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Vol.48 No.4
JournalJournal of the Magnetics Society of Japan
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2024

Hard and Soft Magnetic Materials
Eﬀect of Zr Substitution on the Nd-Based Magnets: Theoretical Investigation of (Zr,Nd)(Fe,Ti)12 Compounds
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N. Y. Merkt, S. Erdmann, T. Klüner, H. İ. Sözen…60

Thin Films, Fine Particles, Multilayers, Superlattices
Characterization of Magnetostrictive Film on Shaft Surface in Inverse Magnetostrictive Torque Sensor Using
Magneto-Optical Kerr Eﬀect

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Journal of the Magnetics Society of Japan
Vol. 48, No. 4

Electronic Journal URL: https://www.jstage.jst.go.jp/browse/msjmag

CONTENTS
Hard and Soft Magnetic Materials
Effect of Zr Substitution on the Nd-Based Magnets: Theoretical Investigation of (Zr,Nd)(Fe,Ti)12
Compounds…………………………………… N. Y. Merkt, S. Erdmann, T. Klüner, H. İ. Sözen

60

Thin Films, Fine Particles, Multilayers, Superlattices
Characterization of Magnetostrictive Film on Shaft Surface in Inverse Magnetostrictive TorqueSensor Using Magneto-Optical Kerr Effect………………………………… K. Ishibashi, M. Sonehara, T. Kodaira, T. Sasaki, and T. Sato

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Copyright ©2024 by the Magnetics Society of Japan.
This article is licensed under the Creative Commons Attribution International License (CC BY 4.0)
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J. Magn. Soc. Jpn., 48, 60-74 (2024)<Paper>Effect of Zr Substitution on the Nd-based Magnets: Theoretical
Investigation of (Zr,Nd)(Fe,Ti)12 Compounds
N. Y. Merkt, S. Erdmann, T. Klüner, H. İ. Sözen

Institute of Chemistry, Carl-von Ossietzky University of Oldenburg., Carl-von-Ossietzky-Straße 9-11 Oldenburg D-26129, Germany

This research aims to identify an alternative solution for the Nd2Fe14B magnet in light of the scarcity of rare-earth(RE) resources. The investigation uses density functional theory (DFT) calculations to assess the effect of partial
substitution of Nd with the transition metal (TM) Zr within the ThMn 12 structure, focusing specifically on the(Zr0.5Nd0.5)Fe11Ti compound. In order to gain a comprehensive understanding, an investigation of intrinsic andmagnetic properties, including saturation magnetization (MS), Curie temperature (TC) and magnetic anisotropy energy(MAE), is carried out on binary to quinary compounds RFe11-yTiy (R: Nd, Zr and Zr0.5Nd0.5, y: 0 ≤ y ≤ 1) and(Zr0.5Nd0.5)Fe10CoTi. The substitution of Ti at different concentrations for thermodynamic stabilization is studied in
ternary and quaternary compounds RFe12-yTiy (0 ≤ y ≤ 1). In addition, the influence of Co on phase stability andintrinsic magnetic properties is studied in the quinary compound (Zr0.5Nd0.5)Fe10CoTi. Special attention is given to thetreatment of the 4f-electrons of Nd and their interaction with the 3d-electrons. Theoretical results are compared withavailable experimental data, although the limited availability of data, especially for Zr-containing compounds, limits
the scope of such comparisons. From the literature and our calculations of binary and ternary compounds, we areencouraged to the quaternary and quinary calculations. Promising magnetic properties of an Nd-lean quaternarycompound suitable for engineering applications have been identified. In particular, for the quaternary compound
ZrNdFe22Ti2, calculated values of |BH|max = 525 kJ/m3 and TC = 783 K are close to those of the Nd2Fe14B magnet.
Keywords: hard magnets, Nd-lean permanent magnets, alternative magnets, zirconium substitution, cobalt
substitution, ThMn12-phase

1. Introduction
In recent years, there has been an growing demand for
low-cost
andresource-efficienthardmagnets,
particularly in the renewable energy and e-mobility
sectors. Currently, the most prevalent magnets are madefrom rare-earth (RE) and transition metal (TM) elements.
The strongest is the Nd2Fe14B compound developed in theearly 1980s.1,2)
The combination of RE and TM elements results inalloys with high saturation magnetization (MS), highCurie temperature (TC) and high magnetocrystalline
anisotropy energy (MAE). The high values for MS and TC
are attributed to the presence of the TM atoms, while RE
elements are responsible for the high anisotropy.
Nevertheless RE elements such as Nd, Dy, Sm and Tb,
as well as some TM elements such as Co are considered
critical for future applications.2–4) In addition, there is a
global monopoly of RE elements, which necessitates thesearch for RE-lean or RE-free hard magnetic
materials.5,6)
Several methods for developing RE-lean magnets havebeen documented in the literature. For instance, grain
size reduction is used to achieve a higher coercive fieldstrength and the critical element Dy can be omitted.7,8)
The coercivity can be significantly increased by applying
the grain boundary diffusion process, which leads to the
reduction of critical RE elements such as Dy or Tb.9,10) In                     Corresponding author: N. Y. Merkt(nico.yannik.merkt@uni-oldenburg.de)60

addition, the partial replacement of the critical RE
elements with more abundant and low-cost elements has
been explored (e.g. see Sakuma et al.11)). So far, thisprocess has mainly been used to produce magnets for low
to medium temperature applications.
We focus on the physical and magnetic properties ofRE-lean permanent magnets based on the ThMn12
prototype structure (also referred to as the 1:12 phase inthis paper). It is worth noting noted that the 1:12 (RE:Fe)
phase has a lower RE content than Nd2Fe14B andexhibits promising magnetic properties.12–14) In ourprevious research we investigated the abundant andinexpensive elements Y and Ce as potential substitutes
for the critical element Nd and their effect on themagnetic properties and thermodynamic phasestabilities of the 1:12 phase.15,16)
In this paper, we consider Zr as a promising
replacement for the RE element Nd. Substituting Zr for
Nd can result in more cost-effective and sustainablemagnets, as Zr is more abundant and affordable thanNd.17,18) Previous studies have reported on the phasestability of compounds with Zr substitution in the 1:12
phase.11,19,20) The studies demonstrated that replacing
Nd with Zr has the potential to generate a saturation
polarization similar to or greater than that of Nd2Fe14B,
as well as an anisotropy field comparable to it.20)
Furthermore, it should be noted that Zr exclusively
substitutes on the 2a sites of the 1:12 phase that are
already occupied by RE elements. Nevertheless, the main
experimental challenge lies in the substitution of Zr in

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these 2a sites, which forms a solid solution with Fe, andcompounds with a Zr concentration of up to 30% havebeen produced so far.21) A possible solution to thisproblem is the selective laser melting method used by
Neznakhin et al.22), as it has successfully produced thecompound (Zr,Sm)Fe11Ti with a 1:1 ratio of Zr to RE.
Based on the selective laser melting method22) and ourtheoretical calculations given in the Sec. 3.1, theconcentration of 50% is considered.
In order to have a systematic understanding of the
intrinsic magnetic properties of Zr-substituted 1:12
compounds, we have started with the binaries andternaries of M2Fe24−yTiy alloys (where M = Zr and Nd andy: 0 ≤ y ≤ 2). Since the 1:12 binary phases are notthermodynamically stable, Ti was used as a stabilizer.23–
26) The Zr substitution and Ti solubility in the quaternarycompounds (Zr,Nd)Fe24−yTiy were studied. Once theequilibrium Ti concentration was found, Co wasadditionally considered as a quinary substitution to
study its effect on magnetic properties and stabilities.
The paper is structured as follows: In Sec. 2, thecomputational details of the theoretical approaches are
explained. Then, in Sec. 3 ab initio based intrinsic
magnetic properties are given. We have started with thesolution enthalpies and formation energies of theconsidered compounds in Sec. 3.1. Physical properties
such as lattice constants and cell volumes are discussed.
In Sec. 3.2 the magnetic properties are given, startingwith the total magnetic moment and the magnetization.
The calculated |BH|max values are presented in Sec. 3.3.
Another important intrinsic magnetic property, theCurie temperature, is calculated using the KorringaKohn-Rostoker (KKR) method and is discussed in Sec. 3.4.
The last magnetic property calculated in this paper, themagnetocrystalline anisotropy energy (MAE), is given inSec. 3.5. Finally, we conclude the paper with remarks anddiscussions in Sec. 4.

2. Computational Details
All theoretical calculations were performed by spinpolarized density functional theory (DFT) using theVienna ab-initio Simulation Package (VASP).27,28) The
projector-augmented wave (PAW) method was used as
implemented in VASP. The exchange correlation effects
were treated within the Perdew-Burke-Ernzerhof(PBE)29) generalized Gradient Approximation (GGA). For
the magnetocrystalline anisotropy energy the local spin
density approximation (LSDA)30) was considered inaddition to the GGA. Optimized structures from GGA
were used in these MAE calculations.
To scan the Brillouin zone we used a Γ-centred k-point
grid of 12×12×10 meshes for the 26-atom supercells used.
The cut-off energy of the plane wave basis was set to
500 eV and the smearing parameter to 0.1 eV. The
convergence criteria within the self-consistent field
scheme was set to 10−5 eV for all calculations except theanisotropy calculations where the criteria was set to10−7 eV. The choice of input parameters gives energy

convergence within an error of less than 1 meV/atom.
The 4f-electrons of Nd have been treated carefully,
although this is a challenge in DFT. The f-electrons of Nd
have been considered in the in-core state, which gives
good mechanical and elastic properties.31) To calculate
the magnetic properties, we went beyond DFT andperformed a DFT+U treatment using the Dudarev
method32) with a Hubbard correction of U = 6 eV. This
Hubbard correction term has been used previously in theliterature for 1:12 Nd-based compounds with goodresults.15,16,33)
Treating the 4f-electrons in the in-core state,hybridization with other orbitals is completely neglected
and instead atomic physics is applied to the Nd-4f states.
Note that only the spin magnetic moment is calculated
from the self-consistent DFT calculations. To obtain the
total magnetic moment, the orbital magnetic moment
must be added. The orbital magnetic moment is made up
of gj ⋅J where gj is the Landé g-factor (gNd = 8/11 34)) andJ is the total angular momentum of 9/2. The total
angular momentum results from Hund’s first rule andthe full spin polarization (S = 3/2) of the three 4f-electronsof the element Nd. The orbital magnetic moment of Nd
can be estimated to be 3.273 µB, using the equation
J = |L−S| (for elements with less than half a full f-shell)
with L = 6. In case of the GGA+U calculations, thecalculated spin fraction of the Nd-4f electrons and thevalue of gj ⋅J were added to avoid double counting for the
spin fraction of the 4f-electrons.
A collinear self-consistent calculation was performed to
calculate the magnetocrystalline anisotropy. The
resulting relaxed structures from the GGA calculations
were used as input for the non-collinear calculations,which take into account the spin-orbit coupling. For each
non-collinear run, the magnetic moment was alignedwith one of the crystallographic directions [001], [010]
and [100].
For the Curie temperature (TC), the Liechtensteinmethod35) was used to determine the exchangeinteraction energies Jij for the ferromagnetic (FM) andlocal moment disorder (LMD) states of the 1:12 phase.
This was achieved by applying the GGA via the KorringaKohn-Rostoker (KKR)36,37) Green’s function method as
implemented in the AkaiKKR38) code, also known as
MACHIKANEYAMA. This method implements theatomic sphere approximation (ASA) using the coherent
potential
approximation(CPA).39,40)
Continuous
concentration changes of both RE and TM sublattices
were considered based on the CPA. Moruzzi, Janak andWilliams (MJW)41) parameterized the local density
approximation (LDA)42,43) which is the basis for all KKR
calculations. In this study the scattering of the system isconsidered up to d-scattering (lmax = 2), i.e. the f-electrons
are placed in the valence state on the basis of the open
core approximation.34,44) The corresponding results of the
GGA-PBE relaxation calculations are used as structural
input for the TC calculations.

