ORE Open Research Exeter
TITLE
On the limits of coercivity in permanent magnets
AUTHORS
Fischbacher, J; Kovacs, A; Oezelt, H; et al.
JOURNAL
Applied Physics Letters
DEPOSITED IN ORE
28 June 2019
This version available at
http://hdl.handle.net/10871/37741
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Appl. Phys. Lett. 111, 072404 (2017); https://doi.org/10.1063/1.4999315 111, 072404
© 2017 Author(s).
On the limits of coercivity in permanent
magnets
Cite as: Appl. Phys. Lett. 111, 072404 (2017); https://doi.org/10.1063/1.4999315
Submitted: 11 May 2017 . Accepted: 05 August 2017 . Published Online: 16 August 2017
J. Fischbacher , A. Kovacs , H. Oezelt , M. Gusenbauer, T. Schrefl , L. Exl , D. Givord , N. M.
Dempsey, G. Zimanyi, M. Winklhofer , G. Hrkac, R. Chantrell, N. Sakuma , M. Yano, A. Kato, T. Shoji,
and A. Manabe
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On the limits of coercivity in permanent magnets
J. Fischbacher,
1
A. Kovacs,
1
H. Oezelt,
1
M. Gusenbauer,
1
T. Schrefl,
1,a)
L. Exl,
2
D. Givord,
3
N. M. Dempsey,
3
G. Zimanyi,
4
M. Winklhofer,
5
G. Hrkac,
6
R. Chantrell,
7
N. Sakuma,
8,9
M. Yano,
8,9
A. Kato,
8,9
T. Shoji,
8,9
and A. Manabe
9
1
Center for Integrated Sensor Systems, Danube University Krems, 2700 Wiener Neustadt, Austria
2
Faculty of Mathematics, Vienna University, 1090 Vienna, Austria
3
CNRS, Institut N
eel, 25 rue des Martyrs, 38042 Grenoble, France
4
Department of Physics, University of California, Davis, California 95616, USA
5
Carl von Ossietzky University of Oldenburg, 26129 Oldenburg, Germany
6
College of Engineering, University of Exeter, Exeter EX4 4QF, United Kingdom
7
Department of Physics, University of York, York YO10 5DD, United Kingdom
8
Toyota Motor Corporation, 1200 Mishuku, Susono, Shizuoka 410-1193, Japan
9
Technology Research Association of Magnetic Materials for High-efficiency Motors (Mag-HEM)
Higashifuji-Branch, 1200 Mishuku, Susono, Shizuoka 410-1193, Japan
(Received 11 May 2017; accepted 5 August 2017; published online 16 August 2017)
The maximum coercivity that can be achieved for a given hard magnetic alloy is estimated by
computing the energy barrier for the nucleation of a reversed domain in an idealized microstructure
without any structural defects and without any soft magnetic secondary phases. For Sm
1–z
Zr
z
(Fe
1–y
Co
y
)
12–x
Ti
x
based alloys, which are considered an alternative to Nd
2
Fe
14
B magnets with a
lower rare-earth content, the coercive field of a small magnetic cube is reduced to 60% of the
anisotropy field at room temperature and to 50% of the anisotropy field at elevated temperature
(473 K). This decrease of the coercive field is caused by misorientation, demagnetizing fields, and
thermal fluctuations. Published by AIP Publishing. [http://dx.doi.org/10.1063/1.4999315]
Permanent magnets are an important material for energy
conversion in modern technologies. Wind power and hybrid
and electric vehicles require high performance permanent
magnets. In motor applications, the magnet should retain a
high magnetization and coercive field at an operating tempera-
ture around 450 K. At this temperature, the magnetization and
the anisotropy field of Sm
1–z
Zr
z
(Fe
1–y
Co
y
)
12–x
Ti
x
are higher
than those of Nd
2
Fe
14
B.
1
In addition, the rare earth to transi-
tion metal ratio of the SmFe
12
based magnets is lower.
Therefore, magnets based on this phase are considered as a
possible alternative to Nd
2
Fe
14
Bmagnets.
