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Emergence of a multidomain regime and spatiotemporal chaos in Gunn diodes under impact ionization conditions
5 March 2001
Physics Letters A 280 (2001) 312–317 www.elsevier.nl/locate/pla
Emergence of a multidomain regime and spatiotemporal chaos in Gunn diodes under impact ionization conditions Hidetaka Ito a,∗ , Yoshisuke Ueda b a Department of Electrical Engineering, Kansai University, Suita, Osaka 564-8680, Japan b Department of Complex Systems, Future University-Hakodate, Hakodate, Hokkaido 041-8655, Japan
Received 25 September 1998; received in revised form 8 January 2001; accepted 26 January 2001 Communicated by A.R. Bishop
Abstract Nonlinear spatiotemporal dynamics of carrier densities in Gunn diodes under impact ionization conditions is numerically investigated using a set of model partial differential equations. Numerical results show that a multidomain regime emerges as a result of the decrease in domain size caused by impact ionization, and that the spatiotemporal evolution of the domains becomes chaotic in the presence of strong impact ionization. 2001 Published by Elsevier Science B.V. Keywords: Multidomain patterns; Spatiotemporal chaos; Gunn oscillation; Impact ionization
1. Introduction Instabilities in semiconductors have attracted a wide variety of experimental, numerical and theoretical studies because these instabilities provide a wealth of complex nonlinear spatiotemporal dynamics [1]. The presence of space charges in semiconductors induces the formation of electric-field domains, their local interaction, and also global interaction through the electric field. The present Letter focuses on the spatiotemporal dynamics in Gunn diodes under impact ionization conditions, in which a multidomain regime emerges and the interaction among the domains leads to spatiotemporal chaos. The Gunn diode is an active microwave element exhibiting self-sustained current oscillations when the
* Corresponding author.
E-mail address: [email protected] (H. Ito).
applied DC voltage exceeds a characteristic threshold value. These oscillations are the result of a negative differential mobility of electrons that leads to electrical breakup of the crystal into traveling high-field domains that form at the cathode and disappear at the anode. For typical GaAs Gunn diodes, the energy required for impact ionization is considerably higher than that required for the electron transfer from the conduction band minimum to the low-mobility satellite valleys; therefore, the effect of impact ionization is negligible. On the other hand, the effect of impact ionization is manifest on the Gunn effect in narrow-gap semiconductors such as InSb. For example, irregular domain motion was observed experimentally in InSb Gunn diodes [2]. This irregularity was later studied ˇ both analytically and numerically by Cenys et al., who explained these oscillations in terms of deterministic chaos resulting from the competition between the Gunn effect and a plasma wave instability, the lat-
0375-9601/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 7 1 - 8
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
ter arising from impact ionization [3]. Their investigation, which was primarily based on a successful linear stability analysis for small-amplitude oscillations, focused on the onset of chaos. In this Letter, we investigate Gunn oscillations in the presence of impact ionization from a different viewpoint in order to clarify the formation of multiple domains and their chaotic behavior in a largeamplitude regime. In order to show explicitly how the multidomain regime is related to the single-domain regime observed in ordinary Gunn oscillations, we formulate a numerical experiment that allows hypothetically controlling the impact ionization rate.
313
Table 1 Model equations Continuity equations for the carrier densities: ∂n = − ∂ (nv (E)) + D ∂ 2 n + g (E)n − γ np, n n ∂x 2 n ∂t ∂x ∂p ∂ (pv (E)) + D ∂ 2 p + g (E)n − γ np. = p p ∂x 2 n ∂t ∂x
Poisson equation: ∂E = n − p − n . d ∂x
Electron and hole velocities: β vn (E) = µn E+(E) β , 1+(E) µp E (E < Ep ), vp (E) = µp Ep (E Ep ).
Impact ionization rate: gn (E) = g0 exp[−(Eg /E)2 ].
