KEK seminar2020 Dec. 10
onlineAtomistic approach to the Nd-Fe-B magnet-- Coercivity at finite temperatures -Seiji Miyashita
Physical society of Japan (JPSJ)
ISSP, The University of Tokyo
Elements Strategy Initiative Center for Magnetic Materials(ESICMM)

Collaborators
Yuta Toga, Masamichi Nishino, Taichi Hinokihara, Ismail. E. Uysal,
Akimasa Sakuma, Takashi Miyake ,Hisazumi. Akai, Satoshi Hirosawa

Magnets
Ferromagnets
Coercivity

High performance magnet
Nd2Fe14B (Masato Sagawa)

Motor (cars)
Properties at high temperatures T (T<Tc)
Rare-earth elements (Dy)Mechanism of coercivity at finite temperatures

Nd2Fe14B magnet at finite temperatures
Thermodynamic properties

𝐴 𝑇

Tr 𝐴𝑒 −𝛽ℋ=
Tr 𝑒 −𝛽ℋ

All properties can be obtained by an atomistic Hamiltonian 𝓗.

Hˆ = Hˆ ex + Hˆ a + Hˆ cry + Hˆ z ,Hˆ ex = − 2 J ij S i  S j ,i, j( )2Hˆ a = − DiA Siz ,iHˆ cry =  Alm r i Olm ,iFirst-principles calculation(KKR method by Akai)
First-principles calculation (Y. Miura et al.,J. Appl. Phys. 115, 17A765 (2014)).
M. Yamada et al., Phys. Rev. B 38, 620 (1988).

l ,mHˆ z = − hi S izi

Y. Toga, M. Matsumoto, S. Miyashita, H. Akai, S. Doi, T. Miyake, and A. Sakuma,Phys. Rev. B 94, 174433 (2016).

Atomic model of Ne2Fe14B magnet
Magnetization M(T)
Reorientation phase transition
Ferromagnetic phase transition

Anisotropy energies

Y. Toga, M. Matsumoto, S. Miyashita, H. Akai, S. Doi, T. Miyake, and A. Sakuma: P.RB 94, 174433 (2016).

Domain wall profile
Type I: Bloch type wall

Type II: Neel type wall
M. Nishino, Y. Toga, S. Miyashita, H. Akai, A. Sakuma, and S. Hirosawa, Phys. Rev. B 95, 094429 (2017).

Thermodynamic properties : magnetization
Reorientation phase transition T~150K
Magnetization M(T)

Tilted anisotropy
mim
Ndの異方性エネルギー Hˆ cry =  Al r Ol ,il ,mz

A. V. Andreev, et al. Sov. Phys. JETP 63(3) 608 (1986).
S. Hirosawa, et al. J. Appl. Phys 59 (1986)873.

3: Nd2Fe14B, 4:Nd2Fe14BH3.8

Atomic model of Ne2Fe14B magnet
Magnetization M(T)
Reorientation phase transition
Ferromagnetic phase transition

Anisotropy energies

Y. Toga, M. Matsumoto, S. Miyashita, H. Akai, S. Doi, T. Miyake, and A. Sakuma: P.RB 94, 174433 (2016).

Domain wall profile
Type I: Bloch type wall

Type II: Neel type wall
M. Nishino, Y. Toga, S. Miyashita, H. Akai, A. Sakuma, and S. Hirosawa, Phys. Rev. B 95, 094429 (2017).

Dynamical response (FMR)
Power Spectrum:S =1
Si ,  = x, y, zN site i

1 t 0 +TI ( ) =  dt S  (t ) S  (t +  )T t0

M. Nishino, S. Miyashita: PRB 100 (2019) 020403(R).

Stochastic LLG(Landau-Lifshitz-Gilbert) equationdM i i=−M
H−M i  (M i  H eff,i ),ieff,i22dt
1 + i(1 +  i ) M i

Heff,i → Heff,i + ξ fl, ii = 1, 2,, N

i (t ) = 0, i (t ) j ( s ) = 2 Di ij  (t − s ).

i
Mi

J. L. Garcia-Palacios and Francisco J. Lazaro: Phys. Rev. B58 (1998) 14937.
T. A. Ostler, et al.: Phys. Rev. B84, 024407 (2011).
R. F. L. Evans, et al: J. Phys. Condens. Matter 26 (2-14) 103202.
M. Nishino and S. Miyashita, Phys. Rev. B91, 134411(1-13) (2015).

