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Phase diagrams for periodically driven Gunn diodes
PHYSICA
Physica D 66 (1993) 143-153 North-Holland
SDI: 0167-2789(93)E0024-U
Phase diagrams for periodically driven Gunn diodes Erik Mosekilde, Jesper Skovhus Thomsen, Carsten Knudsen and Rasmus Feldberg System Dynamics Group, Physics Laboratory 111, The Technical University of Denmark, DK-2800 Lyngby, Denmark
A simulation model of the electron transfer mechanism for bulk negative differential conductivityin n-GaAs is applied to investigate the formation and propagation of subsequent high field domains in the presence of an external microwave field. Phase diagrams for the distribution of modes are drawn for two different values of the non-equivalent intervalley relaxation time. The domain mode is found to entrain with the microwavesignal in a variety of different frequency-locked solutions. At higher microwave amplitudes, period-doublingand other forms of mode-convertingbifurcations take place. In this region, spatio-temporal chaos may also be observed, as indicated by a positive value of the largest Lyapunov exponent. At still higher amplitudes, transitions to delayed, quenched, and limited space charge accumulation modes occur. The Dq-curve characterizing the multifractal structure is calculated for a typical chaotic solution.
1. Introduction
It is well known [1-3] that application of sufficiently high drift fields to n-type GaAs and a n u m b e r of other I I I - V and I I - V I compound semiconductors can give rise to a far-fromequilibrium phase transition in which the spatially homogeneous distribution of conduction electrons becomes unstable and a spontaneous formation of traveling high field domains (Gunn domains) takes place. In the external circuit, the formation and propagation of these domains give rise to current oscillations with a frequency determined by the sample length and the domain velocity. The phase transition is caused by a bulk negative differential conductivity ( B N D C ) associated with the transfer of electrons from the highmobility conduction band minimum to a set of low-mobility satellite valleys situated approximately A = 0 . 3 5 e V higher in the conduction band [4,5]. In thermal equilibrium and at low drift fields, only the low-lying F-minimum is populated, and the homogeneous electron distribution in space is maintained through dielectric relaxation and through diffusion. However as the drift field is increased, the electron gas
starts to heat up and, at a threshold field of about 3.5 kV/cm (for GaAs), the average electron energy becomes high enough for a population of the satellite valleys to begin. The effective density of states in these valleys exceeds that of the conduction band minimum by a factor of the order of 60, and at sufficiently high drift fields, the majority of electrons therefore occupy states in the satellite valleys. In the intermediate range of fields (i.e., between 3.5 kV/cm and approximately 8.0 kV/cm), the average electron mobility decreases with increasing field. This destabilizes the homogeneous electron distribution in space, and traveling high field domains are spontaneously generated with depletion of carriers in the leading edge and accumulation in the trailing edge. The time constant controlling the growth of this instability
T_
noelOvo/OF [
(1)
is typically of the order of I ps. Here, e is the static dielectric constant, n o the thermal equilibrium electron concentration, and e the elementary charge. Ovd/OF is the slope of the average drift velocity versus field characteristic in the
0167-2789/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved
144
E. Mosekilde et al. / Phase diagrams ]br periodically driven Gunn diodes
interval with bulk negative differential conductivity. In practice, the domains usually originate from inhomogeneities at the cathode contact. Stationary conditions are reached when the voltage drop across the domain has become large enough to force the remaining crystal into the Ohmic regime. The domain then travels at a velocity equal to the carrier drift velocity in the Ohmic regions outside the domain [6]. The Gunn effect has been extensively studied both experimentally and by various analytical and numerical methods. In particular, Copeland [7] has shown that the presence of a microwave signal of sufficient amplitude and frequency can suppress the formation of domains and produce an alternative mode of operation which is referred to as the limited space charge accumulation (or LSA) mode [8,9]. With this mode, worthwhile microwave output powers and reasonable dc-to-rf conversion efficiencies can be obtained at frequencies up to about 120 GHz. The idea is that the microwave field during the negative half-period brings the crystal into the Ohmic regime long enough to fully quench all carrier density fluctuations built up in the preceding positive half-period. At the same time, the duration of the positive half-period is too short to allow carrier density fluctuations to amplify into macroscopic depletion and accumulation zones. Since the dielectric relaxation time in the Ohmic regime is a number of times smaller than the time constant r associated with the growth of inhomogeneities in the unstable regime, this type of operation is possible while exploiting on the average the BNDC for amplification of the microwave signal. At somewhat lower microwave frequencies, yet another mode is possible in which a domain is allowed to form during the positive half-period of the microwave signal, but is completely quenched before reaching the anode contact in the subsequent negative half-period [10]. With this type of operation, the quenched domain mode, 1 : 1 frequency locking is realized between the internally generated domain mode and the
external microwave signal. However, modern theory of nonlinear dynamics [11,12] suggests that the phase diagram is much more complicated than this. For an externally forced selfoscillating system, frequency locking should occur at all rational ratios between the two periods, and at higher forcing amplitudes various forms of mode-converting bifurcations are expected, including transitions to a chaotic dynamics. Using a slightly simplified model of the two valley system, we have previously examined these phenomena in considerable detail [13]. In particular, we have shown how the internally generated domain mode, by adjusting the time of domain formation and the speed of propagation, locks to the externally applied microwave signal to produce a variety of different periodic solutions. In the present paper we provide a more detailed overview of the distribution of these modes in the microwave frequency-microwave amplitude parameter plane. The mode distribution is calculated for two different values of the non-equivalent intervalley scattering time. The shape of the Arnol'd tongues is determined, the mode-converting bifurcations are illustrated, and the largest Lyapunov exponent is calculated to identify chaotic solutions. Although Copeland [7] noticed the possibility of tuning the microwave frequency to various harmonics of the transit time frequency, and although a similar observation was made by Chilton [9], we are not aware of detailed experimental investigations of the mode-locking phenomenon for periodically forced Gunn diodes that can support the present findings. Hopefully, such experiments will be performed in the near future. It may be noticed, however, that deterministic chaos and other highly nonlinear phenomena have been observed in connection with a variety of other semiconductor instabilities [14,15]. In particular, Richter et al. [16,17] have observed different types of intermittency developing during low-temperature impact ionization breakdown in extrinsic germanium. Using a model somewhat similar to the one present in
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
this paper, Bonilla and Higuera [18] have recently analysed the stability of the stationary solutions for finite length GaAs Gunn diodes under various biasing conditions.
tion in these states remains almost Maxwellian, with a temperature close to that of the crystal lattice. Thus vz{F } may be approximated by ve{V} = ~2 F ,
2. T h e t w o - v a l l e y m o d e l
We consider a one-dimensional model of a G a A s Gunn diode of length L. A drift field F d is applied along the crystal (x-direction), and we assume that the local electron concentration n(x, t) is distributed between high and low mobility states with densities nl(x, t) and nz(X, t), respectively. Accompanying the variation in the concentration of free carriers there is a local electric field F(x, t), as determined by Poisson's equation
OF Ox
---
e e
- (n I + n 2 -
no).
