Quasilogarithmic magnetic viscosity in perpendicularly anisotropic ...


Quasilogarithmic magnetic viscosity in perpendicularly anisotropic ...

Journal of Magnetism and Magnetic Materials 324 (2012) 2854–2857

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Journal of Magnetism and Magnetic Materials

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Quasilogarithmic magnetic viscosity in perpendicularly anisotropic

Nd–Fe–B films

Q. Yao a ,R.Grössinger b , W. Liu a,n , W.B. Cui a , F. Yang a , X.G. Zhao a , Z.D. Zhang a

a Shenyang National Laboratory for Materials Science, Institute of Metal Research and International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016,

People’s Republic of China

b Institute for Solid State Physics, Vienna University of Technology, Wiedner Haupstrasse 8-10, A-1040 Vienna, Austria

article info

Article history:

Received 4 January 2012

Received in revised form

19 March 2012

Available online 1 May 2012


Magnetic viscosity

Hard magnetic film


Perpendicular anisotropy

Abstract: The quasilogarithmic magnetic viscosity of the perpendicularly anisotropic Nd–Fe–B films is

determined by measuring the time dependent magnetization in coercive fields at 300 K–4.2 K. The

distribution of energy barrier heights in the films is believed to be the origin of this stretched

exponential magnetic relaxation. Another feature exhibited in the anisotropic Nd–Fe–B films is the

nonmonotonic temperature dependence of the magnetic viscosity coefficients, which are extracted by

the decay slopes of the linear quasilogarithmic magnetic aftereffect curves.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

The relaxation time t is given by 1=f as:

Time dependence of magnetization in ferromagnetic materials,

known as magnetic viscosity or magnetic aftereffect, is of great

practical and theoretical significance. Following the first observation

of the magnetic viscosity by Preisach [1], a physical mechanism

of thermally-activated reversal processes of magnetic

domains was developed by Street and Woolley [2] in order to

interpret the magnetic viscosity. Accordingly, the magnetic

hysteresis was explained as a system of domains and domain

walls caused by a complicated free energy landscape, with many

relative minima corresponding to metastable states. The energy

barriers between the metastable energy minima can be overcome

by thermal agitation, which causes an irreversible reversal process

with magnetization relaxing to the thermodynamic equilibrium

state and a consequent decay of magnetization with time.

The transition frequency of the magnetic moment in this relaxation

process is usually estimated by the Arrhenius–Néel statistical

switching model [3]:

f ¼ f 0 expð E B =k B TÞ ð1Þ

where f 0 is a frequency constant of the order of 1010–1012 Hz, k B

the Boltzmann constant, T the absolute temperature and E B the

energy barrier depending on the external field and temperature.

n Corresponding author.

E-mail address: wliu@imr.ac.cn (W. Liu).


t ¼ f 0 expðE B=k B TÞ ð2Þ

which characterizes the mean time needed to cross a local energy

barrier to another energy state. Obviously, the smaller the

relaxation time t and the higher the transition frequency f , the

higher the rate of the magnetic relaxation.

In most permanent magnetic materials, the magnetization

relaxes very slowly towards its ground state in presence of large

anisotropy energy barriers. Therefore, a constant reverse field

(called holding field H 0 ) is usually applied to a magnetically

saturated sample to accelerate the magnetic viscosity testing in

the hard magnets, via lowering the effective anisotropy energy

barriers. Over a limited range of time and assuming a flat energy

barrier distribution, the experimental data of the magnetization

MðtÞ can be described by a logarithmic time dependence [4,5]

MðtÞ¼M 0 SlnðtÞ ð3Þ

where S is the magnetic viscosity coefficient determining the rate

of the magnetic relaxation and M 0 is the magnetization of the

sample at the start of the viscosity measurement (t ¼ 0). To

prevent the lnðtÞ from going to infinity, a reference time t 0 is

usually introduced to establish a lnðt þt 0 Þ dependence of M:

MðtÞ¼M 0 Slnðt þt 0 Þ ð4Þ

The magnetization decay with time due to the magnetic

viscosity becomes pronounced in nanostructured magnetic

systems [6–7]. Whether in particulate or continuous films, the

thermal loss effect of magnetization is manifested by the

0304-8853/$ - see front matter & 2012 Elsevier B.V. All rights reserved.


Q. Yao et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2854–2857 2855

reduction in magnetic switching unit size. Also the magnetization

should become strongly time-dependent as the grain size and the

activation volume largely decrease in nanocrystalline permanent

magnetic thin films [8–11]. However, there are relatively few

studies for temperature dependence of magnetic aftereffect in

perpendicularly anisotropic magnetic films, particularly at very

low temperature.

In the present work, the time dependent magnetization is

measured over a range of temperatures (300 K–4.2 K) for the

nanocrystalline Nd–Fe–B films with perpendicular anisotropy.

The quasilogarithmic magnetic relaxation is observed and studied

for each of the samples at all temperatures. This is followed by a

discussion of the nonmonotonic temperature dependence of

magnetic viscosity coefficients for these perpendicularly anisotropic

Nd–Fe–B films.

