Vol. 39 No. 5 2015 年 9 月 1 日発行 頒価 ¥2,000 （税込）送料実費

ISSN 2432-0250

Vol.43 No.2
JournalJournal of the Magnetics Society of Japan
Electronic Journal URL: https://www.jstage.jst.go.jp/browse/msjmag

2019

Thin Films, Fine Particles, Multilayers, Superlattices
Fabrication of Noncentrosymmetric Nb/V/Ta Superlattice and its Superconductivity
発行（公社）日本磁気学会 〠101‒0052 東京都千代田区神田小川町2丁目8番地
三井住友海上小川町ビル5階☎(03)5281-0106
印刷（株）国 際 文 献 社 〠169‒0075 東京都新宿区高田馬場3の8の8☎(03)3362-9741

F. Ando, D. Kan, Y. Shiota, T. Moriyama, Y. Shimakawa, and T. Ono…17

Structure and Magnetism of c-Plane Oriented Mn50(Te50-xSbx) Epitaxial Thin Films with NiAs-Type Structure
Y. Ashizawa, S. Saito, M. Tsunoda, and M. Takahashi…21

Near-Infrared Magneto-Refractive Eﬀect for Antiferro-Magnetically Exchange Coupled Co/Ru Multilayer
Film in Transmission Conﬁguration
S. Saito, H. Sato, K. Ooki, K. Akahane, and H. Uchida…25

Anomalous Hall Eﬀect and Anisotropic Magnetoresistance in Perpendicularly Magnetized FePt1-xPdx Films
T. Seki, S. Kikushima, and K. Takanashi…29

Biomagnetism / Medical Applications
Dynamics of Magnetization and Easy Axis of Individual Ferromagnetic Nanoparticle Subject to Anisotropy and
Thermal Fluctuations
S. Ota and Y. Takemura

広告◇計測エンジニアリングシステム株式会社◇東英工業株式会社

…34

43

HP: http://www.magnetics.jp/ e-mail: msj@bj.wakwak.com
Electronic Journal: http://www.jstage.jst.go.jp/browse/msjmag

2

2019

INDEX

Journal of the Magnetics Society of Japan
Vol. 43, No. 2
Electronic Journal URL: https://www.jstage.jst.go.jp/browse/msjmag

CONTENTS
Thin Films, Fine Particles, Multilayers, Superlattices
Fabrication of Noncentrosymmetric Nb/V/Ta Superlattice and its Superconductivity……………………………… F. Ando, D. Kan, Y. Shiota, T. Moriyama, Y. Shimakawa, and T. Ono

17

Structure and Magnetism of c-Plane Oriented Mn50(Te50-xSbx) Epitaxial Thin Films with NiAs-Type
Structure………………………………………………… Y. Ashizawa, S. Saito, M. Tsunoda, and M. Takahashi

21

Near-Infrared Magneto-Refractive Effect for Antiferro-Magnetically Exchange Coupled Co/Ru Multilayer
Film in Transmission Configuration…………………………………………………S. Saito, H. Sato, K. Ooki, K. Akahane, and H. Uchida

25

Anomalous Hall Effect and Anisotropic Magnetoresistance in Perpendicularly Magnetized FePt1-xPdx
Films…………………………………………………………………T. Seki, S. Kikushima, and K. Takanashi

29

Biomagnetism / Medical Applications
Dynamics of Magnetization and Easy Axis of Individual Ferromagnetic Nanoparticle Subject to Anisotropy
and Thermal Fluctuations…………………………………………………………………………………… S. Ota and Y. Takemura

Board of Directors of The Magnetics Society of Japan
President:

K. Takanashi

Vice Presidents:

K. Nakagawa, S. Nakamura

Directors, General Affairs:

Y. Miyamoto, K. Niiduma

Directors, Treasurer:

K. Aoshima, K. Ishiyama

Directors, Planning:

Y. Saito, S. Nakagawa

Directors, Editorial:

K. Kobayashi, T. Ono

Directors, Public Relations:

H. Itoh, S. Greaves

Directors, International Affairs: Y. Takemura, M. Nakano
Auditors:

Y. Suzuki, R. Nakatani

34

INDEX
J. Magn. Soc. Jpn., 43, 17-20 (2019)
LetterFabrication of noncentrosymmetric Nb/V/Ta superlattice and its
superconductivity
F. Ando1, D. Kan1, Y. Shiota1, T. Moriyama1, Y. Shimakawa1,2, and T. Ono1,3

Gokasho Uji, Kyoto 611-0011, Japan
Gokasho Uji, Kyoto 611-0011, Japan
3Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University, 1-3
Machikaneyama-cho Toyonaka, Osaka 560-8531, Japan
1Institute for Chemical Research, Kyoto University,

2Integrated Research Consortium on Chemical Sciences, Kyoto University,

Nb, V, and Ta are all well-known superconducting elements in a simple substance. We report thesuperconductivity of Nb/V/Ta superlattices sputtered onto MgO(100) substrates, in which global inversion symmetry
was broken along the stacking direction. It was found that the superlattices had long-range crystalline coherence withthe well-defined periodicity of the constituent layers, and they exhibited superconducting transition despite thethicknesses of the layers.
Keywords: inversion symmetry breaking, artificial superlattice, epitaxial film, X-ray 2θ/θ measurement,
superconductivity
1. Introduction

2. Experimental Procedure

In recent years, spin-orbit interaction (SOI) has been
attracted intense attentions in the various fields of solidstate physics, such as spintronics1)–3), topological matter4),
and unconventional superconductivity5). In the presenceof inversion symmetry breaking, SOI entangles the spinand orbit degrees of freedom in the electron system andenables the mutual control of electronic and spin
properties. In case of superconductors, the SOI withbroken inversion symmetry significantly forbids theconventional classification of Cooper pairs5). Instead, a
new pairing symmetry, i.e., a mixed spin singlet-triplet
state, is expected to be realized6),7). There have been
severalreports
ofthenoncentrosymmetricsuperconductors where the crystal structure lacks acenter of inversion, such as CePt3Si8), CeRhSi39),10) , andUIr11).
Recently, Rashba effect on two-dimensional (2D)superconductivity was studied using d-wave heavy
fermion superconductor CeCoIn5 sandwiched by two
different nonmagnetic metals12). This previous work
reveals that Rashba effect controlled in the tricolor
structure exerts profound changes in the 2D
superconducting properties as theoretically pointed
out13),14). The finding naturally leads us to an expectationthat an exotic superconductivity can be explored in 3D
artificially engineered superlattice using three kinds ofsuperconductors A, B, and C. As shown in Fig. 1(a), global
inversion symmetry is broken in ABC-type superlattice
and we can easily tune Rashba effect yielded byasymmetric potential gradient 𝛻𝛻𝛻𝛻.
In this study, we at first investigate the growth andconductivity of ABC-type superlattices as a new platform
to investigate an exotic superconductivity. We choose Nb,
V, and Ta as the constituent layers, which are allcommonly-used superconducting elements and havebody-centered cubic (bcc) lattice structure15)–19).

The film deposition was carried out by d.c. magnetron
sputtering in an high vacuum system at a base pressureof 1.0×10–5 Pa or better. Ar was used as sputter gas. Prior
to film growth, the MgO(100) substrate was washed withacetone and heated at 600 °C for 30 minutes in the
sputtering chamber to drive off impurities. Figure 1(b)shows a design of the superlattice structure. Nb, V, andTa layers of the thickness t were repetitively sputteredonto the MgO substrate. The deposition rates were kept
constant at 0.35, 0.21, and 0.44 Å/s for Nb, V, and Ta,
respectively. The MgO substrate was heated at 750 °C
during the deposition as well as the previous report17).
We changed the thickness t from 1.0 nm to 5.0 nm while
fixing the total thickness to be 60 nm. We can expect thatRashba effect on the superconductivity can be controlled
by tuning the thickness t. A 3 nm thick Pt layer depositedat room temperature prevents the oxidation.
To confirm the epitaxial growth of Nb/V/Ta
superlattices, reflection high-energy electron diffraction(RHEED) patterns were observed in-situ before and after
the deposition of the superlattices. The electron beam
was injected along the MgO[100] and MgO[110]azimuthal directions. Figure 1(b) shows the RHEED
patterns of MgO substrate (1st) and Nb/V/Ta superlattice
for t = 1.0 nm (2nd). The observed patterns indicate an
epitaxial growth of the bcc-Nb/V/Ta superlattice with
Nb/V/Ta(100) on the MgO(100) substrate where
Nb/V/Ta[100]‖MgO[100]20),21). The in-plane lattice
constant of the top Ta surface is estimated to be 3.2 Åfrom the RHEED streak distance22). This smaller lattice
constant compared with that of bulk bcc-Ta (3.30 Å) canbe accounted for by a lattice mismatch with theneighboring V layer. The similar RHEED patterns wereobserved for other films (t = 2.0 and 5.0 nm, [Nb (2.0)/V(2.0)/Ta (2.0)]×10 and [Nb (5.0)/V (5.0)/Ta (5.0)]×4).

Journal of the Magnetics Society of Japan Vol.43, No.2, 201917

INDEX

Fig. 1(a) Schematic design of ABC-type superlattice. Red arrow represents asymmetric potential gradient 𝛻𝛻𝛻𝛻due to global inversion symmetry breaking. (b) Film structure sputtered onto MgO (100) substrates. Nb,
V, and Ta were chosen as constituent elements for ABC-type superlattice. RHEED streak patterns wereobserved before and after deposition of [Nb (1.0)/V (1.0)/Ta (1.0)]20 superlattice. Electron beam was
injected along the MgO[100] and MgO[110] azimuthal directions.

Fig. 2

X-ray 2θ/θ measurements for [Nb (1.0)/V (1.0)/Ta (1.0)]20/Pt (3.0). (a) Measured (red) and calculated (blue)
X-ray reflection curves identify interfacial morphology of superlattice. Two distinct fringes resulting
from interference between periodic interfaces indicate highly sharp interfaces with almost no mixture.
(b) X-ray diffraction scan was performed to identify crystalline structure of superlattice. Two sharp
peaks corresponding to MgO(200) and Nb(200) [or Ta(200)] strongly indicate that superlattice was
epitaxially grown with bcc lattice structure on MgO(100) substrate.

3. Results and Discussions
To investigate the interfacial morphology andcrystalline structure of the superlattice, we performed Xray 2θ/θ measurements23) using Philips X'Pert MRD
diffractometer with monochromatic CuKα1 radiation.
Figure 2(a) shows the measured and calculated X-ray
reflection curves of the superlattice (t = 1.0 nm). The
measured curve shows two noticeable Bragg peaks andKiessig fringes24). From the series of the two peak-angles,
it is directly found that the film has a characteristic
length of 2.84 nm. This means that the 2.8-nm-thick
Nb/V/Ta multi-layers are periodically deposited withhighly sharp interfaces, which is a desirable construction
to enhance Rashba effect. The measured curve was fitted
as a function of thicknesses and roughness of Nb, V, andTa layers using a Philips X'Pert reflectivity program. The
calculated curve clearly reproduces two Bragg peaks and18

Kiessig fringes of the measured one. The calculated
thicknesses of Nb, V, and Ta layers are 0.9, 1.0, and 0.9
nm, respectively, which almost agree with the original
design of the superlattice. Mean values of the interfacialroughness obtained from the fit were equal to 0.4-0.9 nm(2-4 atomic layers).
Secondly, we performed the X-ray diffraction (XRD)
scan for the same film (t = 1.0 nm). Figure 2(b) shows theobtained result of XRD scan. The expected angles of thepeaks are calculated using lattice constants listed in theAmerican Institute of Physics Handbook22). A distinct
peak around 56° can be seen in the measured XRD curve,
which corresponds to the expected one from bcc-Nb(200)or bcc-Ta(200). Together with the RHEED 2nd in Fig. 1(b),the XRD curve ensures the epitaxial growth of thedeposited films with bcc lattice structure.
Finally, we conducted electric transport measurements
to characterize the electrical conductivity at low

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX

properties7),10),27)–30).
Acknowledgements This work was supported by JSPS
KAKENHI Grant Numbers 15H05702, 16H05977,
26103002, the Collaborative Research Program of theInstitute for Chemical Research, Kyoto University, andthe Cooperative Research Project Program of the
Research Institute of Electrical Communication, Tohoku
University.
References

Fig. 3Temperature dependence of electric
resistivity in Nb/V/Ta superlattices (t = 1.0,
2.0, and 5.0 nm) and 120-nm-thick Nb, V,Ta single-layer films. Inset shows
photomicrograph of processed film alongwith measurement configuration. All[Nb/V/Ta] superlattices exhibited abrupt
drop in resistivity.

temperature. In order to enhance the 3D nature25),26), we
fabricated the superlattices increasing the total
thickness from 60 nm to 120 nm. Note that there isalmost no difference in RHEED patterns by increasing
the total thickness. The inset of Fig. 3(a) shows aphotomicrograph of the processed film with themeasurement configuration. The films were patterned
into a 50-μm-wide wire structure by a conventional
photolithography and Ar ion milling process. Then, Cr (5nm)/Au (100 nm) electrodes were formed on the films
using a resistance heating evaporation system and a liftoff process. Four-terminal measurements were
performed to probe the temperature dependence of the
sheet resistivity using the Physical Property
Measurement System (PPMS-9T, Quantum Design).
Figure 3(a) shows the measurement results for thesuperlattices (t = 1.0, 2.0, 5.0 nm) with those for 120-nmthick Nb, V, Ta single layer films. The superlattices
exhibit the superconducting transitions, whose
temperatures TC are comparable to those of Nb, V, Ta
single layer films. However, the t dependence of the TC issomehow strange because TC among the superlattice (t =2.0 nm) is the highest of the three films. We need further
study about the physics behind the superconductivity.
4. Conclusion
We study the growth and superconductivity of Nb/V/Ta
superlattices sputtered onto MgO(100) substrates. The
RHEED patterns and X-ray 2θ/θ measurements revealthat the superlattices have long-range crystalline
coherence and the well-defined periodicity of theconstituent layers. Moreover, the superlattices exhibitedsuperconducting transition comparable to the Nb, V, andTa single layer films. The ABC-type noncentrosymmetricsuperlattice presented in this work will be a newplatformforsearchingexoticsuperconductive

