AN ANALYTICAL AND NUMERICAL INVESTIGATION OF THE ...€¦ · CHAOS 2010, 3rd Chaotic Modeling and Simulation International Conference, Chania, Crete, Greece, 1 - 4 June 2010 1 AN

CHAOS 2010, 3rd Chaotic Modeling and Simulation International Conference,

Chania, Crete, Greece,

1 - 4 June 2010

1

AN ANALYTICAL AND NUMERICAL INVESTIGATION OF THE

DISSIPATIVE CHAOS IN SEMICONDUCTOR SUPERLATTICES

Alireza Heidaria , O. Anwar Bégb & P. Oseloka Ezepuec

aComputational Physics, Department of Chemistry, Ferdowsi University of Mashhad, Mashhad 91775-1436, Iran. Email: [email protected]

b Research Leader: Biomechanics, Biotechnology and Computational Magnetohydrodynamics, Mechanical Engineering Program, Department of Engineering and Mathematics, Room 4112, Sheaf Building, Sheffield Hallam University, Sheffield, S1 1WB,

England, UK. Email:[email protected].

c Research Leader: Business Intelligence & Quantitative Modelling, Department of Computing, Room 2211 Harmer Building, Sheffield Hallam University, Sheffield, S1 1WB,

England, UK. Email: [email protected]

Abstract

The transport of electrons in a semiconductor superlattice miniband under the influence of electrical and magnetic fields, which are applied in different directions on the superlattice, is investigated. The time series diagrams and the Lyapunov exponent are computed using the fourth-order Runge-Kutta method. The numerical computations show that for particular values of the parameters, which depend on the superlattice characteristics and the fields applied on them, electrons show chaotic behaviors. In addition, for some other parameter values these behaviors become regular and non-chaotic. The presence of a magnetic field, perpendicular to the electrical field, is shown to reduce the chaotic areas in the motion of the electron. An alteration in electron average energy and velocity is attributed to application of the external fields, carrier scattering from other carriers, and the phonons’ and lattices’ faults. The study has important applications in computational physics and semi-conductor chaotic simulations.

Keywords: Dissipative chaos; Semiconductor superlattices; Miniband; Electrons transport; Lyapunov exponent; fourth-order Runge-Kutta method

PACS: 02.; 02.60.-x; 02.70.-c; 05.; 05.10.-a; 05.45.-a; 05.45.Gg; 05.45.Pq; 05.45.Tp; 05.45.Vx



1 - 4 June 2010

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1. INTRODUCTION

Semiconductor superlattices are formed from the consecutive layering of two kinds of semi-

conductors with different energy gaps on an under-layer which is usually of the same

material as one of the semiconductors itself. They are formed using the molecular beam

fitting method (MBE). The most prevalent of these superlattices are AlxGa1-xAs/GaAs, in

which the combining semiconductor GaAs (with an energy gap of about 1.5eV) is placed, in

the form of the every other layer, between the semiconductor layers of AlxGa1-xAs forming a

ternary alloy. It is worth noting that for AlxGa1-xAs, x<0.4 and it has a direct band gap and

its energy gap is just less than 2eV.

Owing to the fact that in forming superlattices very thin barriers are used [1], electron

wave functions fall on each other from the surrounding wells and the tunneling of particles

from one well to another by moving through the barrier becomes probable. This causes the

quantumized energy levels degeneracy to be broken in each well. For the N levels of each

well, each degeneracy level is divided into N additional levels. As a result, a band with 2N

forms (considering the spin) is formed. The superlattice energy bands, which are formed like

this, are called minibands.

Electron transport in semiconductor superlattices, due to their remarkable

characteristics, has stimulated considerable interest in the engineering physics and

electronic materials fabrication research community for some years. Important early studies

have been communicated by Esaki and Tsu [2] and Price [3], both studies being developed

at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York. Esaki and

Chang [4] later studied electronic transport properties in a GaAs-AlAs periodic structure

which they termed the "superlattice" prepared by a molecular-beam epitaxy. They showed

that the differential conductance in the superlattice direction initially decreases slowly, with a

subsequent dramatic plummet to negative values. This stage is then ensued with an

oscillatory behavior with respect to applied voltages under high strength fields. They

elucidated these observations as the formation and expansion of a high-field domain, further

establishing that the voltage period of the oscillation supplies the requisite energy of the first-

excited band, a phenomenon which correlates strongly with that predicted by theory.