(2)
As before, e is the static dielectric constant for GaAs, and n o is the thermal equilibrium electron concentration. In the numerical calculations we have taken n o = 2.0 × 10el/m 3. For simplicity, the electrons have been assumed to carry a positive charge e. In the presence of the electric field and the carrier density gradients, the two groups of electrons produce currents of density ix(X, t) and jE(x, t), respectively. Assuming collision dominated behavior, these current densities may be expressed as Onl
Jl = nlevl { F} - eDl Ox
(3)
(5)
with [z2 being an appropriate mobility. In the following calculations we have taken D 1 = 0 . 0 2 0 m 2 / s , D 2 =0.0008m2/s, and /z2 = 0.032 m2/Vs. Fawcett et al. [19] have performed detailed calculations of the stationary electron transport properties for GaAs as a function of the drift field. In particular, these authors have calculated the velocity field characteristic vl{F } and the equilibrium fraction of electrons n2.eq/n o occupying states in the higher valleys. These calculations take into account a variety of realistic scattering mechanisms as well as non-parabolicity of the conduction band minimum and wavevector dependence of the periodic part of the electronic Bloch functions. To simplify the present treatment, we have fitted simple mathematical expressions to the numerical results obtained by Fawcett et al. Thus,
=Ip.IF(1-aF2) vI{F}
[ Ol,sat
On 2
J2 = n2 e v 2 { F }
--
eD2 ax
(4)
H e r e , vl{F(x, t)} and v2{F(x, t)} denote the local drift velocities of the two types of carriers, and D 1 and D 2 are appropriate diffusion constants. Because of a relatively efficient scattering within the upper valleys, the electron distribu-
for F-< F~at, for F > F~at
(6)
and n2,eq
no
1 + a exp[A/(kBT o + bE2)]
(7)
H e r e , /z~ = 0 . 5 0 m 2 / V s is the low field mobility associated with electrons in the conduction band minimum, Ol.sat 2.3 × 105 m/s is the high-field saturation drift velocity in this band, and F~at = 4 . 3 × 105 V/m is the field at which this saturation occurs. The parameter a is adjusted so as to provide a smooth transition between the two expressions for Vl{F ). n2,eq is the equilibrium concentration of free carriers in the upper valleys corresponding to the applied electric field F. A = 0.35 eV is the energy difference between the two sets of valleys, T Ois the lattice tempera=
and
145
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
146
ture, and the parameters a =0.33 and b = 6.0 x 10 -3 eV/(V/m) 2 were adjusted so as to fit the numerical results as well as possible, k B is Boltzmann's constant. As calculated from (6) and (7), the overall velocity-field characteristic is in close agreement with experimental results. In the present context, (6) and (7) are considered as local equilibrium relations expressing the local drift velocity v l ( x , t) and the local equilibrium concentration of carriers in the upper valleys ?/2,eq(X, t ) in terms of the local electric field F ( x , t) and the local electron concentration n ( x , t). Finally, the carrier density variations are obtained from the equations of continuity On 1 Ot
10j I e Ox
n2,eq -
n2
(8)
"r
and 01l 2 - -
Ot
1 Oj2 -
e Ox
~"
,
f
F ( x , t) dx = Vdll + A sin(2"rrft)] ,
(11)
0
where Vj = F j L is the applied drift voltage, f a n d A denote the frequency and the relative amplitude of the applied microwave signal, respectively. In all calculations we have assumed the average drift field to be Fa = 4.0 kV/cm. This is well within the B N D C region.
(9)
and
n2(O ,
t) = 0
(lOa)
at the cathode end, and neutrality n,(L,
L
rt2,eq -- 172 +
where the second term in each equation represents the scattering of electrons between the two sets of valleys, r being the non-equivalent intervalley scattering time. While the model assumes that drift and diffusion are in equilibrium with the local field and carrier density gradients, respectively, redistribution of carriers between the two sets of valleys is considered a dynamic process which involves time constants comparable to those controlling the domain dynamics. ~ is presumably the parameter which is known with least precision. For this reason we have investigated the behavior of the model both for ~- = 2 ps and for r-= 10ps. The former value corresponds more closely to experimental results on the relaxation of optically excited electrons [20]. As boundary conditions we have assumed thermal equilibrium n,(O, t) = n o
at the anode end. To facilitate domain formation near the cathode contact, we have introduced a slight variation of the specific resistance in the 10% of the crystal which is closest to the cathode. Here a single period of a sinusoidal perturbation of amplitude 0.01 × 1021/m 3 is applied. Finally, we have used a normalization condition of the form
t) + n 2 ( L , t) = n o
(lOb)
3.
Simulation
results
We have simulated the formation and propagation of Gunn domains in the presence of an external microwave field for more than 3000 combinations of the forcing parameters f and A. To solve the equations of motion the sample is divided into 150 sections, each of length Ax = 80 nm, providing a total of 300 dynamical variables. The spatial derivatives are all approximated by central differences while time derivatives are approximated by forward differences. The numerical integration of the set of 300 locally coupled ordinary differential equations is performed by an eight-stage R u n g e - K u t t a pair of order five and six [21]. The integration scheme includes variable time step and error control. As initial conditions we have assumed thermal equilibrium until, at time t = 0, the homogeneous drift voltage is applied to the crystal. Fig. 1 illustrates the onset of a 2:1 frequency locking as a microwave signal of frequency f = 12.5 G H z and relative amplitude A = 0.20 is applied to a stably oscillating Gunn diode. In the
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes F= 12.5 GHz, E E
A= 0.20
70-"
60. 50" 40. Z W 0 Z hi n" n," :D 0
30. 20'
10: 0
0.0
o
,,,,,,,
'6.'2. . . . . . . . 6.'4. . . . . . . . b.'6. . . . . . . . 6.'8. . . . . . . . i.b TIME (ns)
e
7
Fig. 1. O n s e t of a 2:1 frequency locking b e t w e e n the external microwave signal and the internally generated d o m a i n m o d e . T h e microwave signal is applied at time t = 0.4 ns.
top panel, the upper trace shows the temporal variation of the total current density
Jtot(t)
~F
=j,(x, t) +j:(x, t) + ~ c9-~-'
(12)
and the lower trace shows the corresponding variation in the applied voltage. By including the displacement term e(aF/Ot), jtot(t) becomes independent of x and proportional to the current in the external leads. The lower panel shows the formation and propagation of subsequent highfield domains. In this panel, the vertical axis represents the position along the crystal with the cathode end at the bottom. The time axis is common to both panels. The contour curves in the lower panel demarcate that part of the crystal in which the field at a given time is more than 10% higher than in the rest of the crystal. During the period of undisturbed oscillations, the current density is seen to rise to its Ohmic value of approximately 6 5 A / m m 2, remain Ohmic until a domain has been formed, and then decay towards a saturated value of approximately 40 A / m m 2. The current remains saturated as long as the domain propagates through the crystal to rise to its Ohmic value again as the domain disappears at the anode contact. With the assumed parameters, the period of the internal oscillation is approximately 116 ps corresponding
147
to an undisturbed Gunn frequency of fG,,n = 8.6GHz. When, at t = 0 . 4 n s , the microwave field is applied, the diode first passes through a transient period of approximately 0.3ns' duration. Thereafter, a 2 : l mode-locking is established, and a domain is formed precisely once for every two cycles of the external microwave field. This corresponds to a frequency of 6.25 GHz for the domain mode. By adjusting both the time of domain formation and the speed of propagation, the internal oscillation has thus reduced its frequency by nearly 30%. At the same time, the position at which the domain is first detected has also changed. Fig. 2 shows a phase plot of the stationary 2 : 1 solution. Here, we have plotted simultaneous values of the average electric field Fay and the total current density Jtot. By inspecting the figure, we see how the current drops during the formation of a domain, remains low during a negative swing of the microwave signal, and rises again as the domain exits the crystal. Awaiting the average field to become sufficiently high for a new domain to be formed, the current remains high during a second negative swing of the microwave signal. With a forcing amplitude A = 0.20, the 2:1 frequency-locked solution exists over the interval from f = 1 2 G H z to approximately 18.4GHz F= 12.5 GHz,
A= 0.20
80if-. E E
706050-
Z 14J
a I--Z W
n-
t-,,"-'1 (3
40-
3020
1.0'''2.b'
''3.b'''4.b''
'5.b'''6.b'''7.b
Fay (kV/cm) Fig. 2. Phase plot of the stationary 2:1 frequency-locked solution obtained for f = 12.5 G H z and A = 0.20.
