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Boundary effect induced nonhysteretic transition caused by saddle-node bifurcation on a limit cycle in n-GaAs
SolidStateCommunications, Vol.99, No. 5, pp. 305-309,1996 Copyright@ 1996Publishedby Ekwier Science Ltd
Pergamon
Printed in Great Britain. AU rights reserved
0038.10981% s12.00+ .OO PIISOO38-1098(96)00297-9
BOUNDARY EFFECT INDUCED NONHYSTERETIC TRANSITION CAUSED BY SADDLE-NODE BIFURCATION ON A LIMIT CYCLE IN n-GaAs Yuo-Hsien Shiau *vt and Yi-Chen Cheng t * Institute of Physics, Academia Sinica, Taipei, Taiwan 115, ROC t Department of Physics, National Taiwan University, Taipei, Taiwan 106, ROC (Received 17 April 1996; accepted 10 May 1996 by A. Okiji)
Simulation study of nonlinear behavior of spatio-temporal oscillations of n-type GaAs at room temperatures is studied when the applied dc bias is in the negative differential conductivity regime. Modem MBE technique makes it possible that the electron density in the metal-semiconductor contact at the cathode, n(x = 01, may be controlled experimentally. It is found that when nfx = 0) is varied, the system may have a transition from an oscillating (travelling domains) to a non-oscillating (stationary domains) state. The transition is a global bifurcation known as saddle-node bifurcation on a limit cycle. The bifurcation feature of this transition is studied in detail. Copyright @ 1996 Published by Elsevier Science Ltd Keywords: A. semiconductors, D. bifurcation, D. electric-field domains, E. simulation.
It is well known that self-generated current oscillations in n-GaAs at room temperatures are due to the characteristic of negative differential conductivity (NDC) inherited in the system. Conventional studies of this well-known Gunn effect [l], both theoretically and experimentally [2], are mainly concentrated on the formation and propagation of the high field domains and the associated current oscillations when the applied field is increased to above the threshold field and in the NDC regime. In these studies the control parameter is the applied voltage bias. In this paper, we report a simulation study of the boundary effect on the dynamical behavior of the Gunn oscillations described above. We use the degree of the electron depletion at the cathode as the control parameter and find a nonhysteretic transition from a state of travelling domains to a state of stationary domains. It is a quite different result compared with the conventional Gunn effect, which describes a transition from a state of spatially uniform distribution of electric field to a state of propagation of high field domains when the applied field is increased from the positive differential conductivity regime to the NDC regime. The bifurcation scheme we found near the nonhysteretic transition is a standard one which is called saddle-node bifurcation on a limit cycle [3,4]. Similar bifurcation schemes have also been observed in Josephson junction [5J and p-type
germanium at liquid-helium temperatures [6]. In this paper we report in detail the behavior of the frequency scaling, the amplitude of spontaneous oscillation, and the topology of the constructed attractors using timedelay method near the transtion. All these behaviors fit the bifurcation scheme of the standard saddle-node bifurcation on a limit cycle [3,4]. The boundaries affect the dynamical behavior of the system via boundary conditions. One of the relevant equations in the study of the Gunn effect is the divergence equation (in one-dimension): 3E 1 (n - n0k (1) ax=; where n and no are the concentrations of the conduction electrons and the background effective donors, respectively; E is the permittivity of the sample with an applied field is in the x direction. For convenience of analysis we assume that each electron carries a positive charge e > 0. For the Gunn oscillations the high field domains, and therefore the high/low electron density domains, are formed near the cathode and then travel down to the anode [2]. In this picture the electron density at the cathode is primarily dependent on the interface properties of the metal/semiconductor (MS) contact, not on the dynamical behavior of the system. Therefore we may consider n in Eq. (1) at the cathode is a surface dependent quantity and thus imposes 305
306
BOUNDARY EFFECT INDUCED NONHYSTERETIC TRANSITION
a boundary condition at the cathode. With modem
