Intrinsic Localized modes:A mechanism for the migration of defects

Jesús Cuevas Maraver

Nonlinear Physics GroupUniversidad de Sevilla

Outline

1. Nonlinear lattice dynamics: Phonons vs Intrinsic Localized Modes (Discrete Breathers)

2. A “classical” nonlinear model: Frenkel-Kontorova

3. Stationary and moving breathers

4. Point defects: Impurities, vacancies and interstitials

5. Interaction between discrete breathers and vacancies

6. Double vacancies and interstitials

7. Conclusions

References:• J. Phys. A 35 (2002) 10519• Phys. Lett. A 315 (2003) 364

• A great number of systems can be described by oscillator networks (crystals, biomolecules, Josephson-junctions arrays…)

• The interactions between the oscillators is nonlinear, although most of the times they are approximated by linear functions

• An oscillator network is described by the following Hamiltonian:

21( ) ( , )

2 n n n mn m

H mu V u W u u

On-site potentialKinetic Energy Interaction potential

Nonlinear lattices

2 2 2 20 1

1 1( )

2 2n n n nn

H mu u k u u

Linear lattices

Linear vibrational modes: phonons

2 21

1( ) ( )

2 n n n nn

H mu V u k u u

Nonlinear lattices

Phonons + Intrinsic localized modes (breathers)

Discrete breathers

• Exact periodic and localized solutions of the dynamical equations that exist due to nonlinearity and discreteness.

• They exist as long as two conditions are fulfilled (MacKay-Aubry theorem, 1994):

•The on-site potential is nonlinear•The breather frequency does not resonate with phonons

• They have been generated in Josephson-junction arrays and observed in molecular crystals (PtCl).

• They are speculated to play an important role in:•DNA transcription and denaturation bubbles•The appearance of dark lines of mica muscovite•Reconstructive transformations in layered silicates

• Introduced in 1938 to study the dynamics of dislocations.• It consists of a on-site periodic potential (sine-Gordon):

2 21 2

1 1( ) ( ) ; ( ) 1 cos 2

2 4n n n nn

H mu V u k u u V u u

The Frenkel-Kontorova model

• The particles are located at the bottom of the potential:

Breathers in Frenkel-Kontorova

• The Frenkel-Kontorova model supports discrete breathers due to the nonlinearity of the on-site potential

Mobile breathers

• In some conditions, a static breather can be perturbed leading to a mobile state.

• These solutions are not exact: can only be observed through numerical simulations.

• Contrary to static breather, they are not supported in every nonlinear lattices.

• One of the systems supporting moving breathers is the Frenkel-Kontorova model.

Mobile breathers in Frenkel-Kontorova

Point defects

• The Frenkel-Kontorova is the simplest way of modelling vacancies (an empty well) and interstitials (two particles in the same well).

Moving breathers and vacancies

• We consider a modified Frenkel-Kontorova model with a nonlinear interaction potential:

2 21

2( 1)2 2

1 1( ) ( )

2 2

1 1( ) 1 cos 2 ; ( ) 1

4 2

n n n nn

b x

H mx V x W x x

V x x W x eb

• W(x) is the Morse potential. b is the inverse width of the potential:

b=2

b=1

Moving breathers and vacancies

• We have studied the effect of varying the potential width in the interaction.

• For each value of the potential width, the interaction is studied in function of the kinetic energy of the moving breather.

• When the moving breather reaches the vacancy, the latter can move forwards, backwards or remain at rest. However, there is no correlation between the kinetic energy and the number of vacancy jumps:

Critical values

• Statistical analysis:

• For b>bf the vacancy does not move forwards.

• For bb1<b<bb2 there exist a critical value of the kinetic energy for vacancy movement:

Vacancy moving backwards

Vacancy moving backwards

Vacancy moving forwards

Vacancy moving forwards

Double vacancy

• This configuration needs an narrow interaction potential.• The double vacancy does not move forwards. Instead, it can be

broken.• The observed regimes are now: rest, breaking and forwards

movement (with breaking). The latter needs b to be small enough.• The threshold kinetic energy is also observed.

Double vacancy breaking

Interstitials

• Preliminary results.• A threshold kinetic energy is always observed.

• For b>bc, the interstitial always moves forwards.

• For b<bc, the interstitial can move backwards, forwards or remain at rest.

Interstitial moving forwards

Conclusions

• We have described the interaction between moving breathers and vacancies when the interaction potential width is varied.

• Two critical values of the width exist:•Vacancy forwards movement•Threshold kinetic energy

• These critical values can be determined through the existence and stability analysis of static breathers in the neighborhood of the vacancy.

• More information:

http://www.grupo.us.es/gfnl