Nucleation and avalanches in film with labyrintine magnetic domains

Nuclea'on and avalanches in films with labyrinthine

magne'c domains

Andrea Benassi & Stefano Zapperi

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)

Labyrinthine domains

Phys.Rev.Le*. 92, 077206 (2004)

Phys.Rev.B 71, 104431 (2005)

Deconvolving nucleaAon

they find a 0.5 exponent

Labyrinthine domains

Avalanche staAsAcs taken over different intervals of the hysteresis

loop show different criAcal exponents

Appl.Phys.Le*. 95, 182504 (2009)

A phase field model

V =Ku

4

�m2

2− m4

4

�m =

M(r)

Ms= m(x, y)

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

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/4µ0M2s

γ = d/�4π

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

4 2

b

cd

f a

! = 0.5! = 0.6! = 0.7

h

e

g

b c da

f g he

Two different limit behavors

MulAple nucleaAon and coalescence by bridging

Expansion by branching of a single domain and lateral fa*ening

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!!!

Characteris'c lengths

m(x, y, �d) = sin

�πx

�d

�

m(x, y, �w) = tanh

�x

�w

�


�w =√2� domain wall width


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

Triggering of minor avalanches

The difference between consecuAve magneAzaAon maps allows a direct imaging

of avalanches

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

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



Phase field model reloaded

V = [1− λ(r)]Ku

4

�m2

2− m4

4

�

m = α

�dV

dm+∇2m

�− 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�)� = 2KBT δ(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.

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

Memory effects Hysteresis loop unrolled:

Ame

<m>

Memory effects Hysteresis loop unrolled:

Ame

<Φ>

m = α

�dV

dm+∇2m

�− γ

�dr�

m(r�)

|r− r�|3 + hr(r) + he(t) +R(t)

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�)

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?

Nucleation and avalanches in film with labyrintine magnetic domains

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