1) S. Murakami, N. Nagaosa, and S. Zhang, Science 301, 1348
(2003).
2) J. Sinova, D. Culcer, Q. Niu, N.A. Sinitsyn, T. Jungwirth,and A.H. Macdonald, Phys. Rev. Lett. 92, 126603 (2004).
3) A. Fert, V. Cros, and J. Sampaio, Nat. Nanotechnol. 8, 152(2013).
4) X. Qi and S. Zhang, Rev. Mod. Phys. 83, 1057 (2011).
5) E. Bauer and M. Sigrist, Non-CentrosymmetricSuperconductors: Introduction and Overview(Springer-Verlag Berlin Heidelberg, 2012).
6) L.P. Gor’kov and E.I. Rashba, Phys. Rev. Lett. 87, 37004(2001).
7) P. Frigeri, D.F. Agterberg, A. Koga, and M. Sigrist, Phys.
Rev. Lett. 92, 097001 (2004).
8) E. Bauer, G. Hilscher, H. Michor, C. Paul, E.W. Scheidt, A.
Gribanov, Y. Seropegin, H. Noël, M. Sigrist, and P. Rogl,
Phys. Rev. Lett. 92, 027003 (2004).
9) N. Kimura, K. Ito, K. Saitoh, Y. Umeda, H. Aoki, and T.
Terashima, Phys. Rev. Lett. 95, 247004 (2005).
10) N. Kimura, K. Ito, H. Aoki, S. Uji, and T. Terashima, Phys.
Rev. Lett. 98, 197001 (2007).
11) T. Akazawa, H. Hidaka, T. Fujiwara, T.C. Kobayashi, E.
Yamamoto, Y. Haga, R. Settai, and Y. Onuki, J. Phys.
Condens. Matter 16, L29 (2004).
12) M. Naritsuka, T. Ishii, S. Miyake, Y. Tokiwa, R. Toda, M.
Shimozawa, T. Terashima, T. Shibauchi, Y. Matsuda, and
Y. Kasahara, Phys. Rev. B 96, 174512 (2017).
13) S. Fujimoto, J. Phys. Soc. Jpn. 76, 051008 (2007).
14) D. Maruyama and Y. Yanase, J. Phys. Soc. Jpn. 84,
074702 (2015).
15) B.Y. Jin and J.B. Ketterson, Adv. Phys. 38, 189 (1989).
16) J.R. Rairden and C.A. Neugebauer, Proc. IEEE 52, 1234
(1964).
17) S.M. Durbin, J.E. Cunningham, M.E. Mochel, and C.P.
Flynn, J. Phys. F Met. Phys. 11, L223 (1981).
18) P.R. Broussard and T.H. Geballe, Phys. Rev. B 35, 1664
(1987).
19) M. Gutsche, H. Kraus, J. Jochum, B. Kemmather, and G.
Gutekunst, Thin Solid Films 248, 18 (1994).
20) D.B. Beringer, W.M. Roach, C. Clavero, C.E. Reece, and
R.A. Lukaszew, Phys. Rev. ST Accel. Beams 16, 022001(2013).
21) G. Oya, M. Koishi, Y. Sawada, G. Oya, M. Koishi, and Y.
Sawada, J. Appl. Phys. 60, 1440 (1986).
22) D.E. Gray, American Institute of Physics Handbook(McGraw-Hill Book Company, Inc., New York, 1957).

Journal of the Magnetics Society of Japan Vol.43, No.2, 201919

INDEX

23) W.J. Bartels, J. Hornstra, and D.J.W. Lobeek, in Acta
Crystallogr. Sect. A (1986), pp. 409–588.
24) H. Zabel, Appl. Phys. A 58, 159 (1994).
25) J.I. Kodama, M. Itoh, and H. Hirai, J. Appl. Phys. 54, 4050
(1983).
26) A.I. Gubin, K.S. Il’in, S.A. Vitusevich, M. Siegel, and N.
Klein, Phys. Rev. B 72, 064503 (2005).
27) V.P. Mineev, Low Temp. Phys. 37, 1092 (2011).

2028) C. Lu and S. Yip, Phys. Rev. B 78, 132502 (2008).
29) Y. Tanaka, T. Yokoyama, A. V Balatsky, and N. Nagaosa,Phys. Rev. B 79, 060505(R) (2009).
30) M. Sato and S. Fujimoto, Phys. Rev. Lett. 105, 217001(2010).
Received Nov. 13, 2018; Accepted Jan. 21, 2019

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX
J. Magn. Soc. Jpn., 43, 21-24 (2019)<Paper>

Structure and magnetism of c-plane oriented Mn50(Te50-xSbx) epitaxial
thin films with NiAs-type structure
Y. Ashizawa*, S. Saito, M. Tsunoda, and M. Takahashi

Dept. of Electronic Engineering, Graduate School of Engineering, Tohoku University, 6-6-05, Aoba-yama, Sendai 980-8579, Japan*Present address: College of Science and Technology, Nihon University, 7-24-1 Narashinodai, Funabashi, Chiba 274-8501, Japan

Structure, magnetism, and their correlation for NiAs-type Mn50(Te50-xSbx) pseudo-single crystal films were
investigated by changing the nearest Mn-Mn distance by substituting Te for Sb. The main results were as follows.
First, c-plane oriented Mn50(Te50-xSbx) pseudo-single crystal films were prepared in ranges of Sb composition, x, from0 to 16 and from 33 to 51 by employing a c-plane oriented NiAs-type MnTe under-layer. Second, lattice parameter aof the Mn50(Te50-xSbx) films was nearly constant at about 4.17 Å for 0 ≤ x ≤ 38 and reduced gradually down to 4.13 Åfor x ≥ 38 as x increased. Lattice parameter c decreased with a gentle slope from 6.68 to 6.61 Å for 0 ≤ x ≤ 16 andreduced gradually down to 5.78 Å for x ≥ 23. As a result, dMn-Mn changed from 3.34 Å to 2.89 Å. Third, the saturation
magnetic moments mS at 5 K linearly increased with a different slope as x increased for 0 ≤ x ≤ 11 and 33 ≤ x ≤ 51.
Finally, the coefficient of T 3/2 in Bloch’s law A decreased rapidly for x ≤ 16 as x increased, and it increased graduallyfor 16 ≤ x ≤ 51 as x decreased.
Keywords: NiAs-type structure, Mn-Mn distance, Epitaxial growth, MnTe, MnSb
1. Introduction
The Mn50(Te50-xSbx) ternary system with an
NiAs-type hexagonal structure, which is a double
hexagonal close-packed structure ordered with cation
and anion atoms, is interesting in terms of themagnetism and structure of conductive Mn-compounds.
NiAs-type MnSb with lattice parameters, a = 4.128 Åand c = 5.789 Å1), is a ferromagnetic substance whose
magnetic easy axes <11.0> lie in the c-plane1-3) asshown in Fig. 1 (a), while NiAs-type MnTe is an
antiferromagnetic substance whose spins alignferromagnetically and parallel to the c-plane in eachatomic plane of Mn and are antiferromagneticallycoupled with the adjacent atomic plane4) as shown in
Fig. 1 (b). The a of MnTe, which is equal to 4.142 Å, isvery close to that of MnSb, and the c of MnTe, which isequal to 6.711 Å, is far from that of the MnSb5,6).
Namely, the c-plane distance, corresponding to twice
the nearest Mn-Mn distance, dMn-Mn, seems to decide themagnetism of NiAs-type Mn-compounds. It is thus
expected to control the magnetism of these compounds
by controlling the c-plane distance in theMn50(Te50-xSbx) ternary system.
In this study, we thus examined the structure,magnetism, and their correlation for NiAs-type
Mn50(Te50-xSbx) pseudo-single crystal films with Te
substituted for Sb.
2. Experimental procedureMn50(Te50-xSbx) pseudo-single crystal films withorientation were fabricated by using
simultaneous DC sputtering with Mn, Te, and Sb
targets on a Si(111) single crystal substrate. The
sample compositions of the ternary Mn50(Te50-xSbx)
films are summarized in Table 1. The substrate wasc-planechemically cleaned and dry etched by Ar-ion
bombardment before the film deposition. The thicknessof the Mn50(Te50-xSbx) films was constant at 100 nm. A
B20-type MnSi epitaxial layer was used as a template
for the epitaxial growth of the following MnTe
under-layer and Mn50(Te50-xSbx) films. The MnSi
template layer was formed by a natural reaction

Fig. 1 Spin alignments of (a) NiAs-type MnSb, which isferromagnetic substance, and (b) NiAs-type MnTe, which isantiferromagnetic substance.
Table 1
Compositions of c-plane oriented
NiAs-type Mn50(Te50-xSbx) pseudo-single crystal
films fabricated by DC magnetron sputtering

CMn

CTe

CSb

SampleComposition(at.%)(at.%)(at.%)

0
5
11
16
23
29
33
38
50
51

50
49
48
51
50
50
50
49
49
49

50
46
41
33
27
21
17
13
1
0

0
5
11
16
23
29
33
38
50
51

x

Journal of the Magnetics Society of Japan Vol.43, No.2, 201921

INDEX

between the Si substrate and a 1-nm-thick Mn layer
with an elevated substrate temperature of 623 K3,7). As
an under-layer, a c-plane oriented 50-nm-thick MnTe
layer was formed on the MnSi template layer at 623 K
byhetero-epitaxy.
Ontheunder-layer,theMn50(Te50-xSbx) films were grown epitaxially at 523–623
K with a deposition rate of about 0.1 nm/s.
ThechemicalcompositionoftheternaryMn50(Te50-xSbx) films was determined in X-ray
fluorescence analyses with Mn-Kα, Sb-Lα, and Te-Lαlines. The crystal structures of the films were
investigated by using two kinds of X-ray diffraction(XRD) methods; one was a conventional out-of-plane
method of Bragg-Brentano geometry, and the other was
an in-plane method of Gracing incidence geometry8,9)with a Cu-Kα line. The magnetic properties were
measured by a vibrating sample magnetometer at room
temperature (R.T.) and a superconducting quantum
interference
devicemagnetometeratvarioustemperatures between 5 K and R.T.

and dMn-Mn from x = 16 to 23 imply that the NiAs-type
Mn50(Te50-xSbx) single phase for 16 ≤ x ≤ 23 was notstable.
3.2 Magnetic properties of NiAs-type Mn50(Te50-xSbx)
pseudo-single crystal films
Fig. 4 shows saturation magnetic moments per Mnatom, mS, for the NiAs-type Mn50(Te50-xSbx) epitaxial
films as a function of x. The mS values were derived
from the saturation magnetization of the films at 5 K.
mS can be observed even for x = 5. This shows that mS
was proportional to x with a gentle slope for 0 ≤ x ≤ 16.
With an increase in x from 33 to 51, mS also showed alinear rise up to 3.23 B/Mn, but the slope for 33 ≤ x ≤51 was steeper than that for 0 ≤ x ≤ 16.
To estimate the magnetic transition temperatures of
the NiAs-type Mn50(Te50-xSbx) epitaxial films, thecoefficient of T 3/2, A, in Bloch’s T 3/2 law was deduced inanalyses of the temperature dependence of saturation

3. Results

22

3.4
3.2
3.0
2.8

7.0
6.5
6.0
5.5

4.20

c (Å)

dMn-Mn (Å)

Fig. 2Out-of-plane XRD profiles for ternary
Mn50(Te50-xSbx) films deposited on c-plane oriented
MnTe epitaxial under-layer as function of Sb
composition, x.

a (Å)

3.1 Structural properties of ternary Mn50(Te50-xSbx)
films
Fig. 2 shows XRD profiles of out-of-planemeasurement for the ternary Mn50(Te50-xSbx) films as afunction of Sb composition, x. The diffraction lines
observed at around 2θ = 55° originated from the c-planeof the NiAs-type MnTe under-layers. For 0 < x ≤ 16,these diffraction lines had shoulders on the higher
angle side, as indicated by the small arrows in Fig. 2.
These shoulders were identified as the diffraction fromthe c-plane of the NiAs-type Mn50(Te50-xSbx) layer, andthe diffraction line shifted to a higher angle with theincrease in Sb composition, x. For the sample of x = 23,
another diffraction line suddenly appeared at around62°. For x ≥ 23, this diffraction line also shifted to ahigher angle with the increase in x and finally reached
the angle for the c-plane of the NiAs-type MnSb phase.
Namely, the c-plane oriented NiAs-type Mn50(Te50-xSbx)
films with a gradual change in lattice constant, c, were
fabricated in the respective composition ranges of 0 ≤ x≤ 16 and 23 ≤ x ≤ 51. The Mn50(Te50-xSbx) films withc-plane orientation were grown epitaxially, which was
confirmed by using the in-plane XRD method.
Fig. 3 shows lattice parameters a and c and dMn-Mn for
the NiAs-type Mn50(Te50-xSbx) films with c-planeorientation for 0 ≤ x ≤ 51 as a function of x. Lattice
parameters a and c were derived from the in-plane andout-of-plane XRD profiles, respectively. With theincrease in x, a was almost constant at about 4.17 Å for0 ≤ x ≤ 38 and reduced down to 4.13 Å gradually for x ≥38, whereas, with the increase in x, c slightly decreased
from 6.68 to 6.61 Å for 0 ≤ x ≤ 16, steeply dropped for 16≤ x ≤ 23, and decreased gradually down to 5.78 Å for x ≥23. As a result, dMn-Mn changed from 3.34 to 2.89 Å inaccordance with the change in c. The steep changes in c

4.15
4.10
0

10

20

30

x (at.%)

40

50

Fig. 3 Lattice parameters, a and c, and nearest
Mn-Mn distance, dMn-Mn, for ternary NiAs-type
Mn50(Te50-xSbx) films as function of Sb composition,x.

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX

magnetization at low temperature. Bloch’s law10) is asfollows.
M (T )
 AT 3 2 ,(1)M (5 K )32

where z is the number of nearest neighbors, and S is thespin of Mn. The Curie temperatures, TCcal.s, of theMn50(Te50-xSbx) epitaxial films for 5 ≤ x ≤ 51 were
estimated by substituting the J value from Eq. 2 into
Eq. 3 and were plotted with filled circles in Fig. 6 as afunction of the Mn-Mn distance obtained for Fig. 3. The
estimated TCcal.s were normalized by the Curie
temperature of bulk MnSb at 586 K1,11). The Néel
temperature TN of bulk MnTe at 307 K is also plotted inFig. 6. With the increase in dMn-Mn, TCcal. reduced
monotonically for 2.89 Å ≤ dMn-Mn ≤ 3.01 Å and increased
monotonically for 3.31 Å ≤ dMn-Mn ≤ 3.34 Å.
4. Discussion

In both of the compositional dependences of mS and A,
a discontinuous change existed at the same
compositional region, 16 ≤ x ≤ 23. This suggests that themagnetism of the NiAs-type Mn50(Te50-xSbx) films for 0≤ x ≤ 11 was different from that for 33 ≤ x ≤ 51. The
decrease in mS for 33 ≤ x ≤ 51 with the reduction in x
may be explained as follows. The ferromagneticallyaligned local magnetic moments of the Mn atoms werereduced by the increase in the amount of substituted Te
atoms. From the results of Fig. 4, the slope of mS for 33≤ x ≤ 51 can be formularized as:

mS = 3.23·(x / 50).