2
At high tempera-
ture, thermal fluctuations may reduce the coercive field. In this
work, we numerically compute the reduction of coercivity by
thermal fluctuations in Sm
1–z
Zr
z
(Fe
1–y
Co
y
)
12–x
Ti
x
. For compar-
ison, we also include results for Nd
2
Fe
14
B. The letter is orga-
nized as follows. We first review the different effects that
reduce the coercive field in permanent magnets. Then, we pre-
sent a numerical method for the computation of the coercive
field including thermal fluctuations, which is based on finite
element micromagnetics. We introduce the concept of the acti-
vation volume which is widely used in the experimental analy-
sis of coercivity in permanent magnets. Then, we present
numerical results for Nd
2
Fe
14
BandSm
1–z
Zr
z
(Fe
1–y
Co
y
)
12–x
Ti
x
.
Besides thermal fluctuations, several other effects
reduce the coercive field of modern permanent magnets.
Kronm
€
uller et al.
3
refer to the difference between the anisot-
ropy field of a magnet and its coercive field as a discrepancy
from theory. Aharoni
4
predicted that the coercive field of a
hard magnet decreases with increasing width of surface
defects with zero anisotropy. The corresponding minimum
coercive field is 1/4 of the anisotropy field which is reached
for a defect width greater than 5
ffiffiffiffiffiffiffiffiffi
A=K
p
, where A is the
exchange constant and K is the anisotropy constant. Even
smaller coercive fields may occur if the anisotropy increases
gradually from zero to its maximum value as shown by
Becker and D
€
oring
5
and Hagedorn.
6
In addition to defects, local demagnetizing fields reduce the
coercivity of permanent magnets. Gr
€
onefeld and Kronm
€
uller
7
show that the local demagnetizing field may reach values of the
order of the saturation magnetization, M
s
,neartheedgesofa
hard magnetic grain. The total field which is essential for the
switching of a grain is the sum of the local demagnetizing field
and the external field. Therefore, the local demagnetizing field
leads to a further reduction of coercivity.
A further reduction of the coercive field as compared to
the ideal nucleation field, H
N
¼ 2 K/(l
0
M
s
), may result from
dynamic effects.
8
When the external field or the internal
effective field is changing at a rate much faster than the
energy dissipation in the system, the system cannot follow
fast changes in the energy landscape and thus does not reach
the nearest metastable state. Instead, a path through the
energy landscape that brings the system into a reversed mag-
netic state may be taken. Leineweber and Kronm
€
uller
9
show
that dynamic effects can reduce the ideal nucleation field by
up to 20%.
In this work, we focus on thermal fluctuations and calcu-
late the reduction of coercivity caused by these fluctuations.
Magnetization reversal in a permanent magnet is the process
by which an external field creates a reversed nucleus near
structural defects. Thermal fluctuations assist the formation
of the reversed nucleus and thus reduce the coercive field.
The formation of the nucleus is associated with an energy
barrier. Before magnetization reversal, the system is in a
local energy minimum. With the increasing external field,
a)
Electronic mail: tschrefl@gmail.com
0003-6951/2017/111(7)/072404/5/$30.00 Published by AIP Publishing.111, 072404-1
APPLIED PHYSICS LETTERS 111, 072404 (2017)
the energy barrier that separates the local minimum from the
reversed magnetic state decreases.
10
Taking into account
thermal activation, the system can overcome an energy
barrier, E, within a time s ¼ s
0
exp ðE=ðk
B
TÞÞ.
5
Here, k
B
¼ 1.38 10
23
J/K is the Boltzmann constant. The time con-
stant s
0
is the inverse of the attempt frequency f
0
. Often, it is
assumed that the magnet can overcome an energy barrier of
25k
B
T within the time s ¼ 1 s which gives an attempt fre-
quency of f
0
¼ 7.2 10
10
s
1
.
11
Then, the coercive field is
the critical value of the external field, H, at which the energy
barrier E(H) reaches 25k
B
T.