semiconductor fabrication technology [7], it becomes possible that n in the MS contact can be controlled in the laboratory. The MS contact is known as the Schottky barrier (SB) [8], whose properties may affect the performance of many semiconductor devices. The barrier height at the junction will be affected by the presence of the interface or defect states at the junction. An ideal SB is defined to have a barrier height equal to the initial contact potential difference, which is linearly proportional to the work function of the metal, and there is an entire depletion of the conduction electrons near the contact region in the semiconductor side. The index of an interface S is defined to be equal to d+/dy/, where 4 and q are the barrier height and the metal’s work function respectively [7]. For an ideal SB S = 1. However most of the measured values of S for the MS contacts are nonideal. For example, S is approxmately equal to 0.1 for metal/n-GaAs under variously experimental techniques [9]. This phenomenon indicates “Fermi level pinning” at the surface states, and electrical neutrality will be roughly maintained near the contact region. The main reason for S # 1 is due to the presence of the interface/defect states. Modem technique of molecular-beam epitaxy (MBE) can produce S = 1 for a variety of metal/n-GaAs contacts recently [7]. Therefore, it is reasonable to expect that the electron density n in Eq. (1) at the cathode can be controlled experimentally in a wide range from n = 0 (ideal SB, S = 1) to n = no (electrical neutral, S = 0). In the following we study the dynamical behavior of the Gunn diode with a fixed applied field in the NDC region, and consider n at the cathode as the control parameter. We consider an n-type GaAs with a sample length L at room temperature which is applied by a dc bias V. The applied field E = V/L is the NDC region of the current-density-field characteristic of n-GaAs. Thus the system may undergo spatio-temporal oscillations. We assume that the only relevant spatial variable is x which is in the direction of the field E. The dynamical equation for E(x, 1) can be written as [lo, 111
Vol. 99, No. 5
E, E(x, t) = E, +
c E,,,(t) eimkr,
(3) In where E., = V/L is the applied static uniform field, E,,, are the induced fields where m is an integer extending from --oo to +UJ, and k = 2rrlLo with Ls a length constant. The rate equations for E,,,(t) can be obtained by substituting Eq. (3) into Eq. (2), and expanding v’(E) around the static field E3 to obtain coupled first order ordinary differential equations for E,,, [I I], dEm
-
dt
=
(am - imw)E, + i (J,,, - enov,,)d,,o
am = -inov.i’) -D& -
Bpm
=
&vjpp’ E 2w
v,
=
%%,),
v(P) I
=
E
dEP Es&’ By substituting Eq. (3) into the circuit equation, V = jt E(x, t)dx, we obtain a relation between the Fourier components E,,,, w=z-’
&ML
+ i m+O&(l-e”@)Em=O. C
(5)
Equations (3)--(S) are the basic equations for our model. Equation (4) is a set of infinitely many coupled first order ordinary differential equations involving dE,ldt . When the applied field Es is in the NDC regime, vji) is negative and some of the linear coefficient (rmin Eq. (4) will be positive. This means that some of the mode Em will be unstable and the system
undergoes spatio-temporal oscillations. Before we present the details of the numerical results, we see that: (1) Conventional choice of Lo = L [lo, 13,141 is not an appropriate one for the study of the Gunn effect. A simple reason is that from Eq. (3) one would have PE/axpl,=o = aPE/dxpI,,L for all p = 0,1,2 - . - if & = L. These requirements are rather unphysical be-
where S and b are the effective drift velocity and the diffusion constant of the conduction electrons, respectively; Jlo, is the total current density which is a function of time t only [12]. The dynamical equation (2) is a partial differential equation for the field E (x, t 1. We use the Fourier series expansion method to treat the spatial dependence of
cause in the Gunn effect the cathode (x = 0) and the anode (x = L) have quite di!Terent environments In the cathode the electrons are injected from the metal to the semiconductor, in which there is a Schottky barrier to overcome. But in the anode, the electrons travel from the semiconductor to the metal, where there is no potential barrier for the electrons Therefore boundary characteristics and field profiles are quite different at the cathode and the anode, which implies L,s = L is
Vol. 99, No. 5
BOUNDARY EFFECT INDUCED NONHYSTERETIC TRANSITION
