- Email: [email protected]
Crisis in electrical behavior of the TlInSe2 semiconducting compound
PHYSICA ELgEVIER
Physica D 93 (1996) 157-164
Crisis in electrical behavior of the TlInSe2 semiconducting compound Ch. Karakotsou, A.N. Anagnostopoulos
*
Aristotle Universi~. , of Thessaloniki, Solid State Section 313-1, 54006 Thessaloniki, Greece
Received 31 July 1995; revised 13 November 1995; accepted 24 November 1995 Communicated by Y. Kuramoto
Abstract TllnSe2 is a ternary semiconductor exhibiting S-type I - U characteristics. In their negative differential resistance (NDR) region voltage oscillations of non-periodic nature appear. The corresponding phase portraits consist of two interacting subattractors. Transitions between them become more frequent with increasing current values I. The distribution of the times r, between successive transitions, and the corresponding phase portraits indicate that in this system a crisis takes place accompanied by an intermittent behavior of the corresponding orbits.
1. Introduction TllnSe2 is a member of the family of the ternary semiconductors with general formula T l x l n l - x S e , with x = 0.0, 0.1 . . . . . 0.9, 1.0. Its stoichiometry corresponds to x = 0.5. It has been shown experimentally that most of the members of this family are semiconductors with an S-type current-voltage (l-U) characteristic which includes an extended negative differential resistance (NDR) region [1,2]. Chaotic oscillations appear also in the NDR region of the very similar compounds TlInTe2 and T1GaTe2. They were of deterministic chaotic nature as it was proved by methods of nonlinear dynamics [3,4]. The minimal voltage and current values at which the I - U characteristic is entering the NDR region are termed threshold or critical voltage Vc and threshold or critical current Ic, respectively [4]. In Ref. [1] it was shown that the appearance of this region is accompa-
* Corresponding author.
nied by generally non-periodic voltage oscillations, of considerable amplitudes. In the present report we present measurements of such oscillations at two different levels in TlInSe2 and their interpretation in terms of a crisis of the corresponding attractors. To the best of our knowledge this is one of the very few reports on the appearance of crisis phenomena in bulk semiconductors up to room temperature.
2. Experimental part 2.1. The m a t e r i a l
TlInSe2 samples were grown by direct fusion of their constituent elements, i.e. stoichiometric amounts of high purity elements T1, In and Se were inserted in vacuum scaled quartz ampoules which were subsequently placed in a furnace. The ampoule was kept at a temperature of ~ 1 3 0 0 K for 10 days and then the temperature was gradually decreased [2].
0167-2789/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0167-2789(95)00295-2
158
Ch. Karakotsou, A.N. Anagno stopoulos / Physica D 93 (1996) 157-164
The crystals grown by this method were glossy bundles of black, thin, parallel fibers. Their stoichiometry was examined with a special microanalyzing unit (AN 10.000 LINK) adapted to a scanning electron microscope. The crystals were compositionally homogeneous over their whole volume. The determined fluctuations of the stoichiometry were within the error limits of the instrument, i.e. < 4%. The crystals were found to be p-type semiconductors. This was proved by using the thermal-EMF technique. It is reported that TllnSe2 possesses an indirect gap of about 1.14eV at room temperature [1,5] and its structure can be described as consisting of [In3+Se2-] - chains, which extend along the crystallographic c-axis of the material, i.e. parallel to the fibers. Such negatively charged chains are kept together by T1+ cations, forming a tetragonal lattice of group symmetry D 18 - I 4 / m c m [5,6]. The dimensions of the crystals used in the electrical measurements were ~ 1 x 0.5 x 0.5 mm 3. Contacts were formed by Ag in the form of a conductive paste (Elecolit 336). Their ohmic behavior, and their low resistivity, was proved by the four points technique. A current flow along the c-axis of the samples was provided by a pair of electrodes deposited on two opposite surfaces of the rectangular-shaped samples, insert in Fig. 1. A current source (model 225, Keithley Instr) was used to control the current during the mea-
surements of the S-type I - U characteristics. The corresponding voltage drop was registered by a Keithley voltmeter (model 610 A). Voltage oscillations that appeared in the NDR region of the I - U curves were monitored by a H A M E G (HM 408&HM 8148-2) storage oscilloscope. The samples were mounted on the copper cold-finger of a liquid Helium cryostat (Air Products CSA-202) enabling electrical measurements in the temperature range 10-400 K. A copper cup was used to minimize thermal losses and to shield the samples electrically. Appropriate feed-throughs were used to allow the electrical wiring inside the cryostat. All measurements were performed in a vacuum better than 10 -5 Torr. Coaxial cables were used in the circuitry to avoid influence of external noises. With all these precautions noise was restricted to a level below 0.1%. 2.2. E l e c t r i c a l m e a s u r e m e n t s
In Fig. 1 representative current-voltage characteristics are shown as registered on the same TlInSe2 sample at different temperatures. Voltage oscillations were observed in their NDR region just after exceeding Icr as well as at higher values of I. In this region, the voltage values of the I - U curve were obtained by averaging over long time intervals. From the curves of Fig. 1 it can be concluded that with decreasing temperature
3,10"2. 330 K 300K /V
~2.10"; E 260K
1,10";
o
26o
'
E(V/cm)
46o
66o
Fig. 1. Representative J - E characteristics as registered on TllnSe2 at different ambient temperatures,
V Fig. 2. The real form of the I - U characteristic. The NDR region consists of two branches.
