磁性材料における有限温度磁性、相平衡、異相界面の電子論
2021/12/09
Dept. Mater. Sci. Eng., Tokyo TechYoshihiro Gohda
東京工業大学 物質理工学院
合田 義弘tsoP
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anitio!
elbailaeffects of surfaces & interfaces
Nd permanent magnets
W.F. Li et al., Acta Mater. (2011).

Nd2Fe14B≈5 μm•

surface index of the main phase, Nd2Fe14B•

influence on composition and structure of grain-boundary phase
T.T. Sasaki et al., Acta Mater. (2016).

•Dec. 2021

interfaces: important to prevent domain-wall motion
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design of multiple-phase structures
B. Hallemans, et al., J. Phase Equilibria (1995).
T. Koyama, et al., Sci. Technol. Adv. Mater. (2021).

•microstructures: phase separation between the main phase (Nd2Fe14B) and liquids•

phase equilibria: minimization of the total Gibbs free energy of constituent phases.

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Outline•

I: interplay between vibrational and magnetic
excitations•

II: phase equilibria: free energies from firstprinciples•

III: microstructures in permanent magnets

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I Magnetic phase transition

Ferromagnetism (FM)•Dec. 2021

Curie-Weiss paramagnetism (PM)Typically analyzed by Heisenberg model
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Magnetism-dependent phonons
T. Tanaka and Y. Gohda, npj Comput. Mater. 6, 184 (2020).

bcc Fe40
35Frequency (meV)

30
2520

α = 1.0 (FM)α = 0.8
α = 0.61510

α = 0.4
α = 0.2
α = 0.0 (PM)

50Γ

Ν

Γ

Η

P

Γ Phonon density of states•

Ferromagnetism (FM): high phonon frequencies•

Curie-Weiss paramagnetism (PM): low phonon freq.

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magnetic excess Gibbs energy
excessGmag(T ) = GHeisenberg({Jij(T, Emag(T ))}) + ΔGph(Emag(T ), T )(approximated for fixed volume)•Equilibrium state is obtained by the minimization of the total Gibbs energy.

•Jij depends on reference magnetic state: constant in this study (PM)•If the second term by phonons is large, temperature-dependence ofmagnetic energy Emag(T ) is not available in a simple form.

•very difficult to evaluate directly

Dec. 2021•step 1: independence of Emag and T•

step 2: miminization of the free energy to determine Emag(T )7 /22Y. Gohda

Feedback effect on magnetism
Quantum-spin effect (S = 1.07)

Gmag (T, Emag) + Gph (T, Emag)by rescaled Monte Carlo

0100

Gph

−60
−80−100−120−140Gmag

Gmag + Gph−160

FM −120 −100 −80 −60 −40 −20Emag (meV/atom)

eqEmag

PMPM−100−200750 K−300−400960 K−5001200 K−600−180

Emag = argmin[Gmag(T, Emag)]

FM0K0G (meV/atom)

G (meV/atom)−20−40T = 1000 K−120−100−80−60Emag (meV/atom)−40−20eq
Emag= argmin[Gmag(T, Emag)+Gph(T, Emag)]
Emag

Magnetism-dependent phonons affect magnetic state.
• Stabilization of PM by magnetism-dependent phonons•

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Shift of Curie temperature
T. Tanaka

T. Tanaka and Y. Gohda, npj Comput. Mater. 6, 184 (2020).

Tc
959 K−20Tc
1522 Keq

Emag (meV/atom)−60−80563 K

argmin[Gmag + Gph]

1.00.8

argmin[Gmag]−100Cmag (meV/(K・atom))

1.2

PM −400.6−1200.4

FM −1400.2−1600250 500 750 1000 1250 1500 1750

0*experimental Tc1043 K

Temperature (K)Dec. 2021

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IIPhase equilibria: CALPHAD approach•(sub-)regular solution modelGαNNαα=xiGi + RT
xi ln xi + Gexcess∑∑
i=Ai=A•The first term is Gibbs energy of pure substance(from CALPHAD database)•Third term: excess Gibbs energy as RK seriesNNv(n)
αn
Gexcess =xi xjΩij =xi xj
Li,j (xi − xj)
∑∑∑∑
∑
[]
i=A j>i
i=A j>in=0regular solution, if constantsublattice also considered
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microstructure interfacesz [nm]

Y. Ainai et al., APEX 13, 045502 (2020).

SmFe12(110) / SmCu(100)

SmFe12(110) / Fe(001)d1.210•Dec. 2021

SmFeCueffective exchange coupling between main-phasegrains: 1-12/SmCu/1-12 >> 1-12/Fe/1-12
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finite-temperature effects
Fe(a)bcc FMfcc PMbcc PMbcc FM

0102030
phonon energy (meV)

Magnon energy (meV)

Phonon DOS per atom (arb. unit)(b)10001st
1st & 2nd800
600
400
200040Γ

HPk pathΓNℏω = 2S[J(0) − J(q)]•Dec. 2021

phonons and magnons
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III Stability of RE-TM surfaces
Y. Ainai, YG, et al., APEX 13, 045502 (2020).