=Dik BT

Cocercivity(保磁力)End of the metastable state (spinodal point)

Metastable state (Mean-field picture)
Ising modelℋ= −𝐽 σ<𝑖𝑗> 𝜎𝑖 𝜎𝑗 − 𝐻 σ𝑖 𝜎𝑖𝑚 =tanh 𝛽𝐽𝑧𝑚 + 𝛽𝐻

σ_i=±1，<ij>格子上の最近接対、その個数 ｚFree energy F(m:T,H)
Bragg-Williams-近似
ゆらぎ無視の近似

1
𝐹 𝑚: 𝑇, 𝐻
𝑁1
𝐹 𝑚: 𝑇, 𝐻
𝑁𝐽𝑧= − 2 𝑚2 − 𝐻𝑚+𝑘B 𝑇𝐽𝑧1+𝑚1+𝑚1−𝑚1−𝑚ln+ln2222= 2 𝑚2 − 𝑘B 𝑇ln 2cosh(𝛽 𝑧𝐽𝑚 + 𝐻 )「ゆらぎと相転移」,丸善出版（２０１８）

H SP=  J 1 −11
 ln
J 1+ 1−1
J1− 1−1J

Two-level systems:
Ising model (spin), binary alloy (A,B atoms), spin-crossover systems (HS,LS),
adsorption(on, off), lattice-gas model(occupation, vacancy) , vote (yea, nay), etc.

Hysteresis loop
Mean-field theory (Spinodal point)
𝑚 =tanh 𝛽𝐽𝑧𝑚 + 𝛽𝐻
1𝐽𝑧 2𝐹 𝑚: 𝑇, 𝐻 = − 𝑚 − 𝐻𝑚
𝑁21+𝑚1+𝑚1−𝑚1−𝑚+𝑘B 𝑇 2 ln 2 + 2 ln 2

Nucleation Finite energy barrier−1𝑝nucleation ∝ 𝑒 −𝛽∆𝐸𝑐 , 𝜏 = 𝑝nucleation

Metastable states must relax, if we wait long.

Dynamical spinodal point
Multi-nucleation process(Avrami process, deterministic)
Single-nucleation process (stochastic)
Large Hsmall H

720
400
256
128
64

DSP

deterministic

stochastic

P. A. Rikvold, H. Tomita, S. Miyashita and S. W. Sides: Phys. Rev. E 49, 5080-5090 (1994).

Enery barrier

Néel-Arrhenius process

Néel-Arrhenius hysteresis

Single magnetic model: Anisotropy
Single nano-size particle
Superparamagentism
Uniform rotation (Stoner-Wohlfarth)Ne’el-Brown model
W. Wernsdorfer, et at. PRL 78 (1997) 1791,PRL 79 (1997) 4014.

𝜏 = 1/𝑣

Ni, Co, Dy
BaFeCoTiO

Kramers’ method

current

H. A. Kramers, Physica 7, 284 (1940).
H. Tomita, A. Ito and H. Kidachi, Prog. Theor. Phys. 56, 786 (1976).

T. Mori, S. Miyashita and P. A. Rikvold: Phys. Rev. E 81, 011135 (1-10) (2010).

Quantum tunneling at each crossing
Macroscopic quantum tunneling and quantum–classical phasetransitions of the escape rate in large spin systems.
Quantum tunneling
Wentzel–Kramers–Brillouin (WKB) method (Landau and Lifshitz, 1977)‘‘instanton’’ method (Coleman, 1977, 1985)

Large magnetic spin systems
Only for Single crossing

J. L. van Hemmen and A. Suto, in Quantum Tunneling of Magnetism–QTM’94, ed. By L. Gunter and B. Barbara, NATO Advanced Institutes,
Ser. E. Vol 301 (Kluwer, Dordrecht 1995) p.19.
S. A. Owerre and M. B. Paranjape: Phys. Rep. 546, 1 (2015).

