Post on 16-Dec-2015
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.