Fig. 4 Change in saturation magnetic moments per
Mn atom, mS, for Mn50(Te50-xSbx) epitaxial films asfunction of Sb composition, x. mS were derived fromsaturation magnetization of film at 5 K.

3

A (×10-4)

k 
A  C  B  ,
(2)
 J 
where T is temperature, M(T) is the difference insaturation magnetization at T, M(T), and at 5 K, M(5K),
kB is Boltzmann’s constant, J is an exchange integral,
and C is a coefficient. The change in A as a function of x
is shown in Fig. 5. A decreased rapidly for x ≤ 16 andslightly for 16 ≤ x ≤ 51 with the increase in x.
The transition temperatures were estimated from
Bloch’s T 3/2 law and mean field approximation,
2 zJS ( S  1)
T cal . 
,
(3)
3k B

2

Mn50(Te50-xSbx)

M(T)
M(5K)

= AT 3/2

1
0

10 20 30 40 50

x (at.%)

Fig. 5 Change in A, which is coefficient of T 3/2 in
Bloch’s law, for Mn50(Te50-xSbx) epitaxial films asfunction of Sb composition, x.

(4)Here, 3.23 means the magnetic moment per Mn atom in
MnSb. The reducing tendency of mS is also shown by
the broken line in Fig. 4. For 33 ≤ x ≤ 51, the broken
line, which passes through the origin, is in good
agreement with the experimental trend.
The appearance and increase in mS for 0 ≤ x ≤ 11 withthe rise in x may be explained as follows. The
antiferromagnetic coupling between the adjacentMn-atomic layers still existed up to x = 11; as a result,
ferrimagnetism appeared, and the difference in
sub-lattice moments increased as x increased. In thiscase, the slope for mS is:mS = (5.0–3.23)·(x / 50) = 1.77·(x / 50).

(5)Fig. 6 Change in estimated Curie temperature,Tccal., and estimated Néel temperature, TNcal., ofMn50(Te50-xSbx) epitaxial films as function of nearest
Mn-Mn distance, dMn-Mn. Magnetic transition
temperatures were estimated from mean fieldapproximation and Bloch’s T 3/2 law in analyses oftemperaturedependenceofsaturation
magnetization at low temperature.

Journal of the Magnetics Society of Japan Vol.43, No.2, 201923

INDEX

Here, 5.0 and 3.23 mean the magnetic moment per Mn
atom in MnTe and MnSb, respectively. 1.77 means thedifference of it between MnTe and MnSb. The
increasing tendency of mS is shown by the dotted line in
Fig. 4. For 0 ≤ x ≤ 11, the dotted line, which starts fromthe origin, also corresponded to the experimental trend.
It is thought that the NiAs-type Mn50(Te50-xSbx)
system has ferromagnetic coupling in a short Mn-Mn
distance, dMn-Mn ≤ 3.01 Å, and has antiferromagnetic
coupling in a long Mn-Mn distance, dMn-Mn ≥ 3.31 Å asshown in Fig. 5. Therefore, if TAFcal. is defined as thetransition temperature at which the antiferromagnetic
coupling of spins is destroyed by thermal energy, TCcal.
for dMn-Mn ≥ 3.31 Å can be equal to TAFcal.. TAFcal.s areplotted with open circles on the TNcal. side in Fig. 6.
TAFcal. shows good continuity from binary MnTe toternary Mn50(Te50-xSbx) in the region.
M-H curves of the Mn50(Te50-xSbx) epitaxial films were
measured at 5 K to clarify the difference in magnetismfor each compositional region. The M-H curves of thefilms for x = 11, 16, and 33 are shown in Fig. 7. The
cases of x = 11 and 33 represent the cases offerrimagnetism and ferromagnetism, respectively. In
both cases, the curves showed typical hysteresis loops ofa magnetically single phase. In the case of x = 16, thetransition region, the M-H curve shows steps around H= ±1 kOe. These steps indicated multiple magnetic
phases in the film. This means that the ferrimagnetismand ferromagnetism co-existed in the transition region,
16 ≤ x ≤ 23.
According to these explanations, we may say that themagnetism for 16 ≤ x ≤ 23 cannot be connected smoothlywith that for 0 ≤ x ≤ 11 and x ≥ 33, in particular, thesign of inter-atomic-layer exchange coupling. Thissuggests that the structural instability of NiAs-type
Mn50(Te50-xSbx) epitaxial films is possibly induced by
the discontinuity of their magnetism in the transition
region.
5. Conclusion
Structure, magnetism, and their correlation for
NiAs-type Mn50(Te50-xSbx) pseudo-single crystal films
were investigated by changing the nearest Mn-Mn
distance by substituting Te for Sb.
The c-plane oriented Mn50(Te50-xSbx) pseudo-single
crystal films were prepared in ranges of Sb composition,x, from 0 to 16 and from 33 to 51 by employing a c-planeoriented NiAs-type MnTe under-layer.
Lattice parameter a of the Mn50(Te50-xSbx) films

Fig. 7 M-H curves of Mn50(Te50-xSbx) (x = 11, 16, and33) epitaxial films measured at 5 K.
changed from 4.17 to 4.13 Å, and c changed from 6.68 to
5.78 Å, As result, the nearest Mn-Mn distance, dMn-Mn,changed from 3.34 to 2.89 Å.
The saturation magnetic moments mS of Mn atoms at
5 K linearly increased with a different slope as x
increased, 0 ≤ x ≤ 11 and 33 ≤ x ≤ 51.
The coefficient of T 3/2 in Bloch’s law, A, decreased
rapidly for x ≤ 16 as x increased and increased
gradually for 16 ≤ x ≤ 51 as x decreased.
The
magnetismoftheternary
NiAs-type
Mn50(Te50-xSbx) system changed smoothly from
antiferromagnetism
toferrimagnetism
anddiscontinuously from ferrimagnetism to ferromagnetism
by decreasing dMn-Mn.
References
1) T. Okita and Y. Makino: J. Phys. Soc. Jpn., 25, 120 (1968).
2) Y. Ashizawa, S. Saito, and M. Takahashi: J. Appl. Phys., 91,
8240 (2002).
3) S. Saito, Y. Ashizawa, H. Shoji, and M. Takahashi: J. Magn.
Soc. Jpn., 25, 855 (2001) [in Japanese].
4) N. Kunitomi, Y. Hamaguchi, and S. Anzai: J. Phys., 25, 568(1964).
5) I. Oftedal: Z. Phys. Chem., 128, 135 (1927) [in German].
6) S. Greenwald: Acta Crystallogr., 6, 396 (1953).
7) Y. Ashizawa, S. Saito, and M. Takahashi: J. Appl. Phys., 91,
8096 (2002).
8) W. C. Marra, P. Eisenberger, and A. Y. Cho: J. Appl. Phys.,
50, 6927 (1979).
9) K. Omote and S. Y. Matsuno: Adv. X-ray Chem. Anal., 30,
205, (1998) [in Japanese].
10) F. Bloch: Z. Physik., 61, 206 (1930).
11) H. Ido: J. Phys. Soc. Jpn., 25, 625 (1968).
Received August 02, 2018; Accepted December 25, 201824

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX
J. Magn. Soc. Jpn., 43, 25-28 (2019)<Paper>Near-Infrared Magneto-Refractive Effect for Antiferro-Magnetically
Exchange Coupled Co/Ru Multilayer Film in Transmission Configuration
S. Saito, H. Sato, K. Ooki, K. Akahane, and H. Uchida*

Department of Electronic Engineering, Graduate School of Engineering, Tohoku Univ., 6-6-05, Aoba, Aramaki, Aoba-ku, Sendai,

Miyagi, 980-8579 Japan* Electrical and Electronic Information Engineering, Toyohashi Univ. of Technology,

441-8580 Japan

1-1, Hibarigaoka, Tempaku, Toyohashi, Aichi,

A magneto-refractive effect in a near-infrared (IR) region was investigated with a transmission configuration (TMRE) for antiferro-magnetically coupled Co/Ru multilayer films. Applying dielectric function analysis based on an
expanded Drude model that takes spin-dependent scattering into account, it was found that (1) a [Co(4 nm)/ Ru(0.7nm)]10 multilayer film took parallel to antiparallel magnetization states at H = 14 kOe to 0 Oe, respectively, and (2)that T-MRE curves against H took the maximum in the near-IR region and the minimum in the middle wavelength
IR region. (3) The magnitude and wavelength dependence of T-MRE were well fitted by using the expanded Drude
model with p = 1.5 × 1016, SAL = 0.66, and SAL = 0.125, where SAL and SAL were self-averaged relaxation time andspin asymmetry, and p was the plasma angular frequency of conduction electrons. (4) According to calculation basedon the expanded Drude model, the T-MRE of T /T showed the maximum of 0.3% at around 700 nm, and in the longer
wavelength region, it showed a saturated feature with a value of 1.3% at around 3 m.
Key words: magneto-refractive effect, spin-dependent scattering, expanded Drude model, dielectric constant spectrum
1. Introduction
Much attention has been paid to magnetic sensors with
an optical probe (MSOP), since the sensor head can be
relatively miniaturized and used under vacuum, liquid
atmospheres through narrow space with an optical fiber.
In order to realize MSOP with a simple optical system,
an effect with light intensity change is more promisingthan that with light polarization change. Therefore,
development of materials whose optical properties suchas reflectivity and transmittance vary against magnetic
field (H) is eagerly required. Based on this point, we
focused on magneto-refractive effect (MRE) for antiferromagneticallycoupledferro-/
non-ferromagnetic
multilayer films where we can expect linear response
against H with rotational magnetization process.
Concerning MRE, theoretical studies were reported after
1995 1), and experimental researches after 2002 2).
According to these papers and following reports3-6), themain interest is how to measure giant magnetoresistance(GMR) effect with contactless probe especially for Co-Ag
related granular or Co/Cu related multilayer films,
because the origin of both MRE and GMR is generally
understood to be the spin-dependent scattering ofconduction electrons. However, optical properties andtransport properties are not completely the same;
concerning the MRE with transmission configuration (TMRE), there exists wavelength dependence, and it isdesirable that total thickness of a multilayer film shouldbe thin to obtain enough transmitted light intensity for
detection, which is completely different situation for
GMR. In this sense, material investigations of MRE itself
have not sufficiently carried out. Therefore, in this study,
we experimentally obtained MRE with the transmission

configuration for Co/Ru multilayer films with RKKY-likeinterlayer exchange coupling78) as a typical multilayer
system. Furthermore, we applied dielectric function
analysis, and discussed material guide for enhancing TMRE based on an expanded Drude model taking the spindependent scattering into account.
2. Experimental procedure
The multilayer films were fabricated by dc magnetron
sputtering on glass substrates. A Ti(2 nm)/ Ru(3 nm)
layer was adopted as an underlayer for the purpose ofadhesion of the film to the substrate and the control ofcrystalline sheet texture with atomic closed packed plane
parallel to the film plane. A SiN with 10 nm thick wasused as a capping layer to avoid oxidation. The stacking
structure of the magnetic multilayer was [Co(4 nm)/
Ru(dRu nm)]N with the repetition number N of 10 withchanging dRu from 0 to 1.2 nm. Here, Ru was selected asa nonmagnetic layer material to stabilize antiparallel
magnetical coupling state under zero field with a thinner
film thickness than that of Cu by RKKY-like interlayer
exchange coupling.
Magnetic properties were evaluated by the vibrating
sample magnetometer with the maximum applied field(Hmax) of 14
kOe. Opticalproperties were

Fig. 1 Schematic of system for measuring spectroscopic magnetorefractive effect with transmission configuration. L1 to L4 arecondensing and collimation lens.

Journal of the Magnetics Society of Japan Vol.43, No.2, 201925

INDEX

Fig. 2 Magnetic hysteresis loops for [Co(4 nm)/Ru(dRu nm)]10 multilayer films with dRu of (a) 0, (b) 0.2, (c) 0.4, (d) 0.6, (e) 0.7, (f) 0.8, (g) 1.0, and(h) 1.2 nm.

spectroscopically measured by the ellipsometer (M-2000,
J. A. Woollam) with the wavelength region from 250 to1700 nm. T-MRE was spectrometrically measured by ahand-made system with an electromagnet applying
magnetic field (H) along the film plane direction9) asshown in Fig. 1. Here, the measurement wavelength ()range was 9001650 nm limited by monochrometer (NIRQUEST, Ocean Optics) and Hmax was 14 kOe.
3. Results and discussion
3.1 Magnetic properties
Figure 2 shows typical magnetic hysteresis loops with
the Co/Ru multilayer films with the Ru thickness, dRu of(a) 0, (b) 0.2, (c) 0.4, (d) 0.6, (e) 0.7, (f) 0.8, (g) 1.0, and (h)1.2 nm, respectively. In the loops of (a) and (b), steep
magnetization process can be seen around H = 0 caused
by magnetic domain wall motion with ferromagneticallyinterlayer coupling between Co layers. With increasing
dRu, saturation field becomes beyond 14 kOe, due to theantiferro-magnetic interlayer coupling, while the wallmotion magnetization process remains for the films with(c) and (d). Further increasing dRu at (e) to (h), wallmotion process completely disappears and smoothrotational magnetization process with the scissors type is
expected.

multilayer films was measured and was found to be themagnitude of 0.5 to 1.7 % in the spectral range of
9001650 nm (data not shown). In details, T
monotonously increases with increasing , and T
monotonously decreases with thickening dRu.
Figure 3 shows hysteresis curves in transmittancenormalized by that of under 14 kOe of  at 1550 nm
against H for typical samples with [Co(4 nm)/ Ru(0, 0.4,
0.7 nm)]10, whose magnetic properties are shown in Fig.
2 (a), (c) and (e), respectively. Here, T-MRE was defined
as difference in transmittance between that of under H =14 kOe and 0 kOe normalized by that of under 14 kOe.
As seen in Fig. 3 (a), no change in T against H wasobserved even though the sample shows magnetization
process with ferromagnetic hysteresis shown in Fig. 2 (a).
On the other hand, as seen in Fig. 3 (c) and (e), T changed
symmetrically with the H = 0 axis. Compared with thetransmittance hysteresis curves of (c) and (e), saturation
feature under high H region cannot be observed in (c) but
clearly observed in (e). These facts mean that the shapeof transmittance curves of the present multilayer films
reflects angles between magnetic moments ofneighboring layers such as GMR effect rather thandirection of magnetic moments as shown in Fig. 2.
Figure 4 shows the normalized transmittance mapwith [Co(4 nm)/ Ru(0, 0.4, 0.7)]10 multilayer films (color

dRu:
0 nm
0

10

0.4 nm(c)
H (kOe)

99
-10

100

0

10

T/T14kOe (%)(a)
H (kOe)

99
-10

100

T/T14kOe (%)

100

T/T14kOe (%)

3.2 Optical properties and magneto-refractive effectTransmittance (T) of the [Co(4 nm)/ Ru(01.2 nm)]10

0.7 nmH (kOe)

99
-10

(e)

0

10

Fig. 3 MRE curves of wavelength at 1550 nm with thru beam
configuration for [Co(4 nm)/Ru(dRu nm)]10 multilayer films with dRu of(a) 0, (c) 0.4, and (e) 0.7 nm.