Nd2Fe14B

SmFe12(a)

(b)

(110)

(c)

(100)

Sm(d)

Fe-richSm-richPDOS [states/eV]

γ [eV/Å2]

0.08
0.04

[001]

[100]
•

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Fe 3d
Sm 5d

10−1−2(110)(100)

Miller index(001)−3−4−3−2−10ε–εF [eV]123Surfaces stabilized by rare-earthatoms•

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320.120Fe(e)0.16(001)

Valid for both Nd2Fe14B andSmFe12 surfaces

Y. Gohda

Nd2Fe14B/Nd0.67Fe0.33−xGax zigzag interfaces

Fig. 8. (a) 3-D atom map obtained from the post-sinter annealed low-C sample. (b) 1-D compositional profile obtained from a volume (shown inblue in (a)) including thin Nd-rich grain boundary phase. (c) 3-D atom map obtained from the post-sinter annealed high-C sample. (d) 1-Dcompositional profile obtained from a volume (shown in blue in (a)) including the thin Nd-rich grain boundary phase. (For interpretation of thereferences to colour in this figure legend, the reader is referred to the web version of this article.)

Acta Materialia 84, 506 (2015)

Fig. 9. HAADF-STEM images including the thin Nd-rich grain boundary phase obtained from the post-sinter annealed low-C and high-C samples.
(a), (b) and (c) are samples in which the thin Nd-rich grain boundary phase is aligned 58!, 30! and parallel, respectively, to the (0 0 1) plane of theNd2Fe14B phase in the low-C sample. (d), (e) and (f) are samples in which the thin Nd-rich grain boundary phase is aligned 30! and parallel to the(0 0 1) plane of the Nd2Fe14B phase in the high-C sample.

Y. Ainai, YG, et al., APEX 13, 017006 (2020).
y = 0.01
y = 0.04

Eform [eV/atom]

0.3

y = 0.08
y = 0.13

Low-C

0.2

High-C

0.1
0−0.1

•

Dec. 2021

AlCo600

y = 0.17
Tabley 2.
Magnetic properties of the samples.
= 0.21As-sintered
Annealed
As-sintered
Annealed

Nd0.67Fe0.33–yGay

Remanent magnetization
550l0M (T)Coercivity
l0HC (T)

1.44
1.44
1.44
1.44

1.59
1.85
1.44
1.54

Tc [K]

0.4

Y. Ainai, YG, et al., JJAP-Rapid (2020).

500

450
where A is the carbon-enriched phase with the tetragonalstructure. Pan et al. explained that the A-phase formed asa consequence of the reaction between the carbon andNd2Fe14B phase without considering the presence of the400
phase. However, the Fe particles were not observed0Cu Nd-rich Ga
in the EDS elemental map in Fig. 1b, ruling out thepossibility of this reaction. Instead, the volume fractionof the a-Nd phase decreased with increasing the Nd-carbide

Nd0.67Fe0.33–yCuy
Maximumenergy product(BH)max (MGOe)
51.18
51.04
51.37
51.53

(Fig. 4c and f). Therefore, we conclude that the a-Nd phase
was consumed to form the Nd-carbide.
In both low-C and high-C samples, the post-sinteredannealing did not affect the volume fraction and phase constitution
Nd-rich phases
3c, f and 4b,0.03 of the0.06
0.09 (Figs.0.12
0.15d). The
amount of a-Nd has ay close correlation with the coercivity
since a-Nd phase at triple junctions serves as a pool of Ndfor the formation of the thin grain boundary phase during

MStabilization and suppressedmagnetism of supphase by Ga

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code to calculate Jij in OpenMX
A. Terasawa, M. Matsumoto, T. Ozaki, and Y. Gohda, J. Phys. Soc. Jpn. (2019).
•Post-processing computations using outputs of OpenMX•

Cost-efficient calculation of Jij based on Liechtenstein method andfinite-pole approximation of Fermi function
NPnoX X
1X+0 µ0 GJi0,jR =R̃pRe [P̂i ]⌫µ G+(#,z̃,R)[P̂]
0pj⌫iµ,j⌫
jµ0 ,i⌫ (", z̃p , R) .
2 p=1µ,⌫2i 0 0<latexit sha1_base64="x7QhkqEukuHx8XQvu7GCQKSaoQY=">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</latexit>

G+jµ0 ,i⌫ (", ",<latexit sha1_base64="7WkmL85b4Vh4pWeeOo9ujEpjKOM=">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</latexit>

R) =

Z

d3

✓µ ,⌫ 2jka2⇡◆eik·RX Cjµ0 ,n" (k)Ciµ,n" (k)n" + i⌘"n" (k),G+iµ,j⌫ 0 (#, ", R) =

Zd3

✓ka2⇡

◆eik·R

X Ciµ,n0 # (k)Cj⌫ 0 ,n0 #n0" + i⌘

"n0 # (kcomputational time: 3 h 9 mfor 1562 pairs of 244-atom system(4 node × 24 process,Intel(R) Xeon(R) E5-2680 v3)

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Summary•••

I: interplay between vibrational and magnetic excitations•

feedback effects of magnetism-dependent phonons on magnetic states•significant change in the Curie temperature (more than 500 K for bcc Fe)

II: free energies from first principles•

Phase equilibria by CALPHAD approach•temperature effects in excess Gibbs energy

III: Nd permanent magnets•

rare-earth at surfaces to minimize surface energies•magnetic isolation of main-phase grains by subphase optimization
tisopcstdo•

Dec. 2021!
elbailavaion

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Y. Gohda