Quantum effect :Multiple quantum tunneling with many degree of freedomsResonant Tunneling Phenomena: Single molecular magnets

宮下精二：量子スピン系(岩波書店 2006)

Quantum spinodal Phenomena

scaled gap

Escaled = SEk( S )T. Hatomura, B. Barbara and S. Miyashita: Phys. Rev. Lett. 116 037203 (2016).

いろいろなヒステリシス
保磁力: Variety of hysteresis loops
Mean-field hysteresis

Neel-Arrhenius hysteresis

metastability

Thermal activation

Multi-domain

Dipole-dipole Int.

FORC
(first-order reversal curve)

Ensemble

Microscopic approach
Tr 𝐴𝑒 −𝛽ℋ
𝐴 𝑇 =Tr 𝑒 −𝛽ℋa microscopic Hamiltonianℋ3

いろいろなヒステリシス
保磁力: Variety of hysteresis loops
Mean-field hysteresis

Neel-Arrhenius hysteresis

metastability

Thermal activation

Multi-domain

Dipole-dipole Int.

FORC
(first-order reversal curve)

Ensemble

Microscopic approach
Tr 𝐴𝑒 −𝛽ℋ𝐴 𝑇 =Tr 𝑒 −𝛽ℋa microscopic Hamiltonianℋ3

ベクトル模型：一様回転模型
Stoner-Wohlfarth rotation

Ising model

Ising-like Heisenberg model
Easy-axtis-type magnet

𝑚 = ±1𝐸 = −𝐷𝑚𝑧2 − 𝐻𝑧 𝑚𝑧 − 𝐻𝑥 𝑚𝑥 ,𝑚𝑥2 + 𝑚𝑦2 + 𝑚𝑧2 = 1

Change of the trajectory with Hz D=1,Hx=1

Dynamical process
No adiabatic pass

Stoner-Wohlfarth (SW) process
End of the metastability

E = − Dmz2 − H z mz − H x mx , mx2 + m y2 + mz2 = 1z

SxH2DH=,2/3
2/3
3/ 2(sin  + cos  )i.e., ( 2 D ) 2 / 3 = ( H x ) 2 / 3 + ( H z ) 2 / 3
E. C. Stoner and E. P.Wohlfarth, Phil. Trans. R. Soc. A 240, 599 (1948).

Dynamics of grain magnetization at finite temperature-- real-time approach (stochastic -LLG equation) -dM i i
=−MH−M i  (M i  H eff,i ), H eff,i → H eff,i + ξ fl, i i = 1, 2,, Nieff,i22dt
1 + i(1 +  i ) M i

i (t ) = 0, i (t ) j ( s ) = 2 Di ij  (t − s ).
Magnetization reversal time

400 [K]

i

Mi=

Dik BT反転磁化分布の方法
独立核生成プロセス
逆磁場下における磁化の緩和時間（τ） 初期状態all upでｔ秒間観測

ln( N S / N ) = − pt 𝜏 = 1/𝑝
N : サンプル数
Ns : 磁化反転せず生き残ったサンプル数

H=8T

H=4.1T

deterministic region

stochastic region

Merit: easy to remove
initial region

M. Nishino, I. E. Uysal , T. Hinokihara, S. Miyashita, Phys. Rev. B102, 020413(R) (1-5).

Dependence on damping constant（α）

Energy barrier
Stochasticjump

No Energy barrier
Deterministic
Avrami process

Estimation of the coercive field
Large dependence of 𝜏 on damping constant（𝛼）Dynamical phase transition(sharp increase of the relaxation time 𝜏
between deterministic and stochastic regions)

extrapolationℎ𝑐 ≅ 3𝑇(cf. ℎ𝑐 ≅ 3.3𝑇 Monte Carlo method Toga et al.)M. Nishino, I. E. Uysal , T. Hinokihara, S. Miyashita, Phys. Rev. B102, 020413(R).

Estimations of the relaxation time by the energy barrier
Minimum energy paths (MEP) method

f exp( − E ( H C ) / k BT ) = 1
R. Dittrich, et al., J. Mag. Mag. Mat., 250, L12 (2002).
E. Weinan, et al., J. Chem. Phys. 126, 164103 2007.