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
148
[13]. A b o v e this latter frequency, the formation and propagation of a domain can no longer be c o m p l e t e d within two cycles of the microwave signal, the 2:1 mode-locking breaks down, and the oscillations b e c o m e quasiperiodic. For higher microwave frequencies, intervals exist in which the domain m o d e entrains into 3:1, 4:1, 5:1, etc., frequency-locked solutions. Between the main n : 1 intervals, a multitude of smaller-intervals are found, representing m o r e complex entrainments. Between the 2:1 and 3 : 1 intervals, for instance, we find intervals with 5 : 2, 7 : 3, 9 : 4, and 13 : 6 entrainment. Between the 1 : 1 and 2 : 1 intervals we find solutions with 2 domains for each 3 microwave cycles, 4 domains for each 5 cycles, or 7 domains for each 8 cycles. On the low frequency side of the 1:1 interval, on the other hand, we find solutions with 2 or 3 domains for each cycle of the microwave field. With increasing microwave amplitude, the modelocking intervals tend to broaden, The phase diagram in fig. 3 provides an overview of this variation. H e r e , the Arnol'd tongues delineate regions in p a r a m e t e r space in which particular m o d e - l o c k e d solutions can be observed. In the
intermediate range of forcing amplitudes, period-doubling and other mode-converting bifurcations take place. Finally, as the microwave amplitude becomes sufficiently high, a transition to a different set of modes occurs. If the frequency of the microwave signal is relatively low ( < 3 0 G H z ) and yet higher than the undisturbed Gunn frequency f(; ....... the quenched domain m o d e is found [13]. For microwave fieq u e n d e s below fo . . . . a delayed domain mode occurs in which each positive swing of the microwave signal sets off one domain. Finally, for microwave frequencies above approximately 3 0 G H z , a transition to limited space charge accumulation modes takes place. To examine the bifurcation structure in a little m o r e detail, we have performed a nearly vertical scan in the phase diagram for f = 12 GHz. The results of this are illustrated in fig. 4. It is seen how increase of the forcing amplitude lets us out of the 3 : 2 tongue, through a relatively narrow 7 : 6 tongue, and thereafter into the main 1:1 tongue. However, as we enter this tongue we pass through the 2:2, 4 : 4 and 8:8 solutions of a period-doubling cascade before reaching first a
0.80
Ld d3 D ~-~ [3_
0.70
<~
0.50
(_9 Z
• '3/,
0.60 DELAYED
7" I
0
0.40
~d O L
0.30
~>
I--
0.20
\
bA
Od
0.10 0.00
i 0
'
110
20 FORCING
30 FREQUENCY
40
50
60
(GHz)
Fig. 3. Phase diagram showing the distribution of periodic modes over the microwave frequency/microwave amplitude parameter plane. Interspersed with the periodic solutions we observe solutions with quasiperiodic or chaotic behavior. At high microwave amplitudes, transitions to delayed, quenched, and limited space charge accumulation modes take place.
149
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
0.6O0t ~ ~
1:1
0,595
3:3
chaotic mode and then the 3 : 3 solution of a period-3 window. Only on the other side of this cascade, i.e., for forcing amplitudes higher than A = 0.59, is the simple 1:1 solution obtained. We have also performed a more detailed examination of the region between the 3 : 1 and 4: 1 tongues for forcing amplitudes between 0.51 and 0.56. This has yielded the mode distribution shown in fig. 5. Here. we find periodic attractors belonging to the 7 : 2 , 11:3 and 13:4 tongues. There are also period-doubled solutions such as the 8 : 2 and the 14:4 attractors, and both between the 3 : 1 and the 13 : 4 and between the 7 : 2 and the 4 : 1 solutions regions with deterministic chaos are found. Finally, to examine the transition to LSA oscillations, we have performed a vertical scan in the phase diagram for f = 50.0 G H z from A = 0.39 to A = 0.59. Fig. 6 shows some of the results of this scan. For A = 0.39 and 0.43 we are still within the 6:1 A r n o l ' d tongue, and the solution is periodic. Hereafter an interval follows in which we have been able to identify only quasiperiodic solutions, until at A = 0 . 5 9 , the L S A mode appears.
chaos
0.590 ! ~ "1:3 a., E 0.585 0 c 0 0.580
8:8 4:4
2:2 7:6
3:2
0.570
Fig. 4. Nearly vertical scan through the phase diagram for f = 12GHz. The scan shows the existence of a perioddoubling route to chaos near the edge of the 1 : 1 tongue.
0.575 1
~
'
•
0.550
2
0.5251
0.500 /
28
29
3h0
3~1
3~2
3~3
FORCING FREQUENCY (GHz)
Fig. 5. Detailed mode distribution in the region between the 3:1 and the 4:1 tongues.
150
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
A=0.59 / .d:
k
.... .z
/
A-0.4.5 ,,/
A-0.5!~55/~59
Fig. 6. A vertical scan in the phase diagram for f = 5 0 . 0 G H z reveals the transition from 6:1 frequency locking to LSA oscillations.
As previously noted, the non-equivalent intervalley scattering time ~-, which determines the rate of equilibration between low- and highmobility states, is the least well known parameter of the model. The above simulations were all performed with ~-= 2 ps. It may be of interest, however, to examine how the model behaves for other values of ~-. We have therefore performed a series of simulations to determine the phase diagram for r = 10ps. This larger value of should give a more dynamic behavior with less damping, longer transients and more complicated temporal and spatial behaviors. At the same time, the low field mobility for carriers in the conduction band minimum has been increased from 0.5 m2/Vs to 0.8m2/Vs. This has reduced the speed of domain propagation so that, for the same sample length, the Gunn frequency is now 6.9 GHz. Fig. 7 provides an overview of the results obtained. All of the main Arnol'd tongues can still be observed, but their form is different, and, most significantly, the transition to LSA oscillations no longer occurs. Two different realms may now be distinguished in the phase diagram with a division line approximately at f = 2 0 G H z .