not an appropriate choice. A possible way is to choose Ls = 2L [l 1,151 and this is what we have done in this paper. (2) The dynamical equation (2) is a second order partial differential equation with respect to x, therefore we may have two boundary conditions to specify the solutions We impose both boundary conditions at the cathode (X = 0) and leave the anode as a free end. The reason is that the electrons are injected from the cathode into the sample and then travel to the anode. In the process of travelling down the sample, the electrons undergo spatio-temporal oscillations which are governed by the dynamical equation. Therefore the field at the anode depends very much on the dynamical behavior of the system, and can be approximately considered as a free end. (3) We expect that most of the electrons injected from the cathode into n-GaAs are in the high-mobility central valley because the electrons experience a weak field in the metal. It takes distance for the electrons to be accelerated and transferred to the low-mobility satellite valleys of n-GaAs It is only when there are sizable fraction of the electrons transferred to the lowmobility valleys, the system becomes unstable due to NDC. Therefore it is reasonable to approximate the cathode to be in a static situation. We choose E = l& = V/L and aEl&t = en&/no - 1)/e as the two boundary conditions at the cathode. The f&d field condition E = E, has been used to analyze the dynamical behavior of the Gunn effect in previous studies [16]. The second bounary condition is related to the electron density n at the cathode. As we have mentioned earlier in this paper we may choose n at the cathode as a control parameter because it may be controlled in the laboratory by the modern MBE technique [7]. If n = no it represents a neutral MS contact, and for an ideal Schottky barrier n = 0. The parameter values used for the numerical calculationsare: L =I1 m, v’(E) = [~E+~cI(E/&)~]/[I + (E/&)4], p = 5000 cm2/Vsec, vo = 8.5 x lo6 ctu/sec, & = 4.0 kV/cm, E = 1.1 x lo-i2 CouWcm, no = 10%m3, b =I40 cm2/sec, and L&=4.8kVlcm which is in the NDC regime. With this field it is sufficient to keep the modes E,,, with Iml s 3 and all higher order modes can be neglected as a,,, < 0 for Iml r 3. In Fig. 1 we plot the current density as a function of the control parameter n/no. When the control parameter is in the range 0.73 < n/no I 1, the dynamical system is in a state of current oscillation, but when n/no < 0.73 the system bifurcates to a non-oscillating state. The nature of the bifurcation will be discussed in the following. But first we think it is worthwhile mentioning that the non-oscillating state (solid line in
z
5
u
I
* ---
l
t
17.0 I
0.50
l
’
l
l
307
oscillatin current non-osci d ding current atatic current
’
’
’
’ * OS?5
-
.
I
I
I 1.00
n/n0
Fig. 1. Current density versus control parameter n/no. The asterisk symbol t, the solid line, and the dashed line correspond to oscillating current, non-oscillating current, and static current at the operating point, respectively. The boundary between the oscillating and non-oscillating regimes is n/no=0.73. Fig. I) is not the same as the static state (dashed line in Fig. 1). The difference is that the former is a state with stationary domains (Fig. 2), while the latter is a state of spatially uniform. We characterize the transition from an oscillating to a non-oscillating at n/no = 0.73 as nonhysteretic transition caused by saddle-node bifurcation on a limit cycle [3,4]. This is a global bifurcation and our claim is supported by the plots of the amplitude of oscillation and the frequency squared versus n/no, respectively, in Figs 1 and 3. In the oscillating state (0.73 < n/no I 1) we see that the amplitude of oscillation is independent of the control parameter n/no; while the frequency of oscillation varies as the square root of the control parameter n/no. These are the two signatures of nonhysteretic transition caused by saddle-node bifurcation on a limit cycle [3,4]. The phenomenon that the frequency of the current oscillation decreases as n/m approaches the critical value (or bifurcation point) 0.73 is also known as criiically slowing down [3]. To give more details of the nature of the bifurcation feature of the transition, we analyze series of superposition of attractor reconstructions by the time-delay method near the bifurcation point as shown in Fig. 4. It is obvious to see that these trajectories follow the parabolic lines to approach the diagonal, and finally hit the diagonal at a point. This characteristic is precisely the signature of the standard saddle-node bifurcation on a limit cycle [3]. Finally we remark that the bifurcation features
BOUNDARY EFFECT INDUCED NONHYSTERETIC
TRANSITION
Vol. 99, No. 5
22.50
tiz0.03 tf=0.16
ns ns 21.75
19.50
20.25
Jdt)
21.00
21.75
22.50
(A/mm2)
Fig. 4. Superposition of different attractor reconstructions obtained by the time-delay method (T = 20.6 ps) from spontaneous current oscillations at different value of the control parameter n/no = 0.95 (l), n/no = 0.9 (2), n/no = 0.85 (3), n/no = 0.8 (4), n/no = 0.75 (5). Note that the trajectories approach and hit the diagonal parabolically as shown in the dashed square.
