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Determination of the distribution of transverse magnetic anisotropy in thin films from
the second harmonic of Kerr signal
A. García-Arribas, E. Fernández, I. Orue, and J. M. Barandiaran
Citation: Applied Physics Letters 103, 142411 (2013); doi: 10.1063/1.4824647
View online: http://dx.doi.org/10.1063/1.4824647
View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/103/14?ver=pdfcov
Published by the AIP Publishing
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Determination of the distribution of transverse magnetic anisotropyin thin films from the second harmonic of Kerr signal
A. Garcıa-Arribas,1,a) E. Fernandez,1 I. Orue,2 and J. M. Barandiaran1
1 Departamento de Electricidad y Electr onica and BCMaterials, Universidad del Paıs Vasco UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain2SGIker, Universidad del Paıs Vasco UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain
(Received 11 July 2013; accepted 24 September 2013; published online 3 October 2013)
We describe a method to determine the magnetic anisotropy distribution in thin films based on
Kerr magnetometry, well adapted for single micro- and nanostructures. When the sample is excited
by an ac field of small amplitude, for each value of a longitudinal dc field H , the second harmonic
of the Kerr signal gives the contribution of the corresponding transverse anisotropy field H k ¼ H to
the anisotropy distribution. The method is tested on a Permalloy-based multilayer microstructure,
revealing two anisotropy contributions, one of them deviated from the perfect transverse direction.
This confirms and extends a previous characterization performed by far more sophisticated
methods. VC 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4824647]
Magnetic thin films are used in a growing number of
well-established technological areas, such as microsensors
and high frequency devices, and are also key components in
novel spintronic applications like domain-wall logic,1 race-
track memories,2 and spin nanooscillators,3 to name some.
The precise characterization of all aspects of the magnetic
behavior of the films is fundamental to understand and
improve their properties and usefulness. Special attention
has to be paid to the magnetic anisotropy present in the film,
since it largely determines both its static and dynamic mag-
netic behavior.
Many of the aforementioned applications make use of
soft magnetic materials, predominantly Permalloy (Py). In
bulk state, Py displays a vanishingly small intrinsic (magne-
tocrystalline) anisotropy,
4
which is ascribed to the compen-sation between the anisotropies with opposite sign of Fe
( K 1 ¼ 4:7 104 J=m3) and Ni ( K 1 ¼ 0:5 104 J=m3),
although its fundamental origin is still controversial.5 When
Py is prepared in the form of a thin film by physical vapor
deposition methods, a certain degree of magnetic anisotropy
develops regardless of the preparation conditions,6 due to a
self-shadowing effect on the growing film caused by the
oblique incidence of the incoming atoms over the substrate.7
This effect can be exploited to tailor the magnetic properties
of the film by intentionally tilting the substrate with respect
to the plane of the source. Using the data of a recent work on
Py films,8 one can estimate that each degree of inclined inci-
dence causes the anisotropy constant to increase by approxi-mately 135J/m3. This great sensitivity to the deposition
angle largely complicates the obtention of anisotropy-free
films.
On the other hand, some applications require a well
defined uniaxial magnetic anisotropy, as is the case, for
instance, of magnetic sensors based on the Magneto-
Impedance (MI) effect.9 In Py-based thin films, this can be
achieved at the preparation time by oblique deposition or,
more frequently, by depositing the film under the presence
of a magnetic field, possibly followed by a field-annealing.