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
159
1.0-
L
~
u (v)
0.5 O.O-
U(V) -
~ 50 ms 1ms
0.5
Iso.,v (b)
(al -1.5 0
-1.0
I 10
I
i
I
50
30
I
I
10
0
30
ms)
I
1 ms i
50
t (m 0.8
0,Z~"
u(v) 0.0-
-
0.4-
-
0.8
(c) I
0
10
I
I
30
I
I
50
Fig. 3. Representative voltage oscillations in the NDR region of the same 1 - U characteristic at 300 K for different values of the control parameter I higher than lc.: (a) 0.7 mA; (b) 1 mA; (c) 2 mA for the same registration time of 40ms. Oscillations in the upper and lower levels in these plots, although of considerable amplitudes, are rather suppressed because of the selected scale. For this reason corresponding enlargements are presented as inserts in (b).
T, Icr decreases, Vcr increases and the N D R region b e c o m e s m o r e pronounced.
N D R region, o f two individual branches which cannot
The N D R region o f the experimental I - U curves
be m o n i t o r e d separately, since voltage values switch b e t w e e n them. I - U characteristics have generally the
shown in Fig. 1 is the result o f averaged oscillating
f o r m shown in Fig. 2. R e g i o n (I) is o h m i c and no in-
voltage values. In fact a real I - U curve consists, in its
stabilities occur. In region (II), there are two branches:
160
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
an ohmic stable one and a second exhibiting NDR where instabilities occur. If their amplitudes are high enough to intersect the stable branch, the operation point is transferred there. In this case, the corresponding voltage shows only a transient behavior from the oscillating state to the stable one. In region (III) both branches possess NDR where oscillations occur. If these oscillations have sufficient amplitudes, switching between them takes place. Finally, in region (IV) the two branches are merging to form a single one, where instabilities, but no switching, are observed. So depending on the value of the current flowing through the sample, no voltage oscillations [region (I)], transient voltage oscillations [region (II)], switching between two oscillating states [region (III)] or highfrequency voltage oscillations [region (IV)] can be observed. In Fig. 3 high-frequency voltage oscillations are presented, as measured in the NDR region of the same sample TllnSe2 at 300 K, for different current values in region (III). As it can be deduced from these plots they are consisting of oscillations at two different levels, i.e. for long stretches of time the orbit remains
in one band, where it oscillates non-periodically and occasionally bursts out of it oscillating in a second lower band, returning to the first one, after it has spent some time in the lower one. With increasing current the transitions between bands become more frequent, i.e. the time intervals between successive bursts become shorter. Grebogi et al. consider three types for changes that attractors can undergo as a system parameter is varied [7-11]. The first type leads to the sudden destruction of a chaotic attractor. The second type leads to the sudden widening of a chaotic attractor. In the third type of change two or more chaotic attractors merge to form a single chaotic one, which can be more extended in the phase space than the union of the attractors before the change. The characteristic temporal behavior, which occurs for the attractor-widening and attractormerging crisis, is called crisis induced intermittency. In Refs. [7,8,10] signals quite similar to those of Fig. 3 were presented. They are created by the merging of two chaotic attractors [10]. As the control parameter p is increased both attractors approach their common basin boundary, touch each other at the critical
4000
20O0
dO (V d-~ s)
O-
(a) - 2000
-1000
I
I
I
0
1030
2000
U (mV)
Fig. 4. Phase portraits obtained at different values of 1 of the time series of Figs. 3(a)-(c). With increasing I the transitions between the two subattractors become more frequent.