Using numerical micromagnetics, we compute the
energy barrier as a function of the applied field. We discre-
tize the magnet’s microstructure with tetrahedral finite ele-
ments. Minimizing the energy for varying external fields
gives the magnetic states along the demagnetization curve.
For energy minimization, we apply the non-linear conjugate
gradient method as described by Fischbacher and co-work-
ers.
12
The coercive field obtained from the computation of
the demagnetization curve is H
0
. This is the field at which
the energy barrier is zero. We now want to compute the
energy barrier for a field H < H
0
. We apply the string
method
13
in order to compute the minimum energy path that
connects the local minimum at field H with the reversed
magnetic state. A path is called a minimum energy path, if
for any point along the path the gradient of the energy is par-
allel to the path. In other words, the component of the energy
gradient normal to the path is zero. The magnetization con-
figurations along the path are described by images. Each
image is a replica of the total system. The minimum energy
path over a saddle point is found iteratively. A single itera-
tion step consists of two moves. First, each image is
relaxed
14
by applying a few steps of the conjugate gradient
method, and then, the images are moved along the path so
that the distance between the images is constant. We use an
energy weighted distance and truncate the path
15
so that
there are more images next to the saddle point. We repeat
the computation of the minimum energy path for different
applied fields and obtain E(H). We compute H
c
(T) by the
intersection of the E(H) curve with the line E ¼ 25k
B
T (see
Fig. 1).
Path finding algorithms are well established both
in chemical physics and in micromagnetics.
13
As shown in
Fig. 1, the applied algorithms are self-consistent. The switch-
ing field obtained by a classical micromagnetic method is
equal to the critical field at which the computed energy barrier
vanishes. Please note that the computation of the demagneti-
zation curve by energy minimization
16
and the computation
of the minimum energy path use the same computational grid
and the same numerical minimization algorithm. Thermal
fluctuations at the atomistic level are taken into account by
using temperature dependent intrinsic magnetic properties
such as M
s
(T), K(T), and A(T).
The above numerical scheme takes into account thermal
activation over finite energy barriers. Skomski et al.
17
reported another mechanism of coercivity reduction by ther-
mal fluctuations. Spin waves interact with small soft mag-
netic structural defects which in turn cause a reduction of
coercivity. The corresponding change in coercivity was
found to be less than one percent. In our analysis, this effect
is not taken into account.
We can express the coercive field as
H
c
¼ aH
N
N
eff
M
s
H
f
: (1)
Expression (1) is reminiscent of the micromagnetic equation
3
often used to analyze the temperature dependence of coerciv-
ity in hard magnets. The coefficient a expresses the reduction
in coercivity due to defects, misorientation, and intergrain
exchange interactions.
18
The microstructural parameter N
eff
is related to the effect of the local demagnetization field near
sharp edges and corners of the microstructure. The fluctua-
tion field H
f
gives the reduction of the coercive field by
thermal fluctuations.
19
In this work, we will quantify the dif-
ferent effects that reduce the coercivity according to (1).In
particular, we are interested in the limits of coercivity. By
computing a, N
eff
, and H
f
for a perfect hard magnetic particle
without any defect, we can estimate the maximum possible
coercive field for a given magnetic material and microstruc-
ture. This is especially important considering the current
effort to search for new hard magnetic phases with reduced
rare-earth content.
2
In addition, one might take into account
the thermal fluctuation field to know how much magnetic
anisotropy is enough for a permanent magnet.
20
The coercive
field which would be measured in the absence of thermal
activation is H
0
¼ aH
N
N
eff
M
s
.
The height of the energy barrier as a function of field,
E(H), can be derived from viscosity measurements, series
expansion, or micromagnetic simulations. N
eel
21
derived a
series expansion of the form E ¼ c(H
0
– H)
m
to describe the
field dependence of the energy barrier, where c is a constant.
Analyzing the micromagnetic free energy, Skomski et al.