However, the resulting effective magnetic anisotropy pro-
duced in the sample inevitably exhibits a certain spread in
both intensity values and directions, caused primarily by
shape effects (through non-uniform demagnetizing fields),
imperfections at the borders and surfaces, and possible
local variations in the applied field and inhomogeneities in
sample composition. The determination of the anisotropy
distribution is paramount to fully characterize the magnetic
behavior of the samples and to refine the preparation and
processing methods to improve their MI properties. Also,
accurate mathematical descriptions of the MI response usu-
ally require the introduction of a realistic distribution of an-
isotropy within the model.10 Conversely, the analysis of MI
measurements can be used as a tool to determine the distri-bution of anisotropies.11
Given that the magnetic anisotropy is a decisive parame-
ter, significant effort has been made to accurately determine
it in thin films. Direct torque measurements and different
types of methods based on the analysis of the magnetization
curves with different levels of complexity and performance
have been proposed.12 – 15
When a distribution of anisotropies exists in the sample,
all the methods mentioned above provide either the strongest
or the mean value of the distribution, without giving infor-
mation about its actual size and shape. Mathematically, the
distribution of anisotropies can be expressed by a function
Pð H k Þ that gives the probability of a given anisotropy value K being present in the sample. H k is the anisotropy field
defined as H k ¼ 2 K =l0 M s, where M s is the saturation mag-
netization. For each H k , a magnetic field H applied perpen-
dicularly to the easy axis produces a magnetization M k ð H Þgiven by
M k ð H Þ ¼ M s
H
H k when H < H k
M s when H H k ;
8<: (1)
so the total magnetization of the sample isa)Electronic mail: [email protected]
0003-6951/2013/103(14)/142411/4/$30.00 VC 2013 AIP Publishing LLC103, 142411-1
APPLIED PHYSICS LETTERS 103, 142411 (2013)
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M ð H Þ ¼
ð 11
M k ð H Þ Pð H k ÞdH k
¼ M s
ð H
1
Pð H k ÞdH k þ M s H
ð 1 H
Pð H k Þ
H k dH k : (2)
Experimentally, the intensity distribution of uniaxial
anisotropies having their easy axis perpendicular to a given
direction can be quantitatively determined from the magnet-
ization curve measured along that direction, by calculating
the second derivative of the magnetization with respect to
the applied field16
d 2 M
dH 2 ¼ M s
Pð H Þ
H : (3)
This procedure, although extremely simple, is usually
greatly affected by the noise produced by the numerical dif-
ferentiation. The smoothing necessary to clean the output of
the derivative artificially broadens the distribution and masks
the details of a possible fine structure.
Alternatively, the distribution of anisotropies can also
be obtained from the second harmonic response to an ac ex-
citation of small amplitude.17 The magnetic response of the
sample biased by a dc field H b to a field h of small amplitude
can be expanded in a power series as
M ð H b þ hÞ ¼ M ð H bÞ þ hdM
dH
H b
þ1
2h2d 2 M
dH 2
H b
þ :::: (4)
If h ¼ h0eixt , Eq. (4) can be considered the harmonic expan-
sion of the time-varying magnetization M (t )
M ðt Þ ¼ M b þ h0
dM
dH H b
eixt þ1
2
h20
d 2 M
dH 2
H b
ei2xt þ …; (5)
where M b ¼ M ð H bÞ. M (t ) being an even periodic function,
its harmonic expansion is
M ðt Þ ¼ a0 þ 2X1n¼0
aneinxt ; (6)
with an being the n-th harmonic coefficient. The comparison
of Eqs. (5) and (6) directly links the second harmonic com-
ponent a2 with the second derivative of M ( H ), and a substitu-
tion in Eq. (3) results in
Pð H Þ ¼ 4 H h2
0 M sa2: (7)
According to Eq. (7), in order to measure the weight of a
given value of the transverse anisotropy field H k in the distri-
bution of anisotropies, we must bias the sample with a dc field
H ¼ H k , and measure the second harmonic response to an
exciting field of small amplitude.18 If the magnetic signal pro-
duced by the sample is large enough, it can be detected induc-
tively, using a coil wrapped around the sample, a procedure
which results in a much higher resolution than the second de-
rivative method. For instance, this procedure has been used to
accurately follow the evolution of the distribution of anisotro-
pies in amorphous ribbons under applied stresses19 and in
nanocrystalline samples subjected to thermal annealings.20 It
is relevant that the generation of a second harmonic caused by
the magnetic non-linear response has been commonly used in
extremely sensitive devices such as fluxgate sensors. It has
also been analyzed in magneto-impedance materials,21,22
where the field sensitivity of the second and higher harmonics
is very promising for applications.23
In thin films, specially in patterned samples with micro-
or nanosized lateral dimensions, due to their reduced mass,the magnitude of the magnetic signal is so small that measure-
ments based on inductive methods are unfeasible. The
magneto-optical Kerr effect (MOKE) is normally used for the
magnetic characterization in those cases. A considerable num-
ber of methods have been developed for determining the ani-
sotropy in thin films using magneto-optical techniques.12,24,25
Obviously, the distribution of anisotropies can be obtained
using the second derivative method on the MOKE hysteresis
loop, although it tends to be much noisier than the inductive
one, aggravating the problem stated before.
