Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Development of permanent magnet materials—from the view
point of ﬁrst-principles calculation
Hisazumi AKAI1 and Shinji TSUNEYUKI2
1 The Institute for Solid State Physics, The University of Tokyo
Kashiwa-no-ha, Kashiwa, Chiba 277-8581, Japan
2 Graduate School of Science, The University of Tokyo
Hongo, Bunkyo-ku, Tokyo 113-0033, Japan

Abstract
Theoretical approaches investigating the intrinsic
properties of permanent magnet materials are reviewed. Three topics, namely, possible maximum performance of magnetic materials, highthroughput calculation surveying candidates forthe high-performance magnetic materials, and theﬁnite temperature magnetic properties, are discussed. Emphasis is put on the importance of large
scale ﬁrst-principles as well as high-throughput calculations.

1

Introduction

The properties featuring high-performance permanent magnets such as Nd2 Fe14 B originate from
subtle combination of micro, macro and metallographic structures. This means that intrinsic
properties of permanent magnet materials are notdirectly reﬂected to the properties of permanent
magnets. On the other hand, it is also true that theperformance of a permanent magnet as a whole isseverely limited by the intrinsic properties of constituent materials. For example, the upper bound of
coercivity is the magnetocrystalline anisotropy—
reality is that it is usually only 20∼30% of thelatter. Among those intrinsic properties, important ones are, in addition to the magnetocrystalline
anisotropy, the magnetic moment and Curie temperature. The higher the values, the better the
performance expected. The importance of ﬁrst-

principle calculation arises here: although predicting the properties of permanent magnets is very
diﬃcult task, it would be still possible to predict
better materials for permanent magnets on the basisof ﬁrst-principles calculation. Also, the approachis powerful to analyze the mechanism inherent tocertain speciﬁc electronic and magnetic properties.
In the past decade, many works based on ﬁrstprinciples calculation have been done from theabove point of view [1]. There are two aspects
existing in those works: the ﬁrst one is the accumulation of results of ﬁrst-principles calculations
and their analyses. The second is the developmentof methodology that enables us to perform such attempts. In this paper, we will present some of ourresults as examples of the former, putting emphasis
upon the importance of the latter.
We ﬁrst will discuss the possible maximum performance of permanent magnet on the basis of theﬁrst principles calculation [2]. In spite of the intensive eﬀorts, no essentially novel magnet showing better performance than Nd2 Fe14 B magnet has
been so far invented. Then, one might question if
there exists any permanent magnet materials at allsuperior to the already known ones. Or, might it bebetter to proceed to other directions such as pursuing magnets less expensive but with reasonable performance, and hence, replacing ferrite magnets? In
answering this, we will show that there exists a possible maximum performance that we can expect forpermanent magnets. Fortunately, this maximum
has not be reached by currently used permanent24Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

magnet materials and it is worthwhile seeking further possibilities.
One of ways to seek such possibilities may be
performing ﬁrst-principles calculation over a wide
range of composition space of magnetic materials. Since most of those materials contain morethan several elements, in many cases as solid solutions, the procedure inevitably becomes a highdimensional search requiring high-throughput calculations. A machinery based on the Korringa–
Kohn–Rostoker Green’s function method combined with the coherent potential approximation(KKR-CPA) is quite eﬃcient for this purpose [3] .
For the high throughput calculations, it is desirable to perform calculations for many systems with
diﬀerent compositions in parallel. We have devised
such a system, called HOFMAN, which realizes
this procedure for KKR-CPA eﬃciently and havebeen applied to various systems. Some examples
will be given in a later section.
The problem arising in ﬁrst-principles calculations of rare-earth permanent magnets is a treatment of 𝑓 -states playing a main role in provoking the magnetocrystalline anisotropy. In the local
density approximation (LDA), or the generalized
gradient approximation (GGA), the strongly localized 𝑓 -states cannot be treated properly. So far, noreal solution for this problem exists. As a compromise, one may use the exact-exchange [4] insteadof the LDA/GGA exchange. Since the 𝑓 -statesin rare-earth elements are suﬃciently localized,
the exchange energy for 𝑓 -states may be replacedwith that of local 𝑓 -orbitals. This can be simply done by subtracting self-interaction from theHartree energy—self-interaction correction (SIC)[5]—because the exchange in the present case ismostly the self-exchange.
The permanent magnets in electric vehicles andwind turbines are used mostly at temperatures 100∼200◦ higher than the room temperature. Therefore,
the importance is on the performance at rather hightemperatures. Although ﬁnite temperature magnetic properties of materials is out of the scopeof Kohn–Sham scheme, we can still access thisproblem through perturbative ways. Namely, we