Journal of the Magnetics Society of Japan Vol.48, No.4, 2024

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Fig. 1

Schematic representation of the 2 formula unit (f.u.) (26 atoms) body-center-tetragonal ThMn12 (I4/mmm,
space group number 139) 1:12 phases. (a) Unstable RFe12, (b) 2 Ti atom substituted (7.7 at. %) R2Fe22Ti2(RFe11Ti) and (c) 2 Ti and 2 Co substituted (each 7.7 at. %) supercell RFe10CoTi (R2Fe20Co2Ti2). Ti atoms
are substituted in energetically favorable 8i sites and Co atoms are substituted in energetically favorable
8f and 8j sites.

3. Results and Discussion
3.1 Stability of Ti, Co and Zr substituted 1:12
compounds

Besides the well known Nd2Fe14B magnet45,46), anotherclass of typical compounds that fulfil the technologicalrequirements for hard magnetic applications is in theform of RFe12−yTiy. Figure 1(a) shows a schematic
representation of the RFe12 compound.
RFe12 compounds are known to be thermodynamicallyunstable.23–26) To stabilize a bulk system with the ThMn12
structure, partial substitution of Fe atoms is required. Ti
is a commonly used and typical stabilizing element13,47),
and is therefore used. The effect of Co substitution on
both the stabilization and the magnetic properties isstudied. In addition to these substitutions on the TM site,
we have investigated Zr substitution on the RE site to
make the Nd concentration lean.
Primarily, the phase formation energies of the ternary
phases with X substitution (X = Ti and Co) areconsidered:𝐸𝐸 𝑓𝑓 = (Nd2 Fe24−y Xy ) − 2𝜇𝜇Nd − (24 − 𝑦𝑦)𝜇𝜇Fe − 𝑦𝑦𝜇𝜇X= 2𝐸𝐸 𝑓𝑓 (NdFe12 ) + 𝐸𝐸 𝑠𝑠𝑠𝑠𝑠𝑠 (𝑦𝑦)(1)

E(Nd2Fe24−yXy) is the total energy of the X substituted

1:12 phase. X can be Ti or Co on the TM sublattice. For agiven Ti and Co concentration all possible configurations
have been investigated and the most energeticallyfavorable configuration is taken into account in thisformula. Note that a similar formulation can also be
written for Zr substitution on the RE site. µ represents
the ground state of the considered elements. In case of µFe
this is the body centred cubic (bcc) and ferromagnetic(FM) structure. For the elements Co (µCo), Ti (µTi), Nd

62(µNd) and Zr (µZr) the hexagonal structure (hcp) or, in thecase of Nd, the double-hexagonal close-packed structure(dhcp) has been considered. The elements Ti, Nd and Zr
are treated as nonmagnetic (NM) and Co in the FM state.
The phases are thermodynamically stable (unstable)
for negative (positive) formation energies. Although pure
unaries are considered as the reference, the results are
expressed in terms of the formation energy Ef(NdFe12) ofthe unstable phase NdFe12. Various formulations can beconsidered instead of the base compound. However, due
to the shortage of quaternary phase diagrams, naturalreferences have been considered, which are again basecompounds and unaries:
𝐸𝐸 𝑠𝑠𝑠𝑠𝑠𝑠 (𝑦𝑦) = 𝐸𝐸(Nd2 Fe24−y X y ) − 2𝐸𝐸(NdFe12 ) + 𝑦𝑦(𝜇𝜇Fe− 𝜇𝜇Ti )(2)

Since the solution enthalpy calculations include
reference elements which are critical to the resulting
energies, accurate modelling is required. Table 1
compares the physical and magnetic properties of thepure elements with experimental data.
According to the calculations, the Nd-based 1:12
compound, NdFe12, is thermodynamically unstable witha formation energy of 0.94 eV, in agreement withprevious reports.23–25) However, ZrFe12 was calculated to
be stable with a formation energy of −0.76 eV, in
accordance with the other theoretical work.57) This is not
the case for the 50% substitution of Nd with Zr, thecompound (ZrNd)Fe12, which has a formation energy of
0.16 eV.
As 50% substitution of Nd by Zr is apparently not
sufficient to stabilize the ternary phase, additional
substitution of Fe by Ti is required. For the compounds
containing
one
Ti
atom,
NdFe11.5Ti0.5
and(ZrNd)Fe11.5Ti0.5, the formation energies were calculated
to be −0.03 eV and −0.79 eV, respectively. Comparing

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Table 1 Calculated physical and magnetic properties of the pure elements that are included in solution enthalpy
formalism in Eq. 2. a and c are the lattice constants, B0 is the bulk modulus and mtot is the magnetic
moment (where NM stands for nonmagnetic).

Experimental
Lattice constants
Bulk modulus
(Å)
a
c
a
c
(GPa)
Fe
2.832
2.832
2.668h 48)
2.668h 48)
185.47
Co
2.489
4.036
2.5a 50)
4.1a 50)
212.15
Ti
4.553
2.817
4.60a 51)
2.83a 51)128.93Nd
3.704
11.9353.658d 53)
11.839d 53)
33.84
b
55)
b
55)
Zr
3.235
5.166
3.233
5.146
93.57
aX-ray diffraction (XRD) analysis at 300 K.
bHigh-pressure (ultrasonic) measurements combined with X-ray techniques.
cX-ray diffraction (XRD) analysis at >573 K.
dWith Vibrating Sample Magnetometer at 300 K.
eNeutron Diffraction at 300 K.
fBy measuring the flux change.
Alloy

Lattice constants(Å)

these energies, it can be seen that in case of the Zrsubstituted alloy, the substitution of one Ti atom issufficient to stabilize the compound. In case of theternary compound without Zr, the energy of formation isjust below 0 eV. As Zr has the potential to reduce the Ti
concentration, the magnetic properties are enhanced.
The effect of Ti stabilization has been studied for the
ZrNdFe24−yTiy compounds as given in Fig. 2. It starts
with the substitution of the first Ti atom in each TM site,
i.e. 8i, 8j and 8f, in the supercell consisting of 26 atoms (2formula units (f.u.)). Note that a single Ti atom
corresponds to 3.8 at. %. As in previous work for Ce andNd-based 1:12 compounds15,58), the 8i site is calculated tobe the most energetically favorable site. The solution
enthalpies for the 8j and 8f sites are higher by 0.53 and0.72 eV, respectively. Therefore, only the 8i sublattice
was considered for the substitution of the remaining Ti
atoms.

Fig. 2

Calculated Ti solution enthalpies for(ZrNd)Fe24−yTiy according to Eq. 2. The first
Ti atom is considered in 8i, 8j and 8f sites
one by one. Then the remaining Ti atoms
are substituted on 8i site due to the lower
solution enthalpy and all possible
configurations are considered.

Experimental
Bulk modulus

Magnetic moment(GPa)
159-173d 48)
199 ±6a 50)
102-119a 51)
25.4-34.9d 53)
92-102b 55)(µB/atom)
2.21
1.63
NM
NM
NM

Experimental
Magnetic moment
mtot(µB/atom)
2.22a 49)
1.53f 49)
NMf 52)
NMa 54)
NMg 56)Note that the exclusion of the 8j and 8f sublattices isalso supported by experimental work.59,60) The origin of
the 8i preference is due to the larger Wigner-Seitz radiusof this site compared to the 8f and 8j sites.61) In case of
substitution of the second and third Ti atom, all possible
configurations on 8i sites were considered. The Ti
solubility studies on (ZrNd)Fe24−yTiy give an equilibrium
Ti concentration of 7.7 at. %, which is the same as in caseof
NdFe12−xTix
andCeFe12−xTix.15,16,58,62)The
corresponding structure is shown in Fig. 1(b).
Although Ti stabilizes the Zr substituted 1:12
compounds, it reduces the overall magnetic moment due
to its nonmagnetic (NM) nature. For instance,(ZrNd)Fe22Ti2 has 4.06 µB/f.u. less total magnetic
moment than the Ti-free 1:12 compound. Therefore,
ferromagnetic (FM) Co was considered as anothersubstitutional element for (ZrNd)Fe24−xCox and(ZrNd)Fe22−xTi2Cox to improve the magnetic properties
and to understand its effect on the stability.
Figures 3 and 4 show the site preference of Co
substitutions and equilibrium concentrations for theconsidered 1:12 compounds. As shown, Co prefers to
substitute at 8f and 8j sites for both (ZrNd)Fe24−xCox and(ZrNd)Fe22−xTi2Cox cases. In case of the (ZrNd)Fe24−xCox
compounds it can be seen in Fig. 3 that the first Co atom
is substituted into the 8f site with the solution enthalpyof −0.22 eV. This is followed by the 8j and 8i sites with
−0.21 eV and −0.18 eV, respectively. After keeping thefirst Co atom in the 8f position and replacing the second
Co atom, the lowest energies of the 8j, 8f and 8i sites are
obtained with −0.24 eV, −0.22 eV and −0.17 eV,
respectively. As the equilibrium concentration is at 2 Co
atoms (7.7 at. %), additional Co atoms lead to higher
solution enthalpies. It is also noticeable that the 8 j site
gives the most negative energy for the third Co atom andfrom the fourth to the sixth Co atom the 8f site gives thelowest energies.
As can be seen in Fig. 4 in case of the(ZrNd)Fe22−xTi2Cox compounds, again the substitution of
Co into the 8f site with the solution enthalpy of −0.16 eV,
followed by 8j and 8i with about −0.14 eV, leads to themost stabilized compound. For the second Co atom the 8jsite yield the lowest solution enthalpy with −0.22 eV,
followed by 8f with −0.19 eV and 8i with −0.18 eV. For
the third to sixth Co atoms, the 8j sublattice is theenergetically most favorable site. Since sublattice 8i has

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Table 2 GGA calculated physical and magnetic properties of the most stable Co containing compounds (seeFigs. 3 and 4) starting from ZrNdFe24 and ZrNdFe22Ti2 with increasing Co content.
Lattice constants
(Å)

Alloy

ZrNdFe24
ZrNdFe23Co
ZrNdFe22Co2
ZrNdFe21Co3
ZrNdFe20Co4
ZrNdFe19Co5
ZrNdFe18Co6
ZrNdFe22Ti2
ZrNdFe21CoTi2
ZrNdFe20Co2Ti2
ZrNdFe19Co3Ti2
ZrNdFe18Co4Ti2
ZrNdFe17Co5Ti2
ZrNdFe16Co6Ti2

a

8.419
8.419
8.407
8.417
8.419
8.417
8.410
8.411
8.408
8.391
8.407
8.399
8.402
8.407

b

8.419
8.419
8.423
8.417
8.417
8.417
8.410
8.497
8.495
8.492
8.481
8.469
8.459
8.449

c

4.656
4.659
4.664
4.665
4.672
4.682
4.691
4.704
4.708
4.719
4.715
4.719
4.707
4.698