A b o v e this forcing frequency, 2vf~- > 1, and redistribution of carriers between the two sets of valleys can no longer follow the rapid variations in the microwave field. Here, the phase diagram is relatively simple in structure, although the form of the Arnol'd tongues is interesting because of the narrowing of their widths for A > 0.3. Between the main tongues, we have indicated some of the secondary tongues such as the 7 : 2 and the 9:2 tongues. However, the diode also exhibits entrainment with much more complicated locking-ratios. We have thus found regions with 49:9, 43:8, 41:8, 31:6, 37:8, 37:7, 28:5, etc., mode-locked solutions. In the realm below f = 2 0 G H z , the phase diagram bears some similarity to the phase diagrams obtained for self-oscillating systems when the forcing amplitude is large enough to swing them across a H o p f bifurcation [22,23]. The 1:1 tongue is dominating, and several of the minor tongues disappear at high forcing amplitudes. On the high frequency side of several of the tongues, we can observe a period-doubling structure, although the cascades appear to be incomplete. A n o t h e r interesting phenomenon is the disappearance of the 5:3 tongue in the corner be-
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
151
0.80
Ld a :D
0.70 0.60
CL <
0.50
(-9 Z
0.40 r~ ©
LL L.d
0.30
~-
0.20
Ld 0.10
--]
0.00 10
20
40
30
FORCING FREQUENCY
(GHz)
Fig. 7. P h a s e diagram for a v a l u e of the n o n - e q u i v a l e n t intervalley scattering time o f r = 10 ps. C o m p a r e d with fig. 4, the l o w field mobility has also b e e n increased from 0.5 m2/Vs to 0.8 m2/Vs.
tween the 3:2, 1:1, and 2:1 tongues, and the subsequent reappearance of this tongue on the other side of the 2:1 tongue for A >0.45. In fact, the structure of the diagram in this region is clearly much more complicated than we have been able to reveal in this study. It may be mentioned, though that we have identified several regions with spatio-temporal chaos. Calculating, for instance, the largest Lyapunov exponent for f = 18.20 GHz and A = 0.575 yields h = 2.4 x 10-9/s. We have not found regions with more than one positive Lyapunov exponent. Fig. 8 presents a phase space projection of the chaotic attractor obtained for f = 18.20 GHz and A =0.575. Here, we have plotted n2(½L, t)/n o versus n l ( ½L, t)/n o. Along the backward sloping diagonal the sum of the two carrier densities equals n o. Below this diagonal, there is depletion of carriers (leading edge of the domain), and above the diagonal there is carrier accumulation. The phase space projection clearly illustrates the irregular temporal behavior. The motion is also chaotic in space which can be illustrated by means of a stroboscopic map of, for instance, F( ~ L ) versus F( 3 L). Fig. 9 shows the generalized dimension D o for
0.6
0.5
o
0.4
c
c4 o.3 E
0.2
0.1
0.0
i i r i 0.5
i
0.6
i i i i
i
0.7
' i i i
i
i
0,8
i
;
i i
J 0.9
;
i
i
i 1.0
nl(L/2,t)/no Fig. 8. Phase s p a c e projection o f the stationary solution o b t a i n e d for f = 1 8 . 2 0 G H z and A = 0.575. T h e m o t i o n is chaotic in space as well as in time. T h e spatially chaotic b e h a v i o r can be illustrated by m e a n s o f a s t r o b o s c o p i c m a p of, for instance, F(~L) versus F(asL).
the above chaotic attractor. The Dq-curve was calculated using the method described by Jensen et al. [24]. We first determine the partition function
E. Mosekilde et al. / Phase diagrams )or periodically driven Gunn diodes
152 1.6
1.4
1.2
d~ 1.0
0.8
0.6
0.4
1111114rllllllllllllllll~'ltllllll~lllll
-20.0
- 10.0
0.0
10.0
20.0
q
Fig. 9. Generalized dimension D~ for the chaotic attraclor obtained for f = 18.2 G H z and A = 0.575.
F(q,r):~
p,
'(r),
13)
i=l
where pi(r) is the number of data points within a distance of less than r from the ith data point divided by the total number of data points N. For small r, F( q, r) ~ r T~q). Plotting In(F) versus In(r), the slope 7(q) is found by fitting a straight line in a region where the scaling is linear. Then D u = - r ( q ) / ( q - 1). When only a limited number of data points are available, the selection of the right r-region is difficult, and, as demonstrated by Meisel et al. [25], unfortunate choices can lead to obscure results. Thus by choosing r too small, a gross underestimation of D~t for negative values of q will take place. In the same way, too large values for r will lead to an overestimation of Dq for q < 0 . To compute the generalized dimension we have used 9998 data points produced by sampling from ( n l ( ½ L , t), n 2 ( ½ L , t)) each time n i (½ L ) passes a maximum.
4. Conclusion Our model reproduces all the different modes that have been experimentally observed in con-
nection with Gunn diodes. In addition, the model predicts an infinity of other modes which can arise through mode-locking between an external microwave signal and the internally generated domain mode. Even though mode-locking is generic to forced self-oscillating systems, it is of interest to determine the detailed structure of this p h e n o m e n o n for a particular example such as the Gunn diode. In a practical application as microwave generator, the diode will pass some of the mode-locked solutions as the microwave field gradually builds up after application of the dc-field. Changing the working conditions of the diode may also bring it into the quasiperiodic or mode-locked regions. From a more theoretical point of view, the interest in studying the periodically forced Gunn diode derives from the fact that it is a spatially extended structure. Most analyses of mode-locking to date have been concerned with onedimensional maps or with relatively simple systems of coupled ordinary differential equations. It is clear that the future will see much stronger activity in the field of spatially extended, nonlinear dynamic systems. As a last point, it is worth noticing how the value of the non-equivalent intervalley scattering time influences the phase diagram. An experimental determination of this phase diagram may therefore give important information about the rate of communication between the two sets of valleys. Acknowledgements We would like to thank J. Sturis, Y. Ueda and H. Ito for valuable discussions about the phaselocking structure in periodically forced Gunn diodes. M. Hewitt is acknowledge for helpful comments to the manuscript. References [1] J.B. G u n n , IBM J. Res. Develop. 8 (1964) 141. [2] J.S. Heeks, A.D. Woode and C.P. Sandbank. Proc. I E E E 53 (1965) 554.
E. Mosekilde et al.
Phase diagrams for periodically driven Gunn diodes
[3] E. Pytte and H. Thomas, Phys Rev. 179 (1969) 431. [4] C. Hilsum, Proc. IRE 50 (1962) 185. [5] H. Kroemer, IEEE Trans. Electron Devices ED-13 (1966) 27. ]6[ P.N. Butcher, W. Fawcett and C. Hilsum, Br. J. Appl. Phys. 17 (1966) 841. [7] J.A. Copeland, Proc. IEEE 54 (1966) 1479. [8] T.M. Quist and A.G. Foyt, Proc. IEEE 53 (1965) 303. [9] R.H. Chilton, Proc. IEEE 56 (1968) 1124. [10] J.E Carroll, Electron. Lett. 2 (1966) 141. [I1] M.H. Jensen, P. Bak and T. Bohr, Phys. Rev. A 30 (1984) 1960. [12] M.J. Feigenbaum, L.P. Kadanoff and S.J. Shenker, Physica D 5 (1982) 370. [13] E. Mosekilde, R. Feldberg, C. Knudsen and M. Hindsholm, Phys. Rev. B 41 (1990) 2298. [14] K. Aoki, T. Kobayashi and K. Yamamoto, J. Phys. Soc. Jpn. 51 (1982) 2373. [15] S.W. Teitsworth, R.M. Westervelt and E.E. Hailer, Phys. Rev. Lett. 51 (1983) 825.