can estimate the eﬀects of low lying excitation by
constructing a low energy eﬀective Hamiltonian
through DFT as a problem of the ground state ofconstrained systems [6]. The results of such calculations give rise to a set of parameters for the effective Hamiltonian, from which we can obtain theﬁnite temperature electronic and magnetic properties.
There are several kinds of excitations that haveto be taken into account when discussing ﬁnite
temperature magnetic properties of magnetic materials. Among them are phonons and magnons[7, 8, 9]. Usually, single electron excitations at
the Fermi surfaces do not contribute signiﬁcantly
to the magnetic properties at the temperature range
below the Curie temperature. In the following we
neglect such excitations. The treatment of phonons
is the following: We apply a static approximation
for phonons and treat them as external ﬁeld acting on electrons. This problem can be treated in
the framework of DFT. The statistical average canbe taken, in principle, by performing a functional
integral with suitable probability weights. However, this being practically impossible, we further
introduce an approximation to make it tractable.
We employ a single-site approximation, where thefunctional integral is reduced to a conﬁguration
average using CPA. In this scheme, the eﬀects ofphonons are nothing but those of random displacements of atoms around their equilibrium positions.
The average displacements are estimated from either the Debye model or ﬁrst-principles phonon calculations: no noticeaable diﬀerence between them
was found.
The treatment of magnons is somewhat subtle
because the motion of the magnetic moments has tobe separated from the electron degrees of freedom.
We rely on the fact that the time scale of local spin
ﬂuctuations is much slower than that of remaining dynamics. Then we can expand the magnetic
energy into the local spin ﬂuctuations as∬
1
𝐸 mag [𝑺(𝒓)] = −𝑑 𝒓𝑑 𝒓 ′2
𝑺(𝒓) 𝜒(𝒓) −1 𝜒(𝒓, 𝒓 ′) 𝜒(𝒓 ′) −1 𝑺(𝒓 ′), (1)where 𝑺(𝒓) is the spin density, 𝜒(𝒓) and 𝜒(𝒓, 𝒓 ′)25

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

are the local and non-local susceptibilities, respectively. Such an expansion can be performed in
the framework of constrained DFT as long as theabove separation of degrees of freedom and thestatic approximation are employed. This provides
us with a low energy eﬀective Hamiltonian of thesystem, as is already mentioned. If the longitudinal
ﬂuctuations neglected, it is nothing but the Heisenberg Hamiltonian. Now, we can take the same
approach as used for phonons: instead of random
displacement of atoms, random ﬂipping of the local magnetic moments with probability determined
self-consistently at each temperature is consideredwithin KKR-CPA.
We have studied the eﬀects of phonons andmagnons on ﬁnite temperature magnetism, targeting at permanent magnet materials.
We discuss the possible maximum performanceof permanent magnets in Sec. 2. Several examples of high-throughput calculation are presented
in Sec. 3. Section 4 is devoted for the ﬁnite temperature properties of magnetic materials under the
existences of phonons and magnons. We summarize in Sec. 5.

2against the lattice constant 𝑎 and atomic number
𝑍. The crystal structure is ﬁxed to bcc. Here,the fractional atomic number of a ﬁctitious atom
is used: 𝑍 = 25, 26, and 27 corresponds to Mn,
Fe, and Co atoms, respectively. Whereas 𝑀 is adecreasing function of 𝑍 for a large lattice constant(𝑎 ∼ 3.2 Å), it shows a usual Slater-Pauling type behavior [11] for a lattice constant of bcc Fe (2.867 Å
at room temperature): it takes the maximum valueof 2.35 𝜇B /atom at 𝑍 = 26.2 and decreases to bothsides. The peak position shifts towards larger 𝑍
with decreasing lattice constant. Meanwhile, thepeak height decreases rapidly. These behaviors
are the results of the facts: (1) the position where
the transition from strong to weak ferromagnetism
takes place is shifted, and (2) the position where theinstability of ferromagnetism against volume collapse takes place is shifted, both occurring whenatomic number decreases.