Cell volume

Spin magnetic momentat 2a site(Å3)
165.01
165.15165.13165.24165.55165.83165.91
168.08168.13168.11
168.11167.84167.28166.84(µB/f.u.)-0.50 (mZr2a) -0.26 (mNd2a)-0.50 (mZr2a) -0.26 (mNd2a)-0.50 (mZr2a) -0.26 (mNd2a)-0.50 (mZr2a) -0.26 (mNd2a)-0.50 (mZr2a) -0.26 (mNd2a)-0.51 (mZr2a) -0.26 (mNd2a)-0.50 (mZr2a) -0.26 (mNd2a)-0.49 (mZr2a) -0.27 (mNd2a)-0.49 (mZr2a) -0.26 (mNd2a)-0.49 (mZr2a) -0.26 (mNd2a)-0.49 (mZr2a) -0.26 (mNd2a)-0.48 (mZr2a) -0.26 (mNd2a)-0.47 (mZr2a) -0.25 (mNd2a)-0.47 (mZr2a) -0.25 (mNd2a)a larger Wigner-Seitz radius, the preference for Co in
sublattices 8f and 8j is suspected to produce the morestable crystal structures. The theoretically calculatedsite preference is in agreement with experimental
reports11,19,63), which reported that Co prefers the 8f and8j sites. In addition, for both series of compounds, a trend
is observed that the 8i site generally has higher solution
enthalpies and the 8j and 8f sites are more stable. The Co
equilibrium concentration of 7.7 at. % was calculated forboth compounds. The schematic representation of themost stable quinary phase is shown in Fig. 1(c).
The physical and magnetic properties of the lowest
energyconfigurationsoftheconsideredCo
concentrations
for(ZrNd)Fe24−xCox
and(ZrNd)Fe22−xTi2Cox, are given in Tab. 2. In case of Co
substitution in (ZrNd)Fe24−xCox a slight increase inmagnetic properties is observed. For example, the totalmagnetic moment increases by 0.86 µB from Co-free to 6
Co atoms. In case of substitution in (ZrNd)Fe22−xTi2Cox, adecrease in the total magnetic moment of −0.93 µB is
observed with the substitution of 6 Co atoms. Note that
this increase (decrease) only becomes visible with thesubstitution of the fourth Co atom. Thus, Co at its

Fig. 364

Calculated Co solution enthalpies forchemical compositions with the formula(ZrNd)Fe24−xCox according to Eq. 2.

m2atotFig.

4

Saturation
Maximum
Total magnetic
momentmagnetization energy product
mtot
μ0MS
|BH|max(µB/f.u.)
27.16
27.34
27.34
27.46
27.70
27.95
28.02
23.10
23.19
23.22
23.14
22.95
22.54
22.17

(T)
1.92
1.93
1.93
1.94
1.95
1.96
1.97
1.60
1.61
1.61
1.60
1.59
1.57
1.55

(kJ/m3)732.20740.30740.97745.95756.67767.69770.45510.17513.86515.62511.78505.11490.70477.03Calculated Co solution enthalpies forchemical compositions with the formula(ZrNd)Fe22−xTi2Cox according to Eq. 2.

equilibrium concentration has only a small effect on themagnetic properties. The substitution of a Ti or Co atomin (ZrNd)Fe24 leads to a stabilization of −0.96 eV and−0.22 eV, respectively. Therefore, Co has a lower impact
on the stability of the compound than Ti.
The calculated physical properties of the considered
compounds are given in Tab. 3. A trend of decreasing
lattice constants and cell volumes with Nd2Fe24−yTiy >ZrNdFe24−yTiy > Zr2Fe24−yTiy with 0 ≤ y ≤ 2 can be seen.
This trend is in good agreement with existing theoreticaland experimental values.16,20,60) This behaviour has also
been previously documented by Sakuma et al.11).
Furthermore, an increasing volume with increasing Ti
concentration is observed, which is in agreement to thecalculations of Harashima et al.64). The substitution of Co
in the quaternary compound ZrNdFe22Ti2 shows aminimal increase in cell volume (by 0.03 Å3) and in thelattice constant c (by 0.015 Å), while the lattice constantsa (by 0.020 Å) and b (by 0.005 Å) are slightly reduced. As
mentioned above, a GGA+U treatment was also included
for Nd containing compounds (see values in parenthesis
in Tab. 3), which shows the same trends. In case of thebulk moduli, a softening effect of Ti can be seen. This

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INDEX

Table 3 Calculated lattice constant, cell volume and bulk modulus of the considered alloys using GGA. In case of
Nd containing stable compounds GGA+U has been considered for Nd 4f-electrons with U = 6 eV and isgiven in parenthesis.

Experimental
Lattice constants(Å)

Lattice constants
(Å)

Alloy

Nd2Fe24(NdFe12)
Nd2Fe23Ti(NdFe11.5Ti0.5)
Nd2Fe22Ti2(NdFe11Ti)
Zr2Fe24(ZrFe12)Zr2Fe23Ti(ZrFe11.5Ti0.5)
Zr2Fe22Ti2(ZrFe11Ti)
ZrNdFe24((Zr0.5Nd0.5)Fe12)
ZrNdFe23Ti((Zr0.5Nd0.5)Fe11.5Ti0.5)
ZrNdFe22Ti2((Zr0.5Nd0.5)Fe11Ti)
ZrNdFe20Co2Ti2(Nd0.7Zr0.3)2(Fe0.75Co0.25)23Ti

CellvolumeExperimental
Cell volume

Bulk
modulus

Literature
Bulk modulus

a

b

c

a

c

(Å3)

(Å3)

(GPa)

(GPa)

8.535

8.535

4.671

8.574* 65)

4.907* 65)

170.13

180.37* 65)

129.45127.96* 16)

8.568(8.558)8.593(8.584)
8.312

8.530
(8.533)8.535(8.532)
8.312

4.680
(4.675)4.696(4.703)4.647

8.552* 16)
8.552* 16)
8.574a 66)
8.574a 67)

4.686* 16)(4.692)* 16)
4.907a 66)
4.794a 67)

171.02(171.06)
172.19(172.21)160.54171.73* 16)(171.81)* 16)
176.2a 66)
180.35a 67)

128.74127.17* 16)

128.89127.20* 16)

8.285

8.377

4.677

8.258

8.433

4.708

8.358* 68)

4.715* 68)

8.419

8.419

8.439
(8.439)8.411(8.352)8.391(8.399)8.439(8.440)8.497(8.508)8.492(8.483)144.46162.29139.03163.93139.584.656

165.01132.984.669(4.666)4.704(4.703)4.719(4.721)

166.27(166.20)168.08(167.10)168.11(168.17)130.238.536a 20)

4.770a 20)

130.59133.30173.8a 20)*Theoretical references are represented by *.
aX-ray diffraction (XRD) analysis at 300 K.

effect occurs again not only in the Nd2Fe24−yTiycompounds but also in the Zr2Fe24−yTiy compounds aswell as in the compounds containing both Nd and Zr with
ZrNdFe24−yTiy (y: 0 ≤ y ≤ 2) (see Tab. 3).

3.2 Total magnetic moment and magnetization

The magnetization of the chosen compounds is
calculated distinguishing between the total magnetic
moment and the saturation magnetization. The totalmagnetic moment, expressed in Bohr magnetons (µB) per
formula unit, is the cumulative sum of the magnetic
moments of the individual atoms. The saturation
magnetization, measured in Tesla (T), can be derived
from the density of the compound (ρ), the Avogadro
constant (NA), the molecular weight (M) and the average𝑡𝑡𝑡𝑡𝑡𝑡magnetic moment per atom 𝑚𝑚𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
the compound:
𝑡𝑡𝑡𝑡𝑡𝑡𝜇𝜇0 𝑀𝑀𝑆𝑆 = 𝑚𝑚𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
𝜇𝜇𝜌𝜌∙𝑁𝑁𝐴𝐴

𝑀𝑀𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎.

(3)

The calculated intrinsic magnetic properties, including
the total magnetic moment (mtot), the spin magnetic
moment of the atoms in 2a position (m2aspin) and thesaturation magnetization (µ0MS), are given in Tab. 4.
These values are compared with previous theoretical andexperimental data for analogous compounds. It isnoteworthy that the PBE functional tends tooverestimate the magnetic moments69), placing the
calculated values within the expected range. Thisoverestimation can be assessed in case of NdFe 11Ti bycomparison with literature values. Contrarily, for
Zr2Fe24−yTiy (y: 0 ≤ y ≤ 2) compounds such an assessment
is not possible, since neither theoretical nor experimental
values have been published so far. It should also be notedthat even the relaxed structure does not guarantee thatthe magnetic properties are correctly reproduced.
In case of quaternary ZrNdFe24−yTiy (y: 0 ≤ y ≤ 2) alloys,
it is possible to estimate mtot as both experimental andtheoretical values are available for similar compounds
and fall within the expected range (see Tab. 4). In
contrast, for Zr2Fe24−yTiy compounds, the magnetic
moments of Fe and Ti are close to those of Nd2Fe24−yTiy(y: 0 ≤ y ≤ 2) compounds, considering the structuraldifference only at the 2a sites with Zr at −0.47 µB and Nd
at −0.27 µB.
The GGA calculations, using an in-core treatment for
f-electrons, give reasonable mtot for NdFe11−yTiycompounds. In particular, the calculated mtot for
NdFe11Ti is 24.84 µB, while the experimentally measured
mtot by Skokov et al.33) is 23.43 µB. However, the in-coreapproximation neglects the contribution of the f-electrons.
By including f-electrons and using the GGA+U treatment,
where they are considered to be fully localized, a revised
mtot of 25.04 µB (including the orbital magnetic moment)is calculated. In addition, the GGA+U treatment yields asaturation magnetization of 1.69 T, in agreement withexperimentally
determined
MS of 1.38 T to
1.70 T.33,60,66,67)
In the context of the GGA+U approximation, a reduced
spin magnetic moment is observed with a magnitude of
the Nd ion of −3.30 µB. This reduction, approximately−3.0 µB from the GGA value of −0.27 µB, is attributed tothe influence of the f-electrons. Nevertheless, the results
for both treatments are acceptable compared toexperimental values. It is important to note that thecalculated results are based on the summation of theorbital magnetic moment of the Nd-4f-electrons withgj ⋅J = |L − S| = 3.273 µB.
The substitution of Zr atoms into the 1:12 phase results
in an average reduction of the total magnetic momentfrom 2.25 µB/f.u. (ZrNdFe24−yTiy) to 4.21 µB/f.u.
(Zr2Fe24−yTiy). In GGA this reduction can be attributed tothe comparatively lower magnetic moment of Zr(−0.47 µB) compared to Nd (−0.27 µB). Nevertheless, inthe GGA+U treatment, the inclusion of f-electrons leads
to a reduction in the magnetic moment of Nd to −3.3 µB,
showing an opposite trend.

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Table 4 Calculated spin magnetic moments at rare-earth site m2aspin, total magnetic moments mtot (µB/f.u.),
maximum energy product |BH|max (kJ/m3) and saturation magnetization µ0MS (T) with comparison
against available experimental data. In case of Nd containing compounds GGA+ U with U = 6 eV isapplied to Nd 4f-electrons only and results are given in parenthesis.
Alloy

m2aspinmtot

Experimental

mtot(µB/f.u.)