2. Experimental details

Mo(50 nm)/Nd–Fe–B (x)/Mo(50 nm) films (x¼300 nm, 500 nm,

800 nm, and 6.5 mm) were deposited onto heated (100) oriented Si

substrates in a high vacuum chamber using dc magnetron sputtering

[12]. The Mo overlayer was deposited for corrosion protection

and the Mo underlayer was added to resist the oxidation by

the oxygen atoms diffusing from the SiO 2 surface of the Si

substrate. The chamber was evacuated to a background pressure

better than 1:5 10 5 Pa and all depositions were made at a

constant total Ar gas pressure of 0.7 Pa. A sintered Nd 16 Fe 71 B 13

alloy target and a commercial Mo target of higher than 99.9%

purity were used. The deposition rate of 9.5 Å/s was adopted

during sputtering, calibrated by weighing the films. Pre-sputtering

of the NdFeB target was carried out for 30 min to remove any

oxidized layer on the target surface. During deposition, the

temperature of the substrate was fixed at 923 K.

After deposition, the crystallographic structures and the

perpendicular alignment of the films were analyzed by means of

X-ray diffraction (XRD) using Cu Ka radiation [12]. After saturation

with applying fields H up to 90 kOe, the time-dependent magnetization

MðtÞ of the Nd–Fe–B films were recorded over 1800 s at

300 K–4.2 K in constant reverse fields H 0 equal to the coercivities

H c , using a Quantum Design PPMS-9H physical properties measurement

system (PPMS). The H 0 was applied to maximize the

relaxation rates and the viscosity coefficients. Before the MðtÞ

measurements at all temperatures, H c of the films were determined

by the same PPMS as the fields reducing M to zero on the

demagnetization curves with H up to 90 kOe. All magnetic

measurements were performed by applying H perpendicular to

the film plane and with the diamagnetic signals from the Si

substrates being subtracted.

3. Results and discussion

Fig. 1 shows that the coercivities of the anisotropic Nd–Fe–B

films increase monotonically as the temperature decreases from

300 K to 4.2 K. This is due to the increased anisotropy field of

Nd 2 Fe 14 B phase, despite its spin-reorientation at 135 K [13]. At

300 K, the coercivities of the anisotropic Nd–Fe–B films increase

from 4.2 kOe with 300 nm thickness to 11.4 kOe with 6.5 mm

thickness. Microstructure and magnetic characterizations justify

the films’ strong out-of-plane magnetic anisotropy with the easy

axes of the columnar grains normal to the film plane [12]. With

increasing the film thickness from 300 nm to 6.5 mm, the Nd–Fe–

B films’ perpendicular alignment is weakened gradually.

With H c determined, magnetic aftereffects were measured

for all the films over 1800 s at various temperatures under the

Fig. 1. Temperature dependence of coercivities for the Nd–Fe–B films with film

thickness x¼300 nm, 500 nm, 800 nm and 6.5 mm.

constant reverse fields H 0 ( ¼ H c ), by following the procedure

described in the experimental section. A typical set of these

measurements is given in Fig. 2(a), where the magnetization is

plotted against time for the 6.5 mm anisotropic Nd–Fe–B film at

300 K–4.2 K. It is seen from Fig. 2(a) that the film magnetization

decreases as relaxation progresses at all temperatures, with the

relaxation rates being distinctive at different temperatures. However,

the Nd–Fe–B film magnetization decay is not a complete

linear dependence on the logarithm of time (lnðtÞ) as expressed by

Eq. (3). For example, Fig. 2(b) exhibits the magnetization as a

function of log-time for the 6.5 mm Nd–Fe–B film at 300 K–4.2 K.

It is demonstrated that the film magnetization decays linearly on

the log-time aftereffect curves only in a limited time interval

(around 20–1800 s) at all temperatures. From 0 s to 20 s, the

magnetization relaxes nonlinearly with lnðtÞ, indicating the

nonexponential characteristic of the magnetic viscosity for the

Nd–Fe–B film. Furthermore, upon introducing a reference time

constant (t 0 ¼ 10 s) in log-time, the time behavior of the Nd–Fe–B

film magnetization can be perfectly fitted to Eq. (4) which

establishes a linear dependence of magnetization on lnðt þt 0 Þ.

