Depinning transition for domain walls with an internal degree of freedom

Transcription

1 Depinning transition for domain walls with an internal degree of freedom Vivien Lecomte 1, Stewart Barnes 1,2, Jean-Pierre Eckmann 3, Thierry Giamarchi 1 1 Département de Physique de la Matière Condensée, Genève 2 Physics Department, University of Miami 3 Département de Physique Théorique et Section de Mathématiques, Genève Genève 13th March 2009 V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

2 Outline Introduction Outline 1 Interface Physics Systems Depinning transition Experiments 2 Depinning with internal degree of freedom Modelisation Dynamics V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

3 Magnetic domain wall Interface physics Motivations from Lemerle et al., PRL (1998) from Tatara et al., J. Phys. Soc. Jap (2008) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

4 Magnetic domains Interface physics Motivations V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

5 Interface physics Ferroelectric domain wall Motivations Energy U 0 Atomic displacement Pb Pb Ti O Ti O t V from Paruch et al. J. Appl. Phys, (2006) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

6 Interface physics Ferroelectric domain wall Motivations Energy U 0 Atomic displacement Pb Pb Ti O Ti O 2.5 µm from Paruch et al. J. Appl. Phys, (2006) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

7 Contact line of a fluid Interface physics Motivations from Moulinet, Guthmann and Rolley, Eur. Phys. J. E, (2002) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

8 Interface physics Motivations Common underlying description? Large range of physical scales magnetic/ferroelecric domain walls growth interfaces contact line crack propagation Questions Statics fluctuations, roughness Non-equilibrium dynamics motion of the interface Nature, role of disorder V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

9 Interface physics Motivations Common underlying description? Large range of physical scales magnetic/ferroelecric domain walls growth interfaces contact line crack propagation Questions Statics fluctuations, roughness Non-equilibrium dynamics motion of the interface Nature, role of disorder V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

10 Interface physics Disordered elastic systems Motivations z Elasticity: tends to flatten the interface c dz ( r(z) ) 2 2 Disorder: tends to bend it dz V ( r(z), z ) r(z) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

11 Interface physics Disordered elastic systems Motivations z Elasticity: tends to flatten the interface c dz ( r(z) ) 2 2 Disorder: tends to bend it dz V ( r(z), z ) r(z) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

12 Interface physics Disordered elastic systems Motivations z Elasticity: tends to flatten the interface c dz ( r(z) ) 2 2 Disorder: tends to bend it dz V ( r(z), z ) r(z) Competition btw order and disorder V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

13 Interface physics Motivations Is r(z) containing enough? V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

14 Interface physics Motivations Is r(z) containing enough? Have a look to the dynamics in simple examples. V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

15 Interface physics Dynamics Depinning zero temperature Ú ÐÓ ØÝ v v (f f c ) β f c ÓÖ f V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

16 Interface physics Dynamics Depinning finite temperature Ú ÐÓ ØÝ v T > 0 T = 0 v (f f c ) β Ö Ô Ö Ñ f c ÓÖ f V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

17 Interface physics Dynamics Comparison with experiment: ferromagnetic films [ v(f ) exp U c T ( ) µ ] fc f (creep) from Lemerle et al., PRL (1998) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

18 Interface physics Dynamics Comparison with experiment: ferroelectric films [ v(f ) exp U c T ( ) µ ] fc f (creep) from Paruch et al., PRL (2005) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

19 Interface physics Comparison with experiment Dynamics BUT... V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

20 Interface physics Dynamics Comparison with experiment: ferromagnetic wire [ v(f ) exp U c T ( ) µ ] fc f (creep) from Yamanouchi et al., Science (2007) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

21 Interface physics Dynamics Comparison with experiment: ferromagnetic wire [ v(f ) exp U c T ( ) µ ] fc f (creep) from Yamanouchi et al., Science (2007) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

22 Outline Interface physics Outline 1 Interface Physics Systems Depinning transition Experiments 2 Depinning with internal degree of freedom Modelisation Dynamics V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

23 Bulk model Depinning with inertia V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

24 Bulk model Depinning with inertia Bulk energy { [ E = d d x J ( θ) 2 + sin 2 θ( φ) 2] } + K sin 2 θ + K sin 2 θ cos 2 φ Equation of motion ( t + v s ) Ω = Ω ( δe δω + f + η) Ω ( α t + βv s ) Ω V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

25 Bulk model Depinning with inertia Expanded equations of motion t θ α sin θ t φ + v s` x θ β sin θ x φ = 1 2 K sin θ sin 2φ J ` sin θ x sin 2 θ x φ sin θ t φ + α t θ + v s` sin θ x φ + β x θ = 1 2 K sin 2θ cos 2 φ 1 2 `K + J( x φ) 2 sin 2θ H ext sin θ + J 2 x θ V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