Nd2Fe24(NdFe12)

-0.27

29.67

Nd2Fe23Ti(NdFe11.5Ti0.5)
Nd2Fe22Ti2(NdFe11Ti)-0.27(-3.29)-0.28(-3.30)27.14(27.31)24.84(25.04)Zr2Fe24(ZrFe12)Zr2Fe23Ti(ZrFe11.5Ti0.5)
Zr2Fe22Ti2(ZrFe11Ti)

-0.47

24.98

-0.46

23.13

1.66

-0.45

21.28

1.51

ZrNdFe24
ZrNdFe23Ti

ZrNdFe22Ti2

ZrNdFe20Co2Ti2

Nd2(Fe0.75Co0.25)23Ti(Nd0.7Zr0.3)2(Fe0.75Co0.25)23Ti-0.50 (m2aZr)-0.26 (m2aNd)-0.50 (m2aZr)-0.26 (m2aNd)((-0.50 (m2aZr))(-3.27 (m2aNd))-0.49 (m2aZr)-0.27 (m2aNd)((-0.48 (m2aZr)(-3.31 (m2aNd))-0.49 (m2aZr)-0.26 (m2aNd)((-0.49 (m2aZr)(-3.28 (m2aNd))(µB/f.u.)
27.2* 70)
29.15* 46)
31.20* 65)
21.27a 66)
21.90b 67)
23.43c 33)
24.10* 71)
24.50* 46)
25.24d 60)
26.30* 65)

66

(T)

2.03
1.85
(1.86)
1.68(1.69)

1.81

1.92

27.16

|BH|max(kJ/m3)

Experimental|BH|max(kJ/m3)821.55818* 16)680.30(688.49)562.30(571.39)683* 16)
701* 16)
438* 46)569* 16)575* 16)

Experimental

µ0MS

(T)
1.73* 70)
1.99* 46)
2.01* 65)
1.38a 66)
1.48b 67)
1.58c 33)
1.63* 71)
1.65* 46)
1.70d 60)
1.70* 65)654.10548.87455.191.93* (Zr0.1) 68)
1.91* (Zr0.2) 68)
1.89* (Zr0.3) 68)

732.20

24.77

1.74

599.63(24.82)(1.74)(602.83)

23.10

1.60

510.17(23.28)(1.62)(524.51)

23.22

1.61

515.62(23.28)(1.61)

741* (Zr0.1) 68)
726* (Zr0.2) 68)
711* (Zr0.3) 68)(517.70)

25.41d 63)

1.63d 63)529d 63)

24.43d 63)

1.63d 63)
1.66d 20)529d 63)548d 20)*Theoretical references are represented by*.
aExtraction sample magnetometer analysis at 1.5 K.
bMössbauer measurement at 4.2 K.
cSingle crystal, physical property measurement system (PPMS) measured at 10 K.
dVibrating sample magnetometer (VSM) analysis at 4.2 K.

Fig. 5µ0MS(µB/f.u.)

Total magnetic moments of R2Fe24 yTiy (R: Zr
and Nd; y: 0 ≤ y ≤ 2). Since GGA+U
calculations do not change the results
significantly, only GGA results are given. The
red circles show theoretical data46,65,70,71) for
the alloys from others and the blue triangles
show experimental data11,33,60,66,67) for thealloys. Here all literature values correspondto the same compounds. Only theexperimental
mtot
at
ZrNdFe20Co2Ti2
corresponds
tothe(Zr0.3Nd0.7)2(Fe0.75Co0.25)23Ti compound.

The effect of the Ti substitution on the total magnetic
moments of the compounds studied is shown in Fig. 5. In
both GGA and GGA+U calculations, the totalmagnetization of Ti atoms is approximately −1.1 µB. Thisphenomenon is due to the antiferromagnetic orientationof Ti atoms with respect to the Fe atoms. In particular,
there is an observable trend that the total magnetic
moment decreases with increasing concentration of Ti incompounds. The trend calculated in the mtot values of the
Nd2Fe24−yTiy (y: 0 ≤ y ≤ 2) compounds aligns with thefindings of our previous research.16) For all considered
compounds, there is a consistent decrease in the totalmagnetic moment, averaging 2-3 µB/f.u. per Ti atom.
The effect of Co on the total magnetic moment isrelatively modest, as shown in Tabs. 2 and 4. A
comparison between the mtot of ZrNdFe22Ti2 andZrNdFe20Co2Ti2 shows only an increase of 0.12 µB/f.u. in
GGA. This finding is similar for the GGA+U approach aswell. The effect of Co on the magnetic properties depends
on the presence of Ti in the compound. When combinedwith Ti, a decrease in the magnetic properties can be seen,
while in the absence of Ti, an increase is calculated.

3.3 Maximum energy product |BH|max

A magnet consists of magnetic domains, which aresmall regions in which the magnetic moment is oriented
in only one direction. The arrangement of these domains

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Fig. 6 Theoretical maximum energy product |BH|max values for the considered compounds and the literature
for comparison16,46). In addition to conventional GGA, GGA+U with U = 6 eV has been considered forstable Nd containing alloys. Experimental values75,77) of the most common hard magnets are given as
horizontal dashed lines.

is called the microstructure, which is the origin of theextrinsic magnetic properties.72,73) The maximum energy
product |BH|max is one of the key performance measuresfor permanent magnets. It is an important figure-ofmerit for the strength of permanent magnetic materials
and a magnet should be designed to operate near the|BH|max point in order to maximize its efficiency. In
addition, |BH|max serves as a reflective indicator of theupper limit of magnetic energy that can be stored in freespace by a unit volume of permanent magnet. 74) It isfurthermore the optimal product of the two extrinsic
properties remanence Br and coercivity Hc and can be
estimated from the hysteresis loop. In the case of an ideal
quadratic hysteresis loop, there is a relationship between|BH|max and MS, since in that case the remanence
magnetization Mr is maximum and equals MS(μ0Mr = μ0MS). Thus, |BH|max can theoretically becalculated as follows:72,75)|𝐵𝐵𝐵𝐵|𝑚𝑚𝑚𝑚𝑚𝑚 =𝜇𝜇0 𝑀𝑀𝑆𝑆2,
4(4)

where µ0 is the vacuum permeability (µ0 = 4π∙10−7 NA−2).
The calculated |BH|max values of the selectedcompounds are shown in Tab. 4 and Fig. 6.
Figure 6 shows the |BH|max values of known hardmagnets as a reference. A hard magnetic compound canbe considered as promising once its maximum energy
product exceeds 400 kJ/m3. Elevated |BH|max values arecalculated in Ti-free RFe12 compounds where R
represents Y, Zr and Ce. The substitution of a single Ti
atom by Fe (representing 3.8 at. % in the 2 f.u. supercell)
results in an average reduction in |BH|max of about
120 kJ/m3. The substitution of the second Ti atom
(representing 7.7 at. % in the 2 f.u. supercell) further

reduces the |BH|max by a 100 kJ/m3.
The lack of experimental |BH|max data precludes
direct comparison with theoretical values. However,there are alternative theoretical and experimental
results for analogous compounds that facilitatecomparative assessments. Körner et al.46) used the tightbinding
linear-muffin-tin-orbital
atomic-sphereapproximation (TB-LMTO-ASA) to calculate |BH|max forvarious 1:12 phases. Their calculations for NdFe11Ti
yields a |BH|max of 438 kJ/m3, where our GGA (GGA+U)
calculation finds 562 kJ/m3 (575 kJ/m3). Our current|BH|max calculation is in good agreement with theresearch of Herper et al.76), who calculated 490 kJ/m3.
In case of Zr2Fe24−yTiy (y: 0 ≤ y ≤ 2) compounds, thecalculated |BH|max values are ≥ 450 kJ/m3. However,these ternary |BH|max values are about 130 kJ/m3 lower
than the ternary Nd2Fe24−yTiy (y: 0 ≤ y ≤ 2) compounds.
As an example, the ratio of ZrFe11Ti to NdFe11Ti is given
as 455 kJ/m3 and 562 kJ/m3, respectively. As expected,
the quaternary ZrNdFe24−yTiy compounds fall within therange defined by the |BH|max values of the R2Fe24−yTiycompounds (where R: Nd and Zr, y: 0 ≤ y ≤ 2), making
them promising candidates. A comparison between thecalculated |BH|max of ZrNdFe23Ti (600 kJ/m3) and theexperimentally
determined
valueof(Zr0.3Nd0.7)2(Fe0.75Co0.25)23Ti (548 kJ/m3)20) shows aremarkable agreement.
In case of Co substitution, the effect is challenging todetermine due to the limited literature available. When
comparing the calculated |BH|max of ZrNdFe22Ti2 andZrNdFe20Co2Ti2, there is a minimal difference.
Depending on the method used, there is a slight increasefor GGA or decrease for GGA+U in the maximum energy
product of about 6 kJ/m3. For a comparison withliterature, the compound (Nd0.7Zr0.3)2(Fe0.75Co0.25)23Ti is

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Fig. 7 Calculated Curie temperatures TC for all considered 1:12 phases. The exchange interaction energies havebeen calculated for both ordered ferromagnetic (FM) and disordered local momentum (LMD) states, andthe lattice information comes from the theoretically relaxed calculations (see Tab. 3). The literature data
is taken from16,66,68). Here, all literature values refer to the same compounds. Only the theoretical TC for
ZrNdFe24 corresponds to the (Zr0.3Nd0.7)2Fe24 compound.

considered. However, as this compound contains less Zr
and Ti and more Co, the previous effects of the elements
must be taken into account. Due to the effects of Zr andTi, a higher |BH|max can be assumed for the literaturecompound. The |BH|max of the literature compound of529 63) and 548 kJ/m3 20) are higher than the calculated|BH|max of the compounds ZrNdFe22Ti2 andZrNdFe20Co2Ti2, but lower than that of ZrNdFe23Ti.
Based on this ordering of the literature |BH|max, areduction due to Co can be assumed. This reduction in
combination with Ti can also be seen in Tab. 2.
In Fig. 6, the |BH|max values of well-known hardmagnets are provided as a reference. A comparison
between the well-known hard magnet Nd2Fe14B, with a|BH|max of 688 kJ/m3 31), and the quaternary alloy
ZrNdFe11Ti shows that the calculated |BH|max aresmaller than the theoretical value of Nd2Fe14B.
Furthermore, the quaternary Zr compound can becompared with the Y and Ce quaternaries calculated in
our previous work.16) It is observed that in the case of(R,Nd)Fe11.5Ti0.5 (R: Y, Zr and Ce) alloys, the Ycontaining alloy exhibits the highest |BH|max at
628 kJ/m3, followed by the Ce and Zr-containing alloys.
For the (R,Nd)Fe11Ti (R: Y, Zr and Ce) alloys, a trend
emerges where the Zr-containing alloy has the highest|BH|max at 510 kJ/m3, followed by the Y and Ce alloys.