26

Fig. 4 Color maps of MRE with transmittance configuration plottedagainst wavelength and applied magnetic field for [Co(4 nm)/Ru(dRunm)]10 multilayer films with dRu of (a) 0, (c) 0.4, and (e) 0.7 nm. Whitelines with wavelength of 1550 nm correspond to curves shown in Fig.
3.

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEXExp.
Fitted

-20

40
20

1.0

2.0

3.0

4.0

Photon energy, E (eV)

(B)

Cal.
Fitted5𝜔𝜔� = 1.50×1016 (s-1)
𝛽𝛽SAL = 0.125

0.0
0

-1.0

0.8

1

1.2

-0.5

-1.0

Fig. 5 (A) Dielectric constants and (B) T-MRE spectrum for [Co(4nm)/Ru(0.7 nm)]10 multilayer film. Black solid lines mean experimental
results, and blue and red broken lines are results fitted by using
oscillation model with expansion Drude term.

on line). The vertical and horizontal axes indicate  and
H. The white lines with  = 1550 nm correspond to thecurves shown in Fig. 3 (a), (c) and (e). As seen in Fig. 4(a), no change can be seen, whereas in (b) and (c), theminimum and the maximum appears around H = 0.
Focusing on  dependence of the sample (e), theminimum can be observed at  = 900 nm. With increasing up to 1100 nm, the magnitude of the minimum
decreases. Further increasing of  over 1100 nm, thisminimum changes into the maximum. The magnitude of
the maximum enhances with increasing  in the present
experimental range. In the following section, we selectthe sample (e), because we can realize parallel (H = 14kOe) and antiparallel (H = 0 kOe) magnetization statesin our experimental setup of T-MRE.
3.3 Dielectric function analysis with expanded Drude
model
In order to understand the enhancement phenomenon
and change in the extreme values from the maximum tothe minimum in wavelength dependence of T-MRE, wecarried out spectral analysis for dielectric function.
According to M. Vopsaroiu et al.4) dielectric function for a
GMR multilayer film (self-averaging limit of dielectric
constant SAL), assuming the multilayer to be a single
layer, is written by following formula;(1)

where SAL and SAL are self-averaged relaxation time andspin asymmetry, p is the plasma angular frequency of
conduction electrons,  is the angular frequency of theincident light. M/Ms is the normalized magnetization andtakes 1 for the parallel state (saturation) and 0 for the
antiparallel state. In the eq. (1), the first term is the socalled Drude term and second one is an expansion Drude
term taking spin-dependent scattering into consideration.
Here, eq. (1) is based on the form of complex refractive
index of N = n + i. By applying this equation, we canderive SAL and SAL from fitting analysis to optical and
T-MRE spectra obtained by the experiment.
Figure 5 shows (A) Dielectric constants and (B) T-MRE

-1.5

(A)

2.0

0.12
0.25

1.4

Photon energy, E (eV)

2 𝑀𝑀 2𝛽𝛽� �𝜔𝜔� � −𝑖𝑖𝑖𝑖𝑖𝑖
SAL 𝑀𝑀
SAL � 1 −𝜀𝜀
=1− �s 2 𝑀𝑀 2�,2
SAL𝜔𝜔 1 − 𝑖𝑖𝑖𝑖𝑖𝑖(1 − 𝑖𝑖𝑖𝑖𝑖𝑖) − 𝛽𝛽� �
SALSAL
SAL 𝑀𝑀s𝜏𝜏SAL(fs):0.66𝜔𝜔� = 1.50×1016 (s-1 )𝜏𝜏SAL = 0.66 (fs)Exp.

T/T (%)

60

0

T-MRE
(%)
T/T (%)

Real part,  1

-10

80

Imaginary part,  2(A) SAL = 1 + i 2

1.0

T/T (%)

0𝛽𝛽SAL :

0.021
0.042
0.125

0

(B)

-5

1000 2000 3000 4000 5000

Wavelength (nm)

0.17
0.34
1000 2000 3000 4000 5000

Wavelength (nm)Fig. 6 T-MRE spectra calculated by changing (A) tSAL and (B) bSAL
for [Co(4 nm)/Ru(0.7 nm)]10 multilayer film. In calculation, constantvalues of wp = 1.5×1016 and bSAL = 0.125 for (A) and wp = 1.5×1016 andtSAL = 0.66 for (B) were used.

spectrum for the [Co(4 nm)/Ru(0.7 nm)]10 multilayer film.
Black solid lines mean experimental results, and brokenlines with blue and red color are fitted results by using
an oscillation model with the expansion Drude term.
According to the ellipsometry, application of theoscillation model in fitting shown in Fig. 4 (A) revealedp = 1.5 × 1016 and SAL = 0.66. Next, T-MRE spectrum
was fitted based on eq. (1) shown in Fig. 5 (B) throughcalculations of transmittance of parallel and antiparallelstates, which resulted in SAL = 0.125.
Finally, we expected material guide to enhance T-MRE
by calculation to expand spectral range. Figure 6 shows
calculated T-MRE spectra for virtual multilayer films bychanging (A) SAL around 0.66 and (B) SAL around 0.125.
In the calculation, constant values of p = 1.5 × 1016 andSAL = 0.125 for (A) and p = 1.5 × 1016 and SAL = 0.66 for(B) are used. Here, p = 1.5 × 1016, SAL = 0.66 and SAL =0.125 are the values obtained by fitted results as shownin Fig. 5. For all conditions, the sign of T-MRE of T /Tchanges positive to negative with increasing . As seen in(A) for the condition of SAL = 0.66 which corresponds tothe present [Co(4 nm)/Ru(0.7 nm)]10 multilayer film, T/T shows the maximum of 0.3% at around 700 nm,
whereas in the longer wavelength region, it shows thesaturated feature to the value of 1.3% at around 3 m.
Therefore, the effect of the enhancement of themagnitude of T-MRE in Mid-wavelength IR region(negative sign) was found to be larger than that in Near
IR region (positive sign). With increasing SAL,  where
the sign of T /T changes from the positive to the negative
shifts toward the longer  side as shown in (A). In
contrast, with increasing SAL, degree of enhancement
increases as shown in (B). Therefore, considering of
application of MRE, figure of merit can be increased byferro-/ nonferro-magnetic material combination, layer
structure, and total thickness while realizing antiparallel
magnetization states for thinner crystalline ferro-/ nonferromagnetic multilayers.
4. Conclusion
A magneto-refractive effect in a near-infrared (IR)region was investigated with a transmission
configuration (T-MRE) for Co/Ru multilayer films. As
results, It was found that (1) a [Co(4 nm)/ Ru(0.7 nm)]10

Journal of the Magnetics Society of Japan Vol.43, No.2, 201927

INDEX

multilayer film with antiferro-magnetically coupling
showed the magnitude of T-MRE of about 1% at 1550
nm, (2) the magnitude and wavelength dependence of TMRE were well fitted by an expanded Drude model with
p = 1.5×1016, SAL = 0.66 and SAL = 0.125, where SAL andSAL were self-averaged relaxation time and spin
asymmetry, p was the plasma angular frequency of
conduction electrons, and (3) according to calculation
based on the expanded Drude model, the T-MRE of T /Tshowed a saturated feature with a value of 1.3% ataround 3 m. Further increase of T /T in IR region is
expected to enhance SAL of the multilayer.
References
1) J. C. Jacquet and T. Valet, Mater. Res. Soc., 384, 477 (2015).
2) V. G. Kravets, D. Bozec, J.A. Matthew, S. M. Thompson, H.

28

Menard, A. B. Horn, and A. F. Kravets, Phys. Rev. B, 65,054415 (2002).
3) A. B. Granovskii, M. V. Kuz’michev, and J. P. Clerc, J. Exp.
Theor. Phys., 89, 955 (1999).
4) M. Vopsaroiu, J. A. D. Matthew, and S. M. Thompson, Rev.
Sci. Instrum., 75, 3127 (2004).
5) Jian-Qing Wang, Malak T.Sidney, Jared D. Rokitowski, Nam
H. Kim, and Keqiang Wang, J. Appl. Phys., 103, 07F316(2008).
6) M. Vopsaroiu, M. G. Cain, and V. Kuncser, J. Appl. Phys., 110,056103 (2011).
7) S.S.P. Parkin, N. More, and K.P. Roche, Phys. Rev. Lett., 64,
2304 (1990).
8) P.J.H. Bloemen, H.W. van Kesteren, H.J.M. Swagten, and
W.J.M. de Jonge, Phys. Rev. B, 50, 13505 (1994).
9) Haruhiko Sato, Shin Saito, Shyota Mizuno, Koichi Akahaneand Hironaga Uchida, IEEJ Trans. Fund. Mater., 136, 174(2016).
Received Oct. 15, 2018; Accepted Jan. 17, 2019

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX
J. Magn. Soc. Jpn., 43, 29-33 (2019)<Paper>Anomalous Hall Effect and Anisotropic Magnetoresistance in
Perpendicularly Magnetized FePt 1- x Pd x Films
T. Seki*,** , S. Kikushima*, and K. Takanashi*,*****

Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan

Center for Spintronics Research Network, Tohoku University, Sendai 980-8577, Japan

L10-ordered alloys with high uniaxial magnetic anisotropy (Ku) are key materials for future spintronic devices torealize large-scale integration. In this study, we investigated the anomalous Hall effect (AHE) and anisotropic
magnetoresistance (AMR) effect for FePt1-xPdx (001) epitaxial films. Highly L10-ordered structures and perpendicularmagnetization were obtained for all the compositions. The FePt film exhibited the highest Ku and the largest
anomalous Hall angle (θAHE) among the present FePt1-xPdx samples, and the values of Ku and θAHE were gradually
decreased as the Pd concentration increased. In addition, the composition dependence of the AMR ratio showed asimilar tendency to those for Ku and AHE. We discuss possible scenarios to explain these composition dependencies.
Key words: L10-type ordered alloy, anomalous Hall effect, anisotropic magnetoresistance effect
1. Introduction
For the further progress in magnetic storage
technology, we need to tackle several essential problems
every magnetic material has. One of the inevitableproblems is the thermal instability of magnetization
appearing as the volume of magnet (V) is reduced. Inorder to hurdle this problem, high uniaxial magnetic
anisotropy (Ku) is needed because the thermal stability
factor (Δ) of magnetization is given by Δ = KuV / kBT,
where kB is the Boltzmann constant and T is thetemperature. Thus, high Ku materials are key tomaintain sufficient Δ even for a nanometer-scaled
magnet, enabling us to realize the large-scale
integration of spintronic devices.
L10-ordered alloys are representative high Ku
materials1). Thanks to their high Ku along the c-axis of
L10-ordered structure, perpendicularly magnetizedlayers can be formed when those alloys are grown withthe (001) crystal orientation in the thin film form2-5).
This characteristic is beneficial for perpendicularmagnetic recording and spintronic devices utilizing
perpendicular magnetization as a perpendicularspin-polarizer6). Among the L10-ordered alloys, L10-FePt
and L10-FePd are famous alloys, and are regarded as
candidate materials for ultrahigh density magnetic
storage.
Althoughbothalloyshave
similarcharacteristics, such as high Ku values and structures
being composed of Fe and Pt or Pd with large spin-orbit
interaction, there are also definite differences between
them. The most intriguing difference is the magnitudeof magnetization damping. L10-FePd exhibits a muchsmaller damping constant than that of L10-FePt7,8).
Although that might be partially interpreted by thesmaller spin orbit interaction of Pd than that of Pt, therelationship between the transport properties and thespin orbit interaction has not been totally understood
yet for the Fe-Pt-Pd system 9-12).
In this study, we focus on the anomalous Hall effect(AHE) and anisotropic magnetoresistance (AMR) effect.
The magnitude of AHE is characterized by theanomalous Hall angle (θAHE). The spin-orbit interaction
is regarded as an essential factor for AHE, and several
mechanisms, such as skew scattering, side jump andintrinsic deflection, are responsible for the appearanceof AHE13). The expression of AMR also includes the
spin-orbit coupling constant14). Namely, both AHE and
AMR are phenomena related with the spin-orbit
interaction although they are often discussed in the
different theoretical frameworks.
We prepared FePt1-xPdx (001) epitaxial films andsystematically investigated the structures, magnetic
properties and transport properties. The Pdconcentration (x) dependences of Ku, AHE and AMR arecompared, and possible scenarios to understand thosecomposition dependences are qualitatively discussed.
2. Experimental Procedure
50 nm-thick FePt1-xPdx layers were directly grown on
an MgO (001) single crystal substrate by employing theultrahigh vacuum compatible magnetron sputtering
apparatus. The base pressure of main growth chamber
was below 2 × 10-7 Pa. The main growth chamber was
equipped with three independent sputtering cathodes,
where Fe, Pt, and Pd targets were installed. This
cathode arrangement enabled us to vary the alloy
composition by tuning the input powers for sputtering
cathodes. The growth temperature for the FePt1-xPdx
layers was set at 500ºC. This high temperature process
is indispensable to promote the L10-ordering. We
monitored the film growth by employing reflection
high-energy electron diffraction, which exhibited streak
patterns for all the samples. This indicated that flat
surfaces were obtained. The thin films were patterned
into a Hall cross shape in order to measure AHE andAMR through the use of photolithography and Ar ion
milling. The width of Hall cross was designed to be 20

Journal of the Magnetics Society of Japan Vol.43, No.2, 201929

INDEX(e) 1. The unlabeled sharp peaks come from thereflections of the MgO substrate. All the peaks, denotedby the arrows, are 00l reflections of FePt1-xPdx, whichindicates that all the FePt1-xPdx layers show the (001)epitaxial growth on the MgO (001) substrate. In
addition to the 002 and 004 fundamental peaks, the 001
and 003 superlattice peaks are clearly observed,
suggesting the formation of L10-ordered structure.
The lattice constants of c-axis and a-axis, tetragonality(c/a), and the degree of long-range order (S) aresummarized in Figs. 2 (a), (b), (c) and (d), respectively.
The lattice constants of c-axis were obtained from the(002) peaks in the out-of-plane XRD profile (Fig. 1)
whereas those of a-axis were obtained from the 200
peaks in the in-plane XRD profiles (not shown here). S
is defined as2,3)⎡⎣ I super I fund ⎤⎦
(1)
obs ,
S2 =
S=1
⎡⎣ I super I fund ⎤⎦calc

where Ifund and Isuper are the integrated intensities offundamental and superlattice peaks, respectively, andthe subscripts of obs and calc represent theexperimentally observed values and the calculated
values, respectively. The intensities for the fully orderedstate (S = 1) were calculated with taking into accountatomic scattering factors, Debye-Waller factors, Lorentz
factor, polarization factor, and structure factors2,3). In

Fig. 1 Out-of-plane x-ray diffraction profiles for
FePt1-xPdx films with (a) x = 0, (b) 0.25, (c) 0.5, (d)0.75, and (e) 1.
µm.
The structure was characterized using the x-ray
diffraction (XRD) with Cu-Kα radiation. Magnetization
measurements were carried out at room temperatureusing a superconducting quantum interference device
magnetometer. Transport properties were measured atroom temperature using a physical property
measurement system (PPMS) with the maximum
magnetic field of 90 kOe.
In this study, we set the Pd concentration x in Pt1-xPdx
at 0, 0.25, 0.5, 0.75, and 1, where x × 100 corresponds
to the atomic percent of Pd concentration. The actual
alloy compositions were determined to be Fe48Pt52 (for x= 0), Fe48Pt45Pd7 (for x = 0.25), Fe53Pt22Pd25 (for x = 0.5),
Fe48Pt8Pd44 (for x = 0.75), and Fe51Pd49 (for x = 1) byelectron probe x-ray microanalysis. Hereafter, thedesigned Pd concentration x refers to the sample nameinstead the measured composition for simplicity.
3. Results and Discussion
First, we explain the structure and the magnetic
properties for the FePt1-xPdx films. Figure 1 shows the
XRD profiles for (a) x = 0, (b) 0.25, (c) 0.5, (d) 0.75, and

30

Fig. 2 Lattice constants of c-axis and a-axis,
tetragonality (c/a), and the degree of long-rangeorder (S) as a function of Pd concentration (x).
Dashed lines are guides for eyes.