153
[16] R. Richter, U.Rau, A. Kittel, G. Heinz, J. Peinke, J. Parisi and R.P. Huebener, Z. Naturforsch. 46a (1991) 1012. [17] R. Richter, J. Peinke, W. Clauss, U. Rau and J. Parisi, Europhys. Lett. 14 (1991) I. [18] L.L. Bonilla and F.J. Higuera, Physica D 52 (1991) 458. [19] W. Fawcett, A.D. Boardman and S. Swain, J. Phys. Chem. Solids 31 (1970) 1963. [20] J.A. Kash, J.C. Tsang and J.M. Hvam, Phys. Rev. Lett. 54 (1985) 2151. [21] W.H. Enright, K.R. Jackson, S.P. N~rsett and P.G. Thomsen, ACM Trans. Math. Software 12 (1986) 193. [22] I.G. Kevrekidis, R. Aris and L.D. Schmidt, Physica D 23 (1986) 391. [23] C. Knudsen, J. Sturis and J.S. Thomsen, Phys. Rev. A 44 (1991) 3503. [24] M.H. Jensen, L.P. Kadanoff and A. Libchaber, Phys. Rev. Lett. 55 (1985) 2798. [25] L.V. Meisel, M. Johnson and P.J. Cote, Phys. Rev. A 45 (1992) 6989.
Physica D 66 (1993) 143-153 North-Holland
SDI: 0167-2789(93)E0024-U
Phase diagrams for periodically driven Gunn diodes Erik Mosekilde, Jesper Skovhus Thomsen, Carsten Knudsen and Rasmus Feldberg System Dynamics Group, Physics Laboratory 111, The Technical University of Denmark, DK-2800 Lyngby, Denmark
A simulation model of the electron transfer mechanism for bulk negative differential conductivityin n-GaAs is applied to investigate the formation and propagation of subsequent high field domains in the presence of an external microwave field. Phase diagrams for the distribution of modes are drawn for two different values of the non-equivalent intervalley relaxation time. The domain mode is found to entrain with the microwavesignal in a variety of different frequency-locked solutions. At higher microwave amplitudes, period-doublingand other forms of mode-convertingbifurcations take place. In this region, spatio-temporal chaos may also be observed, as indicated by a positive value of the largest Lyapunov exponent. At still higher amplitudes, transitions to delayed, quenched, and limited space charge accumulation modes occur. The Dq-curve characterizing the multifractal structure is calculated for a typical chaotic solution.
1. Introduction
It is well known [1-3] that application of sufficiently high drift fields to n-type GaAs and a n u m b e r of other I I I - V and I I - V I compound semiconductors can give rise to a far-fromequilibrium phase transition in which the spatially homogeneous distribution of conduction electrons becomes unstable and a spontaneous formation of traveling high field domains (Gunn domains) takes place. In the external circuit, the formation and propagation of these domains give rise to current oscillations with a frequency determined by the sample length and the domain velocity. The phase transition is caused by a bulk negative differential conductivity ( B N D C ) associated with the transfer of electrons from the highmobility conduction band minimum to a set of low-mobility satellite valleys situated approximately A = 0 . 3 5 e V higher in the conduction band [4,5]. In thermal equilibrium and at low drift fields, only the low-lying F-minimum is populated, and the homogeneous electron distribution in space is maintained through dielectric relaxation and through diffusion. However as the drift field is increased, the electron gas
starts to heat up and, at a threshold field of about 3.5 kV/cm (for GaAs), the average electron energy becomes high enough for a population of the satellite valleys to begin. The effective density of states in these valleys exceeds that of the conduction band minimum by a factor of the order of 60, and at sufficiently high drift fields, the majority of electrons therefore occupy states in the satellite valleys. In the intermediate range of fields (i.e., between 3.5 kV/cm and approximately 8.0 kV/cm), the average electron mobility decreases with increasing field. This destabilizes the homogeneous electron distribution in space, and traveling high field domains are spontaneously generated with depletion of carriers in the leading edge and accumulation in the trailing edge. The time constant controlling the growth of this instability
T_
noelOvo/OF [
(1)
is typically of the order of I ps. Here, e is the static dielectric constant, n o the thermal equilibrium electron concentration, and e the elementary charge. Ovd/OF is the slope of the average drift velocity versus field characteristic in the
0167-2789/93/$06.00 © 1993- Elsevier Science Publishers B.V. All rights reserved
144
E. Mosekilde et al. / Phase diagrams ]br periodically driven Gunn diodes
interval with bulk negative differential conductivity. In practice, the domains usually originate from inhomogeneities at the cathode contact. Stationary conditions are reached when the voltage drop across the domain has become large enough to force the remaining crystal into the Ohmic regime. The domain then travels at a velocity equal to the carrier drift velocity in the Ohmic regions outside the domain [6]. The Gunn effect has been extensively studied both experimentally and by various analytical and numerical methods. In particular, Copeland [7] has shown that the presence of a microwave signal of sufficient amplitude and frequency can suppress the formation of domains and produce an alternative mode of operation which is referred to as the limited space charge accumulation (or LSA) mode [8,9]. With this mode, worthwhile microwave output powers and reasonable dc-to-rf conversion efficiencies can be obtained at frequencies up to about 120 GHz. The idea is that the microwave field during the negative half-period brings the crystal into the Ohmic regime long enough to fully quench all carrier density fluctuations built up in the preceding positive half-period. At the same time, the duration of the positive half-period is too short to allow carrier density fluctuations to amplify into macroscopic depletion and accumulation zones. Since the dielectric relaxation time in the Ohmic regime is a number of times smaller than the time constant r associated with the growth of inhomogeneities in the unstable regime, this type of operation is possible while exploiting on the average the BNDC for amplification of the microwave signal. At somewhat lower microwave frequencies, yet another mode is possible in which a domain is allowed to form during the positive half-period of the microwave signal, but is completely quenched before reaching the anode contact in the subsequent negative half-period [10]. With this type of operation, the quenched domain mode, 1 : 1 frequency locking is realized between the internally generated domain mode and the
external microwave signal. However, modern theory of nonlinear dynamics [11,12] suggests that the phase diagram is much more complicated than this. For an externally forced selfoscillating system, frequency locking should occur at all rational ratios between the two periods, and at higher forcing amplitudes various forms of mode-converting bifurcations are expected, including transitions to a chaotic dynamics. Using a slightly simplified model of the two valley system, we have previously examined these phenomena in considerable detail [13]. In particular, we have shown how the internally generated domain mode, by adjusting the time of domain formation and the speed of propagation, locks to the externally applied microwave signal to produce a variety of different periodic solutions. In the present paper we provide a more detailed overview of the distribution of these modes in the microwave frequency-microwave amplitude parameter plane. The mode distribution is calculated for two different values of the non-equivalent intervalley scattering time. The shape of the Arnol'd tongues is determined, the mode-converting bifurcations are illustrated, and the largest Lyapunov exponent is calculated to identify chaotic solutions. Although Copeland [7] noticed the possibility of tuning the microwave frequency to various harmonics of the transit time frequency, and although a similar observation was made by Chilton [9], we are not aware of detailed experimental investigations of the mode-locking phenomenon for periodically forced Gunn diodes that can support the present findings. Hopefully, such experiments will be performed in the near future. It may be noticed, however, that deterministic chaos and other highly nonlinear phenomena have been observed in connection with a variety of other semiconductor instabilities [14,15]. In particular, Richter et al. [16,17] have observed different types of intermittency developing during low-temperature impact ionization breakdown in extrinsic germanium. Using a model somewhat similar to the one present in
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
this paper, Bonilla and Higuera [18] have recently analysed the stability of the stationary solutions for finite length GaAs Gunn diodes under various biasing conditions.
tion in these states remains almost Maxwellian, with a temperature close to that of the crystal lattice. Thus vz{F } may be approximated by ve{V} = ~2 F ,
2. T h e t w o - v a l l e y m o d e l
We consider a one-dimensional model of a G a A s Gunn diode of length L. A drift field F d is applied along the crystal (x-direction), and we assume that the local electron concentration n(x, t) is distributed between high and low mobility states with densities nl(x, t) and nz(X, t), respectively. Accompanying the variation in the concentration of free carriers there is a local electric field F(x, t), as determined by Poisson's equation
OF Ox
---
e e
- (n I + n 2 -
no).