3.4 Curie temperature

In addition to a strong magnetization, a high Curie
temperature TC is another desired intrinsic magnetic
property. The well-known and widely used Nd2Fe14B
magnet has a relatively low experimental TC of 588 K75),
while traditional RE-based permanent magnets havehigher TC (e.g. SmCo5 with 1020 K75)). Finding an Ndlean RE magnet with a better balance between
saturation magnetization and Curie temperature thus

68

deserves extensive efforts.
To calculate TC, an effective spin Heisenberg model ischosen, which is solved in the mean field approximation(MFA). In this Heisenberg model the Hamiltonian of the𝑆𝑆𝑖𝑖 ⃗⃗⃗
𝑆𝑆𝑗𝑗 , where
spin-spin interaction is given by 𝐻𝐻 = − ∑𝑖𝑖≠𝑗𝑗 𝐽𝐽𝑖𝑖𝑖𝑖 ⃗⃗⃗⃗⃗⃗⃗⃗⃗𝑆𝑆𝑖𝑖 and 𝑆𝑆𝑗𝑗 correspond to the spin of place i and j,
respectively. Jij corresponds to the exchange coupling
constant between the two sites i and j and was calculatedwith AkaiKKR.38) The absolute value of the magnetic
ordering temperature TC resulting from MFA is
3
kBTC = 2 ∙J0, where kB is the Boltzmann constant. J0 is thesum of Jij and is therefore the total effective exchange ofa given lattice site connected to all others.
One of the main drawbacks of the evaluation of Jij
when considered the FM configuration, is the systematic
overestimation of TC (approx. 45%). Therefore, as analternative to the FM states, the local moment disorder(LMD) approach is considered as well. In the frameworkof the KKR-CPA method, the concept of LMD state (alsocalled disorder local moment (DLM)78)) is conveniently
used to describe the paramagnetic state of a ferromagnet
above TC.79) Nevertheless, magnetic moments survive
and do not vanish. As an advantage of the KKR based on
Green’s function theory, the exchange interactionenergies for such situations and the Curie temperaturefor both FM and LMD states can be calculated.
The calculated Curie temperatures are shown in Fig. 7,
with the relevant theoretical and experimental datareported in the literature. The horizontal dashed lines
represent the TC of established permanent magnets forcomparative analysis. As shown, the calculated FM
ground state approximation in this work overestimatesfor most of the cases as expected. However, calculating
the TC by considering the LMD as the ground state, theagreement improves significantly. This overestimation isdue to the challenges of describing the delocalized

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electronic state in the magnetism of intermetallics80,81)
based on localized degrees of freedom. In addition, theexchange coupling between local moments is calculated
for the ground state and this assumption does not need
to be valid for high temperatures near TC.
Miyake et al.82) highlighted the accuracy of LMD
calculations is intricately linked to the crystal structureof the 1:12 phase, claiming that in certain cases LMD
values exhibit greater precision than their FM
counterparts. This correlation appears to be due tovariations in spin fluctuation near magnetic transitions,although a full understanding of this phenomenon iscurrently lacking and warrants further investigation.
For the further details about the relationship between
LMD performance and both crystal structure andtransition metals we refer Ref.82). In case of Zr2Fe24−yTiy(y: 0 ≤ y ≤ 2) compounds, the lack of available literature
data necessitates that calculated values be considered asindicative measures of TC only. In the specific case of(Zr0.3Nd0.7)2(Fe0.75Co0.25)23Ti, Suzuki et al.63) approximate
a TC of >840 K, implying that the FM values for
ZrNdFe24−yTiy (y: 0 ≤ y ≤ 2) and ZrNdFe20Co2Ti2
compounds are more likely than the LMD results.
Similar to the trends calculated for mtot and |BH|max,
the Curie temperature shows a decrease with increasing
Ti concentration, as shown in Fig. 7. The Curie
temperature decreases approximately 45 K per Ti atom
(3.8 at. %), which is consistent with the average decrease
found in our previous research16). The theoretical FM(LMD) results show the following trend, with calculated
TC values of 792 (646), 783 (652) and 754 (660) K for
R2Fe22Ti2 (R2: Zr2, ZrNd, Nd2, respectively). In theabsence of comprehensive literature data, the trend of
the calculated compounds is estimated from analogous
compounds such as (Zr0.3Nd0.7)2(Fe0.75Co0.25)23Ti63) andNd2Fe24−yTiy (y: 0 ≤ y ≤ 2).16,82)
The substitution of 3.8 at. % Nd with Zr results in the
average reduction in the TC as 6 K for LMD and theaverage increase as 29 K for FM. The complete
substitution of Nd by Zr (7.7 at. %) increases the average
reduction in the TC to 8 K for LMD and the averageincrease to 51 K for FM. Due to a lack of data, thesetrends cannot be directly compared with the literature.
Nevertheless, Schönhöbel et al.83) experimentallyinvestigated the effects of substitution of Nd and Zr into
SmFe11V. This study showed a decrease in TC for
3.1 at. % substitution of Sm as 32 K for Nd and as 26 K
for Zr. Subsequently, Nd lowers TC more than Zr.
Therefore, a slight increase in TC is expected when Nd issubstituted for Zr. When evaluating the Curie
temperatures, the quaternary compound ZrNdFe22Ti2
with 652 K (LMD) and 783 K (FM) exceeds theexperimental TC (588 K) of Nd2Fe14B, but not thetheoretical TC of 1050 K.84)
A Co substitution of 7.7 at. % in ZrNdFe22Ti2 to
ZrNdFe20Co2Ti2 causes an increase in the Curie
temperature of about 85 K (see Fig. 7). This increase isalso observed by Gjoka et al.85), who measured anincrease of about 50 K per 7.7 at. % Co in
Nd0.4Zr0.6Fe10−xCoxSi2.
In summary, due to the expected overestimation byconsidering the FM states, all considered compounds
have TC values slightly below to above the criticalexperimental temperature of Nd2Fe14B (588 K).75)

Additionally, the theoretical TC of above 950 K86) for
Sm2Co17 can potentially be approached through Co
substitution.

3.5 Magnetocrystalline Anisotropy Energy

The next magnetic property considered in this sectionis the magnetocrystalline anisotropy energy (MAE). The
MAE can be determined from the magnetic force
theorem87) or by calculating the energy difference of themagnetization directions. In this study the latterapproach is used to calculate the MAE. Equation (5) isused to calculate the energy difference from the axis withthe easiest magnetization ([001]) to the axis of thehardest magnetization ([010]). Furthermore, in addition
to being defined as in Eq. (5), which follows the generaldefinition, the MAE is determined between the easiest
direction to magnetize ([001]) and the easiest direction
within the plane perpendicular to the easiest direction([100]). This definition ensures that MAE values are
consistent with uniaxial anisotropy and resulting MAE
values are given in parenthesis. Using GGA+U for
Nd2Fe22Ti2 results in energy differences of 7.2⋅10-4 eV for[010] to [001] and 3.7⋅10-4 eV for [100] to [001]. Thismeans that the highest energy difference is by a change
from [010] and [001]. The equations used to calculate theMAE is as follows:∆𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 = 𝐸𝐸[010] − 𝐸𝐸[001] ,(5)∆𝐸𝐸𝑀𝑀𝑀𝑀𝑀𝑀 = 𝐾𝐾1 ∙ 𝑠𝑠𝑠𝑠𝑠𝑠(𝜙𝜙)2 ,(6)

where E[001] and E[010] are the total energies of themagnetization directions [001] and [010] of the supercell,respectively.
From the total energies and the resulting energy
differences the anisotropy constant K1 and the anisotropy
field Ha can be calculated. The anisotropy energy can be
written as:

where ϕ is the angle between the direction ofmagnetization and the easiest axis to magnetize. In thisformulation, only the dominant term containing K1 ispresented, providing a level of accuracy deemed
appropriate for this particular scenario. All phases
considered within the 1:12 configuration exhibit uniaxial
anisotropy, as indicated by K1 being greater than zero.
The higher order anisotropy constants are considerednegligible due to the significant difference in magnitude
between K1 and K2, with K2 being almost two orders ofmagnitude smaller than K1. Consequently, thecharacterization of magnetocrystalline anisotropy (MCA)
can be effectively captured by the singular parameter K1(for more details see88)). A positive (or negative) value of
K1 indicates uniaxial (or planar) anisotropy.
The anisotropy field Ha corresponds to the upper limitof the coercivity and can be calculated as follows:

𝐻𝐻𝑎𝑎 =2𝐾𝐾1.
𝜇𝜇0 𝑀𝑀𝑆𝑆

Journal of the Magnetics Society of Japan Vol.48, No.4, 2024

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69

INDEX

Table 5 Calculated theoretical magnetocrystalline anisotropy constant K1 according to Eq. 6, anisotropy field Ha
according to Eq. 7 and magnetic hardness factor κ from Eq. 8 for stable compounds with available
experimental data. The values are given for both GGA and LSDA treatments for the exchange–
correlation potential and GGA+U (U = 6 eV) results are reported in parenthesis for Nd contained
compounds. The experimental value of K1 and κ of well known hard magnets are also given forcomparison.
K1

K1

Ha

Ha

Ha

GGA

LSDA

κ

Experimental

0.72* 16)0.35[100]

0.71[100]

0.52* 16)0.88** 16)

2.13[100]
3.27[010]

0.69[100]
1.03[010]

0.84[100]
1.04[010]

1.95[100]
3.01[010]
1.78[100]
2.58[010]

0.46[100]
0.92[010]
0.65[100]
0.87[010]

0.78[100]
0.97[010]
0.74[100]
0.90[010]

K1

GGA
(T)

LSDA
(T)

0.61* 16)0.41[100]

1.68[100]

1.61[100]
2.47[010]

1.45[100]
3.22[010]

1.56[100]
2.41[010]
1.43[100]
2.08[010]

0.69[100]
2.69[010]
1.34[100]
2.44[010]

GGA(MJ/m3)LSDA(MJ/m3)

Nd2Fe22Ti2(NdFe11Ti)

0.34[100]

1.42[100]

Zr2Fe22Ti2(ZrFe11Ti)

1.10[100]
2.44[010]

ZrNdFe22Ti2((Zr,Nd)Fe11Ti)
ZrNdFe20Co2Ti2((Zr,Nd)Fe10CoTi)(Zr0.3Nd0.7)2(Fe0.75Co0.25)23Ti

0.55[100]
2.16[010]
1.08[100]
1.97[010]

0.67[010]

Experimental(MJ/m3)Alloy

1.68[010]

1.41** 16)
1.78b 33)

1.41c 20)0.80[010]

2.00[010]

Experimental(T)

1.86** 16)
1.9a 90)2.0a 90)

1.7c,d 20)

Nd2Fe14B
4.9e 89)
Sm2Co174.2e 89)
SmCo5
17.0e 89)
Alnico 5
0.32e 89)[010]Calculated using directions [001] vs [010].
[100]Calculated using directions [001] vs [100].
*Theoretical GGA+U references.
**Theoretical LSDA references.
aDerived from magnetization curve at 300 K.
bSingle Crystal, Physical Property Measurement System (PPMS) measured at 10 K.
cCalculated from source data.
dCalculated out of experimental µ M using the law of approaching saturation.
0 S
eProperties measured at 300 K.

