Nucleation and avalanches in film with labyrintine magnetic domains

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Information about Nucleation and avalanches in film with labyrintine magnetic domains

Published on January 5, 2017

Author: AndreaBenassi3

Source: slideshare.net

1. Nuclea'on  and  avalanches     in  films  with  labyrinthine   magne'c  domains   Andrea  Benassi  &   Stefano  Zapperi    

2. Outline   Experiments  on  labyrinthine  domains     Our  phase  field  model     Characteris>cs  Lengths  and  avalanche  sta>s>cs         A  new  version  of  the  phase  field  model     Ironing  stripe  domains     Memory  effects               In-­‐plane  magne>za>on:  very  preliminary  results  (yesterday)  

3. Labyrinthine  domains   Phys.Rev.Le*.  92,   077206  (2004)         Phys.Rev.B  71,   104431  (2005)   Deconvolving   nucleaAon   they  find  a  0.5   exponent  

4. Labyrinthine  domains   Avalanche  staAsAcs  taken  over   different  intervals  of  the  hysteresis   loop  show  different  criAcal  exponents   Appl.Phys.Le*.  95,  182504  (2009)  

5. A  phase  field  model   V = Ku 4 m2 2 − m4 4 m = M(r) Ms = m(x, y)

6. A  phase  field  model   V = Ku 4 m2 2 − m4 4 m = M(r) Ms = m(x, y) ∂M(r, t) ∂t = −Γ δH[M(r, t)] δM(r, t) Energy  funcAonal  power  expansion   +   linear  relaAon  between  Ame  and   energy  fluctuaAons   Small  Ame   fluctuaAons   hypothesis  

7. A  phase  field  model   hr(r) = 0 hr(r)hr(r ) = Dδ(r − r ) V = Ku 4 m2 2 − m4 4 m = M(r) Ms = m(x, y) ∂M(r, t) ∂t = −Γ δH[M(r, t)] δM(r, t) Energy  funcAonal  power  expansion   +   linear  relaAon  between  Ame  and   energy  fluctuaAons   Small  Ame   fluctuaAons   hypothesis   2  dimensionless   parameters   = 2 A/Ku α = Ku/4µ0M2 s γ = d/4π

8. A  phase  field  model   hr(r) = 0 hr(r)hr(r ) = Dδ(r − r ) V = Ku 4 m2 2 − m4 4 m = M(r) Ms = m(x, y) ∂M(r, t) ∂t = −Γ δH[M(r, t)] δM(r, t) Energy  funcAonal  power  expansion   +   linear  relaAon  between  Ame  and   energy  fluctuaAons   Small  Ame   fluctuaAons   hypothesis   2  dimensionless   parameters   = 2 A/Ku α = Ku/4µ0M2 s γ = d/4π

9. Two  different  limit  behaviors   Depending  on  the  film   thickness  and  on  the  disorder   strength  we  can  have  two  limit   behaviors   -4 -2 0 -0.5 0 0.5 42 b c d f a γ = 0.5 γ = 0.6 γ = 0.7 h e g b c da f g he

10. Two  different  limit  behavors   MulAple  nucleaAon  and   coalescence  by  bridging   Expansion  by  branching  of  a  single   domain  and  lateral  fa*ening    

11. Characteris'c  lengths   m(x, y, d) = sin πx d m(x, y, w) = tanh x w d = α/γ domain width w = √ 2 domain wall width n nucleation diameter MinimizaAon  of  the  energy  with  respect  to  a   fixed  magneAzaAon  configuraAon  with  one   parameter:   NucleaAon  depends  strongly  on  disorder,  any   analyAcal  theory  is  useless!!!  

12. Characteris'c  lengths   m(x, y, d) = sin πx d m(x, y, w) = tanh x w d = α/γ domain width w = √ 2 domain wall width n nucleation diameter MinimizaAon  of  the  energy  with  respect  to  a   fixed  magneAzaAon  configuraAon  with  one   parameter:   NucleaAon  depends  strongly  on  disorder,  any   analyAcal  theory  is  useless!!!   Avalanches  

13. Triggering  of  minor  avalanches   The  difference  between   consecuAve  magneAzaAon   maps  allows  a  direct  imaging   of  avalanches    

14. Avalanche  sta's'cs   Analysis  of  different  loop  regions:     •  The  maximum  avalanche  size  decreases  as   the  domain  density  reaches  its  maximum   •  NucleaAon  and  bridging,  with  their   characterisAc  size,  affect  the  size   distribuAon           NucleaAon  and  annihilaAon:       •  For  nucleaAon  to  take  place  a  barrier  must   be  overcame,  its  value  goes  as  1/γ •  AnnihilaAon  is  almost  independent  of  the   dipolar  field  strength   •  At  zero  temperature  the  gaussian   distribuAon  is  due  to  the  spaAal  disorder  

