50; P. Cvitanovic, M.H. Jensen, L.P. Kadanoff and I. Procaccia, Phys. Rev. Lett. 55 (1985) 343. [2] J. Stavans, F. Heslot and A. Libchaber, Phys. Rev. Lett. 55 (1985) 569; A.P. Fein, M.S. Heutmaker and J.P. Gollub, Phys. Ser. T 9 (1985) 79; H. Haucke and R. Ecke, Physica D, to be published. [3] H.L. Swinney and J. Maselko, Phys. Rev. Lett. 55 (1985) 2366.
5 October 1987
[ 41 G.A. Held and C. Jeffries, Phys. Rev. Lett. 56 (1986) 1183; S. Martin and W. Martienssen, Phys. Rev. Lett. 56 (1986) 1522; E.G. Gwinn and R.M. Westervelt, Phys. Rev. Lett. 57 (1986) 1060. [5] J. Peinke, A. Mtlhlbach, B. Riihricht, B. Wessely, J. Mannhart, J. Parisi and R.P. Huebener, Physica D 23 (1986) 176; J. Parisi, J. Peinke, B. Rohricht and KM. Mayer, Z. Naturforsch. 42 a (1987) 329; R.P. Huebener, K.M. Mayer, J. Parisi, J. Peinke and B. Rohricht, in: Proc. Int. Conf. on Physics of chaos and systems far from equilibrium, Monterey, 1987, to be published in Nucl. Phys. B; J. Peinke, J. Parisi, B. Rohricht, B. Wessely and K.M. Mayer, Europhys. Lett., submitted for publication. [6] E. Scholl, in: Festkorperprobleme, Vol. 26, ed. P. Grosse (Vieweg, Braunschweig, 1986) p.309; Springer series in synergetics, Vol. 35. Nonequilibrium phase transitions in semiconductors (Springer, Berlin, 1987), to be published. [ 71 D. Jlger, H. Baumann and R. Symanczyk, Phys. Lett. A 177 (1986) 141. [ 81 K.M. Mayer, R. Gross, J. Parisi, J. Peinke and R.P. Huebener, Solid State Commun., to be published.
339
EXEMPLARY LOCKING SEQUENCE OF EXTRINSIC GERMANIUM
PHYSICS LETTERS A
DURING
SELF-GENERATED
5 October 1987
QUASIPERIODICITY
U. RAU, J. PEINKE, J. PARISI, R.P. HUEBENER Physikalisches Institut, Lehrstuhl Experimentalphysik II, Universitiit Tiibingen, D-7400 Tiibingen, FRG
and E. SCHOLL Institut fir Theoretische Physik, Rheinisch- Westfdlische Technische Hochschule, D-5100 Aachen, FRG Received 1 June 1987; accepted for publication 6 August 1987 Communicated by A.P. Fordy
We report on experimental investigations of self-organized quasiperiodic and mode-locked behavior during low-temperature avalanche breakdown of p-doped germanium. Under variation of the applied longitudinal magnetic field, undriven spontaneous current oscillations are demonstrated to undergo an exemplary sequence of distinct locking states via an apparent self-similar emergence of high-order quasiperiodic mixing frequencies.
The observed quasiperiodic behavior of many different nonlinear dissipative systems supports the universal predictions of the circle-map theory [ 11, particularly in the case where one fundamental frequency corresponds to a periodic oscillation driven externally. Quasiperiodicity at two incommensurate frequencies is characterized by the motion of the phase space trajectory on an invariant two-torus, the Poincare cross section of which approximates the form of an unfolded circle. Hence, the deterministic process underlying the continuous dynamics of the toroidal system flow can be modelled by simple discrete 1-D mappings of the circle onto itself. The most convenient “sine” circle map is defined as the periodic function 0 n+ 1=.I&~,> =8, +Q-
(Kl27c) sin 2x8, ,
(1)
where 8 means the iterated angular variable on the circle (modulo 1). The parameters 52 and K denote the frequency ratio and the coupling strength of the two competing oscillatory modes, respectively. Increasing strength of the nonlinear coupling between
the oscillators develops an increasing tendency to lock into commensurate motion where the ratio of the oscillatory frequencies is a rational number. Following the hierarchical Farey-tree ordering, the resonant frequency-locked states form hornlike structures within the (4 K) parameter plane known as “Arnold tongues”. The circle map exhibits universal scaling behavior due to the characteristic self-similar staircase structure of the mode-locking intervals at the critical boundary (K= l), where the map ceases to be an invertible diffeomorphism. So far, a number of these predicted features have been tested in different experimental systems #I. Best conformity with numerical results of the circle map has been obtained in experiments where an external periodic forcing is applied to a single internal oscillatory mode of the system. As an advantage, the amplitude and frequency of the driving force represent adequate control parameters which are simply related to the parameters of the circle map. In the ‘I For experiments on hydrodynamic systems see ref. [ 21; for experiments on chemical systems see ref. [ 31; for experiments on solid-state systems see ref. [4].