Fig. 2. Illustration of stationary domains formation at n/no = 0.72. ti, tf and A? mean the initial time, the final time and equal time interval between the neighbouring curves, respectively.
of standard nonhysteretic transition from a nonoscillating to an oscillating state caused by saddlenode bifurcation on a limit cycle can be described by a two-variable (r, 0) dynamical system [3,4], dr z = r - r3,
de -
38 h
*N 5
25-
N
13 -
Fig. 3. Square of the frequency versus control parameter n/no. The straight line is the best fit for the numerical data (O symbol).
=p-rcose, (7) dt where cc is the control parameter. Equations (6) and (7) predict a critical value for the control parameter CI = Lkr = 1 such that for fl < /Jo, the system is in a state of non-oscillating, and that for p > pEr the system is in a state of oscillating. The characteristics of this transition are exactly the same as we have plotted in Figs 1-3, i.e., (1) the amplitude of the oscillation in the oscillating state is independent of the control parameter (Fig. l), (2) the frequency of the oscillation is proportional to (cc- per)1/2 (Fig. 2), and (3) the trajectories in the time-delay plot of different attractor reconstructions with various values of the control parameter approach the diagonal parabolically (Fig. 3). It is interesting to note that the dynamical behavior we studied in this paper is governed by the dynamical equations (3)-(5) which are quite different from the two-variable dynamical equations (6) and (7) and that the control parameter does not appear explicitly in the dynamical equations (3j(5). Yet the bifurcation features of these two systems are exactly the same as
Vol. 99, No. 5
BOUNDARY EFFECT INDUCED
we have shown in Figs 1-3. Our result is therefore a good example of the principle of universality of the bifurcation phenomena regardless how different the dynamical systems are. This is also the first time that a Gunn diode is shown to have the possibility of observing the global bifurcation of nonhysteretic transition from an oscillating to a non-oscillating state caused by saddle-node bifurcation on a limit cycle. Acknowledgements-This work is supported in part by the National Science Council of the Republic of China under Contract Nos. NSC 85-2112-M-002-005 and NSC 85-2112-M-001-003. REFERENCES
1. J. B. Gunn, Solid State Commun. 1, 88 (1963). 2. M. P Shaw, V V Mitin, E. Scholl, and H. L. Grubin, The Physics of Instabilities in Solid State Electron Devices Plenum, New York (1992), Chap. 5 and references therin. 3. J. M. T Thompson and H. B. Stewart, Nonlinear Dynamics and Chaos Wiley, New York (1993). 4. J. Peinke, J. Parisi, 0. E. Rossler, and R. Stoop, Encounter with Chaos Springer-Verlag, Berlin (1992).
