Modelling exchange bias with MuMax 3

Jonas De Clercq ,
a Arne Vansteenkiste ,
a Kristiaan Temst ,
b Bartel Van Waeyenberge a

a
DyNaMat, Department of Solid State Sciences, Ghent University
b
Institute for Nuclear and Radiation Physics, KU Leuven

Exchange bias Unidirectional shift

When cooling a ferromagnetic / antiferromagnetic bilayer in an external Pinned AFM grains, which have a high anisotropy constant and so are
magnetic field below the Néel temperature TN , an unidirectional shift of almost frozen, cause the unidirectional shift of the hysteresis loop due
the hysteresis loop is found due to the coupling between both layers. This to their coupling with the FM layer. Using an exchange stiffness of
effect is used in GMR read heads under the form of spin valves. A pin = 6.9 × 10−12 J/m and Arot = 1.1 × 10−11 J/m at the interface, we find that
FM
our bias field and coercivity agrees well with the experimental data.

AFM 1
free FM layer ⟨⟨mx,FM⟩
M my,FM
0.5
M
non magnetic spacer
Heb

⟨mFM ⟩
0
H H
pinned FM layer
-0.5
T > TN T < TN AFM layer
-1
exchange biased hysteresis loop spin valve -100 -50 0 50 100
Bext (mT)

In most polycrystalline stacks also a training effect can be seen, i.e. the hysteresis loop of the FM layer
coercivity and bias field decrease for an increasing number of hysteresis
cycles n. For n = 1, the athermal component of the training effect con-
tains the largest contribution.

Athermal training effect

Objective After field cooling, the magnetization of the AFM grains is randomly dis-
tributed in the field cooling direction. Rotatable AFM grains have a
low anisotropy constant (KU = 2.0 × 106 J/m3) and so rotate together with
We proof, by reproducing experimental data[1] for an exchange bi-
the FM layer during the hysteresis loop. Especially those grains with an
ased Co(30nm)/CoO(3nm) bilayer, that we can include these 2 effects in
MuMax3 by considering the presence of pinned and rotatable grains anisotropy axis almost perpendicular to the field cooling direction
in the AFM layer. We also demonstrate there can be an asymmetry in contribute to the athermal training effect as explained below.
the reversal mechanism between the descending branch for n = 1 and
further hysteresis loops. 1
[1] T. Dias, E. Menndez, H. Liu, C. Van Haesendonck, A. Vantomme, K. Temst, J. E. Schmidt, R. Giulian, J. Geshev, Rotatable anisotropy driven training effects in exchange
biased co/coo films, Journal of Applied Physics 115 (2014) 243903.
0.5
⟨mAFM ⟩

-0.5 ⟨⟨mx,AFM⟩
Solving micromagnetism with MuMax3 my,AFM
-1
-100 -50 0 50 100
Bext (mT)
In micromagnetism the evolution of the magnetisation M ⃗ (⃗r,t) in an switching of a rotatable grain hysteresis loop of the AFM layer
effective field H
⃗ e f f is determined by the Landau - Lifshitz equation
( ) Suppose the magnetisation of such a rotatable AFM grain is at position
( ) ∂ ⃗ αγ ( )
1 + α2
M
= −γM
⃗ ×H
⃗ ef f − ⃗ × M
M ⃗ ×H
⃗ ef f (1) (a) after field cooling and that the FM rotates coherently. If the FM has a
∂t Ms small initial positive ⟨my,FM ⟩ component, the AFM grain will rotate counter-
clockwise in the descending branch of the first hysteresis loop and jump
with α the dimensionless damping constant and γ the gyromagnetic ra-
irreversibly towards position (b). After the first hysteresis cycle, the
tio. MuMax3[2], which is a GPU - accelerated open source micromagnetic
AFM grain relaxes towards position (c) and so does not return to its ini-
simulation program, allows us to solve this equation for a ferromagnetic
tial position. This results in a non closed AFM hysteresis loop for n = 1.
system by using a finite difference discretisation.
For n > 1 the grain reversibly switches between (b) and (c). As now
⟨my,AFM ⟩ > 0, the AFM applies a net torque on the FM layer which results in
a lower coercivity and bias field. For FM domains in which ⟨my,FM ⟩ < 0,
the reasoning is similar.

Néel domain wall in FM nanowire magnetic vortex

Micromagnetic model
field cooled n = 2, Bext = 100 mT n = 2, Bext = - 45 mT

The polycrystalline Co and CoO layers are each divided into grains with an
average grain size of 12 nm using a Voronoi tessellation. The anisotropy
axes of the FM grains are distributed around the field cooling direction ac-
cording to a normal distribution with a standard deviation of 10°.
The AFM grains are divided into 2 types, pinned and rotatable, accord-
ing to a ratio of 3:7 respectively. Their anisotropy axes are randomly dis-
tributed in plane. No Zeeman or demagnetization energy was taken into
account for the AFM layer. top row: AFM layer, bottom row: corresponding FM layer