(2)
As before, e is the static dielectric constant for GaAs, and n o is the thermal equilibrium electron concentration. In the numerical calculations we have taken n o = 2.0 × 10el/m 3. For simplicity, the electrons have been assumed to carry a positive charge e. In the presence of the electric field and the carrier density gradients, the two groups of electrons produce currents of density ix(X, t) and jE(x, t), respectively. Assuming collision dominated behavior, these current densities may be expressed as Onl
Jl = nlevl { F} - eDl Ox
(3)
(5)
with [z2 being an appropriate mobility. In the following calculations we have taken D 1 = 0 . 0 2 0 m 2 / s , D 2 =0.0008m2/s, and /z2 = 0.032 m2/Vs. Fawcett et al. [19] have performed detailed calculations of the stationary electron transport properties for GaAs as a function of the drift field. In particular, these authors have calculated the velocity field characteristic vl{F } and the equilibrium fraction of electrons n2.eq/n o occupying states in the higher valleys. These calculations take into account a variety of realistic scattering mechanisms as well as non-parabolicity of the conduction band minimum and wavevector dependence of the periodic part of the electronic Bloch functions. To simplify the present treatment, we have fitted simple mathematical expressions to the numerical results obtained by Fawcett et al. Thus,
=Ip.IF(1-aF2) vI{F}
[ Ol,sat
On 2
J2 = n2 e v 2 { F }
--
eD2 ax
(4)
H e r e , vl{F(x, t)} and v2{F(x, t)} denote the local drift velocities of the two types of carriers, and D 1 and D 2 are appropriate diffusion constants. Because of a relatively efficient scattering within the upper valleys, the electron distribu-
for F-< F~at, for F > F~at
(6)
and n2,eq
no
1 + a exp[A/(kBT o + bE2)]
(7)
H e r e , /z~ = 0 . 5 0 m 2 / V s is the low field mobility associated with electrons in the conduction band minimum, Ol.sat 2.3 × 105 m/s is the high-field saturation drift velocity in this band, and F~at = 4 . 3 × 105 V/m is the field at which this saturation occurs. The parameter a is adjusted so as to provide a smooth transition between the two expressions for Vl{F ). n2,eq is the equilibrium concentration of free carriers in the upper valleys corresponding to the applied electric field F. A = 0.35 eV is the energy difference between the two sets of valleys, T Ois the lattice tempera=
and
145
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
146
ture, and the parameters a =0.33 and b = 6.0 x 10 -3 eV/(V/m) 2 were adjusted so as to fit the numerical results as well as possible, k B is Boltzmann's constant. As calculated from (6) and (7), the overall velocity-field characteristic is in close agreement with experimental results. In the present context, (6) and (7) are considered as local equilibrium relations expressing the local drift velocity v l ( x , t) and the local equilibrium concentration of carriers in the upper valleys ?/2,eq(X, t ) in terms of the local electric field F ( x , t) and the local electron concentration n ( x , t). Finally, the carrier density variations are obtained from the equations of continuity On 1 Ot
10j I e Ox
n2,eq -
n2
(8)
"r
and 01l 2 - -
Ot
1 Oj2 -
e Ox
~"
,
f
F ( x , t) dx = Vdll + A sin(2"rrft)] ,
(11)
0
where Vj = F j L is the applied drift voltage, f a n d A denote the frequency and the relative amplitude of the applied microwave signal, respectively. In all calculations we have assumed the average drift field to be Fa = 4.0 kV/cm. This is well within the B N D C region.
(9)
and
n2(O ,
t) = 0
(lOa)
at the cathode end, and neutrality n,(L,
L
rt2,eq -- 172 +
where the second term in each equation represents the scattering of electrons between the two sets of valleys, r being the non-equivalent intervalley scattering time. While the model assumes that drift and diffusion are in equilibrium with the local field and carrier density gradients, respectively, redistribution of carriers between the two sets of valleys is considered a dynamic process which involves time constants comparable to those controlling the domain dynamics. ~ is presumably the parameter which is known with least precision. For this reason we have investigated the behavior of the model both for ~- = 2 ps and for r-= 10ps. The former value corresponds more closely to experimental results on the relaxation of optically excited electrons [20]. As boundary conditions we have assumed thermal equilibrium n,(O, t) = n o
at the anode end. To facilitate domain formation near the cathode contact, we have introduced a slight variation of the specific resistance in the 10% of the crystal which is closest to the cathode. Here a single period of a sinusoidal perturbation of amplitude 0.01 × 1021/m 3 is applied. Finally, we have used a normalization condition of the form
t) + n 2 ( L , t) = n o
(lOb)
3.
Simulation
results
We have simulated the formation and propagation of Gunn domains in the presence of an external microwave field for more than 3000 combinations of the forcing parameters f and A. To solve the equations of motion the sample is divided into 150 sections, each of length Ax = 80 nm, providing a total of 300 dynamical variables. The spatial derivatives are all approximated by central differences while time derivatives are approximated by forward differences. The numerical integration of the set of 300 locally coupled ordinary differential equations is performed by an eight-stage R u n g e - K u t t a pair of order five and six [21]. The integration scheme includes variable time step and error control. As initial conditions we have assumed thermal equilibrium until, at time t = 0, the homogeneous drift voltage is applied to the crystal. Fig. 1 illustrates the onset of a 2:1 frequency locking as a microwave signal of frequency f = 12.5 G H z and relative amplitude A = 0.20 is applied to a stably oscillating Gunn diode. In the
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes F= 12.5 GHz, E E
A= 0.20
70-"
60. 50" 40. Z W 0 Z hi n" n," :D 0
30. 20'
10: 0
0.0
o
,,,,,,,
'6.'2. . . . . . . . 6.'4. . . . . . . . b.'6. . . . . . . . 6.'8. . . . . . . . i.b TIME (ns)
e
7
Fig. 1. O n s e t of a 2:1 frequency locking b e t w e e n the external microwave signal and the internally generated d o m a i n m o d e . T h e microwave signal is applied at time t = 0.4 ns.
