Основы наноелектроники / Основы наноэлектроники / ИДЗ / Книги и монографии / Наноматериалы, методы, идеи. Сборник научных статей, 2007, c.206

(see figure 4).

The spontaneous transverse field influences naturally the currentvoltage characteristic

jx = jx(Ex, T), which does not depend on the direction of the spontaneous field Ey. For the case of the existense of second order NPT's the function jx is presented in figure 5.

Figure 5. The cur rent-volt age characteristic at T = fix (normalized). The curves correspond to the following values of the temperature: 1) T = 0.05; 2) T = 0.5. The solid lines 1-la and l-2a correspond to stable

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states of Ey, the dotted parts lb and 2b to unstable ones. The lines 1-lb and l-2b describe function Jx(E x ,O, T]. which in 2SL units for E and T

coinsides with 1SL current density (10). The maxima of curves correspond to critical value of applied field Ex = Exc(T).

4. Conclusion

In present paper, an exact distribution function has been found of the carriers in the lowest parabolic miniband of a 1SL. The novel formula for the current density in 1SL contains temperature dependence, which leads to the current maximum shift to the low field side with encreasing temperature, that agrees with available experimental data.

The considered SL model includes NPT's studied earlier. Besides, a novel type NPT has been found, in which the sample temperature plays role of a control parameter. It follows from the temperature dependence of ExC(T), that a novel type NPT2 is possible in the model considered. Indeed, in the field interval E1 < |EX| < E2 the transition "Curie temperature" Tc{Ex) exists and near the transition point the nontrivial transverse field ЕУ(Т) 0 appears, which behaves

asEy ~ T Tc Ex . The estimates of the effects predicted are

reduced, in general, to estimate of electric utuu required. At d = 10"7 cm,

= 10-12 s we have 10-2 eV and units for E and T are 2 • 103 and 100 К respectively. At Ex 600V/cm, the value Tc 70 К is obtained.

References

[1]Asche M, Kostial H and Sarbey О G 1980 J.Phys. C: Solid State Phys. 13 L645

[2]Shmelev G M and Epshtein E M 1992 Fiz. Tverd. Tela 34 2565 (Engl. Transl. 1992 Sov. Phys.

Solid State 34 1375)

|3] Shmelev G M , Epshtein E M, Chaikovskii I A and Matveev A S 2003 Izv. AN, ser. fiz. 67 lilO (in Russian)

[4]Matveev A S 2004 Izv. VGPU (Estestv. г fiz.-mat. nauki) No. 4 18 (in Russian)

[5]Shmelev G M , Epshtein E M and Gorshenina T A 2005 condmat/0503092 6] Romanov Yu A 2003 Phys. Solid State 45 559

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[7]Khadzhi P I 1971 Probability Function (RIO AN MSSR, Kishinev) (in Russian).

[8]Grahn H T , von Klitzing K, Ploog К and Dohler G H 1991 Phys. Rev. B, 43 12004

[9]Bonch-Bruevich V L, Zvyagin I P and Mironov A G 1975 Domain Electrical Instabilities in Semiconductors (New-York: Consultants Bureau)

[10]Shmelev G M and Maglevanny 11 1998 J. Phys.: Condens. Matter 10 6995

[11]Gilmore R 1981 Catastrophe Theory for Scientists and

Engineers (A Wiley-Interscience Publication John Wiley & Sons, New York, Chichester, Brisbane, Toronto)

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Stability of Turing Patterns in a Reaction-Diffusion System with Boundary Control

Eugene V. Kagan

E. V. Kagan is with the Department of Industrial Engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv 69978, Israel (e-mail: kaganevg@post.tau.ac.il ).

Abstract—We consider a two-component reaction-diffusion system. If the equilibrium state of this system is unstable, then there exist dissipative structures – Turing patterns – determined by the parameters of the system.

The paper considers stability of Turing patterns in the reactiondiffusion system with boundary control. The study is based on the Liapunov vector functions, and applies the Sirazetdinov method of stability in measure.

The study is stimulated by the example of reaction-diffusion system, which describes behavior of a two-layer semiconductor structure.

Index Terms—Reaction-diffusion system, Turing patterns, boundary control, stability in measure.

INTRODUCTION

A WIDE class of non-linear physical and chemical systems demonstrates a spatial formation of regular wave structures in initially homogeneous media. An accepted description of such phenomenon is provided by a non-linear parabolic system [12]

u D u f (u,b) ,

t

parameter. Such system is called Turing system. It is assumed that the

parameter b such that if b b0 then the homogeneous state of the system is unstable, and the system behaves as follows [7]:

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if b b0 then the auto-waves and the wave structures appear

without external influence.

In the paper, we consider a two-component (n 2 ) Turing system. Such systems are called reaction-diffusion systems. For reactiondiffusion system, u1 is interpreted as a concentration of activator and u2 - as a concentration of inhibitor.

In 1988, Balkarey, et al. [2] considered a semiconductor realization of the reaction-diffusion system and the possibilities of its application for the data storage. Balkarey, et al. showed that if ij 0 , i, j 1, 2 , then

in the semiconductor structure appear dissipative structures depending on the parameter b. In [4], [5], [6] this system with the boundary control and the methods of its realization were considered.