NONHYSTERETIC
TRANSITION
309
5. T. Van Duzer and C. W. Turner, Principles of Superconductive Devices and Circuits Elsevier. New -York (1981). 6. R. Richter, A. Kittel, K. Pyragas, J. Peinke and J. Parisi, Z. Phvs. B 91, 527 (1993). 7. S. Chang, A: Raisanen, L.‘J. Brillson, J. L. Shaw, I? D. Kirchner, G. D. Pettit, and J. M. Woodall, J: ‘vat Sci. Technol. BlO, 1932 (1992). 8. C. Kittel, Introduction to Solid State Physics, 6th ed. Wiley, Singapore (1988). 9. S. Kurtin, T. C. McGill, and C. A. Mend, Phys. Rev Lett. 22, 1433 (1969). 10. H. Haken, Synergetics, 3rd ed. Springer, Berlin, Heidelberg (1983). 11. Y-H. Shiau, C.-H. Ho, and Y.-C. Cheng, submitted for publication. 12. B. G. Bosch and R. W. H. Eagelman, G‘unnEffect Electronics Pritman Publishing, London (1975). 13. K. Nakamura, Prog Theor Phys. 57, 1874 (1977). 14. Z. Jiang and B. Ma, Appl. Phys. A 52, 10 (1991); Phys. Rev. B44, 11072 (1991). 15. M. Kubicek and M. Marek, Computational Methods in Bifurcation Theory and Dissipative Structures Springer-Verlag, New York (1983). 16. M. Shaw, H. L. Grubin, and P R. Solomon, The Gurus-Hilsum Effect Academic, New York (1979).
Pergamon
Printed in Great Britain. AU rights reserved
0038.10981% s12.00+ .OO PIISOO38-1098(96)00297-9
BOUNDARY EFFECT INDUCED NONHYSTERETIC TRANSITION CAUSED BY SADDLE-NODE BIFURCATION ON A LIMIT CYCLE IN n-GaAs Yuo-Hsien Shiau *vt and Yi-Chen Cheng t * Institute of Physics, Academia Sinica, Taipei, Taiwan 115, ROC t Department of Physics, National Taiwan University, Taipei, Taiwan 106, ROC (Received 17 April 1996; accepted 10 May 1996 by A. Okiji)
Simulation study of nonlinear behavior of spatio-temporal oscillations of n-type GaAs at room temperatures is studied when the applied dc bias is in the negative differential conductivity regime. Modem MBE technique makes it possible that the electron density in the metal-semiconductor contact at the cathode, n(x = 01, may be controlled experimentally. It is found that when nfx = 0) is varied, the system may have a transition from an oscillating (travelling domains) to a non-oscillating (stationary domains) state. The transition is a global bifurcation known as saddle-node bifurcation on a limit cycle. The bifurcation feature of this transition is studied in detail. Copyright @ 1996 Published by Elsevier Science Ltd Keywords: A. semiconductors, D. bifurcation, D. electric-field domains, E. simulation.
It is well known that self-generated current oscillations in n-GaAs at room temperatures are due to the characteristic of negative differential conductivity (NDC) inherited in the system. Conventional studies of this well-known Gunn effect [l], both theoretically and experimentally [2], are mainly concentrated on the formation and propagation of the high field domains and the associated current oscillations when the applied field is increased to above the threshold field and in the NDC regime. In these studies the control parameter is the applied voltage bias. In this paper, we report a simulation study of the boundary effect on the dynamical behavior of the Gunn oscillations described above. We use the degree of the electron depletion at the cathode as the control parameter and find a nonhysteretic transition from a state of travelling domains to a state of stationary domains. It is a quite different result compared with the conventional Gunn effect, which describes a transition from a state of spatially uniform distribution of electric field to a state of propagation of high field domains when the applied field is increased from the positive differential conductivity regime to the NDC regime. The bifurcation scheme we found near the nonhysteretic transition is a standard one which is called saddle-node bifurcation on a limit cycle [3,4]. Similar bifurcation schemes have also been observed in Josephson junction [5J and p-type