2. Model and numerical method We use a spatially one-dimensional model of a Gunn diode with dimensionless variables based on the forˇ mulation of Cenys et al. [3]. This model, presented in Table 1, is based on a set of three nonlinear partial differential equations in the form of the Poisson equation and two continuity equations for carriers, including impact ionization and recombination. The system is assumed to occupy the region 0 x L, where x = 0 and x = L correspond to the cathode and the anode, respectively. To hypothetically control the impact ionization rate, the parameter Eg , which corresponds to the direct energy gap, is allowed to vary independently of other physical parameters. In the present study, the parameter values are fixed as follows: Dn = 0.1, Dp = 0.01, µn = 5, µp = 1.5, β = 4, Ep = 2, g0 = 3, γ = 0.5. The normalized donor density nd is assumed to be 0.2 at the cathode region (0 x < 2) and 1 in the bulk (2 x L). This set of parameter values together with Eg = 1.26 corresponds to n-InSb at 77 K having a donor density of 1013 cm−3 [3]. The applied voltage and the length of the diode are set at V = 33 and L = 30, respectively, yielding an average electric field of V /L = 1.1. In our numerical simulation, the sample is divided equally into partitions of length 0.1, and the spatial derivatives are approximated by central differences, whereas time derivatives are approximated by forward differences. The resulting set of coupled ordinary differential equations is then integrated using the
Applied voltage: L 0 E dx = V . Boundary conditions: n(x, t) − p(x, t) = nd (x), n, p:
electron and hole densities.
vn , vp : E: gn :
p(L, t) = 0.
electron and hole velocities.
electric field. impact ionization rate.
fourth-order Runge–Kutta method setting the time step at 0.002.
3. Numerical results We begin by obtaining an ordinary Gunn oscillation by eliminating the effect of impact ionization. This can be achieved either by fixing gn (E) at zero or by setting Eg to be sufficiently large. Fig. 1(a) shows the post-transient spatiotemporal evolution of the electric field for Eg = 30.0. Since the peak value of the electric field is now less than the threshold value for impact ionization, no holes are generated by impact ionization. Thus, the oscillation is considered to be an ordinary Gunn oscillation, in which a single high-field domain forms at the cathode, propagates through the diode and disappears at the anode. As we decrease Eg , the threshold electric field for impact ionization becomes less than the peak value of the electric field, which leads to the occurrence
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H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
(a)
(b)
(c)
(d)
Fig. 1. Spatiotemporal evolution of the electric field. (a) Eg = 30.0, (b) Eg = 5.0, (c) Eg = 3.0, (d) Eg = 2.95, (e) Eg = 2.0, (f) Eg = 1.6, (g) Eg = 1.4, (h) Eg = 1.26.
of impact ionization in the high-field domain. As discussed in detail later, impact ionization reduces the size of the domain. Fig. 1(b) shows the propagation of the domain for Eg = 5.0. In addition to the decrease in the domain size, we see a slight increase in the propagation speed of the domain. This is due primarily to the slight increase in the electron velocities, both
inside and outside the domain. Note that the decrease in the domain size results in an increase in the electric field outside the domain. The gradual change in the domain size and propagation speed continues until Eg approaches a critical value (≈ 3.0), at which point a rapid change in the spatial profile of the electric field occurs. Figs. 1(c)
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
(e)
(f)
(g)
(h)
315
Fig. 1. (Continued.)
and (d) show the spatiotemporal evolutions of the electric field for Eg = 3.0 and 2.95, respectively. In Fig. 1(c), the oscillatory state is still similar to that of the ordinary Gunn oscillation in that a single domain is dominant at any given time. In contrast, Fig. 1(d) shows that two identical domains propagate simultaneously in the diode. The domains, each of which is smaller than the domain for the ordinary Gunn oscillation, share the applied voltage. This is a consequence
of the effect of impact ionization, which reduces the domain size and facilitates the formation of competing domains. As Eg is decreased further, the domain size becomes even smaller, as shown in Figs. 1(e) and (f) for Eg = 2.0 and 1.6, respectively, and the number of coexisting domains increases. Eventually, the oscillation that was previously periodic loses temporal stability, and the behavior of the system becomes highly irreg-
316
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
ular. Figs. 1(g) and (h) show the spatiotemporal evolution of the electric field for Eg = 1.4 and 1.26, respectively. Although the localized structure due to the domain formation remains present, the domains interact and compete with one another in a complex manner and do not converge to any orderly state. By computing the spectrum of Lyapunov exponents, we have confirmed that these irregular oscillations are chaotic and obtained estimated values of 9.3 and 9.6 for the Lyapunov dimension for Eg = 1.4 and 1.26, respectively. These values are much higher than those
obtained for some models of periodically driven Gunn diodes that do not experience impact ionization, e.g., the model derived by Mosekilde et al. [4–6] and a modified model of the present one, which ignores the effect of impact ionization [7,8]. To examine more closely the decrease in the domain size and the formation of the second high-field domain, we next modify the experiment to integrate the model equations (under impact ionization conditions) using an initial condition that corresponds to a state of domain propagation in the ordinary Gunn os-
(a)
(b)
(c)
(d)
Fig. 2. Distribution of the electric field (solid line), electron density (long-dashed line) and hole density (short-dashed line) for the modified experiment. (a) t = 0, (b) t = 0.8, (c) t = 2.4, (d) t = 3.4.