161
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
7000-
2000-
d~
V (w)
-
3000 -
(b) -
8000 -700
I
-200
[
I
300
800
U(mv) 9000
dU ( v E T -g- )
2ooo-
(c -5000 -500
I
I
-100
- 300
I
700
U (mY)
Fig. 4. (continued)
value p = Pc and finally, for p > Pc, overlap. A direct consequence of this behavior is that an orbit spends a lot of time in the region of one of the two attractors. After such a time interval, the orbit abruptly exits this
region and spends a long time interval in the region of the second attractor and so on. Thus, for appropriate values of the control parameter there is one common attractor on which the trajectories switch between
162
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
different behaviors intermittently [10], i.e. for finite time stretches the trajectories orbit on the individual attractors. The theory developed by Grebogi, Ott and York [7-9] predicts that in such a case, the behavior of the system experiences the following features: (i) The time series consists of characteristic temporal behavior between two states [8]. (ii) For a smooth distribution of initial conditions, the time r describing successive switches between the two states (for a constant value of the control parameter p) is exponentially distributed, according to the expression [10]: e ( r ) = ~ 1 e(-r/(r>),
(1)
(r)
where (r) is the averaged time between successive transition. These features are characterizing every kind of crisis. (iii) For a large class of dynamical systems which exhibits crises the averaged time (r) scales with the control parameter p, according to the following law [10]: (r) ~ (p - pc) - r ,
(2)
where Pc is the value of the control parameter at the crisis and y is a critical exponent characteristic of the system.
The same class of dynamical systems when the intermittent behavior of the orbit is caused by external noise is present in them obey the following law for (r) [12,13]: (r> = cr-×g ((p - p c ) / o ) ,
where a corresponds to the level of noise present in the system and g is a function describing the spectral distribution of the noise. Analyzing our data in terms of the features listed above, we present in Fig. 4 the corresponding phase portraits of the time series of Fig. 3. It is obvious from these plots that there is a common attractor of our system consisting of two individual subattractors with the orbit bursting occasionally between them. With increasing I (which is our control parameter) these transitions become more frequent. A representative distribution P ( r ) vs. r is shown in Fig. 5. This plot was obtained for I = 1 mA with a smooth distribution of the initial conditions. Experimental data are presented by closed circles while the solid line represents a least squares fit. Its slope gives an estimate for 1/(r). The value obtained for (r) from this plot is 0.28, while that obtained by averaging the directly measured r ' s was 0.23. Similar results were obtained also for other values of 1.
4
-
•
•
•
2
1.
0
~0
Q3
(3)
0.6
0.9
..~
1.2
Fig. 5. A representative distribution of In P(r) vs. r as obtained for 1 = 1 mA.
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
163
t n <'t>_l
-1.5
2
-3
|
i
i
-i
0
tn (I'I c ) Fig. 6. A double logarithmic plot of (r) vs. ( l - l c ) .
In Fig. 6 a plot of In(r) vs. ln(l-lc) is presented. The experimental values in this plot are denoted by closed circles, while the solid line represents a least squares fit with Eq. (2). lc is the critical current value in the crisis having an estimated value of 0.63 mA. The critical current value is minimizing the scattering between the experimental points and the line representing Eq. (2), similar to the method reported in Ref. [10]. An exponential law obviously holds, resulting in a line whose slope leads to a critical exponent F ----0.51. Within the error limits of the measurements this value coincides with the theoretically predicted one (½ _< F -< 3) for the case of a strictly dissipative system. However, this is not a decisive evidence that intermittency indicated by Figs. 1 and 2 is a crisis induced one since even traces of noise can severely affect the obtained results [12,13] "bumbing" the system, which is close to the critical point, into and out of the crisis region. Furthermore, the amount of the experimental data of Fig. 6 does not exclude fittings with relation (3) [12,13].