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX

Fig. 4 (a) Saturation magnetization (Ms) and (b)uniaxial magnetic anisotropy energy (Ku) as afunction of x. Dashed lines are guides for eyes.

Fig. 3 Magnetization curves for FePt1-xPdx films
with (a) x = 0, (b) 0.25, (c) 0.5, (d) 0.75, and (e) 1.
Solid (dashed) curves are results with field (H)applied perpendicular (parallel) to the film plane.
the cases of ternary alloys, the above factors for the
Pt-Pd site were calculated assuming the averaged
values with the atomic fractions of Pt and Pd.
The lattice constant of c-axis is gradually reduced as x
is increased while a-axis is slightly increased.
Consequently, c/a shows a slight decrease with x. The
literature values15) are c = 0.3702 nm and a = 0.3842 nm
for FePt, and c = 0.3715 nm and a = 0.3850 nm for FePd,
which lead to c/a = 0.964 for FePt and 0.965 for FePd.
These reported values do not explain the Pdconcentration dependence of lattice constants for thepresent FePt1-xPdx films. The epitaxial strain may berelated with the changes in the lattice constants, which
depends on the composition and the kind of substrate,
although the detailed mechanism has not been
understood yet. It is noted that the continuous change
in the lattice constants suggests the formation of
Fe-Pt-Pd ternary alloy without any phase separation.
From Fig. 2 (d), it is confirmed that highly orderedstates (S ~ 0.8) are achieved at all the compositions.
Figure 3 shows the magnetization curves for (a) x = 0,(b) 0.25, (c) 0.5, (d) 0.75, and (e) 1, where the solid(dashed) curves are the results with field (H) applied
perpendicular (parallel) to the film plane. One sees thatthe easy magnetization axis is in the normal direction to
the film plane for all the films. Namely, all the
FePt1-xPdx layers are perpendicularly magnetized.
However, the saturation field in the in-planemagnetization curve is decreased as x is increased. For
example, for x = 0 (FePt), H = 50 kOe is insufficient tosaturate the magnetization along the in-plane direction

while the magnetization of x = 1 (FePd) saturates evenat the in-plane H = 25 kOe. This indicates that theperpendicular magnetic anisotropy is reduced with x.
Figure 4 displays (a) saturation magnetization (Ms)and (b) Ku as a function of x. The value of Ku wascalculated from the effective magnetic anisotropy (Keff)with the correction of shape anisotropy energy (-2πMs2),
i.e. Ku = Keff + 2πMs2. Keff was determined from the area
enclosed between the perpendicular and in-planemagnetization curves. In Fig. 4(a), Ms does not show theremarkable composition dependence. A previous paper1)
reported Ms = 1140 emu/cm3 and 1100 emu/cm3 for FePt
and FePd, respectively, indicating that no remarkabledifference in Ms between FePt and FePd. This explains
why Ms keeps almost constant regardless of x. On theother hand, Ku shows the significant composition
dependence. The largest Ku among the present samples
is obtained at x = 0, which is Ku = 3.5 × 107 erg/cm3, and
Ku is drastically reduced as x is varied from 0 to 1. The
larger Ku for x = 0 than that for x = 1 is consistent withthe fact that the bulk FePt shows the larger Ku than thebulk FePd1). Our finding is that Ku is monotonicallydecreased with x for the Fe-Pt-Pd ternary alloys anddoes not show a local minimum or maximum value.
Next, let us show the results of AHE measurement for
the Hall devices with the FePt1-xPdx layers. Figure 5 (a)is the transverse resistance (Rxy) as a function of H for x= 0, 0.5, and 1, where H was applied along theperpendicular (z) direction to the device plane. The Hall
resistivity (ρH) is composed of two terms13):ρ H = R0 H + RS M (T, H ) , (2)where M(T,H) is the magnetization averaged over the
sample. R0 and RS represent the ordinary andanomalous Hall coefficients, respectively. All the devices
exhibit clear hysteretic transitions of Rxy. The shapes of
Rxy - H coincide with those of the perpendicularmagnetization curves. This means the Rxy change
originates from the AHE. Figures 5 (b), (c) and (d)summarize the longitudinal conductivity (σxx), thetransverse conductivity (σxy), and θAHE as a function of x.
θAHE is defined as θAHE = σxy / σxx. σxx (σxy) increases

Journal of the Magnetics Society of Japan Vol.43, No.2, 201931

INDEX(decreases) as x is increased. As a result, x = 0 exhibits
the largest θAHE = 0.04, and θAHE shows the monotonic
reduction down to 0.01 at x = 1. This tendency might be
explained by the fact that FePt possesses the spin orbit
interaction larger than FePd16).
Figure 6 displays the AMR effect for the FePt1-xPdx
layers. The longitudinal resistance (Rxx) versus H for x =0 is shown in Fig. 6(a), where the linear background dueto the force effect was subtracted. The solid curve
represents the result with H applied along the in-planecurrent (I), i.e. I // H, while the dashed curve represents
that with the perpendicular H, i.e. I ⟘ H. Thus, theshapes of AMR curves for I // H and I ⟘ H correspond to
the in-plane and perpendicular magnetization curves,
respectively. In this study, Rxx for I // H (I ⟘ H) aredefined as Rxx,l (Rxx,t). As shown in Fig. 6(a), Rxx,l islarger than Rxx,t for x = 0, which means that the sign of
AMR for the FePt is positive. The AMR ratio as afunction of x is shown in Fig. 6(b). The definition of
AMR ratio in this study is AMR ratio (%) = (Rxx,l - Rxx,t)
× 100 / Rxx,t. As x is increased, the AMR ratio isdecreased, however a sign reversal of AMR does notoccur. The FePt and FePd exhibit 0.8 % and 0.4 % AMR
ratios, respectively.

Fig. 6 (a) Longitudinal resistance (Rxx) versus H for
x = 0. Solid curve represents the result with H
applied along the in-plane current (I), i.e. I // H,
while dashed curve represents that with theperpendicular H, i.e. I ⟘ H. (b) AMR ratio as afunction of x. Dashed line is guide for eyes.

Fig. 5 (a) Transverse resistance (Rxy) as a functionof H for x = 0, 0.5, and 1, where H was applied along
the perpendicular direction to the device plane. (b)Longitudinal conductivity (σxx), (c) transverse
conductivity (σxy), and (d) θAHE as a function of x.
Dashed lines are guides for eyes.

32

Here we discuss the composition dependence of θAHE
and AMR ratio for the present FePt1-xPdx layers.
Seemann and co-workers reported the different
scattering mechanisms in AHE between FePt andFePd10). According to their experimental study10), theAHE in FePt is dominated by the intrinsic deflection
whereas the side-jump is a dominant mechanism for the
FePd. He et al. investigated the correlation between the
AHE and the spin-orbit coupling parameter11). They
mentioned that there existed a clear positive correlation
between the intrinsic term and the spin-orbit coupling
parameter. Since the present composition dependence of
θAHE is similar to the composition dependence of
intrinsic term reported in the previous paper11), thescenario that the spin-orbit coupling dominates themagnitude of AHE is applicable for the present result.
In contrast to AHE, the magnitude of AMR is affectedby complicated several factors: spin-orbit coupling
constant (λ), exchange field (Hex), and s-d scattering
process. According the Campbell-Fert-Jaoul (CFJ)
model17), the AMR of a strong ferromagnet is relatedwith proportional to (λ / Hex)2, suggesting that large
spin-orbit interaction gives rise to the increase of AMR.
This may be a simple scenario explaining the present
composition dependence of AMR ratio. However, Kokado
et al. derived the more rigorous model for AMR taking
into account the s-d scattering process14). In order tocompletely and quantitatively explain the present result,
we need the further consideration.

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX

4. Conclusion
We investigated the composition dependence of AHE
and AMR for the FePt1-xPdx (001) epitaxial films withhighlyorderedstructures
andperpendicularmagnetization. Among the present samples, the FePt
film exhibited the highest Ku, the largest θAHE, and thelargest AMR ratio. These values were gradually
decreased as x was increased. We discussed possible
scenarios to explain those composition dependences,suggesting that the spin-orbit interaction plays animportant role to determine the magnitude of thesevalues.
Acknowledgements
The authors thank Mr. Y.
Murakami and Mr. I. Narita for their technicalassistance. This work was partially supported by the
Grant-in-Aid for Scientific Research S (23226001) and
Scientific Research B (16H04487) as well as the
Research Grant from the TEPCO Memorial Foundation.
The device fabrication and structural characterization
were partly performed at Cooperative Research andDevelopment Center for Advanced Materials, IMR,
Tohoku University.
References
1) D. Weller, A. Moser, L. Folks, M. E. Best, W. Lee, M. F.
Toney, M. Schwickert, J. U. Thiele, and M. F. Doerner.
IEEE Trans. Magn. 36, 10 (2000)
2) A. Cebollada, D. Weller, J. Sticht, G. R. Harp, R. F. C.
Farrow, R. F. Marks, R. Savoy, and J. C. Scott, Phys. Rev.
B 50, 3419 (1994)

3) T. Shima, K. Takanashi, Y. K. Takahashi, and K. Hono,
Appl. Phys. Lett. 81, 1050 (2002).
4) S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y.
Shimada, and K. Fukamichi, Phys. Rev. B 66, 024413(2002)
5) M. Ohtake, S. Ouchi, F. Kirino, and M. Futamoto, J. Appl.
Phys. 111, 07A708 (2012)6) T. Seki, S. Mitani, K. Yakushiji, and K. Takanashi, Appl.
Phys. Lett. 88, 172504 (2006)
7) S. Mizukami, S. Iihama, N. Inami, T. Hiratsuka, G. Kim, H.
Naganuma, M. Oogane, and Y. Ando, Appl. Phys. Lett. 98,
052501 (2011)8) S. Iihama, A. Sakuma, H. Naganuma, M. Oogane, T.
Miyazaki, S. Mizukami, and Y. Ando, Appl. Phys. Lett. 105,
142403 (2014)9) S. D. Willoughby, J. M. MacLaren, T. Ohkubo, S. Jeong, M.
McHenry, D. E. Laughlin, S.-J. Choi, and S.-J. Kwon, J.
Appl. Phys. 91, 8822 (2002)
10) K. M. Seemann, Y. Mokrousov, A. Aziz, J. Miguel, F.
Kronast, W. Kuch, M. G. Blamire, A. T. Hindmarch, B. J.
Hickey, I. Souza, and C. H. Marrows, Phys. Rev. Lett. 104,
076402 (2010)11) P. He L. Ma, Z. Shi, G.Y. Guo, J.-G. Zheng, Y. Xin, and S.
M. Zhou, Phys. Rev. Lett. 109, 066402 (2012)12) S. Kikushima, T. Seki, K. Uchida, E. Saitoh, and K.
Takanashi, AIP Adv. 7, 085210 (2017)13) N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N.
P. Ong, Rev. Mod. Phys. 82, 1539 (2010)14) S. Kokado, M. Tsunoda, K. Harigaya, and A. Sakuma, J.
Phys. Soc. Jpn. 81, 024705 (2012)15) M. Futamoto, M. Nakamura, M. Ohtake, N. Inaba, and T.
Shimotsu, AIP Adv. 6, 085302 (2016)16) Y. Kota, and A. Sakuma, J. Phys. Soc. Jpn. 81, 084705(2012)
17) I. A. Campbell, A. Fert, and O. Jaoul, J. Phys. C 3, S95(1970)
Received Oct. 19, 2018; Accepted Dec. 25, 2018

Journal of the Magnetics Society of Japan Vol.43, No.2, 201933

INDEX
J. Magn. Soc. Jpn., 43, 34-41 (2019)<Paper>Dynamics of magnetization and easy axis of individual ferromagnetic
nanoparticle subject to anisotropy and thermal fluctuations
Satoshi Ota and Yasushi Takemura*

Department of Electrical and Electronic Engineering, Shizuoka University, 3-5-1 Johoku, Naka-ku,

Hamamatsu 432-8561, Japan*

Department of Electrical and Computer Engineering, Yokohama National University, 79-5 Tokiwadai, Hodogaya-ku,