0375-9601/87/$ 03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
335
Volume 124, number 6,7
PHYSICS LETTERS A
I
/II d
@
I(t)
Tr--?T--
m
mm2
Fig. I. Scheme of the experimental set-up. The hatched surface area on the Ge sample indicates the plate-capacitor arrangement of the ohmic Al contacts.
following, we address the question whether the above circle-map formalism is directly applicable also to the more general experimental situation where the two competing frequencies arise dynamically within the undriven system. This paper reports on experimental investigations of self-organized quasiperiodic and mode-locked behavior during impact-ionization-induced avalanche breakdown of extrinsic germanium at low temperatures. In our semiconductor system two or more natural intrinsic oscillatory modes are simultaneously present as a consequence of the spatiotemporal nonlinear current transport in the breakdown regime [ 51. Therefore, our experiments deal with an interesting study object for nonlinear dynamics. Depending sensitively upon the applied longitudinal magnetic field which acts as the relevant control parameter, the undriven spontaneous current oscillations are demonstrated to undergo an exemplary sequence of distinct locking states via an apparent self-similar emergence of high-order quasiperiodic mixing frequencies. The resulting scenario is discussed in terms of the simple circle-map formalism. Our experimental system consists of a homogeneously p-doped germanium single crystal (acceptor concentration of about 1OL4 cm-3) having the dimensions of about 0.6x 1.7x2.8 mm3 and carrying ohmic aluminum contacts evaporated on the two smallest opposite surfaces. The sample geometry and the electronic measuring configuration are sketched schematically in fig. 1. To provide the ohmic contacts with an electric field, a dc bias voltage I’,, was 336
5 October 1987
applied to the series combination of the sample and the 100 Q load resistor. A dc magnetic field B parallel to the broad sample surfaces could also be applied by a superconducting solenoid surrounding the semiconductor sample. The resulting electric current Zwas found from the voltage drop at the load resistor. The voltage V was measured along the sample. All experiments were performed in a metal cryostat at 4.2 K temperature, the sample always being kept in direct contact with the liquid-helium bath and carefully protected against external electromagnetic irradiation (visible, far infrared). In extrinsic germanium cooled to liquid-helium temperatures most of the charge carriers are frozen out at the impurities and the material becomes an electric insulator. Applying an electric field of sufficiently high strength (in the range of a few V/cm), impact ionization of the shallow impurities takes place in the bulk of the homogeneously doped semiconductor. The resulting avalanche breakdown persists until1 al impurities are ionized. The underlying nonequilibrium phase transition from a low conducting state to a high conducting state is directly reflected in strongly nonlinear regions of negative differential resistivity [ 61. The corresponding nonlinear curvature of the measured current-voltage characteristic is associated with self-generated current oscillations (with a relative amplitude of about 10e3 in the frequency range 0.1-2 kHz) superimposed upon the steady dc current (of typically a few mA). Fig. 2 shows an example of a current-voltage curve obtained at B = 67 G longitudinal magnetic field. Note the multistable and hysteretic form of the characteristic which can be attributed to the formation of spatially inhomogeneous current density structures developing in the homogeneous bulk semiconductor during avalanche breakdown (cf. refs. [ 5,7,8]). Accordingly, we observed a variety of complex temporal dissipative structures along the descending branch of the current-voltage characteristic - preferably in the regions of negative differential resistance. For instance, we found in regions I and II of fig. 2 various types of two-torus quasiperiodic oscillations and period-doubling cascades (including the onset of chaos as well as period-three and period-five windows), respectively. It is emphasized that slight variations of the control parameters (bias voltage and/or longitudinal
PHYSICS LETTERS A
Volume 124, number 6,7
5 October 1987 Locking states II2 -
f/f0 Ill1
J4
f,
112
LJ/ 15
1.3
1.4
1.5
v/v
Fig. 2. Current-voltage characteristic obtained at the applied longitudinal magnetic field 8=67 G. Multistable oscillatory states were found in the regions I and II along the descending branch of the characteristic.
