the role of exchange bias in domain dynamics control

The Role of Exchange Bias in Domain Dynamics Control

A. Benassi1, M.A. Marioni1, D. Passerone1 and H.J. Hug1,2

1-‐ EMPA Swiss Federal InsHtute for Materials Science and Technology, Dübendorf (Switzerland) 2-‐ Department of Physics, Universität Basel, Basel (Switzerland)

Sample and measurements A perpendicular anisotropy ferromagneHc film (FF) is anH-‐coupled with a thinner anHferromagneHc film (AF) grown on top

Upon cooling below the Neel temperature, the AF becomes ordered except for few atomic layer at the interface, here the defects at the interface give rise to a distribuHon of uncompensated spins (UCS)

Being the Neel temperature of the AF smaller than the Curie temperature of the FF, the presence of the ferromagneHc domains can orient the uncompensated spin at the interface during the cooling. This allow us to fix stably the FF domain structure on the cooled AF.

the domains image is taken through the AF layer because the FF field is orders of

magnitude stronger

the UCS image is taken saturaHng the FF domains with an external field

Schmid e al. PRL 105 197201 (2010) Joshi et al. Appl.Phys.LeY. 98 082502 (2011)

Hext

uncompe

nsated

frustrated

AF

FF

The model system The Landau-‐Lifshitz-‐Gilbert (LLG) equaHon rules the precession of a magneHc dipole in an external field:

∂m

∂t= − γ

1 + ξ2m×

�B+ ξ

�m×B

��

B = − 1

Ms

δH[m]

δm+Q(R, t)

�Q(R, t)� = 0

�Q(R, t)Q(R�, t�)� = δ(t− t�)δ(R−R�)2KBT ξ/Msγ

Bm

precession term

Bm

damping term dissipaHon by

microscopic degrees of freedom

Bm

stochasHc term thermal fluctuaHons

Under the following approximaHons:

•  scalar magneHzaHon uniform along the FF thickness d

•  domain walls smaller than the domain size

•  small FF thickness d

the magneHzaHon in the FF can be described by the following hamiltonian power expansion:

Gilbert IEEE Trans. On MagneHcs 40 3434 (2004) Brown Phys.Rev. 130 1677 (1963) Usadel PRB 73 212405 (2006)

m(r, t) = m(x, y, t)z

H =

�d3R

�−Ku(R)

m2

2+

A

2(∇Rm)2 +

µ0M2s d

8π

�d2R�m(R�)m(R)

|R−R�|3− µ0Msm(Hext −HUCS(R))

�

The model system

∂m

∂t= − γ

1 + ξ2m×

�B+ ξ

�m×B

��

B = − 1

Ms

δH[m]

δm+Q(R, t)

�Q(R, t)� = 0

�Q(R, t)Q(R�, t�)� = δ(t− t�)δ(R−R�)2KBT ξ/Msγ

Bm Bm






Jagla PRB 72 094406 (2005) Jagla PRB 70 046204 (2004)

m(r, t) = m(x, y, t)z

The Landau-‐Lifshitz-‐Gilbert (LLG) equaHon rules the precession of a magneHc dipole in an external field:

precession term

Bm

damping term dissipaHon by

microscopic degrees of freedom

stochasHc term thermal fluctuaHons

H =

�d3R

�−Ku(R)

m2

2+

A

2(∇Rm)2 +

µ0M2s d

8π



�

The model system






m(r, t) = m(x, y, t)z


anisotropy term: its fluctuaHons around an average value provides strong pinning points for the domain walls

Ku(R) = �Ku�(1− P (x, y))

�P (R)� = 0

�P (R)P (R�)� = θδ(R−R�)

H =

�d3R

�−Ku(R)

m2

2+

A

2(∇Rm)2 +

µ0M2s d

8π



�

H =

�d3R

�−Ku(R)

m2

2+

A

2(∇Rm)2 +

µ0M2s d

8π



�

The model system anisotropy term: its fluctuaHons around an average value provides strong pinning points for the domain walls






m(r, t) = m(x, y, t)z

Ku(R) = �Ku�(1− P (x, y))

�P (R)� = 0

�P (R)P (R�)� = θδ(R−R�)


anisotropy term: it represents the energy cost for the domain walls. we do not have real Block or Neel walls, just their projecHon along z

H =

�d3R

�−Ku(R)

m2

2+

A

2(∇Rm)2 +

µ0M2s d

8π



�







m(r, t) = m(x, y, t)z

Ku(R) = �Ku�(1− P (x, y))

�P (R)� = 0

�P (R)P (R�)� = θδ(R−R�)



stray field energy: responsible for the domain formaHon non local term to be treated in reciprocal space

H =

�d3R

�−Ku(R)

m2

2+

A

2(∇Rm)2 +

µ0M2s d

8π



�







m(r, t) = m(x, y, t)z

Ku(R) = �Ku�(1− P (x, y))

�P (R)� = 0

�P (R)P (R�)� = θδ(R−R�)




UCS field: as measured in the experiment

H =

�d3R

�−Ku(R)

m2

2+

A

2(∇Rm)2 +

µ0M2s d

8π



�







m(r, t) = m(x, y, t)z

Ku(R) = �Ku�(1− P (x, y))

�P (R)� = 0

�P (R)P (R�)� = θδ(R−R�)




UCS field: as measured in the experiment

External field: uniform

bu

t Hme de

pend

ent

Pueng this approximate hamiltonian inside the LLG equaHon we obtain the following equaHon of moHon: where everything is now in dimensionless units:

the model contains only three non-‐independent dimensionless parameters related to the material proper.es:

∂m

∂τ= (1−m2)

�α(1− p(r))m− 1

4π

�d2r�

m(r�)

|r− r�|3 + hext(t)− hUCS(r) + q(r, τ)

�+ β∇2

rm

The model system

Jagla PRB 72 094406 (2005) Jagla PRB 70 046204 (2004) Zhirnov Zh.Eksp.Teor.Fiz. 35 1175 (1958)

r = R/d

τ = tγµ0Ms/ξ

hext = Hext/Ms

hUCS = HUCS/Ms

q(r, τ) = Q(R, t)/µ0Ms

�KBT = KBT/µ0M2s d

3dimensionless posiHon dimensionless Hme

dimensionless fields

η = θ/d3β = A/µ0M2s d

2α = �Ku�/µ0M2s

dimensionless temperature dimensionless thermal noise

�p(r)p(r�)� = ηδ(r− r�) dimensionless anisotropy noise

uniaxial anisotropy exchange sHffness anisotropy fluctuaHons (strength on the pinning disorder)



The model system


r = R/d

τ = tγµ0Ms/ξ

hext = Hext/Ms

hUCS = HUCS/Ms










∂m

∂τ= (1−m2)

�α(1− p(r))m− 1

4π

�d2r�

m(r�)


�+ β∇2

rm



The model system


r = R/d

τ = tγµ0Ms/ξ

hext = Hext/Ms

hUCS = HUCS/Ms










domain wall energy domain wall width domain size

domain morphology ∝�

β/α∝

�αβ

∂m

∂τ= (1−m2)

�α(1− p(r))m− 1

4π

�d2r�

m(r�)


�+ β∇2

rm

Model validaHon: from micro to macro The procedure for the determinaHon of the three material parameters is made in such a way to fit both macroscopic and microscopic properHes of the sample:

A good iniHal guess for α,β and η makes the measured domain image at 0 mT a steady state of our equaHon of moHon. A good choice of α,β and η is such that, if we use the measured image as the iniHal condiHon of our equaHon of moHon and we let it evolve in Hme, it will not change.

Benassi et al. (waiHng for PRL rejecHon)


A good iniHal guess for α,β and η makes the measured domain image at 0 mT a steady state of our equaHon of moHon. A good choice of α,β and η is such that, if we use the measured image as the iniHal condiHon of our equaHon of moHon and we let it evolve in Hme, it will not change. The measured field from the UCS distribuHon is also included in the equaHon of moHon and it helps in stabilizing the domain configuraHon.



Than we ramp up the external uniform field and we trim the parameters in such a way to reproduce the correct path to saturaHon looking also at the MFM taken at 100 mT, 200 mT and 300 mT.



Once that the microscopic properHes are well reproduced we can check the macroscopic ones (hysteresis loops). We can sHll trim a bit the model parameters to adjust the fine detail. Eventually we have to go back and control again the microscopic behavior. The loops were measured with a sample cooled in a saturaHng field so the printed UCS distribuHon is different from the previous one.




αinit = 6.25

βinit = 0.85

ηinit = 1.5× 10−4

α = �Ku�/µ0M2s = 6.6 → �Ku� = 3.1× 106J/m3

β = A/µ0M2s d

2 = 0.88 → A = 2.2× 10−10J/m

η = 1.88× 10−4

perfectly in the expected range

too big but our 1D walls are less expensive than a real block wall and A must compensate!

Path to saturaHon: the full dynamics Now that the parameters are fixed, the theoreHcal model allows us to access the full dynamics in Hme, we can thus invesHgate the domain behavior with more than few MFM images


As demonstrated by the experiments the domains, retracHng with increasing external field, will try to avoid the frustrated F/AF coupling regions.

UnmounHng the machinery


Now we can switch off separately the different hamiltonian terms and try to understand and quanHfy their contribuHon:

The UCS distribuHon alone has not enough strength to pin the domains and even at 0 mT the shape of the steady configuraHon is quite different from the original one. The saturaHon occurs too early! The anisotropy fluctuaHons have enough strength to keep the iniHal configuraHon pinned, however, without the help of the UCS local field, the pinning sHll occurs too early.

Now we can also try to predict which is the effect of a different UCS distribuHon on the exchange-‐bias (EB) effect and on the coercivity of the FF. •  As expected, switching off the UCS field the EB effect goes to zero

•  Doubling the average value of the UCS field, without changing the fluctuaHons strength, increases the EB effect without affecHng the coercivity

•  Doubling the fluctuaHons of the UCS field, without shiking its average, increase strongly the coercivity with minor changings in the EB loop shik.

Some predicHons on the macroscopic properHes

calc. 10 K

-0.5

0.0

0.5

1.0

-1.0-0.4 -0.2 -0.1 0.0 0.1 0.2 0.4-0.3 0.3

no UCS

double UCS averagedouble UCS fluctuations


Up to now the fluctuaHons of the uniaxial anisotropy have been considered to be uncorrelated (white noise), however they have something to do with the granularity of the sample. Something more about the strength and the correlaHon of these fluctuaHon can be inferred from Barkhausen noise measurements. The presence of Chromium atoms in the AF decouples the magneHc moment of neighboring grains, increasing the UCS field by the 40%. The model will be used to study this new sample in which the role of the UCS map as been enhanced. ParHcular aYenHon will be given to the return point memory effects. The code is easily parallelizable allowing for the descripHon of lager system or for the coupling of two interacHng magneHzed films

Further developments

Benassi et al. PRB 84 214441 (2011)

the role of exchange bias in domain dynamics control

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