3. Discussion The phase portraits of Fig. 4 clearly indicate that with increasing I-Ic the two subattractors interact, tending to their total merging for higher I values, a
2.50 "V"
2001.50'
1.00"
0.50" 0.00 0.0
!°% 2'°
m
i
2.0
4.0
6.0
8D m
Fig. 7. The slope v of the linear parts of the double logarithmic plot of C(l) vs. I plotted as a function of the embedding dimension m. In the insert the Kolmogorov entropy K as a function of m is shown.
feature characteristic for a crisis induced intermittency. Therefore we suggest that this semiconducting system undergoes an intermittency caused by the merging of the two chaotic subattractors. Since our system is an experimental one, possible presence of very low uncontrolled noise can also trigger it to the switching
164
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
behavior. This fact does not influence the resulting oscillatory behavior, which was proved to be of chaotic
References
nature. The dynamics of the system was checked quantitatively, applying the well-known method of Grassberger and Proccacia, which gave very satisfactory results in similar cases [16,17], to determine the minimum embedding dimension m, fractal dimension D2 and the Kolmogorov entropy [ 14,15]. A representative example is given in Fig. 7, where the slope v of the linear part of the double logarithmic plot of the correlation integral C(1) vs. the hypercube length l is plotted as a function of the embedding dimension m for 1 = 1 mA. v tends to saturate at the value u = 2.12. The corresponding Kolmogorov entropy K is plotted as a function of m, insert in Fig. 7. It tends to the positive and finite value K ----0.425 bits/r. Values of the correlation dimension D2 were calculated for different distances of the critical point lc. The correlation dimension D2 increases slightly with I - l c saturating at a value of 2.434-0.039 for I = 3 mA. We have to mention here, that in the case of the very related fibrous compound TllnTe2 time series of similar form were registered. Their evaluation gave results very close to those reported in the present work [18].
[1] M.E Hanias, A.N. Anagnostopoulos, K. Kambas and J. Spyridelis, Mat. Res. Bull. 27 (1992) 25. [2] Ch. Karakotsou, A.N. Anagnostopoulos, K. Kambas and J. Spyridelis, Mat. Res. Bull. 30 (1995) 795-811. [3] M. Hanias, A.N. Anagnostopoulos, K. Kambas and J. Spyridelis, Phys. Rev. B 47 (1993) 4261. [4] G.J. Abutalybov, S.G. Abdullaeva and N.M. Zeinalov, Soy. Phys. Semicond. 16 (1982) 1348. [5] M. Hanias, A. Anagnostopolous, K. Kambas and J. Spyridelis, Physica B 160 (1989) 154. [6] D. Miiller, J. Eulenberger and H. Hahn, J. Anorg. Allg. Chem. 398 (1973) 207. [7] C. Grebogi, E. Ott, E Romeiras and J.A. Yorke, Phys. Rev. A 36 (1987) 5365. [8] E. Ott, Chaos in Dynamical Systems (Cambridge Univ. Press, Cambridge, 1993) pp. 277-291. [9] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Len. 57 (1986) 1284. [10] W.L. Diuo, S. Rauseo, R. Cawley, C. Grebogi, G.H. Hsu, E. Kostelich, E. Ott, H.T. Savage, R. Seguan, M.L. Spano and V.A. Yorke, Phys. Rev. Lett. 63 (1989) 923. [ll] C. Grebogi, E. Ott and J.A. Yorke, Science 238 (1987) 585. [12] J.C. Sommerer, E. Ott and C. Grebogi, Phys. Rev. A 43 (1991) 1754. [13] J.C. Sommerer, W.L. Ditto, C. Grebogi, E. Ott and M.L. Spano, Phys. Rev. Lett. 66 (1991) 1947. [14] P. Grassberger and I. Procaccia, Phys. Rev. A 28 (1983) 2591. [15] P. Grassberger and I. Procaccia, Physica D 9 (1983) 189. [16] Ch. Karakotsou, A.N. Anagnostopoulos, K. Kambas and J. Spyridelis, Phys. Rev. 46 (1992) 16 144. [17] M.P. Hanias and A.N. Anagnostopoulos, Phys. Rev. 47 (1993) 4261. [18] E Heyder, Diploma thesis, University of Frankfurt am Main (1994).
Acknowledgements The authors are grateful Prof. G. Bleris for helpful edge financial support in 930255 research project of
to Prof. J. Spyridelis and discussions and acknowlthe frame of CIPA-CT the EC.