15. Avalanche  sta's'cs   Different  film  thickness:     •  The  avalanche  cutoff  increases  when  γ  is   decreased,  following  the  corresponding   increase  of  the  domain  width  and   confirming  that  α/γ  is  the  relevant   parameter  controlling  the  size  of  the   scaling  regime   Different  Disorder  strength:     •  The  Larger  D  the  larger  the  external  field  at   which  walls  depin,  the  larger  their  jumps.     •  Increasing  D  the  domains  shape  is  slightly   affected  by  the  disorder  strength  but              is   almost  independent  of  D,   •  NucleaAon  diameter              decreases  with   increasing  D   d = α n nucleation diame

16. Phase  field  model  reloaded   V = [1 − λ(r)] ¯Ku 4 m2 2 − m4 4 ˙m = α dV dm + ∇2 m − γ dr m(r ) |r − r|3 + hr(r) + he(t) + R(t) hr(r) = 0 hr(r)hr(r ) = Dδ(r − r ) Two  new  randomness  sources  means  two  new  physical  parameters  to  be  introduced…     λ(r) = 0 λ(r)λ(r ) = Aδ(r − r ) R(r)R(r ) = 2KBTδ(r − r )δ(t − t )R(r) = 0 Random  field   Temperature  noise   Anisotropy  disorder   Random  field  and  random  anisotropy  has  the  same  effect  on  the  domains  topography,   except  that  the  type  of  domain  dynamics  (nucleaAon/coalescence  or  branching)  seems  to   be  a  bit  more  sensiAve  to  A  than  D.  

17. Phase  field  model  reloaded   V = [1 − λ(r)] ¯Ku 4 m2 2 − m4 4 ˙m = α dV dm + ∇2 m − γ dr m(r ) |r − r|3 + hr(r) + he(t) + R(t) hr(r) = 0 hr(r)hr(r ) = Dδ(r − r ) Two  new  randomness  sources  means  two  new  physical  parameters  to  be  introduced…     λ(r) = 0 λ(r)λ(r ) = Aδ(r − r ) R(r)R(r ) = 2KBTδ(r − r )δ(t − t )R(r) = 0 Random  field   Temperature  noise   Anisotropy  disorder   Random  field  and  random  anisotropy  has  the  same  effect  on  the  domains  topography,   except  that  the  type  of  domain  dynamics  (nucleaAon/coalescence  or  branching)  seems  to   be  a  bit  more  sensiAve  to  A  than  D.  

18. Ironing  stripe  domains   No  disorder  (realizaAon  1)   No  disorder  (realizaAon  2)   Gaussian  disorder   •  The  final  orientaAon  of  the  parallel  stripes  depends  on  the  iniAal  random  configuraAon   •  The  presence  of  disorder  inhibits  the  complete  reorientaAon     OscillaAng  external  field  perpendicular  to  the  film  surface:   he(r) = h0 sin(ωt) ω = 0.0126 Γµ0 ≡ 1 h0 = 2 hsat 4

19. Memory  effects   Hysteresis  loop  unrolled:   Ame   m

20. Memory  effects   Hysteresis  loop  unrolled:   Ame   Φ

21. ˙m = α dV dm + ∇2 m − γ dr m(r ) |r − r|3 + hr(r) + h In-­‐plane  Magne'za'on     Just  modifying  the  dipolar  (stray)  field,  our  scalar  model  seems  to  be  able  to  reproduce  the   domain  dynamics  of  in-­‐plane  films.       Now  the  magneAzaAon  is  assumed  to  be  oriented  only  along  the  x-­‐axis  ranging  in  [-­‐1,+1]  an   External  field  is  applied  along  the  same  axis  to  record  hysteresis  loops.   +γ dr 2(x − x )2 − (y − y )2 |r − r| m(r )

22. Open  Issues:   Which  quanAAes  can  be  used  to  characterize  the  memory  effects  and  the  stripes  domains?       One  Hysteresis  loop  takes  24  hours:   •  Do  we  really  need  to  be  so  slow  in  increasing  the  field?   •  How  many  loops  to  test  memory  effects?   •  (Easy)  ParallelizaAon  will  speed  up  our  calculaAons  by  a  factor  of  4       Up  to  now  we  used  only  white  noise,  does  it  make  sense  to  define  a  characterisAc  length   for  the  noise  correlaAon?       Working  in  reciprocal  space  enable  us  to  deal  with  large  systems  but  we  are  forced  to  use   periodic  boundary  condiAons.  Edge  effects  cannot  be  taken  into  account  in  the  simulaAons       In  the  case  of  a  bubbles  lamce,  can  we  play  with  an  external  oscillaAng  field  in  the  same   way  we  do  for  stripe  domains,  to  try  to  order  the  lamce?          

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