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
cillation. Fig. 2(a) represents the initial distribution of the electric field and electron density for the modified experiment. In Fig. 2, solid, long-dashed and shortdashed lines represent the distribution of the electric field, electron density and hole density, respectively. The initial distribution has been obtained by setting Eg at 30.0 and integrating the model equations until the domain reaches the middle of the diode. Note that the size of the domain has already saturated, and without impact ionization the shape of the domain would remain the same until reaching the anode. We now introduce impact ionization by setting Eg at 2.5 and follow the temporal evolution of the distribution of the electric field and carrier densities. Figs. 2(b)–(d) show the successive distributions of the electric field, electron density and hole density at t = 0.8, 2.4 and 3.4, respectively. At the cathode side of the domain, a cluster of holes is generated by impact ionization because of the large accumulation of electrons and the high electric field. The clusters of electrons and holes move in opposite directions and form an electrical dipole layer that produces an electric field in the direction opposite to the applied external field. Consequently, the electric field at the domain decreases and that outside the domain increases. For the present value of Eg , the increase in the electric field outside the domain leads to the formation of the second high-field domain at the cathode before the first domain disappears at the anode.
4. Conclusion We have investigated the formation of multiple domains and their ordered and spatiotemporally chaotic behavior in Gunn diodes under impact ionization conditions. It has been shown that the emergence of the multidomain regime for the InSb Gunn diode is related to the decrease in the domain size that is caused by impact ionization. In the multidomain regime, individual domains have more autonomy than those in periodically driven ordinary Gunn diodes, which leads to the high dimensionality observed for the chaotic states. The effect of impact ionization has been studied also for GaAs Gunn diodes by Oshio and Yahata [9], and careful comparisons of the results observed for
317
these two cases would be valuable. It should also be noted that Kerner and Osipov have studied a great variety of domain motions in various active media [10]. In particular, thermal diffusion domains in a semiconductor can produce a spatiotemporal pattern that is somewhat similar to the temporary-periodic multidomain regime in the present study although the underlying physical phenomena are quite different to each other [11]. Their results, together with the recent attempt to derive universal equations for describing multidomain patterns [12], should be considered and related to the present study.
Acknowledgements We would like to thank Mr. Keigo Harada, former graduate student of Kyoto University, for helpful discussions. One of us (H.I.) is also grateful to Professor Erik Mosekilde of the Technical University of Denmark and Professor Akira Kumamoto of Kansai University for continuous encouragement and supports.
References [1] F.-J. Niedernostheide (Ed.), Nonlinear Dynamics and Pattern Formation in Semiconductors and Devices, Springer, 1995. [2] S. Porowski, W. Paul, J.C. McGroddy, M.I. Nathan, J.E. Smith, Solid State Commun. 7 (1969) 905. ˇ [3] A. Cenys, G. Lasiene, K. Pyragas, Solid-State Electron. 35 (1992) 975. [4] E. Mosekilde, J.S. Thomsen, C. Knudsen, R. Feldberg, Physica D 66 (1993) 143. [5] H. Ito, E. Mosekilde, Y. Ueda, Trans. IEE Jpn. 113A (1993) 365. [6] H. Ito, Ph.D. thesis, Graduate School of Engineering, Kyoto University (1996). [7] H. Ito, K. Harada, Y. Ueda, in: Proc. 1993 International Symposium on Nonlinear Theory and Its Applications, Honolulu, 1993, p. 1015. [8] K. Harada, Master’s thesis, Graduate School of Engineering, Kyoto University (1994). [9] K. Oshio, H. Yahata, J. Phys. Soc. Jpn. 64 (1995) 1823. [10] B.S. Kerner, V.V. Osipov, Autosolitons, Kluwer Academic, 1994. [11] V.V. Gafiichuk, B.S. Kerner, V.V. Osipov, I.V. Tyslyuk, Sov. Phys. Solid State 31 (1989) 1304. [12] C.B. Muratov, Phys. Rev. E 55 (1997) 1463.
