Electron Theory of Permanent Magnets

Takashi  Miyake　      （AIST,  Tsukuba）

EQPCM ISSP，2013.06.19

Outline

■ Coercivity and magnetocrystalline anisotropy ■ Rare-earth magnets ■ Standard theory ■ An example: NdFe11TiN ■ Interface and magnetic reversal

Ferrite Nd-Fe-B

Nd2Fe14B

µ0Ms=1.61 T, µ0Hc>0.8 T Ku=4.3 MJ/m3, κ=1.54

(BH)max<510 kJ/m3

Tc=312°C

3  

Maximum energy product (BH)max

Br=µ0Mr

(µ0H) Coercivity  (Hc)

B=µo(H+M)

(µ0M)

(BH)max

ーµoH

4  

Conditions for strong magnets

■ Squareness ■ Magnetization ■ Coercivity　　　  

H

µ0M

B = µ0(M+H)

µ0MS/2

-­‐  MS/2

-­‐  HC (BH)max=µ0MS2/4

(BH)max= µ0MS2/4 if HC > MS/2

without using rare-metals Temperature effects

Anisotropy magnetic field

H θ

θ = 0,π

θ = π/2

HA = 2 K1 / (µ0MS)

eas

y ax

is

H

In the Stoner-Wolfarth model,

EA = K1 sin2θ + ・・・

HA

Magnetic Anisotropy Energy ・ Shape anisotropy ・ Crystalline anisotropy

Anisotropy magnetic field

Nd2Fe14B

H  (kOe)

M  (µ

B)

M.  Yamada  et  al.,    PRB  38,  620  (1988)

0.5

1.0

0.0 1950 1960 1970 1980 1990

AlNiCo Ba-ferrite Sm5Co

(SmZr)2(CoFeCu)17 Nd2Fe14B

Lab.

Techn.

Discrepancy to H A

HC / HA HC ∝ HA　　 　　　　

∝ K

Coercivity vs. anisotropy magnetic field

200°C

Fe B Nd Dy

κ(Nd2Fe14B) < κ(Dy2Fe14B) 8  

Hc  /  kO

e

Temperature  /  ℃

(NdDy)2Fe14B

■ Rare-metal ■ (BH)max smaller

Dysprosium substitution

■ Temperature dependence MagnetoCrystalline Anisotropy (MCA) energy Magnetization and Curie temperature ■ Difference between coercivity and anisotropy magnetic field ■ New hard magnets

Challenges

10  

Nd2Fe14B  

Fe  Co   Fe16N2  

Dy2Fe14B  

Tb2Fe14B  

Sm2Fe17N3    SmCo5    

Y2Fe14B  

NdFe11TiN  

YFe11Ti  BaFe12O19    Fe2O3  

FePt  YCo5  

CoPt  

τ-­‐MnAl  

Co3Pt  0  

2  

4  

6  

8  

10  

12  

14  

16  

18  

20  

0   0.5   1   1.5   2   2.5   3  

K 1(M

J/m

3 )  

Js  (T)

Limit  of  Shape  

Anisotropy

µ0Ms (T)

New  magnet

EA = K1 sin2θ + ・・・

a

c

L10-type alloy

Fe (bcc): 〜1 µeV Co (hcp): 〜60 µeV FePt: 〜3 meV/f.u. CoPt: 〜1 meV/f.u.

Transition-metal alloys

                       　　 MS(T) K1(MJ/m3) HA(MA/m) TC(K)

Nd2Fe14B

SmCo5

Pr2Fe14B

Dy2Fe14B

Sm2Co17

Sm2Fe17N3

NdFe11TiN

1.60

1.56

0.712

1.07

1.25

1.54

1.45

4.5

5.5

5.4

17.2

3.2

8.6

6.7

5.3

6.9

11.9

28

5.1

20.7

9.6

586

569

598

1,000

1,193

746

729

2-­‐14-­‐1  family

1-­‐5  family

2-­‐17  family

1-­‐12  family

Rare-earth magnets

Herbst,  RMP  (1991)

Nd2Fe14B

n=1,  m=0:  1-­‐5  family　(CaCu5-­‐type)  n=2,  m=1:  1-­‐12  family  (ThMn12-­‐type)  n=3,  m=1:  2-­‐17  family  (Th2Zn17-­‐type,  Th2Ni17-­‐type)  

Li  and  Cadogan,  JMMM  109,  153  (1992)  

Rn-mT5n+2m series

Larson,  Mazin  and    Papaconstantopoulos,    PRB  67,  214405  (2003)

Outline

■ Coercivity and magnetocrystalline anisotropy ■ Rare-earth magnets ■ Standard theory ■ An example: NdFe11TiN ■ Interface and magnetic reversal

LDA band structure of GdCo5

majority  spin minority  spin

TM RE

3d

5d

4f

spin orbital

・  large  magneazaaon  ・  exchange  coupling  　　　→  Curie  temp.