As shown in Fig. 2(c), the 6.5 mm Nd–Fe–B film magnetization

decreases linearly with lnðtþ10½sŠÞ at 300 K–4.2 K. It may be that

the PPMS records the magnetic decay a short time after the

relaxation starts with H 0 being applied, causing the small negative

M 0 ( 29 to 16 Gs). The M 0 of 12 Gs for the 135 K relaxation

needs to be further investigated. Under the holding reverse field

of H 0 ( ¼ H c ), the magnetic relaxation of the perpendicularly

anisotropic film starts from a energy minimum state of nearly

zero net magnetization, with one spin up for every spin down

which is aligned perpendicular to the film plane. Thereafter,

thermal activation and H 0 jointly drive the film magnetization

to gradually decay with time over local energy barriers. Due to the

fluctuations of the local magnetocrystalline anisotropy or the

pinning of the domain-wall motion, the disorder of energy barrier

heights creates a dispersion of relaxation times for activation

grains in the anisotropic Nd–Fe–B film. This relaxation process

causes the M to deviate from a linear dependence on lnðtÞ, giving

rise to the quasilogarithmic or the stretched exponential

magnetic viscous effects. Accordingly, the magnetic viscosity

coefficients S in the coercive fields are extracted from the slopes

of these linear quasilogarithmic aftereffect curves to characterize

the magnetic relaxation rates at various temperatures.

Another feature appearing in these anisotropic Nd–Fe–B films

is the nonmonotonic temperature dependence of the magnetic

viscosity coefficients S, which has also been detected in CoCr films


Q. Yao et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2854–2857

Fig. 2. (a) time t, (b) log-time ln(t) and (c) log-time ln(tþt 0 ) dependence of magnetization for the 6.5 mm Nd–Fe–B film at 300 K–4.2 K.

Fig. 3. Temperature dependence of the magnetization viscosity coefficients S for

the Nd–Fe–B films with film thickness x¼(a) 300 nm, (b) 500 nm, (c) 800 nm and

(d) 6.5 mm.

Fig. 4. Temperature dependence of the initial magnetic decay rates dM/dt at t¼1 s

for the Nd–Fe–B films with film thickness x¼(a) 300 nm, (b) 500 nm, (c) 800 nm

and (d) 6.5 mm.

before [14–16]. Fig. 3 summarizes the S as a function of temperature

from 300 K to 4.2 K for the anisotropic Nd–Fe–B films. It is

astonishing that the films with thicknesses of 300 nm, 500 nm

and 6.5 mm pass through maxima in the temperature dependence

of S. Only for the 800 nm film, S increases monotonically but

nonlinearly with temperature. The temperatures at which S reach

maxima, are 200 K for the 300 nm film and 135 K for both the

500 nm and the 6.5 mm films. The Arrhenius–Néel law [3]

(expressed in Eqs. (1) and (2)) indicates that, both weakened

thermal agitation and usually strengthened anisotropy energy

barriers are supposed to decrease transition frequency f and

prolong relaxation time t with decreasing temperature. Accordingly,

the peak in the decay slope of the aftereffect curve is always

considered anomalous since one would expect any thermally

activated process to be decelerated at lower temperatures.

The origin of the nonmonotonic temperature dependence of

the magnetic viscosity coefficients S may be interpreted by the

following simple arguments [15–16]. At low temperatures, there

is not sufficient thermal activation for the magnetizations to

fluctuate and hop over the energy barriers. This leads to an

extremely low relaxation rate at 4.2 K. With elevating the temperature

to enhance the thermal energy, the magnetic decay rate

reasonably increases due to the increased transition frequency and

the shortened relaxation time. Furthermore, at sufficiently high

temperature with very large transition frequency, the relaxation is

so rapid that the S determined by fitting the expression of

quasilogarithmic magnetic viscosity actually reflects the slow

relaxation rate after the initial decay of most magnetizations.

Therefore, the decay slope on the M vs. lnðtþt 0 Þ linear curve is

reduced. One can notice in Fig. 2(a) that, for all the temperatures,

the magnetization of the anisotropic Nd–Fe–B films initially

decays with much higher speed than the following slow relaxation.

Fig. 4 plots the temperature dependence of the initial decay

rates (dM=dt) at t¼1 s for the anisotropic Nd–Fe–B films. The

samples’ initial magnetic decay clearly demonstrate a different

temperature behavior from the magnetic viscosity coefficients S,

with the initial relaxation rates monotonically increasing with


Q. Yao et al. / Journal of Magnetism and Magnetic Materials 324 (2012) 2854–2857 2857

4. Conclusions

In summary, the magnetic viscous effects in coercive fields

have been studied for the perpendicularly anisotropic Nd–Fe–B

films over a range of temperatures (300 K–4.2 K). The films’

magnetic relaxation is found to be quasilogarithmic over the

measuring period of 1800 s at all the temperatures. This is

attributed to the presence of the energy barrier disorder in the

films which creates a broad distribution of relaxation times.

Linearly fitting the quasilogarithmic decay curves generates the

magnetic viscosity coefficients S which change non-monotonously

with temperature. The reason for the nonmonotonic

temperature behavior of S is shown by the argument that the

quasilogarithmic decay slopes are defined over a time window

where most magnetization has initially relaxed.


This work has been supported by the National Basic Research

Program (No. 2010CB934603) of China, Ministry of Science and

Technology of China and the National Nature Science Foundation

of China under projects 50931006 and 50971123. One of the

authors (Q. Yao) would like to thank ÖAD for financial support

within the China–Austria cooperation program.


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