Physica D 93 (1996) 157-164
Crisis in electrical behavior of the TlInSe2 semiconducting compound Ch. Karakotsou, A.N. Anagnostopoulos
*
Aristotle Universi~. , of Thessaloniki, Solid State Section 313-1, 54006 Thessaloniki, Greece
Received 31 July 1995; revised 13 November 1995; accepted 24 November 1995 Communicated by Y. Kuramoto
Abstract TllnSe2 is a ternary semiconductor exhibiting S-type I - U characteristics. In their negative differential resistance (NDR) region voltage oscillations of non-periodic nature appear. The corresponding phase portraits consist of two interacting subattractors. Transitions between them become more frequent with increasing current values I. The distribution of the times r, between successive transitions, and the corresponding phase portraits indicate that in this system a crisis takes place accompanied by an intermittent behavior of the corresponding orbits.
1. Introduction TllnSe2 is a member of the family of the ternary semiconductors with general formula T l x l n l - x S e , with x = 0.0, 0.1 . . . . . 0.9, 1.0. Its stoichiometry corresponds to x = 0.5. It has been shown experimentally that most of the members of this family are semiconductors with an S-type current-voltage (l-U) characteristic which includes an extended negative differential resistance (NDR) region [1,2]. Chaotic oscillations appear also in the NDR region of the very similar compounds TlInTe2 and T1GaTe2. They were of deterministic chaotic nature as it was proved by methods of nonlinear dynamics [3,4]. The minimal voltage and current values at which the I - U characteristic is entering the NDR region are termed threshold or critical voltage Vc and threshold or critical current Ic, respectively [4]. In Ref. [1] it was shown that the appearance of this region is accompa-
* Corresponding author.
nied by generally non-periodic voltage oscillations, of considerable amplitudes. In the present report we present measurements of such oscillations at two different levels in TlInSe2 and their interpretation in terms of a crisis of the corresponding attractors. To the best of our knowledge this is one of the very few reports on the appearance of crisis phenomena in bulk semiconductors up to room temperature.
2. Experimental part 2.1. The m a t e r i a l
TlInSe2 samples were grown by direct fusion of their constituent elements, i.e. stoichiometric amounts of high purity elements T1, In and Se were inserted in vacuum scaled quartz ampoules which were subsequently placed in a furnace. The ampoule was kept at a temperature of ~ 1 3 0 0 K for 10 days and then the temperature was gradually decreased [2].
0167-2789/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0167-2789(95)00295-2
158
Ch. Karakotsou, A.N. Anagno stopoulos / Physica D 93 (1996) 157-164
The crystals grown by this method were glossy bundles of black, thin, parallel fibers. Their stoichiometry was examined with a special microanalyzing unit (AN 10.000 LINK) adapted to a scanning electron microscope. The crystals were compositionally homogeneous over their whole volume. The determined fluctuations of the stoichiometry were within the error limits of the instrument, i.e. < 4%. The crystals were found to be p-type semiconductors. This was proved by using the thermal-EMF technique. It is reported that TllnSe2 possesses an indirect gap of about 1.14eV at room temperature [1,5] and its structure can be described as consisting of [In3+Se2-] - chains, which extend along the crystallographic c-axis of the material, i.e. parallel to the fibers. Such negatively charged chains are kept together by T1+ cations, forming a tetragonal lattice of group symmetry D 18 - I 4 / m c m [5,6]. The dimensions of the crystals used in the electrical measurements were ~ 1 x 0.5 x 0.5 mm 3. Contacts were formed by Ag in the form of a conductive paste (Elecolit 336). Their ohmic behavior, and their low resistivity, was proved by the four points technique. A current flow along the c-axis of the samples was provided by a pair of electrodes deposited on two opposite surfaces of the rectangular-shaped samples, insert in Fig. 1. A current source (model 225, Keithley Instr) was used to control the current during the mea-
surements of the S-type I - U characteristics. The corresponding voltage drop was registered by a Keithley voltmeter (model 610 A). Voltage oscillations that appeared in the NDR region of the I - U curves were monitored by a H A M E G (HM 408&HM 8148-2) storage oscilloscope. The samples were mounted on the copper cold-finger of a liquid Helium cryostat (Air Products CSA-202) enabling electrical measurements in the temperature range 10-400 K. A copper cup was used to minimize thermal losses and to shield the samples electrically. Appropriate feed-throughs were used to allow the electrical wiring inside the cryostat. All measurements were performed in a vacuum better than 10 -5 Torr. Coaxial cables were used in the circuitry to avoid influence of external noises. With all these precautions noise was restricted to a level below 0.1%. 2.2. E l e c t r i c a l m e a s u r e m e n t s
In Fig. 1 representative current-voltage characteristics are shown as registered on the same TlInSe2 sample at different temperatures. Voltage oscillations were observed in their NDR region just after exceeding Icr as well as at higher values of I. In this region, the voltage values of the I - U curve were obtained by averaging over long time intervals. From the curves of Fig. 1 it can be concluded that with decreasing temperature
3,10"2. 330 K 300K /V
~2.10"; E 260K
1,10";
o
26o
'
E(V/cm)
46o
66o
Fig. 1. Representative J - E characteristics as registered on TllnSe2 at different ambient temperatures,
V Fig. 2. The real form of the I - U characteristic. The NDR region consists of two branches.