・  strong  SOC  +  crystal-­‐electric  field  　　　→  MCA

Exchange magnetic field

5d 3d

4f

3d↑

3d↓

5d↑

5d↓

EF

E

)(Eρ↑

)(Eρ↓

3d－5d orbital hybridization

spin anti-parallel

Effective Hamiltonian

VH cryex +⋅+⋅= SLSH λ2

A. Sakuma

Exchange magnetic field

5d 3d

4f

5d spin－４ｆ spin

Hund coupling

spin parallel

Effective Hamiltonian

VH cryex +⋅+⋅= SLSH λ2

A. Sakuma

5d 3d

4f

－Hex （3d spin）

4f spin

Exchange magnetic field

Effective Hamiltonian

VH cryex +⋅+⋅= SLSH λ2

A. Sakuma

intra-­‐atomic  e-­‐e  correlaaon

SOC

exchange  magneac  field crystal  

electric  field

L, S

J

MJ

Spin-orbit interaction

light rare-earth （ｎ＜７） ：

0>λ

heavy rare-earth （ｎ＞７） ：

0<λ

Ｇｄ （ｎ＝７） ： 0=λ

Ｌ Ｓ Ｊ＝|L－S|

Ｊ // －Ｓ + =

Ｌ Ｓ

+

Ｊ＝|L＋S|

Ｊ // Ｓ =

Ｊ＝|S|

Ｊ = Ｓ =

Effective Hamiltonian

VH cryex +⋅+⋅= SLSH λ2

A. Sakuma

light  rare-­‐earth  （Nd,  Sm）

4f(S)

3d(S) 5d(S)

4f (J)

hybridizaaon Hund

LS

heavy  rare-­‐earth  （Dy）

4f(S)

3d(S) 5d(S)

4f (J)

hybridizaaon Hund

LS

HJ exJg ⋅− )1(2

)1(2)1()1()1(1

+

+−++++=

JJLLSSJJgJ

light RE : J ＝ || SL − 1<gJheavy RE : J ＝ || SL + 1>gJ

Lande factor

Effective Hamiltonian

VH cryex +⋅+⋅= SLSH λ2

A. Sakuma

Crystal Electric Field (CEF)

∑∑

∑

−

−=

=

i j ji

j

ii

crycry

qe

VV

||||

)(

Rr

r

ｃ

ａａａ

ａｃ

Ｒイオン

q=+1

q=+1

Effective Hamiltonian

VH cryex +⋅+⋅= SLSH λ2

point charge model

R  ion

A. Sakuma

Crystal Electric Field (CEF)