magnetic field) lead to a hysteretic jumping or an intermittent-like switching between different attractors. Regardless of the prevailing attractor, however, the resulting system flow generally displays similar nonlinear behavior. In the following, we concentrate on an exemplary scenario of quasiperiodic and mode-locked behavior associated with a characteristic self-similar emergence of high-order mixing frequencies. For a typical scan through control-parameter space, the longitudinal magnetic field was increased from B= 72 G to B=82 G, while the bias voltage was always kept at the constant value V,= 1.788 V. The resulting overall system flow consists of two competing intrinsic oscillatory modes, the originally incommensurate frequencies (say, fo andf, ) of which are both changing fromf0=1.15 kHz andf,=0.30 kHz (at 8~72 G) to& 1.05 kHz andf; =0.90 kHz (at B=82 G), respectively. The frequency bifurcation diagram of fig. 3 gives a schematic synopsis of the examined transition as a function of the applied magnetic control field. Quasiperiodicity was found to be interrupted by distinct stable resonances locked at the rational frequency ratiosf,lfo= l/2, 213, 314, 415, 516, and 617 over the finite field intervals ABzO.90 G, 0.45 G, 0.15 G, 0.20 G, about 0.05 G, and less than 0.05 G, respectively. Locking states filf< II2 could
a b c BIG
def
80
Fig. 3. Dependence of the two fundamental oscillatory modes (frequencies j& f; ) and their mixing components (frequencies f= ufo+ bf; ) upon the applied longitudinal magnetic field (bias voltage VO=1.788 V). All frequencies of the spectral bifurcation diagram are normalized to&. The letters a-f refer to the corresponding power spectra shown in fig. 4.
not be resolved. At magnetic fields below B=72 G quasiperiodicity becomes unstable (rising noise). Most strikingly, an increasing number of high-order mixing components (a, b) defined as the linear combinations afo + bfi (a, b integers) of the two fundamental frequencies f.and fi gradually arises with increasing magnetic field at distinct lockings of oddnumbered denominators (i.e., 213, 415, 617). It is emphasized that due to the continuous presence of the low-order mixing frequency fo-fithe second fundamental frequency fi is not uniquely defined, since fi and f.-fimay be interchanged. The apparent self-similar formation of stable highorder mixing frequencies beyond the 213 locking state becomes clearly manifest in the different current power spectra of fig. 4 obtained for distinct control parameter values (indicated by the letters a-f in fig 3). Fig. 4a gives the power spectrum of the 213 locking state. Increasing the magnetic field from B= 77.0 G to B=77.3 G leads to a bifurcation of the two spectral lines (0, 1): f;=fol3 and (1, -1): fo-fi =2f,l3 into the four components (0, l), (2, -2) and (1, -l), (- 1, 2), respectively (fig. 4b). Note the emergence of the second-order mixing frequencies 2fo - 2fi and -f. + 2fi that are preserved without further bifurcation up to the 415 locking state at B=78.7 G (fig. 4e). The intervening 314 locking state at B= 77.8 G (fig. 4c) is a result of their cross-over and is not followed by a bifurcation of new mixing 337
Volume 124, number 6,7
PHYSICS LETTERS A a)
IQ 01
d)
PldB 50 -
1
5 October 1987
al
f/kHz b)
0
n
0.S
00 1
1.0
b) f)
kdh Ok---T--
50-
I-23113r31
“J
0.6 Fig. 4. Power spectra of the current I at different magnetic control fields (a) B=77.0 G, (b) B=77.3 G, (c) 8=77.8 G, (d) B=78.3 G, (e) B=78.7 G and (f) B=79.0 G as indicated in fig. 3 (biasvoltage I’,,= 1.788 V).