Maximum performance of permanent magnet materials

We assume that 3𝑑 magnetic ions are the main origin of magnetic polarization (magnetization). On
the other hand, 4 𝑓 ions are of main concern forthe magnetic anisotropy energy (MAE) although
3𝑑–5𝑑 transition metal ions partly contribute to it,in particular, at high temperature. For this reason,
we will discuss MAE in a somewhat diﬀerent way
than we do for the magnetic polarization 𝐽S andCurie temperature 𝑇C . The discussions are basedon all-electron ﬁrst-principles electronic structure
calculations performed within the framework of thelocal density approximation (LDA/GGA) of density functional theory (DFT). We used a KKR-CPA
package (AkaiKKR) [3], and for the calculation of
𝑇C , Liechtenstein’s method [10] was exploited.
Figure 1 shows the overall behavior of magnetic
moment 𝑀 (𝑎, 𝑍) per atom of 3𝑑 elements plotted

Figure 1: Magnetic moment per atom 𝑀 against
the lattice constant 𝑎(Å) and the atomic number 𝑍
of a ﬁctitious atom
The behavior of saturation magnetic polarization 𝐽S shown in Fig. 2 does not obey that of themagnetic moment. The most prominent feature isthat it has a dome-like structure appearing around𝑎 = 2.65 Å and 𝑧=26.4, where 𝐽S takes the maximum value of 2.66 T. Such a feature is not seen in
𝑀. We point out, not going into details, that this isrelated to the fact that in the bcc structure, the interatomic distance between nearest neighbor pairs
is rather small, forming a considerable bonding–26

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

antibonding splitting and the associated pseudogap. Unfortunately, the lattice constant 𝑎 = 2.65 Å
is 7 % too small compared with the equilibrium
lattice constant of bcc Fe. It should be noticed that,
contrary to the behavior of the magnetic moment,
the magnetic polarization increases with decreasing 𝑎 up to the point where the magnetic state starts
to rapidly collapse.

Figure 2: Saturation magnetic polarization 𝐽S ofthe system against the lattice constant 𝑎 (Å) andthe atomic number 𝑍 of a fictitious atom. The
ferromagnetic state is unstable in the region to theleft of the dashed line.
Magnetic polarization takes on a large value at
one of the corner points in the 𝑍-𝑎 plane, 𝑍 = 25
and 𝑎 = 3.2 Å, but this is not real. The truthis that in a broad region of the 𝑍-𝑎 plane—the
region to the left of the dashed line in Fig. 2—
the ferromagnetic state is unstable as will be seen
below. If we combine this fact with the information
given by Fig. 2, we may conclude that a large 𝐽S
is expected only in the vicinity of the dome-likestructure seen in Fig. 2, and the upper limit of 𝐽S
would not exceed ∼ 2.7 T.
Figure 3 shows the overall behavior of magnetictransition temperature 𝑇C . Here, we again see adome-like structure near 𝑍 = 26.5 and 𝑎 = 2.9 Å.
This position approximately coincides with the position of the similar dome-like structure in 𝐽S . This
indicates that if 𝑍 = 26.5 and 𝑎 = 2.9 Å is forcedby some means (crystal structure, chemical composition, pressure, temperature, etc.), 𝐽S ∼ 2.7 T

is achieved. 𝑇C drops rapidly toward the corner in
the 𝑍-𝑎 plane, 𝑍 = 25 and 𝑎 = 3.2 Å, where 𝑇C
becomes negative, meaning that the antiferromagnetic state should be the ground state. Although theregion of stable ferromagnetism does not appear to
be extensive, there is hope in the fact that, in mostof the region where the ferromagnetism is stable,
𝑇C is well above 1000 K, which is much higher
than the usual working temperature range. Now,
we may say that the upper limit of 𝑇C is ∼ 2000
K (if fcc structure were assumed, the upper limit
would be ∼ 1500 K).

Figure 3: Magnetic transition temperature 𝑇C
against the lattice constant 𝑎 (Å) and atomic number 𝑍 of a fictitious atom. The negative value of
𝑇C means that the antiferromagnetic state is morestable than the ferromagnetic state.
There are two origins of MAE: one is spin
orbit coupling and the other is magnetic shapeanisotropy. The magnetic shape anisotropy iscaused by the dipole interactions between two magnetic moments. In the case of rare earths, the former is the main origin of MAE and is well describedwithin each 𝐿𝑆 multiplet by the effective Hamiltonian given by 𝐻so = 𝜆𝑳 · 𝑺, where 𝜆 = ±𝜁/𝑛 is the
spin-orbit coupling constant for the ground state
𝐿𝑆 multiplet. The plus sign is for the less-thanhalf-filled 𝑓 shell and vice versa; 𝜁 is the single
electron spin-orbit coupling constant, 𝑛 the number of unpaired electrons. For the 3𝑑–5𝑑 cases,
we use the single electron spin-orbit interactions
given by 𝐻so = 𝑖 𝜁𝑖 ℓ 𝑖 · 𝒔𝑖 , where the sum is over27