Ha reflects an important property of a permanent magnet
and should be as high as possible. It is important to notethat the values calculated here for K1 and MS are valid
for T = 0 K. This excludes the temperature dependence of
H a.
With the anisotropy constant K1 and the saturation
magnetization MS, the magnetic hardness factor κ can becalculated. This is done using the following equation:89)
𝐾𝐾1𝜅𝜅 = √.
𝜇𝜇0 𝑀𝑀𝑆𝑆2(8)

The resulting κ can be used to estimate a materials
potential as a permanent magnet. A magnetic compound
can be classified as a soft (κ < 0.1), semi-hard (0.1 < κ <1), or hard (κ > 1) magnet. Materials with κ > 1 aresuitable for use as permanent magnets, regardless of
their shape. On the other hand, those in the semi-hardrange (0.1 < κ < 1) have shape and dimensional
limitations, such as Alnico magnets.87)
Previously we found that MAE increases withincreasing Ti concentration.16) Therefore, we use thisfinding here and only consider compounds with high Ti
concentration (7.7 at. %), as these are also the most likely
stable compounds. Table 5 shows the calculated
anisotropy constant (K1), anisotropy field (Ha) andhardness factor (κ) obtained by GGA+U and LSDA.
For the NdFe11Ti compound, the GGA+U calculation
yielded an anisotropy constant of 0.67 (0.34 for [100] vs[001]) MJ/m3, which significantly underestimated theexperimentally measured K1 of 1.78 MJ/m3 by Skokov etal.33). Therefore, we considered LSDA to further
investigate the MAE. The LSDA calculation results a
significant increase in K1 to 1.68 (1.42 for [100] vs70

κ

0.49[010]

0.77[010]

κ

0.71c 20)
1.54e 89)
1.89e 89)
4.40e 89)
0.45e 89)[001]) MJ/m3, which is in better agreement with theliterature.
The substitution of Nd with Zr increases the MAE
compared to NdFe11Ti. The quaternary (Zr,Nd)Fe11Ti has
a K1 potentially exceeding the experimental value of
1.78 MJ/m3 for NdFe11Ti, with a difference of 0.38 MJ/m3
for GGA+U and 0.63 MJ/m3 for LSDA. In case of the Zrbased 1:12 compound, ZrFe11Ti, K1 increases by a further
0.28 MJ/m3 for GGA+U and by 0.06 MJ/m3 for LSDA.
However, in case of [100] vs [001], the calculated K1 ofthe Zr-containing compounds do not reach to theexperimental K1 of NdFe11Ti. The values calculated for(Zr,Nd)Fe11Ti and ZrFe11Ti in the case of GGA+U are
1.23 and 0.68 MJ/m3 lower, respectively. The LSDA K1
are also smaller by 0.22 and 0.17 MJ/m3 than the
1.78 MJ/m3 of NdFe11Ti. The evaluation of the Zrcontaining phase is challenging due to a lack of available
literature. The small difference between the LSDA valueof (Zr,Nd)Fe11Ti (2.41 MJ/m3) and the K1 of ZrFe11Ti(2.44 GGA+U and 2.47 MJ/m3 LSDA) suggests that theaccuracy of the GGA+U method increases with higher Zrcontent. This is also supported by the fact that thedifference between the GGA+U and LSDA values for K1
decreases as the Zr content increases, from 1.01 (1.08 for[100] vs [001]) MJ/m3 for Nd2Fe22Ti2 to 0.25 (1.01 for[100] vs [001]) MJ/m3 for ZrNdFe22Ti2 and 0.03 (0.51 for[100] vs [001]) MJ/m3 for Zr2Fe22Ti2.
The impact of Co on the K1 can be examined by thedifference between (Zr,Nd)Fe11Ti and (Zr,Nd)Fe10CoTi
compounds. In the case of GGA (LSDA) treatment K1
reduces 0.19 (0.33) MJ/m3. Note that this reduction israised by a low Co concentration, i.e. 7.7 at. %. Suzuki etal.63) observed a slight decrease in Ha of about 0.06 MJ/m3
due to the substitution of 2.3 at. % Co in Nd(Fe1−yCoy)11Ti.
Since Ha and K1 are directly proportional (see Eq. 7), a
decrease in K1 can be expected. Using the data presented

Journal of the Magnetics Society of Japan Vol.48, No.4, 2024

INDEXin the study by Suzuki et al.63), a reduction of about
0.3 MJ/m3 can be derived for K1, which shows a good
agreement with the calculated LSDA value following
Eq. (5). In case of following the anisotropy from theeasiest perpendicular to the easiest in plane direction,
this effect is also visible to a lesser extent, with areduction in case of LSDA of 0.13 MJ/m3.
In addition, K1 values between RFe11Ti alloys (R: Y, Zr,
Ce and Nd) shows a clear trend in GGA+U values: Nd(0.67 MJ/m3) < Y (0.92 MJ/m3 16)) < Ce (2.08 MJ/m3 16)) <Zr (2.16 MJ/m3). Corresponding LSDA values show aparallel trend: Nd (1.68 MJ/m3) < Y (1.73 MJ/m3 16)) < Ce
(2.12 MJ/m3 16)) < Zr (2.41 MJ/m3). In particular, Zr
induces the most significant increase in K1. The K1
obtained for Zr-containing compounds is considered to bereasonably, especially when compared to the Cecompound. However, when uniaxial anisotropy isensured, the trends change as Zr then shows smaller
values than Ce by 0.51 MJ/m3 and Y by 0.12 MJ/m3 in thecase of LSDA and a smaller value than Ce by 0.98 MJ/m3
in case of GGA+U.
In the context of this study, which includes quaternarycompounds and draws comparisons with previous
research16), the trend calculated in the GGA+U (andLSDA) treatments is as follows: (Y,Nd)Fe11Ti with 0.91
(1.65)16) < (Ce,Nd)Fe11Ti with 1.35 (1.61)16) <(Zr,Nd)Fe11Ti with 2.16 (2.41) MJ/m3. The position of(Zr,Nd)Fe11Ti in this trend is reversed when the valuesof calculations [100] to [001] are considered.
In Eq. 7 the anisotropy field (Ha) is determined by theanisotropy constant (K1) and the saturation
magnetization (MS). It is noteworthy that, as with K1,
there is a shortage of literature for the comparison of Zrcontaining compounds. In case of NdFe11Ti, comparisons
can be made with the experimental Ha values obtained
by Bouzidi et al.90) (1.9 T) and Akayama et al.60) (2.0 T).
The calculated Ha from the GGA+U treatment (0.80 (0.41
for [100] vs [001]) T) is significantly lower than theliterature value, while the LSDA treatment gives a Ha of
2.00 (1.68 for [100] vs [001]) T, showing good agreement.
For ZrFe11Ti, (Zr,Nd)Fe11Ti and (Zr,Nd)Fe10CoTi, the
GGA+U (and LSDA) treatments following the easiest
perpendicular [001] and the hardest directions [010] giveHa values of 3.22 (3.27), 2.69 (3.01) and 2.44 (2.58) T,
respectively. Using the easiest perpendicular [001] andthe easiest in-plane direction [100], the Ha values are
1.10 (1.61), 0.55 (1.56) and 1.08 (1.43) T, respectively.
These Ha values show similar trends to those calculated
for K1. Analogous trends are evident when comparing the
Y and Ce compounds from our previous research16). For
the quaternary compounds the Ha increases with GGA+U(LSDA) treatment as follows: (Zr,Nd)Fe11Ti with 0.55
(1.56) ([100] to [001]) < (Y,Nd)Fe11Ti with 1.13 (2.33)16) <(Ce,Nd)Fe11Ti with 1.73 (2.30)16) < (Zr,Nd)Fe11Ti with
2.69 (3.01) T ([010] to [001]).
The κ values calculated according to Eq. 8 are shownin Tab. 5. The hardness factor (κ) of NdFe11Ti,
determined using the experimental data of Skokov etal.33), giving a κ of 0.93, which is in good agreement withthe calculated κ of 0.77 (0.71 for [100] vs [001]) for LSDA.
The κ values calculated with LSDA of RFe11Ti (R: Y, Zr,
Ce and Nd) show that NdFe11Ti has the lowest κ with
0.77 (0.71 for [100] vs [001]). This κ value then increases
by 0.27 (0.13 for [100] vs [001]) for ZrFe11Ti, followed by

0.41 for YFe11Ti and finally by 0.54 for CeFe11Ti. In thecase of the GGA+U treatment, κ also increases, startingwith 0.49 (0.35 for [100] vs [001]) for NdFe 11Ti, followedby an increase of 0.22 for YFe11Ti, 0.54 (0.34 for [100] vs[001]) for ZrFe11Ti and finally 0.63 for CeFe11Ti. Thus, Zr
increases the hardness factor compared to Nd, but to alesser extent than Y or Ce.
The calculated κ values of the quaternary compound(Zr,Nd)Fe11Ti are 0.92 (0.46 for [100] vs [001]) (GGA+U)
and 0.97 (0.78 for [100] vs [001]) (LSDA). As there is no
direct literature on the Zr compound, it is classified as asemi-hard magnet. Comparing the quaternarycompounds (X,Nd)Fe11Ti (X: Y, Zr and Ce), (Y,Nd)Fe11Ti
has the lowest κ values for GGA+U at 0.67, followed by(Ce,Nd)Fe11Ti with an increase of 0.16 and (Zr,Nd)Fe11Ti
with an increase of 0.25. In case of LSDA, (Zr,Nd)Fe 11Ti
has the lowest κ value at 0.97, with the values for(Y,Nd)Fe11Ti and (Ce,Nd)Fe11Ti being 0.05 higher.
Looking at the values according to the easiest
perpendicular and easiest in plane direction, Zr has thelowest κ in case of GGA+U and LSDA. In the case of Zr
in particular, the GGA+U method gives a more accurateκ than in case of Y or Ce, as can be seen from thedifferences between the GGA+U and LSDA values.
Specifically, the difference is 0.47 for (Y,Nd)Fe11Ti, 0.19
for (Ce,Nd)Fe11Ti and 0.05 for (Zr,Nd)Fe11Ti. The
difference of (Zr,Nd)Fe11Ti, on the contrary, increases to0.32 when considering the [100] to [001] values, which islarger than that of the Ce-containing compound.
As in case of K1 and Ha, the substitution of Co reduces
κ. For the (Zr,Nd)Fe10CoTi compound, the calculated κvalues are 0.87 (0.65 for [100] vs [001]) for GGA+U and0.90 (0.74 for [100] vs [001]) for LSDA. These values arelower than those of the (Zr,Nd)Fe11Ti compound, with adecrease of 0.05 (an increase of 0.19 for [100] vs [001]) for
GGA+U and a decrease of 0.07 (0.04 for [100] vs [001]) for
LSDA. In absence of comparable literature for K1, Ha andκ for (Zr,Nd)Fe11Ti or (Zr,Nd)Fe10CoTi, these values canonly be approximately compared with the analogue
compound (Zr0.3Nd0.7)(Fe0.75Co0.25)11.5Ti0.5. The literaturesample gives 1.41 MJ/m3 for K1, 1.7 T for Ha and a κ of
0.71.20) To sum up, the κ values of the quaternary andquinary compounds in this study exceeds those of thereferenced literature compounds by following thedirections [001] vs [010]. These discrepancies can be
attributed to a higher Zr content, as well as higher Ti andlower Co concentrations in the selected compounds.
Therefore, a reasonable agreement with the limited
available literature can be expected. When the uniaxial
anisotropy is ensured by following the directions [001] vs[100], the calculated values show good agreement withthe literature.
When comparing the K1, Ha and κ characteristics
between the [001] vs [010] and [001] vs [100] orientations(see Tab. 5 for values), there are noticeable deviations.
However, the [001] vs [100] comparison is essential as it
ensures the consistency of the MAE values with theuniaxial anisotropy. Regarding K1, the evaluation showsthat NdFe11Ti has the closest agreement with theliterature in terms of LSDA for [001] vs [010]. In contrast,
the comparison of (Zr,Nd)Fe11Ti and (Zr,Nd)Fe10CoTi
withtheliteraturecompound(Zr0.3Nd0.7)(Fe0.75Co0.25)11.5Ti0.5 shows a better agreementin case of [001] vs [100]. Nevertheless, the values for

Journal of the Magnetics Society of Japan Vol.48, No.4, 2024

71

INDEX[001] vs [010] remain possible due to the differences
between the calculated compounds and the literaturecompound. Similar observations apply to Ha.
Furthermore, the effect of Zr substitution in bothproperties shows a comparable effect compared to Y and
Ce. Regarding κ, all compounds investigated in bothcontexts are in the range of semi-hard magnets.
Furthermore, it is observed that the Zr substitution
increases κ towards hard magnets. Consequently, K1, Ha
and κ values of the examined Zr-containing compounds
appear promising.