germanium at liquid-helium temperatures [6]. In this paper we report in detail the behavior of the frequency scaling, the amplitude of spontaneous oscillation, and the topology of the constructed attractors using timedelay method near the transtion. All these behaviors fit the bifurcation scheme of the standard saddle-node bifurcation on a limit cycle [3,4]. The boundaries affect the dynamical behavior of the system via boundary conditions. One of the relevant equations in the study of the Gunn effect is the divergence equation (in one-dimension): 3E 1 (n - n0k (1) ax=; where n and no are the concentrations of the conduction electrons and the background effective donors, respectively; E is the permittivity of the sample with an applied field is in the x direction. For convenience of analysis we assume that each electron carries a positive charge e > 0. For the Gunn oscillations the high field domains, and therefore the high/low electron density domains, are formed near the cathode and then travel down to the anode [2]. In this picture the electron density at the cathode is primarily dependent on the interface properties of the metal/semiconductor (MS) contact, not on the dynamical behavior of the system. Therefore we may consider n in Eq. (1) at the cathode is a surface dependent quantity and thus imposes 305
306
BOUNDARY EFFECT INDUCED NONHYSTERETIC TRANSITION
a boundary condition at the cathode. With modem
semiconductor fabrication technology [7], it becomes possible that n in the MS contact can be controlled in the laboratory. The MS contact is known as the Schottky barrier (SB) [8], whose properties may affect the performance of many semiconductor devices. The barrier height at the junction will be affected by the presence of the interface or defect states at the junction. An ideal SB is defined to have a barrier height equal to the initial contact potential difference, which is linearly proportional to the work function of the metal, and there is an entire depletion of the conduction electrons near the contact region in the semiconductor side. The index of an interface S is defined to be equal to d+/dy/, where 4 and q are the barrier height and the metal’s work function respectively [7]. For an ideal SB S = 1. However most of the measured values of S for the MS contacts are nonideal. For example, S is approxmately equal to 0.1 for metal/n-GaAs under variously experimental techniques [9]. This phenomenon indicates “Fermi level pinning” at the surface states, and electrical neutrality will be roughly maintained near the contact region. The main reason for S # 1 is due to the presence of the interface/defect states. Modem technique of molecular-beam epitaxy (MBE) can produce S = 1 for a variety of metal/n-GaAs contacts recently [7]. Therefore, it is reasonable to expect that the electron density n in Eq. (1) at the cathode can be controlled experimentally in a wide range from n = 0 (ideal SB, S = 1) to n = no (electrical neutral, S = 0). In the following we study the dynamical behavior of the Gunn diode with a fixed applied field in the NDC region, and consider n at the cathode as the control parameter. We consider an n-type GaAs with a sample length L at room temperature which is applied by a dc bias V. The applied field E = V/L is the NDC region of the current-density-field characteristic of n-GaAs. Thus the system may undergo spatio-temporal oscillations. We assume that the only relevant spatial variable is x which is in the direction of the field E. The dynamical equation for E(x, 1) can be written as [lo, 111
Vol. 99, No. 5
E, E(x, t) = E, +
c E,,,(t) eimkr,
(3) In where E., = V/L is the applied static uniform field, E,,, are the induced fields where m is an integer extending from --oo to +UJ, and k = 2rrlLo with Ls a length constant. The rate equations for E,,,(t) can be obtained by substituting Eq. (3) into Eq. (2), and expanding v’(E) around the static field E3 to obtain coupled first order ordinary differential equations for E,,, [I I], dEm
-
dt
=
(am - imw)E, + i (J,,, - enov,,)d,,o
am = -inov.i’) -D& -
Bpm
=
&vjpp’ E 2w
v,
=
%%,),
v(P) I
=
E
dEP Es&’ By substituting Eq. (3) into the circuit equation, V = jt E(x, t)dx, we obtain a relation between the Fourier components E,,,, w=z-’
&ML
+ i m+O&(l-e”@)Em=O. C
(5)
Equations (3)--(S) are the basic equations for our model. Equation (4) is a set of infinitely many coupled first order ordinary differential equations involving dE,ldt . When the applied field Es is in the NDC regime, vji) is negative and some of the linear coefficient (rmin Eq. (4) will be positive. This means that some of the mode Em will be unstable and the system