Free-energy landscape method
Wang-Landau method for F(m): (Multi-Canonical MC method)
F. Wang and D. P. Landau, Phys. Rev. Lett. 86, 2050 (2001).

Internal Energy : E

Free energy barrier for metastable stateM

Wang-Landau MC method
𝑤 𝐸 = exp −𝛽𝐸Internal Energy : ESamples all energy regions
Sampling num：H(E)Sampling num：H(E)Samples regions around energy minima

adjust 𝑤 𝐸 to get flat 𝐻(𝐸)Internal Energy : E

F. Wang and D. P. Landau, Phys. Rev. Lett. 86 2050 (2001).

Free energy as a function of M

𝐹 𝑇, 𝐻; 𝑀 = 𝐹 𝑇, 0; 𝑀 − 𝐻𝑀

Metastable state

Free energy barrier

𝐹B (𝑇, 𝐻) = 𝐹 𝑇, 𝐻; 𝑀𝑡 − 𝐹(𝑇, 𝐻; 𝑀𝑏 )Y. Toga, S. Miyashita, A. Sakuma and T. Miyake: Role of atomic-scale thermal fluctuations in
the coercivity, Nature partner journal (npj) Computational Materials (2020) 6:67.

Thermal activated coercivity

Temperature dependence of coercivities

Kronmueller equation

Y. Toga, S. Miyashita, A. Sakuma and T. Miyake: Role of atomic-scale thermal fluctuations inthe coercivity, Nature partner journal (npj) Computational Materials (2020) 6:67.

Large reduction of the coercive field

H. Kronmueller, Phys. Stat. Sol. (b) 144 (1987) 385.

Experimental approach to grain reversal

Takahiro Yomogita, Satoshi Okamoto, Nobuaki Kikuchi, Osamu Kitakami, Hossein Sepehri-Amin, Yukiko
K. Takahashi, Tadakatsu Ohkubo, Kazuhiro Hono, Keiko Hioki, and Atsushi Hattori, to be published in Acta
Mater (2020).

いろいろなヒステリシス
保磁力: Variety of hysteresis loops
Mean-field hysteresis

Neel-Arrhenius hysteresis

metastability

Thermal activation

Multi-domain

Dipole-dipole Int.

FORC
(first-order reversal curve)

Ensemble

Microscopic approach
Tr 𝐴𝑒 −𝛽ℋ
𝐴 𝑇 =Tr 𝑒 −𝛽ℋ
a microscopic Hamiltonianℋ3

Multi-domain structure and coercivity
Thermal demagnetization

Maze pattern in a grain
T. Nakamura, K. Toyoki, M. Suzuki18

Effect of the dipole-dipole interaction
3(rij  S i )(rij  S j ) 1 1 ,Hˆ dip = D S
S−j3  i2riji  j 40 rij Computational cost: (NxN)

Algorithms for faster simulation

Stochastic Cut-off method (SCO)：

M. Sasaki and F. Matsubara: J. Phys. Soc. Jpn. 77 024004 (2008).

Walker’s algorithm：
K. Fukui and S. Todo: J. Comput. Phys. 228 2629 (2009).

A Hybrid method：

CPU/MCSO(NlogN)

T. Hinokihara, M. Nishino, Y. Toga, and S. Miyashita, Phys. Rev. B97, 104427 (2018).

Paramerter (K/J, D/J) depencence of magnetic
patterns in thin films
Multidomain structure (64x64;Lz): thermal-quech

T. Hinokihara and S. Miyashita: arXive:2009.11574. Phys. Rev. B(2021) to be published.

Multi-domain structure and coercivity
Thermal demagnetization

From magnetized state
nucleation

Reversewhole grain atonce–2.5 kOe–0.5 kOe–7.0 kOe

磁化 (arb. unit)–3.5 kOe–1.1 kOe

Maze pattern in a grain
T. Nakamura, K. Toyoki, M. Suzuki

M. Suzuki et al., Acta Mat.106, 155 (2016).

Metastability of ferromagnetic state in large grain?

