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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 

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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:  alf@we.lc.ehu.es

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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142411-4 Garcıa-Arribas et al.   Appl. Phys. Lett. 103, 142411 (2013)

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