22
showed that physically reasonable exponents are m ¼ 3/2 and
m ¼ 2. The numerical algorithm presented above does not
make any prior assumption on how the energy barrier changes
with the field. Instead, we compute E(H) for a finite element
model of a magnetic material numerically. For the analysis of
experimental data, the energy barrier is often expressed by a
linear approximation E(H) ¼ vl
0
M
s
(H
0
– H).
23
The activation
volume v is not necessarily related to a physical volume.
Solving E(H) ¼ 25k
B
T for H gives the coercive field. Thus,
we can write (1) as
24
FIG. 1. Left: Computed demagnetization curve for a Nd
2
Fe
14
B cube at
T ¼ 300 K with an edge length of 40 nm. Right: Energy barrier as a function
of the external field. At the coercive field, the energy barrier crosses the
25k
B
T line.
072404-2 Fischbacher et al. Appl. Phys. Lett. 111, 072404 (2017)
H
c
¼ aH
N
N
eff
M
s
25k
B
T
vl
0
M
s
: (2)
The last term in (2) is proportional to the magnetic viscosity
coefficient
25,26
S
v
¼ k
B
T/(vl
0
M
s
), which can be measured
experimentally. Traditionally, equations of form (2) have
been used to analyze the temperature dependence of the
coercivity.
27,28
The viscosity coefficient can be written as S
v
¼ –k
B
T/
(@E/@H).
11
Thus, we can define the activation volume as
v ¼
1
l
0
M
s
@E
@H
: (3)
In this work, we will use (3) to compute the activation vol-
ume, whereby E(H) is computed by finite element micro-
magnetic simulations.
From the comparison of the numerical results with
Equation (1), we can numerically determine the microstruc-
tural parameters a, N
eff
, and the fluctuation field H
f
:
(1) We compute the demagnetizing curve but we switch
off the demagnetizing effects by neglecting the magne-
tostatic self-energy in the total energy. This gives
H
0
¼ aH
N
and we can derive a ¼ H
0
=H
N
.
(2) We compute the demagnetizing curve taking into
account the magnetostatic energy term. This gives
H
0
¼ aH
N
N
eff
M
s
¼ H
0
N
eff
M
s
and we compute
N
eff
¼ðH
0
H
0
Þ=M
s
.
(3) We compute the coercive field including thermal activa-
tion by E(H
c
) ¼ 25k
B
T. The fluctuation field, H
f
¼ H
0
– H
c
, represents the reduction in coercivity due to ther-
mal activation effects.
We are particularly interested in the limits of coercivity
for a given magnetic material. Therefore, we apply the above
procedure for a perfect, nano-sized hard magnetic cube with-
out any defects. The edge length of the cube is 40 nm.
However, we apply the magnetic field one degree off the easy
axis which is parallel to one edge of the cube. First, we apply
the method for Nd
2
Fe
14
B. Then, we will show the limits of
coercivity for Sm
1–z
Zr
z
(Fe
1–y
Co
y
)
12–x
Ti
x
magnets. Table I
gives the intrinsic magnetic properties used for the
simulations. For the simulation, the mesh size was 1.5 nm.
Without soft magnetic defects, the numerically calculated
reversal field computed without magnetostatic interactions
corresponds to an analytic switching field estimated by
Stoner and Wohlfarth,
31
H
0
¼ f ðw
0
ÞH
N
. Here, w
0
denotes the
angle between the applied field and the negative anisotropy
direction and f ðw
0
Þ¼fcos
2=3
ðw
0
Þþsin
2=3
ðw
0
Þg
3=2
.
32
The
agreement between the finite element results without the
magnetostatic energy term and the Stoner-Wohlfarth switch-
ing field was already shown previously.
12
For Nd
2
Fe
14
Bat
300 K, we obtain l
0
H
0
¼ 6:09 T. The self-demagnetizing
field reduces the coercive field to l
0
H
0
¼ 5.29 T. Finally,
with thermal fluctuations, the coercive field is l
0
H
c
¼ 3.94 T.