Here, we propose a method for determining the anisot-
ropy distribution based on measuring the second harmonic
response of the Kerr signal to an excitation of small
amplitude. The experimental set-up is based on the usual
longitudinal magneto-optical arrangement commonly used to
measure the Kerr hysteresis loops,26 with a laser spot
focussed to a size of 20lm. The sample is magnetized
through the dc field provided by a pair of Helmholtz coils
fed by a bipolar power supply, which is swept stepwise in
forward and reverse directions among values that magneti-
cally saturate the sample (up to 1 kA/m). The magnitude of
the dc field steps determines the fineness at which the anisot-
ropy field values are resolved in the measured distribution.
The ac excitation is produced by a second pair of coils, col-
linear with the previous ones, supplied by a signal generator.The amplitude of the excitation must be selected to be large
enough to produce a measurable signal. Often, it is larger
than the step used for sweeping the dc field so the output is
averaged among neighbor data points, producing an experi-
mental broadening of the distribution. Thus, the resolution
and definition of the distribution obtained are often in com-
promise. The frequency of the ac excitation is not decisive.
Unlike the case of inductive measurements where higher fre-
quencies produce larger signals, the only practical requisite
here is that it remains below the cut-off frequency of the
detecting photodiode (30 kHz in our set-up). The output of
the photodetector is routed to a dynamic signal analyzer
(Agilent 35670A) that determines the amplitude of the sec-ond harmonic component through a numerical Fast Fourier
Transform.
The method has been tested on a Py-based microsized
sample consisting of three 170 nm thick Fe20Ni80 layers
alternated with 6 nm thick Ti layers. This multilayer struc-
ture has demonstrated the ability to produce excellent MI
performance by combining the magnetic softness of thin
FeNi layers with the increased thickness necessary for the
skin effect to become effective at moderate frequencies.27
The material was deposited onto a Si wafer by sputtering
under an applied magnetic field to produce a well-defined
in-plane magnetic anisotropy. The sample was patterned by
the lift-off method into a rectangle 2 mm long and 100lm
142411-2 Garcıa-Arribas et al. Appl. Phys. Lett. 103, 142411 (2013)
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wide. Careful alignment assured that the direction of the
induced anisotropy became transverse to the long direction
of the sample. The sample was subjected to a post-deposition
thermal annealing at 200 8C for 1 h in a 1 T magnetic field,
in order to reinforce the induced transverse anisotropy.
Figure 1 shows the MOKE hysteresis loop measured in
the sample with the magnetic field applied along the sample
length. It has been purposely taken close to the edge of the
sample to include a richer magnetic structure. The shape of
the loop, displaying large remanence and coercivity, hints
at the contribution of large closure domains, as confirmed by
the image shown in Fig. 2. It was taken at the remanent state
using a Kerr microscope (Evico Magnetics GmbH,
Germany), adjusted to provide contrast between sample
areas having the magnetization component along the sample
length in opposite directions. It also reveals well-defined
transverse domains, tilted about 4 from the transversal
direction. Certainly, despite the efforts to precisely align the
magnetic field during the deposition, patterning, and thermal
annealing, a small deviation is conceivable. Besides, when
the induced anisotropy is combined with the longitudinalshape anisotropy, if they are not perfectly perpendicular, the
effective easy axis tilts by an amount that depends on the rel-
ative strength of both anisotropies.28 Depending on the geo-
metric aspect ratio of the sample, this angle can become
rather large as evidenced by Nakai and co-workers.29
The bottom graph in Figure 1 shows the results obtained
with the method of the second derivative (Eq. (3)) for the an-
isotropy distribution. After each numerical derivative, a
smoothing among the nearest five points is performed. While
it is evident that this curve reflects the characteristics of the
magnetization loop, very limited insight can be obtained
about the features of the anisotropy in the sample.