26 Bulk model Depinning with inertia Solitonic solution [Walker 1974] sin 2φ(x, t) = f f W f W = 1 2 αk θ(x, t) = 2 arctan exp { [1 + K K cos2 φ ] 1 2 v = H ext H c [ 1 + K K cos 2 φ ] 1 2 ( K J x vt ) } V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

27 Bulk model Depinning with inertia Rigid wall approximation α t r t φ = f + Landscape + η 1 α t φ + t r = 1 2 K sin 2φ + η 2 Unit of length: J/K V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

28 Bulk model Depinning with inertia Rigid wall approximation α t r t φ = f cos κr + η 1 α t φ + t r = 1 2 K sin 2φ + η 2 Effective model: Position r(t) coupled to phase φ(t). V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

29 Depinning with inertia Large K : φ decouples from r V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

30 Depinning with inertia zero temperature Large K : φ decouples from r Ú ÐÓ ØÝ v α t r = f cos κr f < f c ÐÓ Ð Ñ Ò Ñ f = f c Ñ Ò Ñ Ú Ò v (f f c ) 1/2 f c = 1 ÓÖ f V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

31 Depinning with inertia finite temperature Large K : φ decouples from r Ú ÐÓ ØÝ v α t r = f cos κr + η T > 0 Ø ÖÑ Ð ÖÓÙÒ Ò T = 0 v (f f c ) 1/2 f c = 1 ÓÖ f V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

32 Depinning with inertia zero temperature Smaller K : φ matters α t r t φ = f cos κr α t φ + t r = 1 2 K sin 2φ V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

33 Depinning with inertia zero temperature Smaller K : φ matters Dramatic change in the depinning law: v 1 log(f f c ) Ú ÐÓ ØÝ v v 1 log(f f c ) Ø Ð Ö Ñ v (f f c ) 1 2 f c f c ÓÖ f Depinning at lower critical force: f c < f c Bistability V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

34 Phase space Depinning with inertia In the bistable regime (f c < f < f c ) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

35 Phase space Depinning with inertia Homoclinic bifurcation: f > f c V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

36 Phase space Depinning with inertia Homoclinic bifurcation: f = f c V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

37 Finite temperature Depinning with inertia ( ) ɛ 3 escape time exp T }{{} Arrhenius exp ( AT ) (ɛ ɛ c) 2 } {{ } Trapping probability V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

38 Finite temperature Depinning with inertia v T = 0 T > 0 f f Force-velocity characteristics T 0 + f V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

39 Depinning with inertia Analogy α t r t φ = f cos κr α t φ + t r = 1 2 K sin 2φ m t 2r + α tr = f cos κr Ò ÖØ The phase φ plays the role of a velocity: inertia helps to cross barriers [see also Risken chap.11] V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

40 Depinning with inertia Analogy α t r t φ = f cos κr α t φ + t r = 1 2 K sin 2φ m t 2r + α tr = f cos κr Ò ÖØ BUT... V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

41 Depinning with inertia Analogy α t r t φ = f cos κr α t φ + t r = 1 2 K sin 2φ m t 2r + α tr = f cos κr Ò ÖØ the velocity is unbounded WHEREAS φ is bounded and periodic V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

42 Topological transition Depinning with inertia V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

43 Topological transition Depinning with inertia v ØÓÔÓÐÓ Ð ØÖ Ò Ø ÓÒ W(r) = 1 W(ϕ) = 0 W(r) = 1 W(ϕ) = 1 f Successive regimes characterized by winding numbers W V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

44 Experiment Depinning with inertia SPINTRONICS from Parkin et al., Science (2008) V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

45 Ï Ú ÐÓ ØÝ Ñ» µ Ñ µîµ Experiment Depinning with inertia experiment from Parkin et al., Science (2008) µ µ f Ñ Ò Ø Ð Ç µ f Û V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

46 Ï Ú ÐÓ ØÝ Ñ» µ Ñ µîµ Outlook Depinning with inertia Internal degree of freedom unusual depinning law bistability non-monotonous v(f ) at finite T link with experiments µ µ 200 fû f Ñ Ò Ø Ð Ç µ Perspective Current driven wall Interface with elasticity modified creep law? Experiments periodic patterning V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26

47 Ï Ú ÐÓ ØÝ Ñ» µ Ñ µîµ Outlook Depinning with inertia Internal degree of freedom unusual depinning law bistability non-monotonous v(f ) at finite T link with experiments µ µ 200 fû f Ñ Ò Ø Ð Ç µ Perspective Current driven wall Interface with elasticity modified creep law? Experiments periodic patterning V. Lecomte (DPMC - Genève) Internal degree of freedom & depinning 13/03/ / 26