components. For clarity, see fig. 4d obtained at slightly increased magnetic field B=78.3 G. However, the system flow again bifurcates beyond the 415 locking state via introducing other two third-order mixing frequencies 3fo - 3fi and - 2f,+ 3fi in an analogous self-similar way, as indicated in the power spectrum of fig. 4f obtained at B= 79.0 G. The analogous bifurcation procedure recurs at the higher 5/6 and 6/7 locking states (cf. fig. 3). The pattern of the frequency bifurcation diagram together with the power spectra clearly show an increasing number and strength of high-order mixing frequencies with increasing longitudinal magnetic field. A simple interpretation is possible, assuming that the degree of nonlinear coupling between the two fundamental oscillatory modes of the quasiperiodic state increases with increasing magnetic field. Comparing the experimental locking sequence with the resonance structure of the (9 IQ parameter space of the circle map (1 ), one might presume that our experimental system upon variation of the single control parameter B follows a distinct path through the two-parameter mapping space as indicated schematically in fig. Sa. Thus, variation of the longitudinal magnetic field B simultaneously changes both 338
n
08
1.0
Fig. 5. Mode-locking states obtained for the “sine” circle map. (a) Phase diagram in (Q, K) parameter space. The solid line K= Q indicates an exemplary path followed by the experimental system. The intersection intervals with the Arnold tongues correspond to the experimentally observed locking sequence. (b) Winding number W= (Iln)( 0, - 0,) calculated for n = 500 iterations versus frequency ratio B assuming K=Q.
the frequency ratio 52and the coupling strength K #*. The circles in fig. 5a point out the intersection intervals of the system path (K=Q) with the Arnold tongues. An approximate reproduction of the experimentally observed locking sequence is also demonstrated in the diagram of fig. 5b. Here the “effective” frequency ratio W (termed as winding number) determining the average twist of the phase space trajectory as a consequence of the nonlinear coupling is plotted for the circle map (1) versus the “bare” frequency ratio s2 under the assumption K= 52.
To conclude, we have shown that self-generated quasi-periodic and mode-locked behavior of our semiconductor system can well be modelled by distinct features of the circle-map formalism. The authors thank B. Rohricht, K.M. Mayer, M. x2Note that analogous behavior can be derived from a simple mechanical oscillatory system of two weakly coupled, undamped pendula by increasing, for example, the mass of one pendulum as the relevant control parameter.
Volume 124, number 6,7
PHYSICS LETTERS A
Weise, and O.E. RGssler for discussions. Partial support of this work from the Stiftung Volkswagenwerk is gratefully acknowledged.
References [ 1] M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. A 30 (1984) 1960; T. Bohr, P. Bak and M.H. Jensen, Phys. Rev. A 30 (1984) 1970; P. Bak, T. Bohr and M.H. Jensen, Phys. Ser. T 9 (1985) 50; P. Cvitanovic, M.H. Jensen, L.P. Kadanoff and I. Procaccia, Phys. Rev. Lett. 55 (1985) 343. [2] J. Stavans, F. Heslot and A. Libchaber, Phys. Rev. Lett. 55 (1985) 569; A.P. Fein, M.S. Heutmaker and J.P. Gollub, Phys. Ser. T 9 (1985) 79; H. Haucke and R. Ecke, Physica D, to be published. [3] H.L. Swinney and J. Maselko, Phys. Rev. Lett. 55 (1985) 2366.
5 October 1987
[ 41 G.A. Held and C. Jeffries, Phys. Rev. Lett. 56 (1986) 1183; S. Martin and W. Martienssen, Phys. Rev. Lett. 56 (1986) 1522; E.G. Gwinn and R.M. Westervelt, Phys. Rev. Lett. 57 (1986) 1060. [5] J. Peinke, A. Mtlhlbach, B. Riihricht, B. Wessely, J. Mannhart, J. Parisi and R.P. Huebener, Physica D 23 (1986) 176; J. Parisi, J. Peinke, B. Rohricht and KM. Mayer, Z. Naturforsch. 42 a (1987) 329; R.P. Huebener, K.M. Mayer, J. Parisi, J. Peinke and B. Rohricht, in: Proc. Int. Conf. on Physics of chaos and systems far from equilibrium, Monterey, 1987, to be published in Nucl. Phys. B; J. Peinke, J. Parisi, B. Rohricht, B. Wessely and K.M. Mayer, Europhys. Lett., submitted for publication. [6] E. Scholl, in: Festkorperprobleme, Vol. 26, ed. P. Grosse (Vieweg, Braunschweig, 1986) p.309; Springer series in synergetics, Vol. 35. Nonequilibrium phase transitions in semiconductors (Springer, Berlin, 1987), to be published. [ 71 D. Jlger, H. Baumann and R. Symanczyk, Phys. Lett. A 177 (1986) 141. [ 81 K.M. Mayer, R. Gross, J. Parisi, J. Peinke and R.P. Huebener, Solid State Commun., to be published.
339