Palmier and Chomette [5] studied the transport properties at low applied electric field in

a one-dimensional semiconductor superlattice using two scattering processes, namely the



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deformation potential and the polar optical modes. They applied their model to GaAs/GaAlAs

superlattice systems, calculating the mobility tensor as a function of the superlattice period.

Movaghar [6] developed a theoretical model transport of charge carriers in

semiconductor multiple-quantum-well structures for motion perpendicular to the layers,

based on strong electric fields with a feature enabling the possibility of describing the low-

field ohmic region by using a simple scaling argument. He showed that disorder may

significantly modify the nature of the physical effects predicted on the basis of the 'pure'

eigenstates, with reference to experiments on magneto-transport perpendicular to the layers.

Further interesting experimental studies have been communicated by Peterson et al [7]

and Noguchi et al [8]. Rossi et al [9] presented a seminal study of ultrafast dynamics of

photoexcited carriers in semiconductor superlattices using a Monte Carlo solution of the

coupled Boltzmann transport equations for electrons and holes. They were able in this work

to describe kinetically the pertinent interaction mechanisms including intra-miniband and

interminiband carrier-phonon scattering processes. They identified the significant influence

of carrier–polar-optical-phonon interaction in determining the nature of the carrier dynamics

in the low-density limit, and further showed that the miniband width, in comparison with the

phonon energy, is the fundamental parameter for predicting the existence of Bloch

oscillations.

Alekseev et al [10] presented a robust model for the dynamics of ballistic electrons in a

miniband of a semiconductor superlattice (SSL) under the influence of an external, time-

periodic electric field, using the semiclassical, balance-equation approach, incorporating

elastic and inelastic scattering (as dissipation) and the self-consistent field generated by the

electron motion. They computed for a range of values of the dissipation parameters the

regions in the amplitude-frequency plane of the external field in which chaos arises,

proposing that for terahertz external fields of the amplitudes achieved by industrial free-

electron lasers, chaos may be observable in SSL systems. Scamarcio et al [11] reported

observations of infrared emission between superlattice conduction minibands, for electrons

resonantly injected into the first excited state miniband by an applied electric field make a

radiative transition to the ground state miniband. They described experiments in

AlInAs/GaInAs superlattices, achieving peak luminescence wavelengths of λ= 5 and 7 µm.

Alekseev et al [12] further studied the motion of ballistic electrons in a single miniband

of a semiconductor superlattice driven by a terahertz laser polarized along the growth

direction, investigating zones of complex dynamics, which may include chaotic behavior and

symmetry-breaking. They computed the magnitude of the DC current and voltage that



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spontaneously appears in regions of broken-symmetry for parameters characteristic of real

semiconductor superlattices.

Gómez et al [13] examined the electron transmission probability in semiconductor

superlattices in which barrier height is modulated by a Gaussian profile, showing that these

structures serve as efficient energy band-pass filters and achieving a lower number of

unintentional defects and enhanced performance. They further demonstrated that current-

voltage characteristic presents negative differential resistance with peak-to-valley ratios

which are significantly in excess of conventional semiconductor superlattices. Gmachl et al

[14] quantified Inter-sub-band optical absorption around 1.55 µm in GaN/AlGaN multiple

quantum wells (MQWs) grown by molecular-beam epitaxy. They designed and fabricated

QWs embedded in barriers consisting of a short period superlattice of narrow GaN QWs and

only 65% AlN mole-fraction barriers, refining the electron Bragg confinement which allowed

a peak absorption wavelengths as short as 1.52 µm. This approach also permitted

modulation doping of the structures via doping of the narrow superlattice wells and

subsequent charge transfer into the active well, which led to a decrease in the absorption

line width, from ∼200 to ∼130 meV, for these structures.

One of the most detailed theoretical investigations of electric transport in semiconductor

superlattices has been communicated by Wacker [15]. He has highlighted that such

transport is dominated by pronounced negative differential conductivity and has lucidly

reviewed the standard transport theories for superlattices, i.e. miniband conduction,

Wannier-Stark-hopping, and sequential tunneling, elucidating their relation to each other via

a comparison with a quantum transport model based on nonequilibrium Green functions.