Therefore, we can conclude that in Nd
2
Fe
14
B, the maximum
possible coercive field of a cubic grain is only 60% of the
ideal nucleation field H
N
. The values of a, N
eff
, l
0
H
f
, and
l
0
S
v
are 0.91, 0.5, 1.35 T, and 0.054 T, respectively.
Figure 1 shows the computed demagnetizing curve for
the Nd
2
Fe
14
B cube and the energy barrier as a function of
the external field computed with the intrinsic magnetic prop-
erties at T ¼ 300 K. Static energy minimization for the
decreasing external field gives a switching field of
l
0
H
0
¼ 5.29 T. This is exactly the field at which the energy
barrier reaches zero. The reduction of coercivity owing to
thermal fluctuations is 25%. Using (3), we can compute the
activation volume, v ¼ (4.38 nm)
3
, from the slope of the
E(H) curve. The activation volume can be compared with
the domain wall width, d ¼ p
ffiffiffiffiffiffiffiffiffi
A=K
p
, which is 4.2 nm, giving
v ¼ 1.12d
3
.
33
Figure 2 gives the minimum energy path and
the magnetization configuration at the saddle point of the
energy landscape. At the saddle point, a small nucleus,
which has an extension a, is formed. Interestingly, the vol-
ume of the reversed nucleus, (1/8)(4pa
3
/3), roughly corre-
sponds to the activation volume v as given by (3). For the
small perfect cube, the computed coercivity, the viscosity
coefficient, and the activation volume are higher than the
experimental values found in Nd
2
Fe
14
B based magnets.
For comparison with experiments, we performed a simi-
lar simulation of a granular Nd
2
Fe
14
B ensemble consisting
of 64 polyhedral grains with an average grain size of 60 nm.
We generated the grain structure from a centroid Voronoi
tessellation, using the software tool Neper.
34
The grains of
the Nd
2
Fe
14
B model system were separated by a weakly fer-
romagnetic grain boundary phase with l
0
M
s
¼ 0.5 T. The
thickness of the grain boundary phase was approximately
3 nm. Grain boundaries in hot deformed Nd
2
Fe
14
B magnets
were found to contain up to 55 at. % Fe.
35
The average
TABLE I. Intrinsic magnetic properties used for the simulations. The table
gives the anisotropy constant K(MJ/m
3
), the saturation magnetization l
0
M
s
(T), and the exchange constant A(pJ/m) for different temperatures T(K). For
Nd
2
Fe
14
B, the material properties are taken from Hock
29
and Durst and
Kronm
€
uller.
30
For Sm
1–z
Zr
z
(Fe
1–y
Co
y
)
12–x
Ti
x
compounds, the material prop-
erties are taken from Kuno et al.
1
The exchange constant is estimated.
Material T l
0
M
s
KA
Nd
2
Fe
14
B 300 1.61 4.30 7.7
Nd
2
Fe
14
B 450 1.29 2.09 4.89
SmFe
11
Ti 300 1.26 5.17 10
Sm(Fe
0.75
Co
0.25
)
11
Ti 300 1.42 4.67 10
Sm(Fe
0.75
Co
0.25
)
11.5
Ti
0.5
300 1.58 4.57 10
(Sm
0.8
Zr
0.2
)(Fe
0.75
Co
0.25
)
11.5
Ti
0.5
300 1.63 4.81 10
SmFe
11
Ti 473 1.02 2.80 6.5
Sm(Fe
0.75
Co
0.25
)
11
Ti 473 1.28 2.54 8.1
Sm(Fe
0.75
Co
0.25
)
11.5
Ti
0.5
473 1.45 2.61 8.4
(Sm
0.8
Zr
0.2
)(Fe
0.75
Co
0.25
)
11.5
Ti
0.5
473 1.50 2.79 8.4
FIG. 2. Left: Minimum energy path for a Nd
2
Fe
14
B cube at T ¼ 300 K with
an edge length of 40 nm. Right: Magnetization configuration of the saddle
point with a reversed nucleus of size a.
072404-3 Fischbacher et al. Appl. Phys. Lett. 111, 072404 (2017)