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo     

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all electrons. Table 1 shows the values of the spinorbit coupling constant 𝜁 for some representative
elements.
Assuming that the orbitals are ﬁrmly boundto the lattice, the upper limit of the magnetic
anisotropy constant 𝐾1 for Sm (Sm3+ in Sm-type
Sm element) estimated from the value of 𝜁, together
with the values of ⟨𝐿⟩, is as large as ∼ 1000 MJm−3 .
The upper limit of 𝐾1 for other lanthanides, if scaledby the value of 𝐿, also would be similar to the Sm
case. This value obtained under the assumptionthat the 4 𝑓 orbitals are ﬁrmly bound to the lattice,
however, seems too large: one order of magnitude
smaller, i.e., ∼100 MJm−3 would be realistic.



Table 1: Absolute values of the single electron
spin-orbit coupling 𝜁 (eV) of element materials Fe,
Sm, and Pt. 𝜁 at the Fermi energy are given.



f
0.001
0.163
0.001



d
0.068
0.218
0.762



p
0.231
1.36
2.72



Fe
Sm
Pt

s
8.18
10.0
44.2



𝜁 (eV)

scheme is not implemented in the present version of
KKR-CPA—full-potential KKR-CPA codes (FPKKR) exist but they do not suit the high-throughput
calculation because of the heavy computational demands. On the other hand, the pseudo-potential
codes can calculate the ordered system quite eﬃciently although their accuracy has to be checkedfrom time to time through comparisons with theresults obtained by other reliable method such as
FPKKR and FLAPW. Considering the above, oneof the practical ways to do is to combine KKR-CPA
and pseudo-potential codes: use pseudo-potential
codes for the end points, i.e., ordered alloys, ofcomposition space and use KKR-CPA to interpolate whole the region of composition space. A
computational system HOFMAN constructed under such a strategy is now running.



3



High-throughput calculation

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The purpose of high-throughput calculation in thepresent context is to generate data of intrinsic
properties of magnetic materials spreading overmulti-dimensional composition space. This enable us to construct permanent magnet materials
databases that might be used to ﬁnd new candidates of high-performance permanent magnet
materials. Given a structure and a set of components, e.g., Sm(Fe1−𝑥 Co 𝑥 )12 (N1−𝑦 S 𝑦 ), we typically need calculations for 100∼1000 diﬀerent systems. Since these systems are compositionally disordered, usual band structure calculation is not sensible. An eﬃcient way to manage this problem is
to use the KKR-CPA method [3] , which can takea conﬁgurational average of disordered systems
rather accurately. Unfortunately, the full-potential







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       

Figure 4: Magnetic polarization (T) (top) and 𝑇C(K) (bottom) of Sm(Fe1−𝑥 Co 𝑥 )12 (N1−𝑦 S 𝑦 ).
An example of the usage of HOFMAN system is seen in Fig. 4, where a new alloy system

28

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Sm(Fe1−𝑥 Co 𝑥 )12 (N1−𝑦 S 𝑦 ) is examined. The results show that the Curie temperature rises by replacing Fe with Co by ∼40% whereas it decreases
the magnetic polarization. On the other hand, replacing N with S turned out not to be promising as
for the magnetic properties.
1.0

Co concentration x

0.8

0.7
Sm2(Fe1-xCox)17Ny
JS (T)

0.75
0.8

4

0.85
0.9

0.6

0.95
1.05
1.1
1.15
1.2

0.0
0.0

0.5

1.0
1.5
2.0
N contents y

2.5

3.0

1320

1300

1.0

0.8

0.6

0.4

0.2

0.0

Sm2(Fe1-xCox)17Ny
TC(K)01282601 0124
1220
120080111160 1140
1120
1100
1060 1080
1040

0.0
1.0

0.5

10
00 10
20

Co concentration x

Finite temperature properties ofpermanent magnet materials

1
0.4

0.2

1.0
1.5
2.0
N contents y

2.5

3.0

Sm2(Fe1-xCox)17Ny
3

K1 (MJ/m )