4. Conclusion
Based on positive results for Nd-lean magnets15,63), we
investigated (Nd,X)Fe12−yTiy X=Zr compounds for theirmagnetic properties. Previous results suggested Y and Ce
as suitable substitutes.16) Here, we investigated Zr as analternative to Nd, as it is more abundant and lessexpensive.17,18) Despite the lack of direct experimental
data, our theoretical results agree well with theliterature on similar compounds. The literature also
discusses the effect of Co on phase stability and magnetic
properties when substituted with Zr.11,19,20,63) Our
calculations give us confidence in the calculated
properties for Nd-lean quaternary and quinary
compounds.
An accurate description of the RE-4f-electrons isessential for predicting the intrinsic magnetic properties.
The highly localized Nd-4f-electrons pose a challenge tothe accurate calculation of this property using available
DFT exchange correlation functionals. To address this,the DFT+U approach with a Hubbard U value of 6 eV
was used as a correction scheme, as previously done inliterature for 1:12 Nd-based compounds, with favorable
results.15,16,33)
The methodology of our study successfully predicts theoverall magnetization trend of 1:12 phases with NdFe11Ti
exhibiting the highest magnetization, followed by
ZrNdFe11Ti and ZrFe11Ti. For NdFe11Ti, µ0MS ranges
from 1.38 to 1.70 T, which agrees well with our GGA(GGA+U) calculations of 1.68 (1.69) T. With Zr
substitution µ0MS reaches 1.60 (1.62) T for ZrNdFe11Ti
and 1.51 T for ZrFe11Ti using GGA (GGA+U). When
compared with (Zr0.3Nd0.7)(Fe0.75Co0.25)11.5Ti0.5, a good
agreement can be seen with the experimentally foundµ0MS value of 1.63 T.63) Our GGA (GGA+U) calculations
show a high |BH|max of 510 (525) kJ/m3 for ZrNdFe11Ti,
exceeding Sm2Co17B with 294 kJ/m3.75) Notably,
ZrNdFe11.5Ti0.5, a compound that includes less Ti, has a|BH|max of 600 (603) kJ/m3 which is smaller than thetheoretical |BH|max of Nd2Fe14B with 688 kJ/m3.31) It
was also found that the effect of Co on the magnetic
properties depends on the presence of Ti in the compound.
In combination with Ti, a decrease of the magnetic
properties (mtot, |BH|max) is observed, while an increaseis calculated in the absence of Ti.
The mean field approximation (MFA) computes theCurie temperature, consistently overestimating it in the
ferromagnetic (FM) state, but quantitatively inagreement with the existing literature. In particular,
REFe11Ti compounds show superior results in the local
moment disordered (LMD) state, supported by
NdFe12−yTiy (y: 0 ≤ y ≤ 1) compounds.82) FM-based
calculations tend to overestimate by about 45% for72

ternary compounds. The calculated Curie temperaturesfor (Zr,Nd)Fe11Ti are 783 K (FM) and 652 K (LMD),
which are higher than the experimental TC (588 K) for
Nd2Fe14B but lower than the theoretical TC of 1050 K. A
7.7 at. % Co substitution in (Zr,Nd)Fe11Ti to(Zr,Nd)Fe10CoTi increases TC by about 85 K.
The magnetocrystalline anisotropy constant K1 in Ndbased compounds was insufficiently described bygeneralized gradient approximation (GGA) methods, asreported in previous studies.16,33,91) The local spin density
approximation (LSDA) showed a better agreement withthe experimental results. The GGA+U method gave moreaccurate results with increasing Zr concentration, where
Zr(Co) induced an increase (decrease) inmagnetocrystalline anisotropy. The hardness factor κ,
which indicates a material’s potential as a permanent
magnet, is derived from K1 and the saturation
magnetization µ0MS. For the 50% Nd-lean (Zr,Nd)Fe11Ti
compound a theoretical κ of 0.97 (0.78 in case of [100] vs[001]) has been calculated, classifying it as a semi-hardmagnet. To further increase κ, nitrogenation has been
proposed, as shown in previous research.11, 16)
In conclusion, this study demonstrates that thesubstitution of the critical rare-earth (RE) element Nd
with the more abundant Zr in ThMn12 compounds yields
promising RE-lean magnets with magnetic properties
comparable to or only slightly inferior to other Nd-leanmagnets. Consequently, based on our theoretical
calculations,
the
newly
proposedquaternary(ZrNd)Fe24−yTiy compounds emerge as potential andpromising permanent magnets.

Acknowledgements Calculations were performed at

the HPC Cluster CARL, located at the University of
Oldenburg (Germany) and funded by the DFG, UnitedStates through its Major Research Instrumentation
Program (INST 184/157-1 FUGG) and the Ministry of
Science and culture (MWK) of the Lower Saxony State.

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Received Feb. 11, 2024; Revised Mar. 30, 2024; Accepted May 01,
2024

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Copyright ©2024 by the Magnetics Society of Japan.
This article is licensed under the Creative Commons Attribution International License (CC BY 4.0)
http://creativecommons.org/licenses/by/4.0/

J. Magn. Soc. Jpn., 48, 75-80 (2024)<Paper>

Cha r a c t e r i z a t i o n o f M a g ne t o s t r i c t i v e F i l m o n S ha f t S ur f a c e i n I nv e r s e
M a g ne t o s t r i c t i v e T o r q ue S e ns o r U s i ng M a g ne t o - o p t i c a l K e r r Ef f e c t
K. Ishibashi*,**, M. Sonehara**, T. Kodaira*, T. Sasaki*, and T. Sato***Sensortronics Laboratory, Tamagawa Seiki Co., Ltd., 1020 Kega, Iida-shi, Nagano-ken, 395-8520, Japan** Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano-shi, Nagano-ken, 380-8553, Japan

Methods for evaluating the characteristics of magnetostrictive materials include dynamic magnetic domain
observation using a magneto-optical Kerr effect microscope, obtaining B-H curves using a vibrating sample typemagnetometer that vibrates a magnetized sample with a DC magnetic field, and so on. Generally, magnetic domains
are observed in magnetic materials on flat surfaces. The authors previously researched and developed an inverse
magnetostrictive torque sensor in which a magnetostrictive material is deposited on a cylindrical shaft surface.
Therefore, to accurately evaluate the characteristics of a torque sensor, it is necessary to observe the magnetic
domains while torque is applied on a curved surface. This paper explains that method and presents the results ofdynamic domain observation of magnetostrictive materials on the shaft surface using a newly developed
magneto-optical Kerr effect microscope.
Keywords: magneto-optical Kerr effect, magnetostrictive torque sensor, magnetic domain observation
1 . I n tro d u cti o nCurrently, in response to global energy challenges suchas mitigating global warming, the automotive industry
is accelerating the adoption of next-generation vehicles
like electric vehicles (EV) and plug-in hybrid vehicles(PHV) 1). Moreover, the electrification trend extends tovarioussectors
includingrobotics,
householdappliances, drones, and even aircraft. While these
initiatives primarily target energy conservation andCO2 reduction, optimizing the efficiency of electricmotors is paramount.
Traditionally, motor control relies on detecting the
motor's rotational axis angle and regulating the optimalcurrent flow for each angle2). The precision of angle
sensors3,4,5,6) and current sensors7), which detect therotational position of the shaft8), significantly influences
motor controllability. However, when the motortransfers power to the shaft, various disturbance load
torques such as friction, torsion, and gear effects are
generated, limiting control accuracy with conventional
angle and current sensors. Therefore, achieving
high-efficiency motor control requires the detection andfeedback of actual shaft torque, inclusive ofdisturbances9). This paper focuses on an inverse
magnetostrictive torque sensor10,11,12,13) utilizing theinverse magnetostrictive effect14,15), deemed suitable formotor control. The verification results of an evaluation
method for magnetostrictive material films, crucial indetermining torque sensor performance, are presented.
Figure 1 (a) illustrates the configuration of theinverse magnetostrictive torque sensor employed in this
Corresponding
author:K.
Ishibashi
kazuyuki-ishibashi@tamagawa-seiki.co.jp).

(e-mail:paper. In the case of an inverse magnetostrictive torquesensor, a magnetostrictive material film is deposited on
the surface of the shaft. When torque is applied to theshaft, both tensile and compressive stresses are inducedon the magnetostrictive film adhered to the shaft's
surface. A winding coil is positioned on themagnetostrictive film as a detection unit to monitorchanges in the characteristics of the magnetostrictive
material16). As torque is applied, both tensile andcompressive
stressesareexertedonthemagnetostrictive material, resulting in alterations of its
magnetic properties (permeability) in each direction.
(a)

(b)

Fig. 1 Principle of inverse magnetostrictive torquesensor detection. (a) Relationship between stressdirection and magnetization direction of inverse
magnetostrictive torque sensor. (b) Coil inductanceschange and output voltage characteristics.

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Therefore, by arranging two coils (detection coil 1 andcoil 2) according to the direction of stress, changes in
the magnetic properties of the magnetostrictive
material can be detected as variations in coilinductance, as shown in Figure 1 (b). By converting thisdifference into voltage, it is possible to detect thedirection and magnitude of applied torque. Note that
Figure 1 (b) illustrates the principle and is not based on
actual measurement data.
The authors have previously conducted evaluations of
sensor characteristics using electroplating as themagnetostrictive film of the magnetostrictive torquesensor17). Through this experience, they have recognized
the importance of the change in magnetic properties ofthe magnetostrictive film when torque is applied, as a
crucial characteristic of torque sensors. However, atpresent, a method for properly evaluating themagnetostrictive material formed on the surface of thetorque sensor shaft has not been established. The
magnetostrictive film material used as a torque sensor
needs to be evaluated non-destructively, reproducing
the direction of the magnetic field and the state of
torque application on the curved surface of the axis tomatch the shape of the torque sensor. However,conventional magnetic property evaluation methods
such as B-H analyzers are not configured to obtainmagnetic properties when torque is applied to themagnetostrictive material on the axial surface. This
paper focuses on magneto-optical Kerr effectmicroscopy, which has the advantage of confirming
magnetic properties non-destructively. The authors
conducted observations of the movement of magnetic
domain walls18) and the observation of magnetic
domains according to the operating conditions of thetorque sensor, dynamically evaluating the magnetic
properties of the magnetostrictive film material 19).
2 . Exp e ri m e n ta l M e th o d
2 . 1 E xp e ri m e n ta l d e vi ce s tru ctu re
Figure 2 (a) depicts the configuration of themagneto-optical Kerr effect microscope utilized in thisstudy. This apparatus is equipped with a mechanism toexert external torque on the shaft designated for thetorque sensor and allows for arbitrary adjustment of theexternal magnetic field angle. Figure 2 (b) illustrates
the configuration of the magneto-optical Kerr effectmicroscope (NEOARK CORPORATION: BH-753-TSC)
employed for shaft evaluation in this research. Table 1
outlines the device specifications. The magnetic field
applied during magnetic domain observation ranges
from ±24 kA/m (±300 Oe), and due to equipment
limitations, only positive torque can be applied to theshaft, with a maximum of 100 Nm.
2 . 2 Sa m p l e s h a f t o ve rvi e w
Figure 3 illustrates a sample for magnetic domain
observation. This sample is a thin-film magnetostrictive
20,21)
sample(Fe-Si-B Amorphous Metal Ribbon

76

22)

Material : Hitachi Metals Metaglas Ⓡ 2605SA1,
Thickness: 0.025 mm), chosen for its high permeability
and low hysteresis characteristics suitable for torquesensing applications. The thin film was affixed to a
40-mm stainless steel shaft using adhesive. With thissetup, magnetic domain observation was conducted, andthe magnetization curve was determined based on thecontrast ratio of the magnetic domain image for eachapplied torque.
(a)

(b)

Detector (CCD camera)cameraMagnetic field
direction (+45o)

Applying torque(0 to +50 Nm)

Shaft

Fig. 2 Overview of Kerr effect microscope. (a) Device
function overview. (b) Magneto-optical Kerr effectphotomicrograph for shaft evaluation.
Table 1 Kerr effect microscope specifications.
Item

Specification

Observation
cameraNumber of pixels: 1280×960
Gradation: 12 bit

Objective lens

10 times

Field of view size

480×360 μm

Max magnetic field±23,873 A/m (±300 Oe)

Max torque+100 Nm

Fig. 3 Sample shaft overview.