Yokohama 240-8501, Japan

In order to design magnetic nanoparticles that are optimized for biomedical applications such as hyperthermia
treatment and magnetic particle imaging, their magnetization dynamics should be analyzed. Through conventional
analytical methods related to magnetization dynamics such as susceptibility and the magnetization curve, the net
properties of magnetization have been previously evaluated. However, an examination of net properties does notyield information related to the stochastic dynamics due to thermal fluctuations and systematic dynamics influenced
by the anisotropy with respect to single particles. In this study, the time evolution of the magnetization and easy
axis of an individual magnetic nanoparticle was numerically simulated using the Landau-Lifshiz-Gilbert equation interms of magnetic nanoparticles with fixed and rotatable easy axes, solids and liquids, respectively. The
superparamagnetic and ferromagnetic regimes of the Langevin dynamics were clearly discerned through evaluationof the effects of anisotropy and thermal fluctuation, and an index for determining the transition point between theseregimes provided. In particular, effects related to the core volume and the anisotropy constant associated with theanisotropy energy were assessed using the Stoner-Wohlfarth model.
Key words: magnetic nanoparticles, anisotropy energy, thermal energy, easy axis
1. Introduction
Magnetization dynamics in magnetic nanoparticles(MNPs) were important to develop biomedical
application. MNPs are used as a heat source for
hyperthermia 1) and as a tracer for magnetic particle
imaging (MPI) 2). In general, with respect tohyperthermia, heat dissipation is derived from theenergy losses such as hysteresis and magnetic
relaxation losses. The hysteresis loss is due to thepresence of the ferromagnetic regime of the MNPs,
which in turn is derived from the anisotropy of themagnetization along the easy axis. The magnetic
relaxation loss is related to the hysteresis associatedwith the time delay of the magnetization response fromthe applied field. This hysteresis loss that is due toanisotropic effects and magnetic relaxation also affects
the harmonic response of the magnetization that is
crucial for MPI.
In order to evaluate magnetization dynamics, the ACsusceptibility 3)‒6) and AC magnetization curve 7)‒9) havebeen previously determined. In addition, the specificloss power and the intrinsic loss power 10)‒13) for
hyperthermia, the magnetic particle spectrometry
signal 14), and the harmonics of the magnetization 15)‒17)for MPI have been assessed. Theoretical and numericalsimulations focusing on AC susceptibility and AC
magnetization curves have also revealed the dynamics
of magnetization 18)‒21) and the easy axis 22),23) withrespect to magnetic relaxation.
Magnetic relaxation is divided into Néel and
Brownian relaxations, referring to relaxations of themagnetization and the easy axis, respectively. In termsof conventional theory, magnetic relaxation occurs inaccordance with the theory of the effective relaxation
time τeff, given by 1/τeff= 1/τN + 1/τB, where τN and τB are

34

the Néel and Brownian relaxation times, respectively 18).
This conventional theory indicated the dominance of one
type of relaxation, with one relaxation time shorter thanthe other. However, the superimposition of the Néel and
Brownian relaxations has been observed using
22)
numerical
simulations
andmagnetization
measurements 24). In our previous study, increases inmagnetization derived from the Brownian relaxation in
addition to the Néel relaxation were measured when
applying a pulse field 25). Moreover, the dynamics of theeasy axis of MNPs in a liquid was estimated using thesubtracted magnetization curve indicating thedifference between the measured magnetization curvein a solid and that in a liquid 26).
Through an analysis of susceptibility and themagnetization curve, their dynamics in terms ofindividual particles remain unclear, although the net
properties of the magnetization and easy axis areevaluated. García-Palacios and Lázaro observed the
trajectories of individual magnetization using theLangevin-dynamics approach, which indicated thepresence of magnetization reversal properties associatedwith anisotropy and thermal fluctuation 27). Lopez-Diazet al. showed that the transition of magnetism depended
on particle size, in terms of the switching responses oftypical individual magnetizations 28). In this study,
using numerical simulation methods that employ theLandau-Lifshiz-Gilbert (LLG) equation, the dynamics ofmagnetization and the easy axis were evaluated byfocusing on single particle trajectories in conditions with
fixed and rotatable easy axes, i.e., a solid and a fluid,
respectively. The effects of the uniaxial anisotropy along
the easy axis and thermal fluctuation at the transition
from the superparamagnetic regime represented bystochastic Langevin dynamics to the ferromagnetic

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX

regime given by the Stoner-Wohlfarth model 29) wereassessed. This single particle model reveals the effects
of anisotropy and thermal fluctuations in the
superparamagnetic and ferromagnetic regimes.
2. Numerical simulation
2.1 Simulated materials
Monodispersed single-domain MNPs were consideredin the numerical simulation. The interparticle dipole
interaction was ignored, and the hydrodynamic
diameters dh of the MNPs was equal to their core
diameters dc. The effective uniaxial anisotropy
constants Ku were 8, 9.8, 12, 14.7, and 18 kJ/m3 with dcof 11 nm. In addition, with respect to the case for which
Ku was equal to 12 kJ/m3, dc of 9.61, 10.3, 11.8, and 12.6
nm were used. These parameters were considered forthe evaluation of the magnetization dynamics anddepended on the ratio of the anisotropy energy to thethermal energy, ξ= KuVM/kBT, where VM is the volume
of a core particle, kB is the Boltzmann constant (1.38 ×
10−23 J/K), and T is the temperature, set to 300 K (Table1). The saturation magnetic moment Ms was 269 kA/m,
which was taken from the information related to thematerial measured in Ref. 24).
2.2 Simulation methods
The dynamics of magnetization in the case of arotatable easy axis such as the MNPs in a fluid are
calculated using the modified LLG equation21),23). In a
fluid, the dynamics of the easy axis is considered inaddition to the magnetization dynamics. The orientationof the easy axis is calculated using the unit vector andthe angular velocity of the easy axis, n and ω,
respectively, as follows:𝑑𝑑𝑑𝑑� 𝛚𝛚 � 𝑑𝑑.
𝑑𝑑𝑑𝑑

(1)

𝑑𝑑𝑑𝑑 𝜇𝜇0 𝑀𝑀s 𝑉𝑉M 𝑑𝑑𝑑𝑑�
� 𝜇𝜇0 𝑀𝑀s 𝑉𝑉M 𝑑𝑑 � �𝐇𝐇ex � 𝐇𝐇th � � 𝜅𝜅𝛚𝛚 � Гth ,
𝑑𝑑𝑑𝑑
𝛾𝛾
𝑑𝑑𝑑𝑑

(2)

The differential equation of the angular velocity of theeasy axis is given by 21)
Θwhere Θ is the moment of inertia, μ0 is the permeabilityof free space, γ is the gyromagnetic ratio, m is themagnetic moment normalized by the saturation
magnetic moment, Hex is the excitation field, Hth isthermally fluctuating field, 𝜅𝜅 is the friction coefficient,Γth is random torque due to thermal fluctuation. γ isestimated to be γ= μ0MsVM(1+α2)/(2ατNkBT) where τN=
π1/2τ0exp(ξ)/2ξ1/2 30). 𝜅𝜅 is expressed as 𝜅𝜅 = 6ηVH where ηis the medium viscosity, set to 0.89 mPa·s, and VH is theparticle hydrodynamic volume. Hth and Γth have
Gaussian distributions with a zero mean. The varianceof Hth and Γth satisfied the following equations:〈𝐻𝐻th ,𝑖𝑖 �𝑡𝑡�𝐻𝐻th ,𝑗𝑗 �𝑡𝑡𝑡�� �

2𝛼𝛼𝑘𝑘B 𝑇𝑇𝛿𝛿 𝛿𝛿�𝑑𝑑 � 𝑑𝑑𝑡�,1 � 𝛼𝛼 2 𝛾𝛾𝛾𝛾0 𝑀𝑀s 𝑉𝑉M 𝑖𝑖𝑖𝑖

(3)

〈Г𝑖𝑖 �𝑡𝑡�Г𝑗𝑗 �𝑡𝑡𝑡�� � 2𝜅𝜅𝑘𝑘B 𝑇𝑇𝑇𝑇𝑖𝑖𝑖𝑖 𝛿𝛿�𝑑𝑑 � 𝑑𝑑𝑡�.
(4)
In Eqs. (3) and (4), i and j are the Cartesian indices ofdifferent particles, and δij is the Kronecker delta
function, and δ is the Dirac delta function.
Here, with respect to ω, the inertial effects can beignored, and μ0MsVM/γ is extremely small. Thus, Eq. (2)is rewritten as the equation of ω, as follows:

Table 1 The anisotropy constant Ku and the core
diameter dc of the numerically simulated MNPs. Ku
was equal to 8, 9.8, 12, 14.7, and 18 kJ/m3 with dc=
11nm. For Ku=12 kJ/m3, dc was 9.61, 10.3, 11, 11.8,
and 12.6 nm, where the ratios of the anisotropy
energy to the thermal energy ξ= KuVM/kBT were 1.35,
1.65, 2.02, 2.47, and 3.03, respectively.

Ku [kJ/m3](dc= 11 nm)
dc [nm](Ku= 12 kJ/m3)
ξ

8

9.8

12

14.7

18

9.61

10.3

11

11.8

12.6

1.35

1.65

2.02

2.47

3.03

1
𝛚𝛚 � �𝜇𝜇0 𝑀𝑀s 𝑉𝑉M 𝑑𝑑 � �𝐇𝐇ex � 𝐇𝐇th � � Гth �.
𝜅𝜅

(5)

𝑑𝑑𝑑𝑑
� 𝛚𝛚 � 𝑑𝑑.
𝑑𝑑𝑑𝑑

(6)

When the magnetization is entirely bound to the easy
axis represented by Eq. (1), the dynamics of themagnetization is given by

In addition to the term influenced by the rotation of theeasy axis in Eq. (6), the magnetization dynamics isrepresented by the damping process according to theconventional LLG equation 31), as follows:
𝛾𝛾
𝛚𝛚
𝑑𝑑𝑑𝑑
� 𝛚𝛚 � 𝑑𝑑 �
𝑑𝑑 � �𝐇𝐇eff � 𝐇𝐇th � �
1 � 𝛼𝛼 2
𝛾𝛾
𝑑𝑑𝑡𝑡
�

𝛚𝛚
𝛼𝛼𝛼𝛼
𝑑𝑑 � �𝑑𝑑 � �𝐇𝐇eff � 𝐇𝐇th � ��,𝛾𝛾
1 � 𝛼𝛼 2(7)where α is damping parameter (α= 0.1), Heff is effective
field composed of Hex and anisotropy field (Han). The
anisotropy
field
isestimated
to
be
Han=
2Ku(m·n)n/(μ0Ms). ω/γ indicates the field derived fromthe rotation of the easy axis.
In the case of the model with a fixed easy axis, theterm including ω in Eq. (7) was omitted. For thenumerical simulations, the differential equations weresolved with the Runge‒Kutta method. In the numericalsimulations used for the analysis of the net
magnetization dynamics, there were 29440 particles.
The excitation AC field was applied along the x-axis
with the amplitude Hmax of 8 kA/m and the frequency fof 10 kHz except for the frequency dependence analysis.
2.3 Time evolution of magnetization and easy axis
orientations
The magnetization and easy axis orientations wereobserved during the half cycle for a phase of 0‒π in an
applied AC field, Hmaxcos(2πft), where t is the time. The
chosen time interval was 100 ns. Because the dynamics
of the individual magnetization and easy axis are
subject to fluctuations caused by thermal disturbances
and therefore show variance, it is necessary to confirm
the statistical accuracy of these simulations. With
respect to the statistical analysis, the magnetizationand easy axis orientations were evaluated in one cycle.
The number of particles N was 100.

Journal of the Magnetics Society of Japan Vol.43, No.2, 201935

INDEX

Fig. 1 Time evolutions of individual magnetization orientation in the spherical model for the MNP with Ku= 12
kJ/m3 and dc= 11 nm, with the easy axis fixed along (a-i) θ= 0 rad, (a-ii) θ= π/3 rad, and (a-iii) θ= π/2 rad in the x-z
plane. The lines over the y-z plane perpendicular to the applied field along the x-axis indicate the reversal of themagnetization, and other lines represent the rotation of the magnetization. (b) Magnetization curve and (c)distribution of the magnetization orientation along the applied field direction on the x-axis for the MNPillustrated in (a-i)‒(a-iii). The statistical analysis was conducted in one cycle of the applied field, with a timeinterval of 100 ns (N= 100).

Fig. 2 Time evolutions of individual magnetization orientations in the spherical model for the MNP with (a)Ku= 8, (b) 9.8, (c) 14.7, and (d) 18 kJ/m3 with dc= 11 nm, with the easy axis fixed along θ= 0 rad.
3. Results and discussions
3.1 Dynamics of magnetization with a fixed easy axis
3.1.1 Effect of easy axis direction
Figure 1 shows the time evolutions of the individual
magnetization orientation in the three-dimensionalmodel with respect to a MNP of Ku= 12 kJ/m3 and dc= 11
nm. The easy axis was oriented so that the angle
between the easy axis and the applied field was θ= 0,
π/3, and π/2 rad in the x-z plane in Figs. 1(a-i)‒(a-iii),
respectively. On the basis of the Stoner-Wohlfarthmodel, the magnetization was distributed around theeasy axis and along the direction of the applied field
because of low potential energy 20),29).
Figure 1(b) shows the magnetization curve of the
MNP with Ku= 12 kJ/m3 and dc= 11 nm. A slight
coercivity indicates ferromagnetism. The distribution of
the magnetization along the same direction as theapplied field is evidenced in Fig. 1(c). With decreasing θ,
the magnetization distribution was strongly aligned
along the direction of the applied field, owing to the
reduced anisotropy energy barrier 9),32),33). Two nodes
connected by a line show the continuous trajectory of
the magnetization in Fig. 1(a). The lines over the y-z
plane perpendicular to the applied field indicate themagnetization reversal, while other lines represent therotation of the magnetization. Random oscillations in
the magnetization including frequent reversals androtations were due to thermal fluctuations resulting
from the superparamagnetism.

36

3.1.2 Dependence of the anisotropy constant and core
diameterFigure 2 shows the time evolutions of the individual
magnetization depending on Ku, with dc= 11 nm. The
magnetization reversals that are due to thermal
fluctuations were suppressed by the large anisotropy
energy. Moreover, in contrast to the spreading of themagnetization along the spherical surface in thesmall-Ku regime, the distribution of the magnetization
was concentrated on the direction of the easy axis forlarge ξ. The magnetization was strongly bound by theanisotropy.
The ratio of the magnetization reversal in the
trajectories was confirmed in Fig. 3. According to thetime evolution shown in Fig. 2, the magnetization
reversal that was due to the thermal noise wassuppressed by the large anisotropy energy. When dc wasincreased with Ku= 12 kJ/m3, the magnetization
reversal ratio decreased because of the increase in theanisotropy energy, which is similar to the case in which
Ku increased with dc= 11 nm. However, themagnetization reversal ratios for each ξ were
statistically larger than for the case of large dc despite
the same ξ values (Table 1). The p-values were p < 0.01
in ξ ≤ 2.47 and p < 0.1 in ξ= 3.03.
The potential energy Ep is given by
𝐸𝐸p � �u 𝑉𝑉M sin2 �𝜃𝜃 𝜃 𝜃𝜃� � �0 𝑀𝑀𝑠𝑠 𝑉𝑉M |𝐇𝐇ex |cos�𝜑𝜑�,(8)where φ is the angle between the magnetization and theapplied field 29). The first and second terms in Eq. (8)are the anisotropy and Zeeman energies, respectively.