Physics Letters A 280 (2001) 312–317 www.elsevier.nl/locate/pla
Emergence of a multidomain regime and spatiotemporal chaos in Gunn diodes under impact ionization conditions Hidetaka Ito a,∗ , Yoshisuke Ueda b a Department of Electrical Engineering, Kansai University, Suita, Osaka 564-8680, Japan b Department of Complex Systems, Future University-Hakodate, Hakodate, Hokkaido 041-8655, Japan
Received 25 September 1998; received in revised form 8 January 2001; accepted 26 January 2001 Communicated by A.R. Bishop
Abstract Nonlinear spatiotemporal dynamics of carrier densities in Gunn diodes under impact ionization conditions is numerically investigated using a set of model partial differential equations. Numerical results show that a multidomain regime emerges as a result of the decrease in domain size caused by impact ionization, and that the spatiotemporal evolution of the domains becomes chaotic in the presence of strong impact ionization. 2001 Published by Elsevier Science B.V. Keywords: Multidomain patterns; Spatiotemporal chaos; Gunn oscillation; Impact ionization
1. Introduction Instabilities in semiconductors have attracted a wide variety of experimental, numerical and theoretical studies because these instabilities provide a wealth of complex nonlinear spatiotemporal dynamics [1]. The presence of space charges in semiconductors induces the formation of electric-field domains, their local interaction, and also global interaction through the electric field. The present Letter focuses on the spatiotemporal dynamics in Gunn diodes under impact ionization conditions, in which a multidomain regime emerges and the interaction among the domains leads to spatiotemporal chaos. The Gunn diode is an active microwave element exhibiting self-sustained current oscillations when the
* Corresponding author.
E-mail address: [email protected] (H. Ito).
applied DC voltage exceeds a characteristic threshold value. These oscillations are the result of a negative differential mobility of electrons that leads to electrical breakup of the crystal into traveling high-field domains that form at the cathode and disappear at the anode. For typical GaAs Gunn diodes, the energy required for impact ionization is considerably higher than that required for the electron transfer from the conduction band minimum to the low-mobility satellite valleys; therefore, the effect of impact ionization is negligible. On the other hand, the effect of impact ionization is manifest on the Gunn effect in narrow-gap semiconductors such as InSb. For example, irregular domain motion was observed experimentally in InSb Gunn diodes [2]. This irregularity was later studied ˇ both analytically and numerically by Cenys et al., who explained these oscillations in terms of deterministic chaos resulting from the competition between the Gunn effect and a plasma wave instability, the lat-
0375-9601/01/$ – see front matter 2001 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 0 7 1 - 8
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
ter arising from impact ionization [3]. Their investigation, which was primarily based on a successful linear stability analysis for small-amplitude oscillations, focused on the onset of chaos. In this Letter, we investigate Gunn oscillations in the presence of impact ionization from a different viewpoint in order to clarify the formation of multiple domains and their chaotic behavior in a largeamplitude regime. In order to show explicitly how the multidomain regime is related to the single-domain regime observed in ordinary Gunn oscillations, we formulate a numerical experiment that allows hypothetically controlling the impact ionization rate.
313
Table 1 Model equations Continuity equations for the carrier densities: ∂n = − ∂ (nv (E)) + D ∂ 2 n + g (E)n − γ np, n n ∂x 2 n ∂t ∂x ∂p ∂ (pv (E)) + D ∂ 2 p + g (E)n − γ np. = p p ∂x 2 n ∂t ∂x
Poisson equation: ∂E = n − p − n . d ∂x
Electron and hole velocities: β vn (E) = µn E+(E) β , 1+(E) µp E (E < Ep ), vp (E) = µp Ep (E Ep ).
Impact ionization rate: gn (E) = g0 exp[−(Eg /E)2 ].