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3 . Exp e ri m e n ta l R e s u l ts
3 . 1 M a gn e ti c d o m a i n i m a ge o b s e rva ti o n re s u l ts
The direction of the magnetic field applied to theshaft is +45o. Magnetic domain images were observedwith applied torques of 0 Nm, 10 Nm, and 20 Nm.
Figures 4, 5, and 6 show images for each appliedmagnetic field and torque. When torque is applied, thesurface of the magnetostrictive film in the +45 omagnetic field application direction experiencescompression. The boundary between contrast shades in
the magnetic domain image represents the domain wall,and it is evident that the domain wall moves with eachapplied magnetic field. In Figures 4, 5, and 6, theapplied magnetic field around 400 A/m to 700 A/m,
where the movement of the domain wall (change incontrast) is most pronounced, is highlighted.
Furthermore, since the applied magnetic field cannot be
arbitrarily set due to device resolution, the appliedmagnetic fields under each condition are not uniform.
3 . 2 M a gn e ti z a ti o n cu rve co m p a ri s o n a n d co n s i d e ra ti o n
Figure 7 illustrates a graph measuring themagnetization curve from the contrast ratio of themagnetic domain images. The vertical axis represents
the contrast ratio, indicating the ratio of gradation
when the contrast around ±1000 A/m at 0 Nm is set to±1. The slope of the magnetization curve represents themagnetic permeability, and in this experiment, theslope becomes gentler at +10 Nm and +20 Nm, resulting
in a decrease in magnetic permeability.
1). +667 A/m

2). +621 A/m

1). +665 A/m

2). +602 A/m+10 Nm Torque

Shaft direction
3). +539 A/m

4). +455 A/m

5). +390 A/m

6). +408 A/m

Fig. 5 Magnetic domain image with +10 Nm torque.
1). +657 A/m

2). +602 A/m+20 Nm Torque

Shaft direction

Shaft direction
3). +552 A/m

4). +489A/m

3). +528 A/m

4). +467 A/m

5). +429 A/m

6). +377 A/m

Domainwall

5). +428 A/m

6). +393 A/mFig. 4 Magnetic domain image with zero torque.

Fig. 6 Magnetic domain image with +20 Nm torque.

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Figure 7 (b) is a graph showing the variation incontrast ratio from 400 A/m to 700 A/m within thegraph of Figure 7 (a), where the change is notable.
Additionally, in Figure 7 (c), the rate of change of theslope of the magnetization curve for each applied torqueis depicted. This graph confirms the reproducibility of
the applied torque direction, in the sequence of 0 Nm,
10 Nm, 20 Nm, 10 Nm, and 0 Nm. In Figure 7 (c), as thetorque increases, the rate of change of the slope of themagnetization curve decreases, and when the torquereturns to 0 Nm, the slope reverts to its original value.
Magnetic demagnetization is performed for eachmeasurement, hence magnetic hysteresis due to torqueapplication is not considered. Additionally, the contrast
ratio at 0 Nm serves as the reference point, set at 100%.
The reason for the variation in the slope of themagnetization curve upon applying torque is believed tostem from the phenomenon where the applied torquecompresses the magnetostrictive material in the
direction perpendicular to the external magnetic field,
leading to a decrease in permeability as torque isapplied.
(a)Next, using the same evaluation axis, the slopechange of the magnetization curve due to the contrast
ratio of the magnetic domain image was confirmed at
multiple points. Figure 8 shows the measurement
points of the shaft. Four locations were extracted at
90-degree intervals with respect to the axis rotation
direction, and magnetic domain images were observedwith a magneto-optical Kerr effect microscope for eachapplied torque. Figure 9 shows a plot of the rate ofchange in the slope of the magnetization curve for eachapplied torque at each point. At any point on the axis,
the slope of the graph became gentle when +50 Nm
torque was applied. At each observation point, the rate ofchange in the slope of the 0-degree point is approximately±10%, while at the 90-degree point, it is about ±40%,resulting in an average rate of change of approximately±20%. The differences observed at each observation point
are thought to be due to the non-uniformity in themagnetic properties of the amorphous ribbon foil.
Therefore, it is considered possible to improve theaccuracy of the torque sensor by arranging multiple
detection units and averaging the acquired characteristic
results when constructing a torque sensor using materials
similar to those used in this experiment.

To compare with the experiment using themagneto-optical Kerr effect microscope, Figures 10 and
11 present the results of confirming the change in coil

Measuring
points0 deg

90 deg270 deg(b)

Magnetostrictive
film

180 deg
Evaluation shaft

Measuring points(4 places every 90 degree)

Fig. 8 Measurement points on the shaft.

(c)Fig. 7 Magnetization curve for each applied torque.
(a) Magnetization curve measured from contrast ratio.
(b) Enlarged graph around 400 A/m to 700 A/m.
(c) Magnetization curve slope change rate comparison.
The arrows indicate the sequence for torque application.

78

Fig. 9 Magnetization curve slope change ratecomparison.

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inductance using the same evaluation shaft. Figure 10
depicts the positioning of the detection coil on theevaluation shaft, with the core featuring the detection
23)
coil winding oriented at +45o in the axial direction .
The core material utilized was permalloy sintered metal,
with an air gap of l = 0.1 mm between the core and theshaft. The coil itself consisted of magnet wire with awire diameter of φ0.1 mm and N = 30 turns. During
measurement, the excitation frequency was set to 52
kHz with a current of 10 mA. The coil inductance L atthat time can be calculated using the following
equation.

L=μN 2S/l ・・・(1)In the equation, μ represents the magnetic
permeability, and S is the cross-sectional area of thecore. Parameters other than μ are fixed values, so L is
directly proportional to μ. Therefore, the change in coilinductance L is proportional to the change in μ that
occurs when torque is applied to the shaft. Figure 11
illustrates the trend of coil inductance L when torque isapplied. At +50 Nm, the surface of the magnetostrictive
film of the detection coil oriented at +45 o undergoes
compression, leading to a decrease in coil inductance L.
This indicates a reduction in magnetic permeability μ at
+50 Nm. Furthermore, when the torque is returned to 0
Nm, the inductance L reverts to its original value. This
trend mirrors the change in the slope of themagnetization curve estimated from the contrast ratio
observed with the magneto-optical Kerr effectmicroscope, as demonstrated in Figure 7 or Figure 9.
Additionally, the starting point is set at 100% for 0 Nm.
In this scenario, the uppermost point at 0 Nm serves as
the starting point.

Fig. 10 Detection coil arrangement model.

However, there was a discrepancy in the rate ofchange between the observed alteration in magnetic
permeability using the magneto-optical Kerr effectmicroscope and the change in magnetic permeability
measuredvia
coil
inductance.
While
the
magneto-optical Kerr effect microscope measurements
showed a change of approximately 10 to 40%, thealteration in coil inductance indicated a result of about
0.3% to 0.4%, significantly diverging from the former.
This disparity is attributed to the fact that the external
magnetic field generated by the detection coilmeasurement method is lower than the ±1000 A/m
range of the Kerr effect microscope. When calculatedwith a coil turn number of 30 turns, a current of 10 mA,
and a core length of 10 mm, the generated external
magnetic field is only about 1/30 of ±30 A/m. Therefore,
it is conceivable that the change in magnetic
permeability during torque application is minimal.
4 . Su m m a ry
The trend of the change in the magnetization curve
slope measured from the contrast ratio using amagneto-optical Kerr effect microscope for different
applied torques was the same as the result of
measuring with a coil, but the rate of change itself wasdifferent. Moreover, in this evaluation method, a means
to quantitatively evaluate permeability has not been
established. However, in inductance evaluation using acoil, there are significant error factors such as
variations in the coil winding, fluctuations in themagnetic properties of the core, and variations inmounting position. Additionally, increasing the external
magnetic field strength, which is the variable in thisexperiment, requires enlarging the detection unit,
thereby imposing limitations on evaluating thematerial's intrinsic properties. This measurement
method, utilizing a magneto-optical Kerr effectmicroscope, is less susceptible to these factors and issuitable for evaluating the torque sensor shaft in its
product state. In the future, by quantifying changes inmagnetic permeability, this evaluation method could
serve as a benchmark for assessing single-material
evaluation methods, contributing to the development of
optimal materials for torque sensors.
Acknowledgments This paper was supported by a
subsidized project of the New Energy and Industrial
Technology Development Organization (NEDO).
Re f e re n ce s

Fig. 11 Graph of coil inductance change rate. The
arrows indicate the sequence for torque application.

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22, 2024

Journal of the Magnetics Society of Japan Vol.48, No.4, 2024

INDEX

Editorial Committee Members・Paper Committee Members
S. Yabukami and T. Taniyama (Chairperson), N. H. Pham, D. Oyama and M. Ohtake (Secretary)
H. Aoki
M. GotoT. GotoK. Hioki
S. Inui
M. Iwai
Y. Kamihara
H. Kikuchi
T. Kojima
H. KuraK. Masuda
Y. Nakamura
K. Nishijima
T. Nozaki
T. SatoT. Suetsuna
K. Suzuki
Y. Takamura
K. ThamT. Tanaka
N. Wakiya
S. Yakata
A. YaoS. Yamada
M. Yoshida
N. Adachi
S. Haku
K. Kobayashi
Y. SatoI. Tagawa
S. Yoshimura

K. Bessho
S. Honda
T. MakiS. Seino
K. Tajima

M. DoiS. Isogami
S. Muroga
M. Sekino
M. Takezawa

T. DoiT. Kawaguchi
M. NaoeY. Shiota
T. Takura

T. Hasegawa
T. KawaiT. Narita
T. Shirokura
S. Tamaru

K. ItoA. Kuwahata
E. Shikoh
M. Toko
R. Hashimoto
N. Kikuchi
M. Sakakibara
S. Sugahara
T. Yamazaki

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編 集 委 員・論 文 委 員
薮上 信（理事） 谷山智康（理事） Pham NamHai（幹事）
青木英恵
伊藤啓太
乾
成里
岩井守生
後藤太一
後藤
穣
佐藤
拓
仕幸英治
Kim Kong Tham
仲村泰明
西島健一
山田晋也
吉田征弘
脇谷尚樹

大竹 充（幹事） 小山大介（幹事）
神原陽一
菊池弘昭
藏
裕彰
末綱倫浩
鈴木和也
高村陽太
野崎友大
日置敬子
増田啓介

桑波田晃弘
田中哲郎
八尾
惇

小嶋隆幸
都甲
大
家形
諭

安達信泰
白倉孝典
土井達也
槙
智仁

菊池伸明
田倉哲也
白
怜士

佐藤佑樹
田島克文
別所和宏

塩田陽一
田丸慎吾
本多周太

磯上慎二
菅原
聡
土井正晶
室賀
翔

川井哲郎
清野智史
直江正幸
山崎貴大

川口昂彦
関野正樹
成田正敬
吉村
哲

小林宏一郎
田河育也
橋本良介

榊原
満
竹澤昌晃
長谷川 崇

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Journal of the Magnetics Society of Japan
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