∑∑

∑

−

−=

=

i j ji

j

ii

crycry

qe

VV

||||

)(

Rr

r

ｃ

ａａａ

ａｃ

Ｒイオン

q=+1

q=+1

Effective Hamiltonian

∑∑⇒i ml

iiiml

ml

li zyxQAr

,),,(

VH cryex +⋅+⋅= SLSH λ2

point charge model

Spherical Harmonics

A. Sakuma

OBgH ml

ml

mlexJ ∑+⋅−=

,)1(2 HJ

Stevens’ operator-equivalent method

ArB ml

lJ

ml ><=θ

γθβθαθ JJJ === 642 ,,)()( 33 rfrrfdrr l

ll

l==∫>=<

Aml

2/)(2/)(

)1(3

4444

2222

2222

202

JJOJJO

JJO

JJJOyx

z

−+

−+−

+=

−=

−=

+−=

ArJJK J02

21 )2/1(3 〉〈−−= αMCA energy

CEF parameter

Stevens’ factor

A. Sakuma

ArJJK J02

21 )2/1(3 〉〈−−= α

Depend on R ion

CEF surrounding the R ion

CEF parameter 4f electrons

CEF

J

A02

α J

〉〈r2 A. Sakuma

ArJJK J02

21 )2/1(3 〉〈−−= α

102×α J JNd

Dy

-0.64

-0.63

9/2

15/2

Sm +4.13 ５/2

Ce　　　　 　Pr　　　 Nd　　　　Pm　 Sm Gd

Tb　　　　 　Dy　　　 Ho　　　　Er　 Tm Yb Lu

opposite sign

L

A. Sakuma

ArJJK J02

21 )2/1(3 〉〈−−= α

102×α J JNd

Dy

-0.64

-0.63

9/2

15/2

Sm +4.13 ５/2

Ce　　　　 　Pr　　　 Nd　　　　Pm　 Sm Gd

Tb　　　　 　Dy　　　 Ho　　　　Er　 Tm Yb Lu

L

A. Sakuma

ｃ

ａａａ

ａｃ

Ｒイオン

q=+1

q=+1

)11(||)(33

02

caelattA −=

a  <  c　→ 002 >A

a  >  c　→ 002 <A

0)etc.DyNd,(0 1 >⇒<∴ KJα

0)etc.ErSm,(0 1 >⇒>∴ KJα

R ion

ArJJK J02

21 )2/1(3 〉〈−−= α

A. Sakuma

ｃ

ａａａ

ａｃ

Ｒイオン

q=+1

q=+1

Smの４ｆ 電子雲　ｑ＜０

ｃ

ａａａ

ａｃ

Ｒイオン

q=+1

q=+1

Nd（α< 0） Sm(α>0)

0)11(||)(33

02 >−=

caelattA

a  <  c  A. Sakuma

ｃ

ａａａ

ａｃ

Ｒイオン

q=+1

q=+1

ｃ

ａａａ

ａｃ

Ｒイオン

q=+1

q=+1

01 <K01 >K

Nd（α< 0） Sm(α>0)

A. Sakuma

Valence (5d, 6s) electron cloud

4f electron cloud

ｃ

ａ ａ

ａ

ａ

ｃ

q  =+1  

q  =+1  

４ｆ electrons

.)(.)( valAlattAA ml

ml

ml +=

CEF Aml

A. Sakuma

Density Functional Theory

■ Reasonably good for M and Tc of 3d metals ex.) exchange coupling Jij → Classical Heisenberg model ■ MCA energy numerical problem for 3d metals difficult to treat 4f electrons (LDA+U?, SIC? , ・・・)

Takahashi,  Ogura  and  Akai,  J.  Phys.:  Cond.  Mat.  (2007)

Slater-Pauling curve

First-principles calculation of CEF parameters

Kohn-Sham equation

real spherical harmonics

CEF parameters

4f  atomic  orbital

Outline

■ Coercivity and magnetocrystalline anisotropy ■ Rare-earth magnets ■ Standard theory ■ An example: NdFe11TiN ■ Interface and magnetic reversal

RFe11Ti

Hu  et  al.,  J.  Phys:  Cond.Mat.  1,  755  (1989)  

■ RFe12 is unstable ■ RFe12-xMx (M=Al,Ti,V,Cr,Mo,W,… ) ■ Drastic change by N-doping ex.) uniaxial anisotropy in NdFe11TiN

NdFe12 vs. NdFe12N

NdFe12 -83 K NdFe12N +413 K

A20 〈r2〉

c Fe Fe

Fe Fe

spd electrons

f electron

Fe Fe

Fe Fe

N

N

A.Sakuma, JPSJ 61, 4119 (1992)

R: 2a Fe: 8i, 8j, 8f Ti: 8i N: 2b

NdFe12 vs. NdFe11Ti

NdFe12 -83 K NdFe11Ti +54 K

A20 〈r2〉

c Fe Fe

Fe Fe

spd electrons

f electron

Fe Fe

Fe Fe

Ti

(1)  Grain boundary phase: microscopic structures and composition (2)  Magnetic properties at interfaces (3) Relation between MCA and coercivity

main phase：Nd2Fe14B grain boundary phase：Fe-Cu-Nd amorphous, Nd oxides, …

Sintered magnet

Nd2Fe14B surface

Moriya, Tsuchiura and Sakuma, J. Appl. Phys. 105, 07A740 (2009); Tanaka et al., J. Appl. Phys. 109, 07A702 (2011)

=

Landau-­‐Lifshitz-­‐Gilbert  (LLG)  equaaon

Micromagnetics

■ Ab-inotio evaluation of parameters ■ Spatial variation ■ Large-scale simulation (coarse graining, massive-parallel calc.)

Summary

■ Coercivity and magnetocrystalline anisotropy ■ Rare-earth magnets ■ Standard theory ■ An example: NdFe11TiN ■ Interface and magnetic reversal