Vashaee and Shakouri [16] developed a comprehensive theory for nonisothermal

electron transport perpendicular to multilayer superlattice structures is presented, computing

the current–voltage characteristics and the cooling power density using Fermi–Dirac

statistics, density-of-states for a finite quantum well and the quantum mechanical reflection

coefficient. They further compared their calculations with the experimental dark current

characteristics of quantum well infrared photodetectors achieving excellent correlation over a

wide temperature range for a variety of superlattice structures. This study superceded

previous investigations in that lateral momentum conservation was satisfied for the case of

electron transport in planar semiconductor barriers.

Perales et al [17] obtained comprehensive solutions of the drift-diffusion model equations

for miniband transport in strongly coupled superlattices, basing their analysis on a single-

miniband Boltzmann–Poisson transport equation with a BGK (Bhatnagar–Gross–Krook)



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collision term founded on a consistent Chapman–Enskog expansion. The reduced drift-

diffusion equation was solved numerically and travelling field domains and current

oscillations presented. This work achieved a wide spectrum of frequencies and agreed well

with experiments on GaAs/AlAs superlattices. An important outcome of the articles

discussed is the identification of the probability of negative differential guidance in presence

of the strong electrical fields. The negative differential guidance is formed because of the

electron motion inside the miniband and somewhere in the momentum in which the electron

effective mass is negative.

During recent years, semiconductor superlattices have been widely used in fabricating

electronic devices [18-20]. Due to this, researchers have become interested in investigating

the electron motion in superlattices and determining the locations of chaotic motion.

Therefore, identifying the places in which electron behavior is chaotic is critical in the

efficient and robust fabrication of modern semiconductor devices. Chaos can be formed in

electron motion in semiconductor superlattices as a result of their non-isotropic effective

mass and the application of external fields.

Articles written so far have, often, dealt with the electron behavior in superlattices under

the influence of one external electrical field applied in the direction of the crystal growth [21-

30]. We extend this approach here to analyze the transport of electrons in a semiconductor

superlattice miniband under the influence of electrical and magnetic fields, which are applied

in different directions on the superlattice. This work has important and immediate

applications in semiconductor and solid state physics industries and has thus far not

appeared in the engineering physics literature.

2. THEORETICAL FUNDAMENTALS

In this article, carrier transport is investigated in semiconductor superlattices in the presence

of magnetic and electrical fields and in the form of the one-miniband model. Here, electron

elastic scattering is considered from lattice impurities (the relaxation time approximation) and

the scattering resulted from other electrons. In addition, the self-consistent electrical field

caused by the electron’s charge density is, also, considered. The external electrical field

which is applied perpendicularly on the superlattice layers takes the form:

ˆext 0 E zE = E cos teΩ

(1)



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in which, 0E is the domain and EΩ is the electrical field frequency. In addition, the external

magnetic field is defined as follows:

( )ˆ ˆext 0 B y zB = B cos t sin e +cos eθ θΩ

(2)

where 0B is the domain, BΩ is the magnetic field frequency, andθ is the angle which the

magnetic field builds with z –axis (the superlattice layers growth direction). Owing to the fact

that electrons move freely in layers plain (x, y plain), the energy relation for the electrons

which belong to one miniband of a semiconductor superlattice can be written in the form [1,

21]:

( ) ( )2 2 zx y*

p a1 1P = p + p + 1- cos

2m 2ε

∆ h

(3)

in which the Tang-Best model is used for computing energy in the z direction, andP k= h is

the crystal momentum,*

em =0.07m is the effective mass of an electron moving in the x and y

directions, em is the mass of the free electron, a is the superlattice alternating, and∆ is the

miniband width. Using the Lorentz force equation, one can derive the energy time changes

and the electron velocity. Also, the self-consistent field time changes can be defined using

Ampere’s electromagnetic law. Finally, the normalized equations of the electron motion in

one miniband of a superlattice may be presented, thus:

( )z 0w= -Ev - w - wεγ& (4)

( )x

0 Bx y z v x*

-eB cos tv = v cos - v sin - v

mθ θ γ

Ω&

(5)

y

0 By x v y*

eB cos tv = v cos - v

mθ γ

Ω&

(6)

( )z

2 2 *2 2

z x 0 B x y v z2 2

e a a mv = - E+ v B cos tsin -w+ v +v - v

2 4θ γ

∆ ∆Ω

&

h h (7)



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z s E s E EE = v - E+ cos t - sin tα αω ωΩ Ω Ω& (8)

in which εγ is the energy relaxation frequency, ivγ is the electron velocity relaxation frequency

in the direction of ( )i = x, y,z , α is the self-consistent field relaxation frequency,

and 0s

eaE=ω

h, which is famously known as the Stark frequency. Also, w and 0w are the

electron energy and the equilibrium energy, respectively, which, the latter, is defined as:

1

BB0

0

B

I2k Tk T

w = -4

I2k T

∆

∆ ∆ (9)

in which 1I and 0I are the zero and first-order reformed Bessel functions, T is the lattice

temperature, and Bk is the Boltzmann constant. In Eqs. (4) and (8), all relaxation coefficients

are normalized by dividing into

1/22 2

E 2

2 e naπω

∆≡ h

. These equations show that a change in

the electrons average energy and velocity is attributable to the application of the external

fields, carrier scattering from other carriers, and the phonon and lattice faults. It is worth

noting that the external fields are of insufficient strength to be able to modify the band

structure; therefore, in all of the above-mentioned computations, the dispersion Eq. (3) is

used.

3. NUMERICAL RESULTS

In the numerical computations, we integrate from motion Eqs. (4) and (8) using the highly

efficient, well-tested fourth-order Runge-Kutta method with optimized steps and then draw

the velocity time series diagram and the largest Lyapunov exponent. This numerical

technique has been used successfully in numerous complex multiphysical problems. Takhar

et al [31] studied hydromagnetic heat transfer in porous media using Runge-Kutta methods.

Bég et al [32] investigated vorticity diffusion effects in permeable convection with Runge

Kutta and difference schemes. Further studies in thermophysical and geophysical fluid



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dynamics employing variations of Runge-Kutta quadrature have been communicated by

Chamkha et al [33] and Takhar et al [34]. In the context of solid state physics, an excellent

summary of Runge-Kutta methods has been given by Parechi and Russo [35]. Kuo et al [36]

used a fourth order R-K method to simulate passive Q-switching with solid-state saturable

absorbers in laser engineering. Prodan and Nordlander used Runge-Kutta numerical

quadrature to study electronic structure of metallic nanoshells for a large nanoshell (of 10 nm

in diameter) containing more than 2.5×104 conduction electrons. Becker et al [38] employed

R-K methods to investigate the Cameron bands luminescence from a new

thermoluminescence (TL) system (CO-doped solid Ar) - with selective excitation by

synchrotron radiation and thermally stimulated luminescence methods. They showed that

activation energies and the frequency factors of charge traps are very efficiently computed

by numerical fitting of the TL emission spectrum with a Runge-Kutta solver. Maiorov et al

[39] studied kinetics of plasma particles around a stationary dust grain in the presence of an

ion flow is studied using a three-dimensional molecular dynamics simulation method, based

on a Runge-Kutta numerical solver. Edwards and Burnett [40] developed a highly efficienct

Runge-Kutta method for solving the transient nonlinear Schrödinger equation with an

external potential, achieving excellent accuracy for the case of a dilute Bose-condensed

assembly of trapped neutral atoms where the potential varies on the same scale as the

condensate. Cranfield et al [41] have also successfully used Runge-Kutta numerical

techniques in perturbation analysis of oscillating magnetic fields in the radical pair

mechanism. In the present study we shall consider superlattice parameters like [42]:

-8 21 -3 -21a=10 m,n= 3×10 m , = 35.2×10 J∆ (10)

In experiments done in the references [43, 44], the areas of the damping parameters are

determined asE

0 0.2αω

≤ ≤ , E

0.2εγω

≤ , andE

0 υγω

≤ . First, we investigate the electron

behavior in the presence of the electrical field and in the absence of the magnetic field. Fig.

(1) shows velocity time series for values s

E

=1.6ωω

, E

E

=0.2ωΩ

,E

=0.01αω

, zv

E E

= =0.02εγ γω ω

,

andyxvv

E E

= =0γγ

ω ω. Also, this figure shows electron velocity for these chaotic parameters

values because the velocity changes in a non-normalized (non-periodic) way. One can be



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sure of the presence of chaos by determining the Lyapunov exponent. By computing the

distance between two routes, with very close initial conditions, one can compute the

Lyapunov exponent. If the divergence rate of the two routes is in the form of an exponent

function with a positive exponent, the motion is chaotic. Consider two close routes with a

distance 0d , we compute nd distance between these routes numerically, and in cases

where n

0

d

d is larger than a value between 2 to3 , we have, again, normalize nd to 0d . The

Lyapunov exponent is defined in this form:

( )= lim

τσ σ τ

→∞ (11)

in which:

( )N

n

n=1 0

d1ln

dσ τ

τ

≡

∑ (12)

N

n

n=1

τ τ≡ ∆∑ is the normalized time and nτ∆ is the (normalized) time distance between the n th

step and ( )n -1th

step. Also, time is normalized toE

1

ω. In numerical computations with

infinite time, this is interpreted to mean reaching a time in which the diagram ( )σ τ reaches a

rather continuous tend on the basis of the time.