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
159
1.0-
L
~
u (v)
0.5 O.O-
U(V) -
~ 50 ms 1ms
0.5
Iso.,v (b)
(al -1.5 0
-1.0
I 10
I
i
I
50
30
I
I
10
0
30
ms)
I
1 ms i
50
t (m 0.8
0,Z~"
u(v) 0.0-
-
0.4-
-
0.8
(c) I
0
10
I
I
30
I
I
50
Fig. 3. Representative voltage oscillations in the NDR region of the same 1 - U characteristic at 300 K for different values of the control parameter I higher than lc.: (a) 0.7 mA; (b) 1 mA; (c) 2 mA for the same registration time of 40ms. Oscillations in the upper and lower levels in these plots, although of considerable amplitudes, are rather suppressed because of the selected scale. For this reason corresponding enlargements are presented as inserts in (b).
T, Icr decreases, Vcr increases and the N D R region b e c o m e s m o r e pronounced.
N D R region, o f two individual branches which cannot
The N D R region o f the experimental I - U curves
be m o n i t o r e d separately, since voltage values switch b e t w e e n them. I - U characteristics have generally the
shown in Fig. 1 is the result o f averaged oscillating
f o r m shown in Fig. 2. R e g i o n (I) is o h m i c and no in-
voltage values. In fact a real I - U curve consists, in its
stabilities occur. In region (II), there are two branches:
160
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
an ohmic stable one and a second exhibiting NDR where instabilities occur. If their amplitudes are high enough to intersect the stable branch, the operation point is transferred there. In this case, the corresponding voltage shows only a transient behavior from the oscillating state to the stable one. In region (III) both branches possess NDR where oscillations occur. If these oscillations have sufficient amplitudes, switching between them takes place. Finally, in region (IV) the two branches are merging to form a single one, where instabilities, but no switching, are observed. So depending on the value of the current flowing through the sample, no voltage oscillations [region (I)], transient voltage oscillations [region (II)], switching between two oscillating states [region (III)] or highfrequency voltage oscillations [region (IV)] can be observed. In Fig. 3 high-frequency voltage oscillations are presented, as measured in the NDR region of the same sample TllnSe2 at 300 K, for different current values in region (III). As it can be deduced from these plots they are consisting of oscillations at two different levels, i.e. for long stretches of time the orbit remains
in one band, where it oscillates non-periodically and occasionally bursts out of it oscillating in a second lower band, returning to the first one, after it has spent some time in the lower one. With increasing current the transitions between bands become more frequent, i.e. the time intervals between successive bursts become shorter. Grebogi et al. consider three types for changes that attractors can undergo as a system parameter is varied [7-11]. The first type leads to the sudden destruction of a chaotic attractor. The second type leads to the sudden widening of a chaotic attractor. In the third type of change two or more chaotic attractors merge to form a single chaotic one, which can be more extended in the phase space than the union of the attractors before the change. The characteristic temporal behavior, which occurs for the attractor-widening and attractormerging crisis, is called crisis induced intermittency. In Refs. [7,8,10] signals quite similar to those of Fig. 3 were presented. They are created by the merging of two chaotic attractors [10]. As the control parameter p is increased both attractors approach their common basin boundary, touch each other at the critical
4000
20O0
dO (V d-~ s)
O-
(a) - 2000
-1000
I
I
I
0
1030
2000
U (mV)
Fig. 4. Phase portraits obtained at different values of 1 of the time series of Figs. 3(a)-(c). With increasing I the transitions between the two subattractors become more frequent.