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

x-axisz-axisd = 11 nm
c3

K = 12 kJ/m
u

40
20
0

8

10 12 143 16 18
K [kJ/m ]

9

10

u

11 12
d [nm]

13cFig. 3
Ratio of the magnetization reversal
represented over the y-z plane along the x-axis andthe x-y plane along the z-axis for the all themagnetization trajectories. Ku and dc for the MNPs
were shown in Table 1. The easy axis was fixed on thex-axis. The statistical analysis was conducted within
one cycle of the applied field with a time interval of
100 ns (N= 100).

s

Magnetization [M/M ](a) 0.5
0.25

Ku [kJ/m3]
9.8
14.7
18

3.1.3 Transition in the magnetic regime
As shown in Fig. 2, the change in the anisotropy
energyinducesthetransition
fromthesuperparamagnetic regime, the stochastic aspect of theLangevin dynamics, to the ferromagnetic regime, theirenergetically systematic aspect shown by the
Stoner-Wohlfarth model 28). The Langevin equation
describing the system of time evolution for themagnetization is generalized asdc [nm]
10.3
11.8
12.6

0

-0.25
-0.5

(b)

5

-4
0
4
8
-8
-4
0
4
8 -8
Magnetic field [kA/m]
Magnetic field [kA/m]
Mmax
Hc

dc= 11 nm

Ku= 12 kJ/m3

0.5
0.4

2

0.2

1

0.1

[M/M ]

c

0.3

max

Mmax
Hc

3

𝑑𝑑𝑑𝑑�𝑡𝑡�� ��𝐦𝐦�𝑡𝑡�� � ��𝑡𝑡�,
𝑑𝑑𝑑𝑑

M

H [kA/m]

4

Ku= 12 kJ/m3

dc= 11 nms

08

10 12 14 3 16 18
K [kJ/m ]
u

9

10

11 12
d [nm]

13

Hc in ξ= 3.03 was particularly large. In a solid, thehysteresis is due to pinning of the magnetization to theeasy axis and the phase delay derived from the Néel
relaxation. In ξ= 3.03, because the Néel relaxation timeis short compared to the frequency of the applied field,
the hysteresis is dominantly due to pinning of themagnetization by the large anisotropy energy barrier.
For the same ξ values, Hc was equal with each other forξ ≤ 2.47. However, Hc in Ku= 18 kJ/m3 and dc= 11 nm
was larger than that in the case of Ku= 12 kJ/m3 and dc=
12.6 nm in ξ= 3.03, because Zeeman energy enough forthe magnetization to reverse is similar with each other.
The magnetization overcoming the energy barrier was
reduced by the small Zeeman energy in the case of ξ=
3.03. The decrease of Mmax for Ku ≥ 12 kJ/m3 and dc= 11
nm was also associated with the large anisotropy energy
barrier and the unchanged Zeeman energy. Mmax
increased with increasing dc where Ku= 12 kJ/m3
because of the presence of high Zeeman energy.

0cFig. 4 (a) Magnetization curve and (b) the maximal
magnetization Mmax, and coercivity Hc depending on
Ku and dc, shown in Table 1. The easy axis was fixed
on the x-axis.
An increase of dc enhances the amplitude of both theanisotropy and Zeeman energies, whereas Ku affects
only the anisotropy energy. The effect of dc on the
Zeeman energy is similar with that of |Hex| in the
Stoner-Wohlfarth model 34). Thus, the effectively large|Hex| in addition to the large anisotropy inhibitedmagnetization reversal due to thermal noise in the caseof a large dc for the same ξ. The magnetization reversal
along the z-axis was also evaluated in Fig. 3. However,
the reversal ratio was independent of the anisotropy
energy because the easy axis associated with theuniaxial anisotropy and the applied field wereperpendicular to the z-axis, and only random thermal
fluctuations affected the magnetization oscillation.
Figures 4(a) and (b) show the magnetization curves,and the dependence of the maximal magnetization Mmax
and coercivity Hc on Ku and dc. Hc was increased by thelarge anisotropy energy. The hysteresis associated with

(9)

where the function A(m(t)) represents the systematic
term that represents the potential energy and thefunction F(t) is the term that describes fluctuations thatare due to thermal disturbance 27),35). Where thefluctuating term dominates, the magnetization
dynamics
clearlyplace
thesysteminthesuperparamagnetic regime.
Figure 5 shows the dependence of the energy ratio
Ep/kBT on φ by Eq. (8). The large anisotropy in the
superparamagnetic
regimeprevents
themostenergetically stable magnetization from reversion to themetastable condition by overcoming the anisotropy
energy barrier through thermal fluctuations. Here, the3
2B

x-axisz-axisp

60

E /k T

Magnetization reversal
ratio [%]

INDEX

1

3

(K [kJ/m ], d [nm])= 0 raduc(8, 11)
Thermal fluctuation(12, 11)(12, 9.61)
Metastable

The most
stableEnergybarrier0

Easy axism|H |= 6 kA/m
ex

0

/3

2/3
 [rad]Fig. 5 Potential energy normalized by the thermalenergy, Ep/kBT with respect to φ with the easy axis
fixed along θ= 0 rad. (Ku, dc) were (8 kJ/m3, 11 nm),
(12 kJ/m3, 11 nm), and (12 kJ/m3, 9.61 nm), where ξ=
1.35, 2.02, and 1.35, respectively. The intensity of
the applied field |Hex| was 6 kA/m.

Journal of the Magnetics Society of Japan Vol.43, No.2, 201937

INDEX

Fig. 6 Time evolutions of the individual orientations of the (a‒d) magnetization and (e‒h) rotatable easy axis in
the spherical model with respect to the MNP with (a, e) Ku= 9.8, (b, f) 12, (c, g) 14.7, and (d, h) 18 kJ/m 3 with dc=
11 nm.

3.2 Magnetization dynamics with a rotatable easy axis
3.2.1 Trajectories of individual magnetization and easy
axis
Figure 6 shows the time evolutions of the individual
magnetization and the rotatable easy axis depending on
Ku for dc= 11 nm. For Ku ≤ 12 kJ/m3 (ξ ≤ 2.02), as aneffect of thermal fluctuations, the easy axis was
randomly rotated along the spherical surface, whilefluctuating magnetization reversal occurred. For Ku ≥14.7 kJ/m3 (ξ ≥ 2.47), both the magnetization and theeasy axis were rotated along the spherical surface.
The anisotropy energy bound the magnetization to the38

rotatable easy axis, as in the case of the fixed easy axis.
The angle between the magnetization and the easy axis
ψ was evaluated for the individual particles. ψ was
calculated from the vector product between the unit
vector of the magnetization and the easy axis, |m × n|.
Origin symmetry is applicable for the easy axis (Eq. (8)).
The averaged value of ψ shown in Fig. 7 indicates thatψ was reduced with a large anisotropy energy with theincrease of both Ku and dc. Therefore, considering themagnetization dynamics in a fluid, the reduction in theanisotropy energy barrier implies that the particles
exhibitsuperparamagnetic
behaviorintheferromagnetic regime, which is empirically confirmed by
the reduced coercivity in a fluid compared with that in asolid on the DC magnetization curve 24). This may be
distinguished from the superparamagnetic regime as
the effect of thermal fluctuations is reduced in the caseof MNPs due to the large anisotropy energy.
/4

d = 11 nm

3

K = 12 kJ/mc

 [rad]energy barrier is defined as the maximal value of Ep/kBT
and is enlarged along with the increments of Ku. At thesame ξ value, with large dc and an unchanged energy
barrier height, Ep/kBT is increased and decreased in themetastable and the most stable conditions, respectively(Fig. 5). When dc was increased, the magnetic moment
was monotonically increased in ξ ≤ 3.03 because
energetic stability was enhanced and reduced in themost stable and metastable conditions, respectively (Fig.
4(b)). The inverse reversal into the metastable condition
that was due to the thermal disturbance in the
superparamagnetic regime was suppressed by the highenergy barrier, which led to an increased probability ofbound magnetization in the most stable condition. By
contrast, in the ferromagnetic regime at large Ku, thehigh energy barrier inhibits magnetization reversal
from the metastable to the most stable conditions
because both the minimal and secondary minimal
potential energies were unchanged, which results in a
reduced Mmax. Thus, the plotted dependence of Mmax on
Ku showed a peak at Ku= 12 kJ/m3 (Fig. 4(b)). The valueof ξ at the Mmax peak is the index of the energetic
threshold of the transition from the superparamagnetic
regime to the ferromagnetic regime in the Langevindynamics described by Eq. (9).

u

/6

/12

0

8

10 12 143 16 18
K [kJ/m ]
u

9

10

11 12
d [nm]
c13

Fig. 7 Angle between the magnetization and theeasy axis ψ depending on the anisotropy energy for
the Ku and dc values shown in Table 1. The easy axis
was rotatable. ψ was calculated using the expression|m × n|. The statistical analysis was conducted in
one cycle of the applied field with a time interval of
100 ns (N= 100).

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEXs

Magnetization [M/M ]0.3

3

K [kJ/m ]
u
9.8
14.7
18

d [nm]
c
10.3
11.8
12.6

0
-0.3
-0.6
6

(b)

5

3

d = 11 nm

K = 12 kJ/m

c

-8

u

-4
0
4
8
-4
0
4
8 -8
Magnetic field [kA/m]
Magnetic field [kA/m]

d = 11 nm

3

K = 12 kJ/m

c

u

0.5

M
max
H

4

0.4
0.3

M
max
H

0.2

1
0

s

2

c

[M/M ]

Hc [kA/m]

c

3

0.6

max

3.3 Frequency dependence of magnetization and easy
axis dynamics
Figure 10(a) shows the time evolutions of theindividual magnetization at f= 1 kHz and 100 kHz. The
time evolution at f= 10 kHz is shown in Fig. 1(a-i). The
magnetization reversals that are due to thermal
fluctuations was suppressed with the increasing field
frequency, which is indicated in Fig. 10(b). The
frequency dependence of the magnetization curves inFig. 10(c) shows the increasing Hc with increasing field
frequency due to the phase delay of the magnetization
associated with the Néel relaxation. In the highfrequency, the magnetization dynamics correspond totheferromagnetic
regime.
The
remanent
magnetizations without overcoming the energy barrier
along the easy axis in the high frequency are(a) 0.6

M

0.1
8

10 12 14 16 18
K [nm]

9

10

u

11 12
d [nm]

130cFig. 8 (a) Magnetization curve and (b) the maximal
magnetization Mmax, and coercivity Hc depending on
the anisotropy energy for each Ku and dc value shownin Table 1. The easy axis was rotatable.
0.5

d = 11 nm
c3

K = 12 kJ/m
u

0.4

M,max/kBT

3.2.2 Properties of net magnetization
Figures 8(a) and (b) show magnetization curves, andtheir corresponding Mmax and Hc, which increased withincreasing ξ. High dc values corresponded to a high Mmax,
for the same ξ values. This was indicated by the large
Zeeman energy that corresponded to large dc, as in thecase of the fixed easy axis (Eq. (8)). In the case of therotatable easy axis, because the MNPs exhibit
superparamagnetic behavior, the Hc derived from thehysteresis that was due to the anisotropy energy barrier
is marginal. At high frequencies, the increase in Hc isassociated with the phase delay derived from themagnetic relaxations. The magnetic torques ΓM=
μ0MsVM|m×Hex| that were included in the first term ofEq. (5) depended on both Ku and dc. Figure 9 shows theratio of the maximum value of the magnetic torqueΓM,max to the thermal energy, ΓM,max/kBT. ΓM,maxincreased with increasing Ku (ξ ≤ 2.02) and saturated in
2.02 ≤ ξ ≤ 3.03. The dependence of magnetic torque on
Ku has been confirmed by Carrey and Hallali 36). ΓM is
proportional to VM and increases for large dc. The large
anisotropy energy enhances ΓM and affects the
Brownian relaxation time. Yoshida and Enpuku tookaccount of the AC field amplitude Hac associated withmagnetic torque, describing an effective Brownian
relaxation time, τB,eff= τB/[1+0.07(μ0MsHac/kBT)]1/2,
where τB= 3ηVH/kBT 37). Mamiya and Jeyadevan also
indicated the effect of magnetic torque on the Brownian
relaxation time 22). According to these conventional
studies, the Brownian relaxation time is reduced due tothe large magnetic torque, which induces a small Hc.
Usadel also showed that the phase delay of themagnetization in a viscous liquid was reduced in thecase of a large field amplitude 21). In Fig. 8, Hc wasnegligible for small Ku (≤ 9.8 kJ/m3) where dh= dc= 11
nm, which indicates that the phase delay of themagnetization derived from the Brownian relaxation
was negligible. Thus, Hc increased for large Ku wheredc= 11 nm, which was due to the phase delay of themagnetization derived from only the Néel relaxation. In
addition, Hc increased with increasing dc at Ku= 12
kJ/m3, which was due to the phase delay related to the
Néel relaxation 18) because Hc values were the same forthe same ξ (≤ 2.47). On the other hand, in the case forwhich Ku= 12 kJ/m3 and dh= dc= 12.6 nm, Hc was larger
than that when Ku= 18 kJ/m3 and dc= 11 nm because of
the phase delay associated with Brownian relaxation.

0.3
0.2
0.1
0 8

10 12 143 16 18
K [kJ/m ]
u

9

10

11 12
d [nm]

13cFig. 9 Maximal magnetic torque normalized by thethermal energy, ΓM,max/kBT, depending on theanisotropy energy, for each Ku and dc value shown in
Table 1.
represented as the increases of the phase delay and Hc
in the net magnetization properties.
In the case of the rotatable easy axis, the time
evolutions of the individual magnetization and easy axis
at f= 1 kHz and 100 kHz are shown in Fig. 11(a). The
time evolution at f= 10 kHz is shown in Fig. 6(b, f).
Figure 11(b) shows the frequency dependence of themagnetization curves and the mean orientation of theeasy axis, <cos θ>. At f= 1 kHz, Hc of the magnetization
curve and the hysteresis of the easy axis was negligible.
The slight Hc confirmed at 10 kHz was derived from the
Néel relaxation because this Hc value was similar to the
Hc in the case of the fixed easy axis. At 100 kHz, Hc inthe case of the rotatable easy axis was larger than forthe case of the fixed easy axis, which is described as thephase delay derived from the Brownian relaxation. In
particular, the minimum value of <cos θ> at 100 kHz
was higher than 0.5, which indicates that the easy axis
was steadily oriented along the applied field direction 23).
The time evolution of the easy axis also illustrated therotation with the orientation toward the specificdirection at 100 kHz, whereas the easy axis was

Journal of the Magnetics Society of Japan Vol.43, No.2, 201939

INDEX

Fig. 10 Time evolutions of the individual magnetization orientation in the spherical model for the MNP with
Ku= 12 kJ/m3 and dc= 11 nm with the easy axis fixed along θ= 0 at the field frequency of (a-i) 1 kHz and (a-ii) 100
kHz. (b) Ratio of the magnetization reversal represented over the y-z plane along the x-axis and (c) magnetization
curve in the frequency of 1‒100 kHz. The statistical analysis was conducted in one cycle of the applied field, witha time interval of 100 ns (N= 100).