top panel, the upper trace shows the temporal variation of the total current density
Jtot(t)
~F
=j,(x, t) +j:(x, t) + ~ c9-~-'
(12)
and the lower trace shows the corresponding variation in the applied voltage. By including the displacement term e(aF/Ot), jtot(t) becomes independent of x and proportional to the current in the external leads. The lower panel shows the formation and propagation of subsequent highfield domains. In this panel, the vertical axis represents the position along the crystal with the cathode end at the bottom. The time axis is common to both panels. The contour curves in the lower panel demarcate that part of the crystal in which the field at a given time is more than 10% higher than in the rest of the crystal. During the period of undisturbed oscillations, the current density is seen to rise to its Ohmic value of approximately 6 5 A / m m 2, remain Ohmic until a domain has been formed, and then decay towards a saturated value of approximately 40 A / m m 2. The current remains saturated as long as the domain propagates through the crystal to rise to its Ohmic value again as the domain disappears at the anode contact. With the assumed parameters, the period of the internal oscillation is approximately 116 ps corresponding
147
to an undisturbed Gunn frequency of fG,,n = 8.6GHz. When, at t = 0 . 4 n s , the microwave field is applied, the diode first passes through a transient period of approximately 0.3ns' duration. Thereafter, a 2 : l mode-locking is established, and a domain is formed precisely once for every two cycles of the external microwave field. This corresponds to a frequency of 6.25 GHz for the domain mode. By adjusting both the time of domain formation and the speed of propagation, the internal oscillation has thus reduced its frequency by nearly 30%. At the same time, the position at which the domain is first detected has also changed. Fig. 2 shows a phase plot of the stationary 2 : 1 solution. Here, we have plotted simultaneous values of the average electric field Fay and the total current density Jtot. By inspecting the figure, we see how the current drops during the formation of a domain, remains low during a negative swing of the microwave signal, and rises again as the domain exits the crystal. Awaiting the average field to become sufficiently high for a new domain to be formed, the current remains high during a second negative swing of the microwave signal. With a forcing amplitude A = 0.20, the 2:1 frequency-locked solution exists over the interval from f = 1 2 G H z to approximately 18.4GHz F= 12.5 GHz,
A= 0.20
80if-. E E
706050-
Z 14J
a I--Z W
n-
t-,,"-'1 (3
40-
3020
1.0'''2.b'
''3.b'''4.b''
'5.b'''6.b'''7.b
Fay (kV/cm) Fig. 2. Phase plot of the stationary 2:1 frequency-locked solution obtained for f = 12.5 G H z and A = 0.20.
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
148
[13]. A b o v e this latter frequency, the formation and propagation of a domain can no longer be c o m p l e t e d within two cycles of the microwave signal, the 2:1 mode-locking breaks down, and the oscillations b e c o m e quasiperiodic. For higher microwave frequencies, intervals exist in which the domain m o d e entrains into 3:1, 4:1, 5:1, etc., frequency-locked solutions. Between the main n : 1 intervals, a multitude of smaller-intervals are found, representing m o r e complex entrainments. Between the 2:1 and 3 : 1 intervals, for instance, we find intervals with 5 : 2, 7 : 3, 9 : 4, and 13 : 6 entrainment. Between the 1 : 1 and 2 : 1 intervals we find solutions with 2 domains for each 3 microwave cycles, 4 domains for each 5 cycles, or 7 domains for each 8 cycles. On the low frequency side of the 1:1 interval, on the other hand, we find solutions with 2 or 3 domains for each cycle of the microwave field. With increasing microwave amplitude, the modelocking intervals tend to broaden, The phase diagram in fig. 3 provides an overview of this variation. H e r e , the Arnol'd tongues delineate regions in p a r a m e t e r space in which particular m o d e - l o c k e d solutions can be observed. In the
intermediate range of forcing amplitudes, period-doubling and other mode-converting bifurcations take place. Finally, as the microwave amplitude becomes sufficiently high, a transition to a different set of modes occurs. If the frequency of the microwave signal is relatively low ( < 3 0 G H z ) and yet higher than the undisturbed Gunn frequency f(; ....... the quenched domain m o d e is found [13]. For microwave fieq u e n d e s below fo . . . . a delayed domain mode occurs in which each positive swing of the microwave signal sets off one domain. Finally, for microwave frequencies above approximately 3 0 G H z , a transition to limited space charge accumulation modes takes place. To examine the bifurcation structure in a little m o r e detail, we have performed a nearly vertical scan in the phase diagram for f = 12 GHz. The results of this are illustrated in fig. 4. It is seen how increase of the forcing amplitude lets us out of the 3 : 2 tongue, through a relatively narrow 7 : 6 tongue, and thereafter into the main 1:1 tongue. However, as we enter this tongue we pass through the 2:2, 4 : 4 and 8:8 solutions of a period-doubling cascade before reaching first a
0.80
Ld d3 D ~-~ [3_
0.70
<~
0.50
(_9 Z
• '3/,
0.60 DELAYED
7" I
0
0.40
~d O L
0.30
~>
I--
0.20
\
bA
Od
0.10 0.00
i 0
'
110
20 FORCING
30 FREQUENCY
40
50
60
(GHz)
Fig. 3. Phase diagram showing the distribution of periodic modes over the microwave frequency/microwave amplitude parameter plane. Interspersed with the periodic solutions we observe solutions with quasiperiodic or chaotic behavior. At high microwave amplitudes, transitions to delayed, quenched, and limited space charge accumulation modes take place.
149
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
0.6O0t ~ ~
1:1
0,595
3:3
chaotic mode and then the 3 : 3 solution of a period-3 window. Only on the other side of this cascade, i.e., for forcing amplitudes higher than A = 0.59, is the simple 1:1 solution obtained. We have also performed a more detailed examination of the region between the 3 : 1 and 4: 1 tongues for forcing amplitudes between 0.51 and 0.56. This has yielded the mode distribution shown in fig. 5. Here. we find periodic attractors belonging to the 7 : 2 , 11:3 and 13:4 tongues. There are also period-doubled solutions such as the 8 : 2 and the 14:4 attractors, and both between the 3 : 1 and the 13 : 4 and between the 7 : 2 and the 4 : 1 solutions regions with deterministic chaos are found. Finally, to examine the transition to LSA oscillations, we have performed a vertical scan in the phase diagram for f = 50.0 G H z from A = 0.39 to A = 0.59. Fig. 6 shows some of the results of this scan. For A = 0.39 and 0.43 we are still within the 6:1 A r n o l ' d tongue, and the solution is periodic. Hereafter an interval follows in which we have been able to identify only quasiperiodic solutions, until at A = 0 . 5 9 , the L S A mode appears.
chaos
0.590 ! ~ "1:3 a., E 0.585 0 c 0 0.580
8:8 4:4
2:2 7:6
3:2
0.570
Fig. 4. Nearly vertical scan through the phase diagram for f = 12GHz. The scan shows the existence of a perioddoubling route to chaos near the edge of the 1 : 1 tongue.
0.575 1
~
'
•
0.550
2
0.5251
0.500 /
28
29
3h0
3~1
3~2
3~3
FORCING FREQUENCY (GHz)
Fig. 5. Detailed mode distribution in the region between the 3:1 and the 4:1 tongues.
150
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
A=0.59 / .d:
k
.... .z
/
A-0.4.5 ,,/
A-0.5!~55/~59
Fig. 6. A vertical scan in the phase diagram for f = 5 0 . 0 G H z reveals the transition from 6:1 frequency locking to LSA oscillations.