For the sake of an improved investigation of the kind of motion,

( )σ τ is drawn in Fig. (2)

for Fig. (1) parameters values. Also, in the numerical solution of the equations and in

drawing ( )σ τ , the initial conditions are:

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )

1 2

1 2

1 2

1 2

x x

y y

z z

1 s 2 1 0

w 0 = w 0 = -1

v 0 = v 0 = 0

v 0 = v 0 = 0

v 0 = v 0 = 0

E 0 = ,E 0 = E 0 +ω δ

(13)



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Here, δ is a small value of magnitude 0.001and, also, indexes1and 2 show the

routes1and 2 , respectively. As can be seen in Fig. (2),

( )σ τ shows a constant value as big

as0.03 for large times ( )>10000τ . Therefore, by a good approximation, one can say that

the Lyapunov exponentσ is positive. The Lyapunov exponent’s positivity highlights the

presence of chaos in the absence of a magnetic field.

One can observe that in the absence of the magnetic field the electron motion for the

utilized parameters in Figs. (1) and (2) is chaotic. Now, we change the parameters as:

E

E

=1ωΩ

(14)

s

E

=0.1ωω

(15)

yxvv

E E

= =0γγ

ω ω (16)

E

=0.01αω

(17)

zv

E E

= =0.01εγ γω ω

(18)

Fig. (3) shows the electron mean velocity time series for the new parameters. As can be

seen from this figure, velocity changes periodically and there is no sign of chaos. In this

diagram, the fundamental frequency is the external field frequency with a greater period

which is on it, due to the non-linear equations.

Fig. (4) shows ( )σ τ for new parameters. As can be seen from this figure,

( )σ τ shows a

negative constant value for large times. Therefore, the Lyapunov exponent is negative,

which shows a regular motion (non-chaotic) for these new parameters. The above-

mentioned results are equivalent to the reference results [43] which have investigated the

chaos in superlattices just in presence of one electrical field. This emphasizes the accuracy

of the Eqs. (4) to (8).



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11

In this section, we deal with the numerical solution of the motion equations in the

presence of a magnetic field with a domain of-120

E

B= 10 Ts

ωand a frequency of B

E

=0.1ωΩ

. In

order to solve the equations numerically, we consider the same parameters values used in

Fig. (1) (i.e., the values which cause the electron’s motion to be chaotic). We aim to

investigate how applying a magnetic field affects the chaotic motion of the electron. We

apply the magnetic field perpendicularly on the electrical field and in the direction of y on the

superlattice. Figs. (5) and (6) show the electron velocity time series and ( )σ τ in the presence

of the magnetic field, respectively.

As can be seen in these figures, the electron velocity is periodic for the large enough

time and, also, ( )σ τ shows the constant and negative value for the large time which

represents the negative Lyapunov exponent. Therefore, chaos, in presence of the magnetic

field, fades. It is worth noting that when we consider the magnetic field parallel to the

electrical field, the electron motion remains chaotic, which can be seen in Figs. 1 and 2.

For the sake of reaching a better view of the qualitative behavior of the system, we draw

the phase space diagram for the electrical field domain on the basis of its frequency. We do

this by changing the electrical field domain and frequency by a 0.01step and, then, compute

the Lyapunov exponent for them. Figs. (7) and (8) are drawn for relaxation

parametersyxvv

E E

= =0γγ

ω ω, zv

E E

= =0.1εγ γω ω

, andE

=0.01αω

. The dark areas show the chaotic

areas and the white ones deal with the periodic motion. Fig. (7) is drawn in the absence of

the magnetic field and Fig. (8) is considered in the presence of the magnetic field.. It is

observed that the magnetic field application (-120

E

B= 10 Ts

ωand B

E

=0.1ωΩ

), which is as large

as the electrical field, causes the chaos to be eliminated on the left side of the diagram.