undergoes spatio-temporal oscillations. Before we present the details of the numerical results, we see that: (1) Conventional choice of Lo = L [lo, 13,141 is not an appropriate one for the study of the Gunn effect. A simple reason is that from Eq. (3) one would have PE/axpl,=o = aPE/dxpI,,L for all p = 0,1,2 - . - if & = L. These requirements are rather unphysical be-
where S and b are the effective drift velocity and the diffusion constant of the conduction electrons, respectively; Jlo, is the total current density which is a function of time t only [12]. The dynamical equation (2) is a partial differential equation for the field E (x, t 1. We use the Fourier series expansion method to treat the spatial dependence of
cause in the Gunn effect the cathode (x = 0) and the anode (x = L) have quite di!Terent environments In the cathode the electrons are injected from the metal to the semiconductor, in which there is a Schottky barrier to overcome. But in the anode, the electrons travel from the semiconductor to the metal, where there is no potential barrier for the electrons Therefore boundary characteristics and field profiles are quite different at the cathode and the anode, which implies L,s = L is
Vol. 99, No. 5
BOUNDARY EFFECT INDUCED NONHYSTERETIC TRANSITION
not an appropriate choice. A possible way is to choose Ls = 2L [l 1,151 and this is what we have done in this paper. (2) The dynamical equation (2) is a second order partial differential equation with respect to x, therefore we may have two boundary conditions to specify the solutions We impose both boundary conditions at the cathode (X = 0) and leave the anode as a free end. The reason is that the electrons are injected from the cathode into the sample and then travel to the anode. In the process of travelling down the sample, the electrons undergo spatio-temporal oscillations which are governed by the dynamical equation. Therefore the field at the anode depends very much on the dynamical behavior of the system, and can be approximately considered as a free end. (3) We expect that most of the electrons injected from the cathode into n-GaAs are in the high-mobility central valley because the electrons experience a weak field in the metal. It takes distance for the electrons to be accelerated and transferred to the low-mobility satellite valleys of n-GaAs It is only when there are sizable fraction of the electrons transferred to the lowmobility valleys, the system becomes unstable due to NDC. Therefore it is reasonable to approximate the cathode to be in a static situation. We choose E = l& = V/L and aEl&t = en&/no - 1)/e as the two boundary conditions at the cathode. The f&d field condition E = E, has been used to analyze the dynamical behavior of the Gunn effect in previous studies [16]. The second bounary condition is related to the electron density n at the cathode. As we have mentioned earlier in this paper we may choose n at the cathode as a control parameter because it may be controlled in the laboratory by the modern MBE technique [7]. If n = no it represents a neutral MS contact, and for an ideal Schottky barrier n = 0. The parameter values used for the numerical calculationsare: L =I1 m, v’(E) = [~E+~cI(E/&)~]/[I + (E/&)4], p = 5000 cm2/Vsec, vo = 8.5 x lo6 ctu/sec, & = 4.0 kV/cm, E = 1.1 x lo-i2 CouWcm, no = 10%m3, b =I40 cm2/sec, and L&=4.8kVlcm which is in the NDC regime. With this field it is sufficient to keep the modes E,,, with Iml s 3 and all higher order modes can be neglected as a,,, < 0 for Iml r 3. In Fig. 1 we plot the current density as a function of the control parameter n/no. When the control parameter is in the range 0.73 < n/no I 1, the dynamical system is in a state of current oscillation, but when n/no < 0.73 the system bifurcates to a non-oscillating state. The nature of the bifurcation will be discussed in the following. But first we think it is worthwhile mentioning that the non-oscillating state (solid line in
z
5
u
I
* ---
l
t
17.0 I
0.50
l
’
l
l
307
oscillatin current non-osci d ding current atatic current
’
’
’
’ * OS?5
-
.