161
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
7000-
2000-
d~
V (w)
-
3000 -
(b) -
8000 -700
I
-200
[
I
300
800
U(mv) 9000
dU ( v E T -g- )
2ooo-
(c -5000 -500
I
I
-100
- 300
I
700
U (mY)
Fig. 4. (continued)
value p = Pc and finally, for p > Pc, overlap. A direct consequence of this behavior is that an orbit spends a lot of time in the region of one of the two attractors. After such a time interval, the orbit abruptly exits this
region and spends a long time interval in the region of the second attractor and so on. Thus, for appropriate values of the control parameter there is one common attractor on which the trajectories switch between
162
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
different behaviors intermittently [10], i.e. for finite time stretches the trajectories orbit on the individual attractors. The theory developed by Grebogi, Ott and York [7-9] predicts that in such a case, the behavior of the system experiences the following features: (i) The time series consists of characteristic temporal behavior between two states [8]. (ii) For a smooth distribution of initial conditions, the time r describing successive switches between the two states (for a constant value of the control parameter p) is exponentially distributed, according to the expression [10]: e ( r ) = ~ 1 e(-r/(r>),
(1)
(r)
where (r) is the averaged time between successive transition. These features are characterizing every kind of crisis. (iii) For a large class of dynamical systems which exhibits crises the averaged time (r) scales with the control parameter p, according to the following law [10]: (r) ~ (p - pc) - r ,
(2)
where Pc is the value of the control parameter at the crisis and y is a critical exponent characteristic of the system.
The same class of dynamical systems when the intermittent behavior of the orbit is caused by external noise is present in them obey the following law for (r) [12,13]: (r> = cr-×g ((p - p c ) / o ) ,
where a corresponds to the level of noise present in the system and g is a function describing the spectral distribution of the noise. Analyzing our data in terms of the features listed above, we present in Fig. 4 the corresponding phase portraits of the time series of Fig. 3. It is obvious from these plots that there is a common attractor of our system consisting of two individual subattractors with the orbit bursting occasionally between them. With increasing I (which is our control parameter) these transitions become more frequent. A representative distribution P ( r ) vs. r is shown in Fig. 5. This plot was obtained for I = 1 mA with a smooth distribution of the initial conditions. Experimental data are presented by closed circles while the solid line represents a least squares fit. Its slope gives an estimate for 1/(r). The value obtained for (r) from this plot is 0.28, while that obtained by averaging the directly measured r ' s was 0.23. Similar results were obtained also for other values of 1.
4
-
•
•
•
2
1.
0
~0
Q3
(3)
0.6
0.9
..~
1.2
Fig. 5. A representative distribution of In P(r) vs. r as obtained for 1 = 1 mA.
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
163
t n <'t>_l
-1.5
2
-3
|
i
i
-i
0
tn (I'I c ) Fig. 6. A double logarithmic plot of (r) vs. ( l - l c ) .
In Fig. 6 a plot of In(r) vs. ln(l-lc) is presented. The experimental values in this plot are denoted by closed circles, while the solid line represents a least squares fit with Eq. (2). lc is the critical current value in the crisis having an estimated value of 0.63 mA. The critical current value is minimizing the scattering between the experimental points and the line representing Eq. (2), similar to the method reported in Ref. [10]. An exponential law obviously holds, resulting in a line whose slope leads to a critical exponent F ----0.51. Within the error limits of the measurements this value coincides with the theoretically predicted one (½ _< F -< 3) for the case of a strictly dissipative system. However, this is not a decisive evidence that intermittency indicated by Figs. 1 and 2 is a crisis induced one since even traces of noise can severely affect the obtained results [12,13] "bumbing" the system, which is close to the critical point, into and out of the crisis region. Furthermore, the amount of the experimental data of Fig. 6 does not exclude fittings with relation (3) [12,13].