Fig. 11 Time evolutions of the individual (a-i, iii) magnetization and (a-ii, iv) rotatable easy axis orientation in
the spherical model for the MNP of Ku= 12 kJ/m3 and dc= 11 nm at the field frequency of (a-i, ii) 1 kHz and (a-iii,iv) 100 kHz. (b-i) Magnetization curve, (b-ii) easy axis orientation <cos θ>, and (c) ψ in the frequency of 1‒100
kHz. The statistical analysis was conducted in one cycle of the applied field, with a time interval of 100 ns (N=100).
randomly orientated over the spherical surface at 1 kHz
and 10 kHz. At 100 kHz, the magnetization wasoriented along the easy axis toward the specificdirection and was rotated on the spherical surface,which is similar to the case of the large ξ at 10 kHz in
Fig. 6. The transition from the superparamagnetic
behavior in the low frequency to the ferromagnetic
behavior in the high frequency was observed in the caseof the rotatable easy axis. Though ψ increased withincreasing frequency, the increased value was slight in
Fig. 11(c), which indicates that the magnetization
dynamics show the ferromagnetic regime regardless ofthe field frequency at Ku= 12 kJ/m3 and dc= 11 nm. It isobserved that the frequency no longer affected themagnetic regime whereas the magnetic behavior was
influenced by the frequency.
4. Conclusion
The dynamics of individual particle magnetization andthe easy axis were evaluated for the three-dimensional

40

trajectories of a single MNP using numerical simulation.
In this paper, the effects of the core size and anisotropyof MNPs on the magnetization and easy axis dynamics
were revealed in the same anisotropy energy. For the
MNP in a solid (fixed easy axis) model, the ratio of theanisotropy energy to the thermal energy acts as theindex of the threshold for the transition from thesuperparamagnetic to the ferromagnetic regime,
considering the effects of stochastic thermal fluctuations
and the potential energy of the MNP based on the
Stoner-Wohlfarth model. In a fluid (rotatable easy axis),
depending on the anisotropy energy, the dynamics ofthe magnetization and the easy axis changed from thatof steep reversal to rotation on the spherical surface,which was associated with the transition from thesuperparamagnetic to the ferromagnetic regime. The
net magnetization shown in the magnetization curvesindicates superparamagnetic behavior in the case of therotatable easy axis, while the ferromagnetic regime wasclearly observed when examining the trajectories of the

Journal of the Magnetics Society of Japan Vol.43, No.2, 2019

INDEX

single MNP. The specific magnetic behavior in a fluid
was clearly distinguished from the magnetic regime by
the observation of the individual MNP. The magnetic
regime in a fluid was illustrated as the difference
between the reversal and rotation in the magnetization
dynamics. Moreover, it is also statistically revealed thatthe traceability of the easy axis to the magnetization ina fluid is enhanced in large anisotropy energy. The
transition of the magnetic behavior depending on theapplied field frequency was also confirmed. However, it
is of note that the magnetic regime based on thepotential energy was no longer influenced by thefrequency. This evaluation of the dominance ofsuperparamagnetic and ferromagnetic regimes in thestochastic Langevin dynamics of the system that
depends on the anisotropy energy and thermal
fluctuations contributes to the optimal material designof MNPs for use in biomedical applications such as
hyperthermia treatment and MPI.
Acknowledgements This work was partially supportedby the JSPS KAKENHI Grant Numbers 15H05764,
17H03275, 17K14693.
References
1) A. Jordan, R. Scholz, P. Wust, H. Fähling, R. Felix: J.
Magn. Magn. Mater., 201, 413–419 (1999).
2) B. Gleich and J. Weizenecker: Nature, 435,
1214–1217 (2005).
3) P. C. Fannin, B. K. P. Scaife, and S. W. Charles: J.
Phys. D: Appl. Phys., 21, 533–534 (1988).
4) R. Hergt, R. Hiergeist, I. Hilger, W. A. Kaiser, Y.
Lapatnikov, S. Margel, U. Richter: J. Magn. Magn.
Mater., 270, 345–357 (2004).
5) M. Suto, Y. Hirota, H. Mamiya, A. Fujita, R. Kasuya,
K. Tohji, and B. Jeyadevan: J. Magn. Magn. Mater.,
321, 1493–1496 (2009).
6) S. B. Trisnanto and Y. Kitamoto: Phys. Rev. E, 90,032306 (2014).
7) H. Kobayashi, A. Hirukawa, A. Tomitaka, T. Yamada,
M. Jeun, S. Bae, and Y. Takemura: J. Appl. Phys.,107, 09B322 (2010).
8) D. Cabrera, J. Camarero, D. Ortega, and F. J. Teran:J. Nanopart. Res., 17, 121 (2015).
9) S. A. Shah, D. B. Reeves, R. M. Ferguson, J. B.
Weaver, and K. M. Krishnan: Phys. Rev. B, 92,094438 (2015).
10) S. Dutz, J. H. Clement, D. Eberbeck, T. Gelbrich, R.
Hergt, R. Müller, J. Wotschadlo, M. Zeisberger: J.
Magn. Magn. Mater., 321, 1501–1504 (2009).
11) M. Kallumadil, M. Tada, T. Nakagawa, M. Abe, P.
Southern, Q. A. Pankhurst: J. Magn. Magn. Mater.,
321, 1509–1513 (2009).
12) E. Kita, T. Oda, T. Kayano, S. Sato, M. Minagawa,
H. Yanagihara, M. Kishimoto, C. Mitsumata, S.
Hashimoto, K. Yamada, and N. Ohkohchi: J. Phys.

D: Appl. Phys., 43, 474011 (2010).
13) S. Ota, T. Yamada, and Y. Takemura: J. Nanomater.,
2015, 836761 (2015).
14) S. A. Shah, R. M. Ferguson, and K. M. Krishnan: J.
Appl. Phys., 116, 163910 (2014).
15) D. Eberbeck, F. Wiekhorst, S. Wagner, and L.
Trahms: Appl. Phys. Lett., 98, 182502 (2011).
16) T. Yoshida, N. B. Othman, and K. Enpuku: J. Appl.
Phys., 114, 173908 (2013).
17) S. Ota, R. Takeda, T. Yamada, I. Kato, S. Nohara,
and Y. Takemura: Int. J. Magn. Part. Imag., 3,1703003 (2017).
18) R. E. Rosensweig: J. Magn. Magn. Mater., 252,
370–374 (2002).
19) P. C. Fannin, C. N. Marin, C. Couper: J. Magn.
Magn. Mater., 322, 1677–1681 (2010).
20) J. Carrey, B. Mehdaoui, and M. Respaud: J. Appl.
Phys., 109, 083921 (2011).
21) K. D. Usadel: Phys. Rev. B, 95, 104430 (2017).
22) H. Mamiya and B. Jayadevan: Sci. Rep., 1, 157(2011).
23) T. Yoshida, S. Bai, A. Hirokawa, K. Tanabe, and K.
Enpuku: J. Magn. Magn. Mater., 380, 105−110(2015).
24) S. Ota, T. Yamada, Y. Takemura: J. Appl. Phys., 117,
17D713 (2015).
25) S. B. Trisnanto, S. Ota, and Y. Takemura: Appl.
Phys. Express, 11, 075001 (2018).
26) S. Ota and Y. Takemura: Appl. Phys. Express, 10,085001 (2017).
27) J. L. García-Palacios and F. J. Lázaro: Phys. Rev. B,
58, 14937–14958 (1998).
28) L. Lopez-Diaz, L. Torres, and E. Moro: Phys. Rev. B,
65, 224406 (2002).
29) E. C. Stoner and E. P. Wohlfarth: Philos. Trans. R.
Soc. London, Ser. A, 240, 599 (1948).
30) W. T. Coffey and Y. P. Kalmykov: Langevin equation
3rd ed., pp. 126‒130 (World Scientific, Singapore,2012).
31) W. T. Coffey and Y. P. Kalmykov: J. Magn. Magn.
Mater., 164, 133–142 (1996).
32) T. Yoshida, Y. Matsugi, N. Tsujimura, T. Sasayam,
K. Enpuku, T. Viereck, M. Schilling, F. Ludwing: J.
Magn. Magn. Mater., 427, 162–167 (2017).
33) R. Takeda, S. Ota, T. Yamada, Y. Takemura: J.
Magn. Soc. Jpn., 42, 55–61 (2018).
34) A. Sukhov and J. Berakdar: J. Phys.: Condens.
Matter, 20, 125226 (2008).
35) G. E. Uhlenbeck and L. S. Ornstein: Phys. Rev., 36,
823–841 (1930).
36) J. Carrey and N. Hallali: Phys. Rev. B, 94, 184420(2016).
37) T. Yoshida and K. Enpuku: Jpn. J. Appl. Phys., 48,
127002 (2009).
Received Oct. 11, 2018; Accepted Jan. 8, 2019

Journal of the Magnetics Society of Japan Vol.43, No.2, 201941

INDEX

Editorial Committee Members・Paper Committee Members
K. Kobayashi and T. Ono (Director), T. Kato, K. Koike and T. Taniyama (Secretary)
A. Fujita
H. GotoH. Hashino
S. Honda
S. Inui
S. Kasai
A. Kikitsu
H. Kikuchi
T. Kimura
T. Kubota
T. Nagahama
H. Naganuma
M. Naoe
M. Ohtake
N. PhamT. SatoT. SatoK. Sekiguchi
M. Sekino
T. Shima
M. Sonehara
T. Tanaka
S. YamadaK. Yamamoto
H. Yuasa
N. Adachi
S. Isogami
Y. Kota
T. Narita
M. Takezawa
S. Yoshimura

K. Bessho
K. KamataT. MakiD. Oyama
T. Takura

M. DoiH. KatoE. Miyashita
J. Ozeki
M. Tsunoda

T. DoiK. KatoT. Morita
T. Saito
S. Yabukami

Y. Kanai
K. Miura
T. Sasayama
Y. Shiratsuchi

T. Hasegawa
T. KodaS. Muroga
S. Seino
T. Yamamoto

N. Inaba
S. Kokado
H. Nakayama
K. Tajima
K. Yamazaki

Notice for Photocopying
If you wish to photocopy any work of this publication, you have to get permission from the following organization to which licensing of copyright clearance is delegated by the copyright owner.
〈All users except those in USA〉
〈Users in USA〉
Japan Academic Association for Copyright Clearance, Inc.（JAACC）
Copyright Clearance Center, Inc.
6–41 Akasaka 9-chome, Minato-ku, Tokyo 107–0052 Japan
222 Rosewood Drive, Danvers, MA01923 USA
Phone 81–3–3475–5618 FAX 81–3–3475–5619 E-mail: info@jaacc.jp
Phone 1–978–750–8400 FAX 1–978–646–8600

編 集 委 員・論 文 委 員
小林宏一郎（理事） 小 野 輝 男 （理事）加 藤 剛 志 （幹事）小 池 邦 博 （幹事）谷 山 智 康 （幹事）
乾
成里
大竹
充
葛西伸哉
金井
靖
喜々津哲
菊池弘昭
木村
崇
笹山瑛由
佐藤
拓
佐藤
岳
嶋
敏之
白土
優
関口康爾
関野正樹
直江正幸
永沼
博
長浜太郎
橋野早人
PHAM NAMHAI
藤田麻哉
山田晋也
山本健一
湯浅裕美
安達信泰
古門聡士
土井正晶
薮上
信

磯上慎二
小田洋平
中山英俊
山崎慶太

稲葉信幸
齊藤敏明
成田正敬
山本崇史

小瀬木淳一
清野智史
長谷川崇
吉村
哲

小山大介
田倉哲也
別所和宏

加藤宏朗
竹澤昌晃
槙
智仁

加藤和夫
田島克文
宮下英一

窪田崇秀
曽根原誠
本多周太

後藤博樹
田中哲郎
三浦健司

鎌田清孝
角田匡清
室賀
翔

神田哲典
土井達也
森田
孝

複写をされる方へ
本会は下記協会に複写に関する権利委託をしていますので，本誌に掲載された著作物を複写したい方は，同協会より許諾を受けて複写して下さい．但し（社）日本複写権センター（同協会より権利を再委託）と包括複写許諾契約を締結されている企業の社員による社内利用目的の複写はその必要はありません．（社外頒布用の複写は許諾が必要です．）
権利委託先：一般社団法人学術著作権協会〒107–0052 東京都港区赤坂 9–6–41 乃木坂ビル
電話（03）3475–5618 FAX（03）3475–5619 E-mail: info@jaacc.jp
なお，著作者の転載・翻訳のような，複写以外の許諾は，学術著作権協会では扱っていませんので，直接本会へご連絡ください．

本誌掲載記事の無断転載を禁じます．

Journal of the Magnetics Society of Japan
Vol. 43 No. 2 (通巻第 302 号) 2019 年 3 月 1 日発行

Vol. 43 No. 2 Published Mar. 1, 2019by the Magnetics Society of JapanTokyo YWCA building Rm207, 1–8–11 Kanda surugadai, Chiyoda-ku, Tokyo 101–0062
Tel. +81–3–5281–0106
Fax. +81–3–5281–0107
Printed by JP Corporation Co., Ltd.
Sports Plaza building 401, 2–4–3, Shinkamata Ota-ku, Tokyo 144–0054
Advertising agency: Kagaku Gijutsu-sha

発行：（公社）日本磁気学会

101–0062 東京都千代田区神田駿河台 1–8–11 東京YWCA会館 207 号室

製作：ジェイピーシー 144–0054 東京都大田区新蒲田 2–4–3 スポーツプラザビル401 Tel.（03）6715–7915
広告取扱い：科学技術社

Copyright

111–0052 東京都台東区柳橋 2–10–8Ⓒ 2019 by the Magnetics Society of Japan

武田ビル 4F

Tel.（03）5809–1132