I
I
I 1.00
n/n0
Fig. 1. Current density versus control parameter n/no. The asterisk symbol t, the solid line, and the dashed line correspond to oscillating current, non-oscillating current, and static current at the operating point, respectively. The boundary between the oscillating and non-oscillating regimes is n/no=0.73. Fig. I) is not the same as the static state (dashed line in Fig. 1). The difference is that the former is a state with stationary domains (Fig. 2), while the latter is a state of spatially uniform. We characterize the transition from an oscillating to a non-oscillating at n/no = 0.73 as nonhysteretic transition caused by saddle-node bifurcation on a limit cycle [3,4]. This is a global bifurcation and our claim is supported by the plots of the amplitude of oscillation and the frequency squared versus n/no, respectively, in Figs 1 and 3. In the oscillating state (0.73 < n/no I 1) we see that the amplitude of oscillation is independent of the control parameter n/no; while the frequency of oscillation varies as the square root of the control parameter n/no. These are the two signatures of nonhysteretic transition caused by saddle-node bifurcation on a limit cycle [3,4]. The phenomenon that the frequency of the current oscillation decreases as n/m approaches the critical value (or bifurcation point) 0.73 is also known as criiically slowing down [3]. To give more details of the nature of the bifurcation feature of the transition, we analyze series of superposition of attractor reconstructions by the time-delay method near the bifurcation point as shown in Fig. 4. It is obvious to see that these trajectories follow the parabolic lines to approach the diagonal, and finally hit the diagonal at a point. This characteristic is precisely the signature of the standard saddle-node bifurcation on a limit cycle [3]. Finally we remark that the bifurcation features
BOUNDARY EFFECT INDUCED NONHYSTERETIC
TRANSITION
Vol. 99, No. 5
22.50
tiz0.03 tf=0.16
ns ns 21.75
19.50
20.25
Jdt)
21.00
21.75
22.50
(A/mm2)
Fig. 4. Superposition of different attractor reconstructions obtained by the time-delay method (T = 20.6 ps) from spontaneous current oscillations at different value of the control parameter n/no = 0.95 (l), n/no = 0.9 (2), n/no = 0.85 (3), n/no = 0.8 (4), n/no = 0.75 (5). Note that the trajectories approach and hit the diagonal parabolically as shown in the dashed square.
Fig. 2. Illustration of stationary domains formation at n/no = 0.72. ti, tf and A? mean the initial time, the final time and equal time interval between the neighbouring curves, respectively.
of standard nonhysteretic transition from a nonoscillating to an oscillating state caused by saddlenode bifurcation on a limit cycle can be described by a two-variable (r, 0) dynamical system [3,4], dr z = r - r3,
de -
38 h
*N 5
25-
N
13 -
Fig. 3. Square of the frequency versus control parameter n/no. The straight line is the best fit for the numerical data (O symbol).
=p-rcose, (7) dt where cc is the control parameter. Equations (6) and (7) predict a critical value for the control parameter CI = Lkr = 1 such that for fl < /Jo, the system is in a state of non-oscillating, and that for p > pEr the system is in a state of oscillating. The characteristics of this transition are exactly the same as we have plotted in Figs 1-3, i.e., (1) the amplitude of the oscillation in the oscillating state is independent of the control parameter (Fig. l), (2) the frequency of the oscillation is proportional to (cc- per)1/2 (Fig. 2), and (3) the trajectories in the time-delay plot of different attractor reconstructions with various values of the control parameter approach the diagonal parabolically (Fig. 3). It is interesting to note that the dynamical behavior we studied in this paper is governed by the dynamical equations (3)-(5) which are quite different from the two-variable dynamical equations (6) and (7) and that the control parameter does not appear explicitly in the dynamical equations (3j(5). Yet the bifurcation features of these two systems are exactly the same as
Vol. 99, No. 5
BOUNDARY EFFECT INDUCED
we have shown in Figs 1-3. Our result is therefore a good example of the principle of universality of the bifurcation phenomena regardless how different the dynamical systems are. This is also the first time that a Gunn diode is shown to have the possibility of observing the global bifurcation of nonhysteretic transition from an oscillating to a non-oscillating state caused by saddle-node bifurcation on a limit cycle. Acknowledgements-This work is supported in part by the National Science Council of the Republic of China under Contract Nos. NSC 85-2112-M-002-005 and NSC 85-2112-M-001-003. REFERENCES
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