18

Paramerter (K/J, D/J) depencence ofmagnetic patterns in thin films
Multidomain structure (64x64;Lz): field-quench

T. Hinokihara and S. Miyashita: arXive:2009.11574.

Hysteresis of large (multi-domain) grains
Random start

ferro start

D/J: strength of the dipole-dipole interaction (size effect)
K/J: strength of the anisotropy energy.
T. Hinokihara and S. Miyashita: arXive:2009.11574.

Collapse of metastability by DDI

T. Hinokihara and S. Miyashita: arXive:2009.11574.

Hysteresis with multidomain
one single particle

elliptic Co particle of 300 nm x 200 nm x 35 nm

Worfgang Wernsdorfer KIT Lecture,
PRB 53, 3341 (1996 ) and PRB 53, 6536 (1996).

A. Hubert and R. Schaefer: Magnetic Domains (Springer (1998)
C. Kooy and U. Enz: Philips Res. Repts 15, 7 (1960)

Simplified mean-field analysisA
B
Bℋ = −𝐽 ෍ 𝑺A
𝑖 ∙ 𝑺𝑗 − 𝐽 ෍ 𝑺𝑖 ∙ 𝑺𝑗 +𝐷 ෍
<𝑖𝑗>
<𝑖𝑗>
𝑖𝑗A𝑺A
𝑖 ∙ 𝑺𝑗 − 3 𝑺𝑖 ∙ 𝒓 𝑺𝑗 ∙ 𝒓

𝑟3− ෍ 𝐻 ∙ 𝑺𝑖𝑖𝑥 = 𝑆𝑖𝑧A𝑦 = 𝑆𝑖𝑧B
𝑀 = 𝑆𝑖𝑧A + 𝑆𝑖𝑧B
𝐹 𝑥, 𝑦 =𝐽𝑧 2𝑥 − 𝑘B 𝑇ln 2cosh2 𝛽𝐽𝑧𝑥 − 𝛽𝐷𝑦 + 𝛽𝐻
2𝐽𝑧+ 𝑦 2 − 𝑘B 𝑇ln 2cosh2 𝛽𝐽𝑧𝑦 − 𝛽𝐷𝑥 + 𝛽𝐻
2M. Nishino and S. Miyashita, Phys. Rev. B 88, 014108 (2013).

いろいろなヒステリシス
保磁力: Variety of hysteresis loops
Mean-field hysteresis

Neel-Arrhenius hysteresis

metastability

Thermal activation

Multi-domain

Dipole-dipole Int.

FORC
(first-order reversal curve)

Ensemble

Microscopic approach
Tr 𝐴𝑒 −𝛽ℋ𝐴 𝑇 =Tr 𝑒 −𝛽ℋa microscopic Hamiltonianℋ3

Distribution of hysteresis loop
(interacting Preisach model)

K. Toyoki, private communication

34

Preisach model and FORC analysis
Hysteresis of each grain

hysteron

𝐻1𝐻2

Distribution P(Ui, Vi) and FORC(First-order reversal curve)
𝑃 𝐻1 , 𝐻2 = 𝑀 𝐻1 + 𝜖1 , 𝐻2 + 𝜖2 + 𝑀 𝐻1 , 𝐻2−𝑀 𝐻1 + 𝜖1 , 𝐻2 − 𝑀 𝐻1 , 𝐻2 + 𝜖2
𝜕 2 𝑀 𝐻1 , 𝐻2=𝜕𝐻1 𝜕𝐻2

FORC analysis for archaeology

F. Matau, et al.: J. Archaeological Science 40 (2013) 914. 37-2

Interacting Preisach modelc(i) Axis of anisotropy

Magnetization of a grain
M (i)−𝐻0 (i)
Coercivity of a grain

𝐻0 (i)𝑎𝑧H𝑎𝑥𝑎𝑦hysteronGrain size（cluster）

Interaction among the grains
Dipole-dipole interaction (moving model)e.g., C. R. Pike, et.al. PRB 71, 134407 (2005)

Interaction with the grain boundary

22

Trial to a renormalized Hamiltonian for finite
temperatures to study larger system.
Continuum model: Temperature dependence of stiffness constant.
15A (pJ/m)

𝐸cont105A x (T ypeI )Exchange Stiffness const.
Magnetic Anisotropy const.