6
7.5

-3

0.4

4.5

3

0

0.6

1.5

-1.5

0.8
Co concentration x

where the content of N 𝑦 takes 0 ≤ 𝑦 ≤ 3 (Fig.
5). Here the magnetic anisotropy constant 𝐾1 wasalso calculated. In this case, 𝑇C monotonicallyincreases as the concentration of Co increases. Although the addition of N lowers 𝑇C , it enhances
the uniaxial magnetic anisotropy 𝐾1 . This fact was
already known for Sm2 Fe17 N 𝑦 but addition of Co
further enhances this tendency is a new ﬁnding.

10 9
.5

0.2

0.0
0.0

0.5

1.0
1.5
2.0
N contents y

2.5

3.0

Figure 5: Magnetic polarization (T) (top) and𝑇C (K) (middle), and 𝐾1 (MJ/m3 ) (bottom) of
Sm(Fe1−𝑥 Co 𝑥 )12 N 𝑦 .
Another example is for Sm2 (Fe1−𝑥 Co 𝑥 )17 N 𝑦 ,There are many cases that there exist experimental
data of magnetic polarization 𝐽S at several diﬀerent
temperatures around the room temperature. What
we would like to do is to estimate the temperaturedependence of 𝐽S in whole the range of temperature below 𝑇𝐶 from these rather restricted data.
One of the ways to do this is the following: First,
the Curie temperatures, which are not necessarily known experimentally, are estimated by ﬁtting
the temperature dependence of magnetic moment
by Kuz’min’s empirical formula [12]. Meanwhile,the theoretical Curie temperature and 0K magnetic
moment are obtained by ﬁrst-principles calculation
using KKR-CPA. Then theoretical temperature dependence of magnetization is obtained again using
Kuz’min’s formula. Finally, the temperatures are
scaled using experimentally estimated Curie temperatures. Examples of such estimation of temperature dependence of 𝐽S are depicted in Fig. 6 for
SmFe12 family with several diﬀerent compositions.
The theoretical estimations reasonably ﬁt the experimental data and also provide the extrapolation to
the whole range of temperature.
Before entering the discussion of the temperature dependence of magnetic properties including the eﬀects of phonons and magnons,
it might be useful to see the eﬀects ofmagnon excitations to the Curie temperature.
Sucheﬀects comes in through the parameters∬𝑑 𝒓𝑑 𝒓 ′ 𝑺(𝒓) 𝜒(𝒓) −1 𝜒(𝒓, 𝒓 ′) 𝜒(𝒓 ′) −1 𝑺(𝒓 ′) = 𝐽𝑖 𝑗
in eq. (1). Here the integral are performed withineach atomic cell surrounding 𝑖-th or 𝑗-th atom.
There are two limiting cases in the calculations of29

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo
Calc.
Calc. lmd
Expt.

R2Fe14B
1000
TC (K)𝐽𝑖 𝑗 : one is the calculation at the ground state andanother is the calculation above 𝑇C where the directions of local moments align randomly without any
short-range order. Which is more realistic depends
on the characteristic of the spin-ﬂuctuations. The
former is realistic for the system where the local
ﬂuctuation of the magnetic moments does not playa role. The latter is suitable for the opposite cases.

500

0

La

Ce

th]

Pr

Nd Sm Gd Tb Dy Ho Er
Rare-earth elements

Tm Yb

Lu

Y

Figure 7: Calculated 𝑇C of 𝑅2 Fe14 B compared withexperiments.

2.0(R1-xZrx)(Fe0.75Co0.25)11+yTi1-y

1.5

1000

1.0
SmFe11Ti
Sm(Fe0.75Co0.25)11Ti
Sm(Fe0.75Co0.25)11.5Ti0.5(Sm0.8Zr0.2)(Fe0.75Co0.25)11.5Ti0.5
Nd2Fe14B
SmFe11Ti
Sm(Fe0.75Co0.25)11Ti
Sm(Fe0.75Co0.25)11.5Ti0.5(Sm0.8Zr0.2)(Fe0.75Co0.25)11.5Ti0.5
Nd2Fe14B

0.5

0.0

Calc.
Calc. lmd
Expt.