As previously noted, the non-equivalent intervalley scattering time ~-, which determines the rate of equilibration between low- and highmobility states, is the least well known parameter of the model. The above simulations were all performed with ~-= 2 ps. It may be of interest, however, to examine how the model behaves for other values of ~-. We have therefore performed a series of simulations to determine the phase diagram for r = 10ps. This larger value of should give a more dynamic behavior with less damping, longer transients and more complicated temporal and spatial behaviors. At the same time, the low field mobility for carriers in the conduction band minimum has been increased from 0.5 m2/Vs to 0.8m2/Vs. This has reduced the speed of domain propagation so that, for the same sample length, the Gunn frequency is now 6.9 GHz. Fig. 7 provides an overview of the results obtained. All of the main Arnol'd tongues can still be observed, but their form is different, and, most significantly, the transition to LSA oscillations no longer occurs. Two different realms may now be distinguished in the phase diagram with a division line approximately at f = 2 0 G H z .
A b o v e this forcing frequency, 2vf~- > 1, and redistribution of carriers between the two sets of valleys can no longer follow the rapid variations in the microwave field. Here, the phase diagram is relatively simple in structure, although the form of the Arnol'd tongues is interesting because of the narrowing of their widths for A > 0.3. Between the main tongues, we have indicated some of the secondary tongues such as the 7 : 2 and the 9:2 tongues. However, the diode also exhibits entrainment with much more complicated locking-ratios. We have thus found regions with 49:9, 43:8, 41:8, 31:6, 37:8, 37:7, 28:5, etc., mode-locked solutions. In the realm below f = 2 0 G H z , the phase diagram bears some similarity to the phase diagrams obtained for self-oscillating systems when the forcing amplitude is large enough to swing them across a H o p f bifurcation [22,23]. The 1:1 tongue is dominating, and several of the minor tongues disappear at high forcing amplitudes. On the high frequency side of several of the tongues, we can observe a period-doubling structure, although the cascades appear to be incomplete. A n o t h e r interesting phenomenon is the disappearance of the 5:3 tongue in the corner be-
E. Mosekilde et al. / Phase diagrams for periodically driven Gunn diodes
151
0.80
Ld a :D
0.70 0.60
CL <
0.50
(-9 Z
0.40 r~ ©
LL L.d
0.30
~-
0.20
Ld 0.10
--]
0.00 10
20
40
30
FORCING FREQUENCY
(GHz)
Fig. 7. P h a s e diagram for a v a l u e of the n o n - e q u i v a l e n t intervalley scattering time o f r = 10 ps. C o m p a r e d with fig. 4, the l o w field mobility has also b e e n increased from 0.5 m2/Vs to 0.8 m2/Vs.
tween the 3:2, 1:1, and 2:1 tongues, and the subsequent reappearance of this tongue on the other side of the 2:1 tongue for A >0.45. In fact, the structure of the diagram in this region is clearly much more complicated than we have been able to reveal in this study. It may be mentioned, though that we have identified several regions with spatio-temporal chaos. Calculating, for instance, the largest Lyapunov exponent for f = 18.20 GHz and A = 0.575 yields h = 2.4 x 10-9/s. We have not found regions with more than one positive Lyapunov exponent. Fig. 8 presents a phase space projection of the chaotic attractor obtained for f = 18.20 GHz and A =0.575. Here, we have plotted n2(½L, t)/n o versus n l ( ½L, t)/n o. Along the backward sloping diagonal the sum of the two carrier densities equals n o. Below this diagonal, there is depletion of carriers (leading edge of the domain), and above the diagonal there is carrier accumulation. The phase space projection clearly illustrates the irregular temporal behavior. The motion is also chaotic in space which can be illustrated by means of a stroboscopic map of, for instance, F( ~ L ) versus F( 3 L). Fig. 9 shows the generalized dimension D o for
0.6
0.5
o
0.4
c
c4 o.3 E
0.2
0.1
0.0
i i r i 0.5
i
0.6
i i i i
i
0.7
' i i i
i
i
0,8
i
;
i i
J 0.9
;
i
i
i 1.0
nl(L/2,t)/no Fig. 8. Phase s p a c e projection o f the stationary solution o b t a i n e d for f = 1 8 . 2 0 G H z and A = 0.575. T h e m o t i o n is chaotic in space as well as in time. T h e spatially chaotic b e h a v i o r can be illustrated by m e a n s o f a s t r o b o s c o p i c m a p of, for instance, F(~L) versus F(asL).
the above chaotic attractor. The Dq-curve was calculated using the method described by Jensen et al. [24]. We first determine the partition function
E. Mosekilde et al. / Phase diagrams )or periodically driven Gunn diodes
152 1.6
1.4
1.2
d~ 1.0
0.8
0.6
0.4
1111114rllllllllllllllll~'ltllllll~lllll
-20.0
- 10.0
0.0
10.0
20.0
q
Fig. 9. Generalized dimension D~ for the chaotic attraclor obtained for f = 18.2 G H z and A = 0.575.
F(q,r):~
p,
'(r),
13)
i=l
where pi(r) is the number of data points within a distance of less than r from the ith data point divided by the total number of data points N. For small r, F( q, r) ~ r T~q). Plotting In(F) versus In(r), the slope 7(q) is found by fitting a straight line in a region where the scaling is linear. Then D u = - r ( q ) / ( q - 1). When only a limited number of data points are available, the selection of the right r-region is difficult, and, as demonstrated by Meisel et al. [25], unfortunate choices can lead to obscure results. Thus by choosing r too small, a gross underestimation of D~t for negative values of q will take place. In the same way, too large values for r will lead to an overestimation of Dq for q < 0 . To compute the generalized dimension we have used 9998 data points produced by sampling from ( n l ( ½ L , t), n 2 ( ½ L , t)) each time n i (½ L ) passes a maximum.
4. Conclusion Our model reproduces all the different modes that have been experimentally observed in con-
nection with Gunn diodes. In addition, the model predicts an infinity of other modes which can arise through mode-locking between an external microwave signal and the internally generated domain mode. Even though mode-locking is generic to forced self-oscillating systems, it is of interest to determine the detailed structure of this p h e n o m e n o n for a particular example such as the Gunn diode. In a practical application as microwave generator, the diode will pass some of the mode-locked solutions as the microwave field gradually builds up after application of the dc-field. Changing the working conditions of the diode may also bring it into the quasiperiodic or mode-locked regions. From a more theoretical point of view, the interest in studying the periodically forced Gunn diode derives from the fact that it is a spatially extended structure. Most analyses of mode-locking to date have been concerned with onedimensional maps or with relatively simple systems of coupled ordinary differential equations. It is clear that the future will see much stronger activity in the field of spatially extended, nonlinear dynamic systems. As a last point, it is worth noticing how the value of the non-equivalent intervalley scattering time influences the phase diagram. An experimental determination of this phase diagram may therefore give important information about the rate of communication between the two sets of valleys. Acknowledgements We would like to thank J. Sturis, Y. Ueda and H. Ito for valuable discussions about the phaselocking structure in periodically forced Gunn diodes. M. Hewitt is acknowledge for helpful comments to the manuscript. References [1] J.B. G u n n , IBM J. Res. Develop. 8 (1964) 141. [2] J.S. Heeks, A.D. Woode and C.P. Sandbank. Proc. I E E E 53 (1965) 554.
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Phase diagrams for periodically driven Gunn diodes
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