J Cuevas, C Katerji, B Sánchez-Rey and A Álvarez

Non Linear Physics Group of the University of SevilleDepartment of Applied Physics I

February 21st, 2003

Influence of moving breathers on vacancies migration

• Previous: Experimental evidence

• Model

• Vacancy

• MB & vacancy

• Results

Outline

X

X

X

X

X radiation (Ag)

Experimental evidence: defects migrate

sample (Si)

X

X

X

X

X

XX

X

XX

XX

X

XX

layer (Ni)

1

2

43

-1 0 1 2 3 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1Morse potentials

x

W(x

)

-0.5 0 0.5 1 1.5 2 2.5 3 3.50

0.01

0.02

0.03

0.04

0.05

0.06sine-Gordon potential

x

V(x

)

The model

Frenkel-Kontorova + anharmonic interaction

21)(exp2

1 axbxW

1L 1a

L

xLxV

2cos1

4 2

2

Plane waves (phonons)

The Hamiltonian:

1

'2

2

1 nn

nnn xxWCxVxH

02 11'2 nnnnn xxxCbxx

2sin4 22

02 q

Cph

tqnin

phextx 0

'2CbC

9.0bb

bT2

nnnnn

nn axxbaxxbC

L

xLxH 2

1

2

1

'

2

22 1)(exp

2

11)(exp

2

12cos1

42

1

The linearized equations:

Dispersion relation

2sin4 2'22

02 q

Cbph

Frecuency breather:

,,

1 ,, 0

vn1vn 1vn

The vacancy

nx

nv 22

2

1

2

1 vK

0,,0,

2

1,0,

2

1,0,,00

v

0vv

The moving breather

Energy added to the breather:

0 0.13

03.02

Gaussian distribution

K Forinash, T Cretegny and M Peyrard.

Local modes and localization in a multicomponent nonlinear lattices. Phys. Rev. E, 55:4740, 1997.

Analysis:

• b (Morse potential)

•

The moving breather

Three types of phenomena

i) The vacancy moves backwards.

ii) The vacancy moves forwards.

iii) The vacancy remains at its site.

vn1vn 1vn

MB

vn

i) The most probably is that the vacancy moves backwards

vn1vn 1vn

MB

ii) But the vacancy can move forwards

vn

vn1vn 1vn

MB

iii) And the vacancy can stay motion-less

C=0.4

remainsforwards

backwards

Probability

Probability of the vacancy’s movement

C=0.5

backwards

forwards remains

Probability

Probability of the vacancy’s movement

vn1vn 1vn

b

bMorse potentials

The coupling forces

C=0.5

backwards

forwards remains

Probability

Probability of the vacancy’s movement

C=0.4

remainsforwards

backwards

Probability

Probability of the vacancy’s movement

C=0.4

bifurcation

11 vv nn uu

vn

C=0.5

bifurcation

11 vv nn uu

vn

C=0.4

nx

Amplitude maxima of a vacancy breather

bifurcation

C=0.5

nx

Amplitude maxima of a vacancy breather

bifurcation

Conclusions

• The moving breathers can move vacancies.

• The behaviour is very complex and it depends of the values of the coupling.

• The changes of the probabilities of the different movements of the vacancies are related with the existence of bifurcations.

• Is there a more rich complexity from 1-dim to 2-dim ?

Thank you very much.