2. Model and numerical method We use a spatially one-dimensional model of a Gunn diode with dimensionless variables based on the forˇ mulation of Cenys et al. [3]. This model, presented in Table 1, is based on a set of three nonlinear partial differential equations in the form of the Poisson equation and two continuity equations for carriers, including impact ionization and recombination. The system is assumed to occupy the region 0 x L, where x = 0 and x = L correspond to the cathode and the anode, respectively. To hypothetically control the impact ionization rate, the parameter Eg , which corresponds to the direct energy gap, is allowed to vary independently of other physical parameters. In the present study, the parameter values are fixed as follows: Dn = 0.1, Dp = 0.01, µn = 5, µp = 1.5, β = 4, Ep = 2, g0 = 3, γ = 0.5. The normalized donor density nd is assumed to be 0.2 at the cathode region (0 x < 2) and 1 in the bulk (2 x L). This set of parameter values together with Eg = 1.26 corresponds to n-InSb at 77 K having a donor density of 1013 cm−3 [3]. The applied voltage and the length of the diode are set at V = 33 and L = 30, respectively, yielding an average electric field of V /L = 1.1. In our numerical simulation, the sample is divided equally into partitions of length 0.1, and the spatial derivatives are approximated by central differences, whereas time derivatives are approximated by forward differences. The resulting set of coupled ordinary differential equations is then integrated using the
Applied voltage: L 0 E dx = V . Boundary conditions: n(x, t) − p(x, t) = nd (x), n, p:
electron and hole densities.
vn , vp : E: gn :
p(L, t) = 0.
electron and hole velocities.
electric field. impact ionization rate.
fourth-order Runge–Kutta method setting the time step at 0.002.
3. Numerical results We begin by obtaining an ordinary Gunn oscillation by eliminating the effect of impact ionization. This can be achieved either by fixing gn (E) at zero or by setting Eg to be sufficiently large. Fig. 1(a) shows the post-transient spatiotemporal evolution of the electric field for Eg = 30.0. Since the peak value of the electric field is now less than the threshold value for impact ionization, no holes are generated by impact ionization. Thus, the oscillation is considered to be an ordinary Gunn oscillation, in which a single high-field domain forms at the cathode, propagates through the diode and disappears at the anode. As we decrease Eg , the threshold electric field for impact ionization becomes less than the peak value of the electric field, which leads to the occurrence
314
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
(a)
(b)
(c)
(d)
Fig. 1. Spatiotemporal evolution of the electric field. (a) Eg = 30.0, (b) Eg = 5.0, (c) Eg = 3.0, (d) Eg = 2.95, (e) Eg = 2.0, (f) Eg = 1.6, (g) Eg = 1.4, (h) Eg = 1.26.
of impact ionization in the high-field domain. As discussed in detail later, impact ionization reduces the size of the domain. Fig. 1(b) shows the propagation of the domain for Eg = 5.0. In addition to the decrease in the domain size, we see a slight increase in the propagation speed of the domain. This is due primarily to the slight increase in the electron velocities, both
inside and outside the domain. Note that the decrease in the domain size results in an increase in the electric field outside the domain. The gradual change in the domain size and propagation speed continues until Eg approaches a critical value (≈ 3.0), at which point a rapid change in the spatial profile of the electric field occurs. Figs. 1(c)
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
(e)
(f)
(g)
(h)
315
Fig. 1. (Continued.)
and (d) show the spatiotemporal evolutions of the electric field for Eg = 3.0 and 2.95, respectively. In Fig. 1(c), the oscillatory state is still similar to that of the ordinary Gunn oscillation in that a single domain is dominant at any given time. In contrast, Fig. 1(d) shows that two identical domains propagate simultaneously in the diode. The domains, each of which is smaller than the domain for the ordinary Gunn oscillation, share the applied voltage. This is a consequence
of the effect of impact ionization, which reduces the domain size and facilitates the formation of competing domains. As Eg is decreased further, the domain size becomes even smaller, as shown in Figs. 1(e) and (f) for Eg = 2.0 and 1.6, respectively, and the number of coexisting domains increases. Eventually, the oscillation that was previously periodic loses temporal stability, and the behavior of the system becomes highly irreg-
316
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
ular. Figs. 1(g) and (h) show the spatiotemporal evolution of the electric field for Eg = 1.4 and 1.26, respectively. Although the localized structure due to the domain formation remains present, the domains interact and compete with one another in a complex manner and do not converge to any orderly state. By computing the spectrum of Lyapunov exponents, we have confirmed that these irregular oscillations are chaotic and obtained estimated values of 9.3 and 9.6 for the Lyapunov dimension for Eg = 1.4 and 1.26, respectively. These values are much higher than those
obtained for some models of periodically driven Gunn diodes that do not experience impact ionization, e.g., the model derived by Mosekilde et al. [4–6] and a modified model of the present one, which ignores the effect of impact ionization [7,8]. To examine more closely the decrease in the domain size and the formation of the second high-field domain, we next modify the experiment to integrate the model equations (under impact ionization conditions) using an initial condition that corresponds to a state of domain propagation in the ordinary Gunn os-
(a)
(b)
(c)
(d)
Fig. 2. Distribution of the electric field (solid line), electron density (long-dashed line) and hole density (short-dashed line) for the modified experiment. (a) t = 0, (b) t = 0.8, (c) t = 2.4, (d) t = 3.4.