0

200

400
600
Temperature (K)TC (K)JS (T)

R2Co14B

500

0

800

1000

th]

La

Pr Nd Sm Gd Tb
Rare-earth elements

Y

Figure 8: Calculated 𝑇C of 𝑅2 Co14 B compared withexperiments.
Figure 6: Temperature dependence of magnetization of various systems of SmFe12 family.
Figures 7 and 8 show the results of calculation of
𝑇C of 𝑅2 (Fe,Co)14 B (𝑅 = La, Ce,. . . , Lu, Y). The
results under the above two options are comparedwith the experiments. For the Fe based cases, thecalculation assuming that the local ﬂuctuation isdominant (red) gives good agreement with the experiments (blue) whereas the assumption that thelong range ﬂuctuation determined 𝑇C (green) seems
very reasonable for Co cases.
The above discussion is only for 𝑇C withmagnons excitations. For the ﬁnite temperature
properties below 𝑇C and also with not only magnons
but also phonons need more elaobrate treatment as
mentioned in Sec. 1.
Table 2: Calculated 𝑇C of Nd2 Fe14 B
phonons only plus magnons
0K
500K

1062K
1198K

561K
548K

It was fond that, in the case of bcc Fe, there is asigniﬁcant eﬀect of electron–phonon scattering on
the Curie temperature 𝑇C , while that of magnons is
not prominent (not shown). On the other hand, theeﬀect of magnons, becomes remarkable in the caseof Nd2 Fe14 B permanent magnet materials. Some
examples of calculated 𝑇C of Nd2 Fe14 B are given
in Table 2. The ﬁrst column indicates the temperature in which the average displacement due tophonons is calculated. The experimental value of
𝑇C is 585K.
Figure 9 shows the calculated magnetic polarization of Nd2 Fe14 B. Both the eﬀects of phononsand magnons are included. The theoretical curveshows a weak ﬁrst-order transition at the magnetictransition temperature, which occurs as a resultof electron–phonon coupling. The temperature isscaled so that 𝑇C reproduces the experimental valueof 585K although the scaling factor is close to 1.
The calculated results are fairly consistent with experiments, is concluded.

30

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

References40

M (T) (µB/f.u.)

Nd2Fe14B M(T)30

[1] See, e.g., T. Miyake, Y. Harashima,
T. Fukazawaa, and H. Akai, Sci. Tech. Adv.
Mater.Sci. Tech. Adv. Mater. 22, 543 (2021)

20
thoeryexperiment
Kuz'min

10

0

0

100

200

[2] H. Akai, Scr. Mater. 154, 300 (2018).
300

400

500

600

[3] H. Akai, AkaiKKR,
http://kkr.issp.u-tokyo.ac.jp/(2002).

Temperature (K)Figure 9: Calculated and experimental magnetic
polarization vs. temperature of Nd2 Fe14 B[4] T. Kotani, Phys. Rev. B 50, 14816 (1994).

5[5] A. Filippetti, C.D. Pemmaraju, S. Sanvito,
P. Delugas, D. Puggioni, and V. Fiorentini,
Phys. Rev. B 84, 195127 (2011) andreferences therein.

Summary

We discuss the theoretical approached to investigate the intrinsic properties of permanent magnet materials, focusing on three subjects: possible maximum performance of magnetic materials,
high-throughput calculation to survey the candidate of the high-performance magnetic materials
in multi-dimensional composition space, and the ﬁnite temperature magnetic properties. Admittedly,
we have not yet succeeded in inventing any new
permanent magnet materials that realize the performance exceeding that of Nd2 Fe14 B, the best permanent magnet materials so far known. However, we
put emphasis on the fact that the remaining composition space not yet searched is immensely large
and we still have much chance to get a leap. For
this, large scale ﬁrst-principle calculation such as
reviewed here is one of most powerful approaches.
In addition, it is clear that schemes fully equippedwith machine learning technique provide us witha complementary approach for the development of
new permanent magnet materials[6] P.H. Dederichs, S. Blügel, R. Zeller, and
H. Akai, Phys. Rev. Lett. 53, 2512 (1984).
[7] N. H. Long, M. Ogura, and H. Akai, Phys.
Rev. B 85, 224437 (2012).
[8] S. Kou and H. Akai, Solid State Commun.
296, 1 (2018).
[9] H. Shinya, S. Kou, T. Fukushima,
A. Masago, K. Sato, H. Katayama-Yoshida,and H. Akai, Appl. Phys. Lett. 117, 042402(2020).
[10] A. I. Liechtenstein, M. Katsnelson,
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