3. Discussion The phase portraits of Fig. 4 clearly indicate that with increasing I-Ic the two subattractors interact, tending to their total merging for higher I values, a
2.50 "V"
2001.50'
1.00"
0.50" 0.00 0.0
!°% 2'°
m
i
2.0
4.0
6.0
8D m
Fig. 7. The slope v of the linear parts of the double logarithmic plot of C(l) vs. I plotted as a function of the embedding dimension m. In the insert the Kolmogorov entropy K as a function of m is shown.
feature characteristic for a crisis induced intermittency. Therefore we suggest that this semiconducting system undergoes an intermittency caused by the merging of the two chaotic subattractors. Since our system is an experimental one, possible presence of very low uncontrolled noise can also trigger it to the switching
164
Ch. Karakotsou, A.N. Anagnostopoulos/Physica D 93 (1996) 157-164
behavior. This fact does not influence the resulting oscillatory behavior, which was proved to be of chaotic
References
nature. The dynamics of the system was checked quantitatively, applying the well-known method of Grassberger and Proccacia, which gave very satisfactory results in similar cases [16,17], to determine the minimum embedding dimension m, fractal dimension D2 and the Kolmogorov entropy [ 14,15]. A representative example is given in Fig. 7, where the slope v of the linear part of the double logarithmic plot of the correlation integral C(1) vs. the hypercube length l is plotted as a function of the embedding dimension m for 1 = 1 mA. v tends to saturate at the value u = 2.12. The corresponding Kolmogorov entropy K is plotted as a function of m, insert in Fig. 7. It tends to the positive and finite value K ----0.425 bits/r. Values of the correlation dimension D2 were calculated for different distances of the critical point lc. The correlation dimension D2 increases slightly with I - l c saturating at a value of 2.434-0.039 for I = 3 mA. We have to mention here, that in the case of the very related fibrous compound TllnTe2 time series of similar form were registered. Their evaluation gave results very close to those reported in the present work [18].
[1] M.E Hanias, A.N. Anagnostopoulos, K. Kambas and J. Spyridelis, Mat. Res. Bull. 27 (1992) 25. [2] Ch. Karakotsou, A.N. Anagnostopoulos, K. Kambas and J. Spyridelis, Mat. Res. Bull. 30 (1995) 795-811. [3] M. Hanias, A.N. Anagnostopoulos, K. Kambas and J. Spyridelis, Phys. Rev. B 47 (1993) 4261. [4] G.J. Abutalybov, S.G. Abdullaeva and N.M. Zeinalov, Soy. Phys. Semicond. 16 (1982) 1348. [5] M. Hanias, A. Anagnostopolous, K. Kambas and J. Spyridelis, Physica B 160 (1989) 154. [6] D. Miiller, J. Eulenberger and H. Hahn, J. Anorg. Allg. Chem. 398 (1973) 207. [7] C. Grebogi, E. Ott, E Romeiras and J.A. Yorke, Phys. Rev. A 36 (1987) 5365. [8] E. Ott, Chaos in Dynamical Systems (Cambridge Univ. Press, Cambridge, 1993) pp. 277-291. [9] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Len. 57 (1986) 1284. [10] W.L. Diuo, S. Rauseo, R. Cawley, C. Grebogi, G.H. Hsu, E. Kostelich, E. Ott, H.T. Savage, R. Seguan, M.L. Spano and V.A. Yorke, Phys. Rev. Lett. 63 (1989) 923. [ll] C. Grebogi, E. Ott and J.A. Yorke, Science 238 (1987) 585. [12] J.C. Sommerer, E. Ott and C. Grebogi, Phys. Rev. A 43 (1991) 1754. [13] J.C. Sommerer, W.L. Ditto, C. Grebogi, E. Ott and M.L. Spano, Phys. Rev. Lett. 66 (1991) 1947. [14] P. Grassberger and I. Procaccia, Phys. Rev. A 28 (1983) 2591. [15] P. Grassberger and I. Procaccia, Physica D 9 (1983) 189. [16] Ch. Karakotsou, A.N. Anagnostopoulos, K. Kambas and J. Spyridelis, Phys. Rev. 46 (1992) 16 144. [17] M.P. Hanias and A.N. Anagnostopoulos, Phys. Rev. 47 (1993) 4261. [18] E Heyder, Diploma thesis, University of Frankfurt am Main (1994).
Acknowledgements The authors are grateful Prof. G. Bleris for helpful edge financial support in 930255 research project of
to Prof. J. Spyridelis and discussions and acknowlthe frame of CIPA-CT the EC.