In 2003, Ishikawa, et al. [3] addressed a reaction-diffusion system, which is controlled by the bifurcation parameter b b1 ,b2 , where b1 and b2 are determined by mutually independent Wiener processes, and showed that the stable dissipative structures in the system are still existing.

The study of the reaction-diffusion system with stochastic boundary conditions is conducted in 1994 by Sowers [11] who considered the system with the white noise on the boundaries (in the Neumann form). The similar study of the system with the boundary conditions in the Dirichlet form was treated in 2002 by Alòs and Bonaccorsi [1].

In the paper, we consider reaction-diffusion system, which describes the two-layer semiconductor behavior with the boundary control. The goal of the paper, in opposite, to the previous works, is to study stability of dissipative structures on the characteristic of the control processes.

Considered problem and methods

The considered reaction-diffusion system is presented in the following form:

where u1 and u2 are interpreted as non-equilibrium concentrations of activator and inhibitor, correspondently. Thus, the initial conditions for the system (1) are:

We suppose that boundary conditions are given as follows:

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which correspond to the quadric recombination of the electrons and holes with stationary concentrations u10 and u20 , find parameters of the

processes and such that the solutions of the system are stable autowaves.

For stability analysis of the system (1), we apply the generalization of the Liapunov methods, suggested by Sirazetdinov [10], and called stability in measure.

The idea of the method is following. Let u :D T R2 , D R2 , T 0, , be a process. We suppose that the domain D is bounded, and the values on its bounds are determined by boundary conditions (3). To study stability of the process u 0 relatively to permanent disturbances, e.g., processes and , consider real measures 0 0 u ,t and

u ,t ,t , which are defined on the set of processes u, and a real

d 0, 0 0 , and for each process u the measure is continuous on t.

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- for every solution of the system such that 0 0 for t t0 0 , and

-for every disturbance such that d d ,

We consider the process u u1, u2 0, 0 as an initial equilibrium state and require that it is unstable in measures for the boundaries (3). Then we consider the dissipative structures for the system, and consider their stability in measures.

Approximate solution of the system

Suppose that the solution of the system (1) is given by the following formal series:

n 0

such that the initial and the boundary conditions hold true. For certain approximate methods for solving system (1) in the form (5) see, e.g. [8].

Let us consider solution (5) of the system (1) with the functions f1 and f2 as given by (4). Let

zn exp i nt j n x k n y ,

multiplication of the series (5) [13]. For each s 1, 2 , this equation is equivalent to the system of one equation for Re zn , and the equations for all possible rearrangements of i, j and k.

Suppose that space and time derivations of us,n are equal to zero. This

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supposition is equivalent to the one used in the method of slowly varying amplitudes [8]. Then from (6) we obtain:

From these formulas, using the initial and boundary conditions (2) and (3), and a requirement of convergence of the series (5), the certain values of the amplitudes and phases can be obtained. Initial condition

(2) in this case, has a form

and for the boundary conditions series (5) must converge to the corresponding functions and .

Stability of the process in measures

Let us define the measures 0 , and d as follows:

because of the initial condition (2), at the time t 0 both obtain constant

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an initial state of the system applied to silicon structure is instable. The similar reasons hold true for germanium, as well.

If functions u1 and u2 are (quasi-)periodic at t, then there exist functions f1 and f2 , not necessary linear, such that 0 for t 0. In fact, if the result u1 u2 of multiplication of the series (5) correspond to a periodic function, then functions f1 and f2 (4) with constant velocities v1 and v2 are periodic, and 0 .

Measure characterizes a current state of the system. If functions u1

determine a bound for the measure . Thus, the process is unstable in these measures.

Numerical simulations trialed for the system with constant boundaries and corresponding to silicon coefficients show that if f1 f2 0 , then the process diverges exponentially, and during 100 time-steps the concentrations u1 and u2 grow from ~1010 to ~10100 . If, in opposite,

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with the silicon parameters demonstrates complex behavior (sec. 0, Fig. 1). The initial state is broken and the traveling waves appear. For the germanium parameters (sec. 0, Fig. 2), the system leaves an initial state and converges to the stable state.

Stability of Turing patterns

Let us consider the measure . For stability of the system this measure has to be bounded, and the bound is determined by the measures 0 and d .

If the process is stable and if 0 , then the state of the system is

non-homogeneous, and wave solutions exist. Nevertheless, as shown by simulations of the system with silicon parameters (sec. 0, Fig. 1), the states of the system periodically change.

For stability of Turing patterns, in addition to the stability of the

The first condition means that the state of the system is nonhomogeneous, i.e. it corresponds to some, may be irregular, pattern on the surface, and the second condition requires from this state to be constant in time.

Let us apply measure to the series (5) and functions f1 and f2 given by (4). Using the formulas for us,n and taking into account that for

and u2 , respectively, we obtain:

u1 u2 1 v1 1u10u20 1 n2 n2 i n 1u2 2 ,

n

u2 u1 2 v2 2u10u20 2 n2 n2 i n 2u1 2 .

measure can be written in two equivalent forms:

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