H. Ito, Y. Ueda / Physics Letters A 280 (2001) 312–317
cillation. Fig. 2(a) represents the initial distribution of the electric field and electron density for the modified experiment. In Fig. 2, solid, long-dashed and shortdashed lines represent the distribution of the electric field, electron density and hole density, respectively. The initial distribution has been obtained by setting Eg at 30.0 and integrating the model equations until the domain reaches the middle of the diode. Note that the size of the domain has already saturated, and without impact ionization the shape of the domain would remain the same until reaching the anode. We now introduce impact ionization by setting Eg at 2.5 and follow the temporal evolution of the distribution of the electric field and carrier densities. Figs. 2(b)–(d) show the successive distributions of the electric field, electron density and hole density at t = 0.8, 2.4 and 3.4, respectively. At the cathode side of the domain, a cluster of holes is generated by impact ionization because of the large accumulation of electrons and the high electric field. The clusters of electrons and holes move in opposite directions and form an electrical dipole layer that produces an electric field in the direction opposite to the applied external field. Consequently, the electric field at the domain decreases and that outside the domain increases. For the present value of Eg , the increase in the electric field outside the domain leads to the formation of the second high-field domain at the cathode before the first domain disappears at the anode.
4. Conclusion We have investigated the formation of multiple domains and their ordered and spatiotemporally chaotic behavior in Gunn diodes under impact ionization conditions. It has been shown that the emergence of the multidomain regime for the InSb Gunn diode is related to the decrease in the domain size that is caused by impact ionization. In the multidomain regime, individual domains have more autonomy than those in periodically driven ordinary Gunn diodes, which leads to the high dimensionality observed for the chaotic states. The effect of impact ionization has been studied also for GaAs Gunn diodes by Oshio and Yahata [9], and careful comparisons of the results observed for
317
these two cases would be valuable. It should also be noted that Kerner and Osipov have studied a great variety of domain motions in various active media [10]. In particular, thermal diffusion domains in a semiconductor can produce a spatiotemporal pattern that is somewhat similar to the temporary-periodic multidomain regime in the present study although the underlying physical phenomena are quite different to each other [11]. Their results, together with the recent attempt to derive universal equations for describing multidomain patterns [12], should be considered and related to the present study.
Acknowledgements We would like to thank Mr. Keigo Harada, former graduate student of Kyoto University, for helpful discussions. One of us (H.I.) is also grateful to Professor Erik Mosekilde of the Technical University of Denmark and Professor Akira Kumamoto of Kansai University for continuous encouragement and supports.
References [1] F.-J. Niedernostheide (Ed.), Nonlinear Dynamics and Pattern Formation in Semiconductors and Devices, Springer, 1995. [2] S. Porowski, W. Paul, J.C. McGroddy, M.I. Nathan, J.E. Smith, Solid State Commun. 7 (1969) 905. ˇ [3] A. Cenys, G. Lasiene, K. Pyragas, Solid-State Electron. 35 (1992) 975. [4] E. Mosekilde, J.S. Thomsen, C. Knudsen, R. Feldberg, Physica D 66 (1993) 143. [5] H. Ito, E. Mosekilde, Y. Ueda, Trans. IEE Jpn. 113A (1993) 365. [6] H. Ito, Ph.D. thesis, Graduate School of Engineering, Kyoto University (1996). [7] H. Ito, K. Harada, Y. Ueda, in: Proc. 1993 International Symposium on Nonlinear Theory and Its Applications, Honolulu, 1993, p. 1015. [8] K. Harada, Master’s thesis, Graduate School of Engineering, Kyoto University (1994). [9] K. Oshio, H. Yahata, J. Phys. Soc. Jpn. 64 (1995) 1823. [10] B.S. Kerner, V.V. Osipov, Autosolitons, Kluwer Academic, 1994. [11] V.V. Gafiichuk, B.S. Kerner, V.V. Osipov, I.V. Tyslyuk, Sov. Phys. Solid State 31 (1989) 1304. [12] C.B. Muratov, Phys. Rev. E 55 (1997) 1463.