4. CONCLUSIONS

In this article, we have shown that in presence of the electrical field and in the absence of

the magnetic field, the electron motion in the superlattices, for some parameters related to

the superlattice and the external field, can be chaotic. In addition, for some other

parameters, it can be periodic or non-chaotic. However, in presence of the orthogonal



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electrical and magnetic fields, chaos can be eliminated and the electron motion in

superlattices, for some parts of the phase space, becomes regular or non-chaotic.

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16

6. FIGURE CAPTIONS

Figure 1. The electron velocity in the absence of the magnetic field on the basis of time

for parameters zv

E E

= =0.02εγ γω ω

, E

E

=0.2ωΩ

, s

E

=1.6ωω

,yxvv

E E

= =0γγ

ω ω, and

E

=0.01αω

. The

electron motion does not have a particular period and is chaotic.

Figure 2. Diagram ( )σ τ in the absence of the magnetic field for the parameters of Fig.

(1). It can be seen that ( )σ τ shows a constant and positive value of 0.03 for >10000τ ,

which puts a true mark on the presence of chaos.

Figure 3. The electron velocity time series in the absence of the magnetic field for the

parameters s

E

=0.1ωω

, E

E

=1ωΩ

,yxvv

E E

= =0γγ

ω ω, zv

E E

= =0.01εγ γω ω

, andE

=0.001αω

. The electron

motion is periodic (non-chaotic).

Figure 4. Diagram ( )σ τ in the absence of the magnetic field for the parameters of Fig.

(3). The Lyapunov exponent’s being negative shows the regular and non-chaotic motion.

Figure 5. The electron velocity time series in the direction of z, in presence of the

magnetic field perpendicular on the electrical field, and on the basis of the normalized time

for the parameters of Fig. (1).In presence of the magnetic field, the electron motion becomes

regular and non-chaotic.

Figure 6. Diagram ( )σ τ in presence of the magnetic field, perpendicular on the electrical

field. The presence of the magnetic field causes the annihilation of chaos.

Figure 7. Phase space diagram in the absence of the magnetic field. The dark areas

show the chaotic motion and the white areas represent the periodic (non-chaotic) motion.

Figure 8. Phase space diagram in presence of the magnetic field. The dark areas show

the chaotic ones and the white areas represent the periodic (non-chaotic) motion.



1 - 4 June 2010

17

7. FIGURES

Figure 1. The electron velocity in the absence of the magnetic field on the basis of time for

parameters zv

E E

= =0.02εγ γω ω

, E

E

=0.2ωΩ

, s

E

=1.6ωω

,yxvv

E E

= =0γγ

ω ω, and

E

=0.01αω

. The electron

motion does not have a particular period and is chaotic.

Figure 2. Diagram ( )σ τ in the absence of the magnetic field for the parameters of Fig. (1). It

can be seen that ( )σ τ shows a constant and positive value of 0.03 for >10000τ , which puts

a true mark on the presence of chaos.



1 - 4 June 2010

18

Figure 3. The electron velocity time series in the absence of the magnetic field for the

parameters s

E

=0.1ωω

, E

E

=1ωΩ

,yxvv

E E

= =0γγ

ω ω, zv

E E

= =0.01εγ γω ω

, andE

=0.001αω

. The electron

motion is periodic (non-chaotic).



1 - 4 June 2010

19

Figure 4. Diagram ( )σ τ in the absence of the magnetic field for the parameters of Fig. (3).

The Lyapunov exponent’s being negative shows the regular and non-chaotic motion.

Figure 5. The electron velocity time series in the direction of z, in presence of the magnetic

field perpendicular on the electrical field, and on the basis of the normalized time for the

parameters of Fig. (1).In presence of the magnetic field, the electron motion becomes

regular and non-chaotic.



1 - 4 June 2010

20

Figure 6. Diagram ( )σ τ in presence of the magnetic field, perpendicular on the electrical

field (the presence of the magnetic field causes the annihilation of chaos).

Figure 7. Phase space diagram in the absence of the magnetic field. The dark areas show

the chaotic motion and the white areas represent the periodic (non-chaotic) motion.



1 - 4 June 2010

21

Figure 8. Phase space diagram in presence of the magnetic field. The dark areas show the

chaotic ones and the white areas represent the periodic (non-chaotic) motion.

AN ANALYTICAL AND NUMERICAL INVESTIGATION OF THE ...€¦ · CHAOS 2010, 3rd Chaotic Modeling and Simulation International Conference, Chania, Crete, Greece, 1 - 4 June 2010 1 AN

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