A z (T ypeI I )

00200

400
600
Temp. (K )

800

1000

Y. Toga, M. Nishino, S. Miyashita, T. Miyake, and A. Sakuma, Phys. Rev. B98, 054418 (2018).

Renormalized Hamiltonian with a variable on-site magnet

Choice of interaction form is difficult

Nd2Fe14B magnet at finite temperatures
Thermodynamic properties

Response function

𝐴 𝑇

Tr 𝐴𝑒 −𝛽ℋ=
Tr 𝑒 −𝛽ℋ

1 t 0 +T
I ( ) =  dt S  (t ) S  (t +  )T t0
All properties can be obtained by a microscopic Hamiltonian ℋ.

Coercivity : relaxation from a metastable stateSingle grain : Arrhenius law:
LLG, Minimum energy path, Free-energy method
Dipole-dipole interaction for large size:
Stochastic cutoff method, Time quantified MC method
Ensemble of grains :Preisach model, FORC diagram
Effects of grain boundary phase and surface properties

論文リスト:Finite temperature property, Atomic specification
新規開発アルゴリズム
LLG: finite temperature (西野) 、MC: Constrained, Free energy landscape, (栂)MC: dipole-dipole interaction, SCO, TQMC (檜原)論文
[1] M. Nishino and S. Miyashita, Phys. Rev. B91, 134411(1-13) (2015).
[2] S. Mohakud, S. Andraus, M. Nishino, A. Sakuma, S. M., Phys. Rev. B 94, 054430 (2016).
[3] Y. Toga, M. Matsumoto, S. Miyashita, H. Akai, S. Doi, T. Miyake, and A. Sakuma: Phys. Rev. B 94, 174433 (2016).
& Phys. Rev. B 94, 219901 (2016).
[4] M. Nishino, Y. Toga, S. M., H. Akai, A. Sakuma, and S. Hirosawa, Phys. Rev. B 95, 094429 (2017).
[8] Y. Toga, M. Nishino, S. Miyashita, T. Miyake, and A. Sakuma, Phys. Rev. B98, 054418 (2018) .
[5] T. Hinokihara, M. Nishino, Y. Toga, and S. Miyashita, Phys. Rev. B97, 104427 (2018).
[6] S. Hirosawa, M. Nishino, S. Miyashita, Adv. Nat. Sci.: Nanosci. Nanotechnol. 8 (2017) 013002.
[7] S. Miyashita, M. Nishino, Y. Toga, T. Hinokihara, T. Miyake, S. Hirosawa and A. Sakuma,Scripta Materialia, 154,259 (2018).
[8] M. Nishino, S. Miyashita: Phys. Rev. B100, 020403(R) (2019).
[9] I. E. Uysal, M. Nishino, S. Miyashita, Magnetic fields threshold for nucleation and depinning inof the neodymium permanent magnet Nd2Fe14B, Phys. Rev. B101,094421 (2020).
[10]Y. Toga, S. Miyashita, A. Sakuma and T. Miyake, Temperature dependence of the coercive force – activationenergy for magnetization reversal –, Nature Partner Journal Computational Materials 6, 67 (2020).
[11]M. Nishino, I. E. Uysal , T. Hinokihara S. Miyashita, Dynamical aspects of magnetization reversal in the Neodymium
permanent magnet by a stochastic LLG simulation at a finite temperature -- real time dynamics and a quantitative
estimation of coercive force --, B102,020413(R) (2020)[12]T. Hinokihara and S. Miyashita: Multidomain structure and coercive force, Phys. Rev. B(2021) to be published.
T. Hinokihara, Y. Okuyama, M. Sasaki, and S. Miyashita: Time Quantized Monte Carlo Method forLong-range Interacting Systems 投稿中
M. Nishino, I. E. Uysal , S. Miyashita: Effect of the magnetic anisotropy of the surface neodymium atoms on the coercivity 投稿中
S. Miyashita, et al, Atomistic theory of thermally-activated magnetization processes in25
rare earth permanent magnets投稿中

Thank you very much