The distribution of transverse anisotropy obtained by the
second harmonic of the Kerr signal is shown in Fig. 3. The
dc field is swept with a variable step size, which is finer
(4 A/m) in the peak regions. The four peaks corresponding to
a full loop sweep are displayed. A 900 Hz excitation ac field
with a 20 A/m (rms) amplitude is used, which is larger than
the dc step, so some smoothing effect is expected, equivalent
to having an instrumental broadening of the order of the ex-
citation amplitude. The measurement of the second harmonic
at the signal analyzer is averaged 20 times for each field
point to enhance the signal-to-noise ratio. The distribution is
measured at exactly the same place in the sample as the loop
shown in Fig. 1.In contrast to the second derivative method, in the dis-
tribution obtained from the second harmonic, the peaks are
very well defined with a very small hysteresis. This is
because, in the second harmonic method, large domain wall
movements, produced when stepping the bias field, do not
contribute to the signal. According to the theory, the meas-
ured curve should match the distribution of intensities of
FIG. 1. (a) MOKE hysteresis loop of a multilayer [Fe20Ni80(170nm)/
Ti(6nm)]2 /Fe20Ni80(170 nm) structure. (b) Anisotropy distribution deduced
from the hysteresis loop by the second derivative method (Eq. (3)).
FIG. 2. Domain structure in the remanent state obtained by Kerr micros-
copy. The borders of the 100 lm wide sample are clearly seen at the top and
at the bottom of the figure.
FIG. 3. Anisotropy distribution obtained from the second harmonic of the
Kerr response using the method proposed here. The inset shows the decom-
position of the measured distribution in three separated contributions.
142411-3 Garcıa-Arribas et al. Appl. Phys. Lett. 103, 142411 (2013)
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anisotropies with easy axes perpendicular to the direction
of the measuring field. However, a single anisotropy
not perfectly perpendicular produces a similar apparent
distribution.16 It is therefore not possible, in principle, to
establish unambiguously if the result is due either to a dis-
tribution of intensities of perfectly transverse anisotropies,
or to the existence of easy axes deviating a certain angle
from the perpendicular, or to a mixture of both. However,
thanks to the resolution achieved by the second harmonicmethod, a deeper insight is possible. As shown in the inset
of Fig. 3, the peaks of the measured distribution are satis-
factorily fitted using three gaussian contributions. This
strongly suggests the existence of two main anisotropy con-
stants corresponding to anisotropy field values of 208 A/m
(contribution G1) and 241 A/m (contributions G2 and G3).
The width of the contributions G1 and G2 matches the
experimental broadening expected for the amplitude of the
excitation used in the experiment, so they most probably
correspond to perfectly transversal anisotropies with well-
defined values. The broader G3 contribution is probably
due to not-perfectly transversal easy axes. The width of the
apparent distribution produced by an anisotropy not aligned
with the transverse direction can be quantified using the
Stoner-Wolfart model to calculate the magnetization curve
and then Eq. (3) to obtain the distribution. Using these
results, we can conclude that the width of contribution G3
corresponds to an easy axis tilted 4.5. If G1 and G2 were
also produced by tilted anisotropies, their widths would
only be consistent with deviations from the perpendicular
not greater than 1.
The four peaks measured in the full dc field sweep are
not identical due to small hysteresis effects and the noise
that affects the measurement. However, all peaks can be fit-
ted in a similar fashion and three anisotropy contributionsare needed to account for the broadening at the base (G3),
the narrow top (G2), and the shoulder that makes the peak
asymmetric (G1), although best-fit parameters vary slightly.
Evidently, the contribution G3 is congruous with the
tilted magnetic domains observed in the sample. The analysis
of the results obtained with the method presented here not
only reveals this feature but also uncovers the existence of
other perfectly transverse anisotropies. Different anisotropies
could have been generated during the fabrication (deposition)
and processing (patterning and annealing) of the sample, and
further investigation is needed to clarify this point. We have
presented a very powerful tool to do so, demonstrating that a
relatively simple experiment, which uses a quasi-conventionalKerr set-up, can produce an insightful description of the con-
figuration of the anisotropy in thin films even in the case of a
single microstructure. Besides, the potential of this method
can be still significantly improved using state of the art equip-
ment,30 so we believe that it will help to enhance the perform-
ance of magnetic micro- and nanodevices.
We acknowledge the financial support from the Spanish
(Project No. MAT2011-27573-C04-03) and Basque (Projects
Etortek-Actimat and Saiotek-PE12UN025) Governments.
Dr. J. Feutchwanger provided useful comments.
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