Micromagnetics for Spintronics

Micromagnetics for Spintronics

André THIAVILLE

Laboratoire de Physique des Solides Univ. Paris-Sud & CNRS

91405 Orsay, France

International School on Spintronics & Spin-Orbitronics, Fukuoka 16-17 dec. 2016, A. Thiaville 1

F

F

Mechanics of continuous media

What does Micromagnetics ?

Micromagnetics = mechanics of continuous magnetic structures International School on Spintronics & Spin-

Orbitronics, Fukuoka 16-17 dec. 2016, A. Thiaville

2

Why use Micromagnetics ?

International School on Spintronics & Spin-Orbitronics, Fukuoka 16-17 dec. 2016,

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L

231,0

2

0 s

demech

MEE

0,

2

demechE

LAE

stable monodomain state for

3

231

22

0 LL

AM

s

Demagnetising factor N

NL

With anisotropy, the transition size increases too

When not to use Micromagnetics ? nm-size objects


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The progressive development of Micromagnetics

1926 1931 1931 1932 1935 1944 1946 1948 1955 1963 1969 . . .

P. Weiss F. Bitter K. Sixtus, L. Tonks F. Bloch L. Landau, E. Lifchitz L. Néel C. Kittel W. Döring T.L. Gilbert W.F. Brown A.E. LaBonte

Concept of magnetic domains Observation of domain (walls) Dynamics of domain walls Bloch wall Effective field, LL dynamic equation Bloch walls for different anisotropies Single domain particles Domain wall dynamic structure, mass Gilbert damping, LLG equation Micromagnetics Asymmetric Bloch wall: numerical micromagnetics


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What is Micromagnetics ?


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0

0.2

0.4

0.6

0.8

1

m

my

mx

position x

atomic spins continuous distribution

Basic assumptions of Micromagnetics

x

y

mTMM s

)(

1) Fixed magnetization modulus

2) Slow variations at the atomic scale -> continuous model trm ,

1m

« micromagnetization »


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i j jiSSJE

Basic magnetic energy terms

H

Exchange

Anisotropy Demagnetising field

Applied field


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Micromagnetic equations : statics

Dss HmMHmMmKGmAE

00

2

2

1)()(

exchange anisotropy applied field demagnetizing field

+ boundary conditions

Statics : minimise Brown equations

effective field exchangeanisodemagappliedeff HHHHH

mM

A

s

0

2

m

E

MH

s

eff

0

1

0

mxH eff

0

nm

rdEV

3

NB « functional derivative »


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MdivHdiv D

0

DHrot

0appHdiv

jHrot app

+ boundary conditions

Demagnetizing field : magnetostatic approximation

MHB

0 with 0Bdiv

and jHrot

demagnetizing field applied field

nMnHHD

ext

D

)(

int

0)(int

tHHD

ext

D

appD HHH

volumic magnetic charges

surfacic magnetic charges


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Basic micromagnetic characteristic lengths

KA Bloch wall width parameter

A=10-11 J/m, K=102 – 105 J/m3 = 1 - 100 nm

2

0

2

sM

A

exchange length

Ms= 106 A/m = some nm

2

2

0

2

sM

KQQuality factor

Q > 1 hard material Q << 1 soft material


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The Bloch wall : length Easy axis z

y x

q m

x

y

)(cos

)(sin0

xθ

xθm

No demag energy 0mdiv

qq 22

sinKdxdAE

qq ,0 -5 -4 -3 -2 -1 0 1 2 3 4 50

0.25

0.5

0.75

11


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The magnetic vortex : length

0//

rxry

m0mdiv

2D magnetization

0 nm

Divergence of the exchange energy 22

/ rAmAEech

3D magnetization

)(cos

/)(sin

/)(sin

r

rxr

ryr

m

qq

q 0mdiv

0 nm

qq

222

/sindrdrAE

ech

Ansatz

θM

E s

dem

2

2

0 cos2

International School on Spintronics & Spin-

Orbitronics, Fukuoka 16-17 dec. 2016, A. Thiaville

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Observation by Magnetic force microscopy (MFM)

NiFe (50 nm)

T. Okuno

JM Garcia

240nm 240nm

Topography Magnetic image


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The Néel wall

Thin film without anisotropy, or small in-plane anisotropy

Bloch wall

Néel wall

+ + +


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Walls in soft thin films

Symmetric Néel wall

Asymmetric Néel wall

Asymmetric Bloch wall


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2D instability of the Néel wall : cross-tie

- - - - - - -

+ + + + + + +

electron holography image

A. Tonomura et al., Phys. Rev. B25 6799 (1982)

map of the magnetic charges


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Magnetization dynamics

/ML

Angular momentum dynamics

gyromagnetic ratio (>0)

meg

gB

2

dtLd

m

H

HmMs

0

mHdtmd

0

..102.25

00IS

28 GHz/ T

Can be found directly from quantum mechanics


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Micromagnetic equations : dynamics

dtmdmmH

dtmd

eff 0

Landau-Lifshitz-Gilbert (LLG)

mHmmH effeff

2

0

1

Effective field exchangeanisotropydemagappliedeff HHHHH

mM

A

s

0

2

m

E

MH

s

eff

0

1

: Gilbert damping parameter

(solved form of LLG)


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Properties of the magnetization dynamics

0.2)(

2

dtmdm

dt

md

Conservation of the magnetization modulus

2

00

00

)/(.

..

dtmdMmxH

dtmdM

dtmdxmHM

dtmdHM

dtdE

seff

s

effs

effs

Decrease of the energy with time : the magnetic system is not isolated

1)

2)


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J. Miltat, Nano Spin School, Cargese 05-2005

Numerical Micromagnetics


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Finite differences : Finite elements :

Two types of numerical schemes


- easy to code - well adapted to demag field calculation (FFT) - most used

- can handle any shape - can easily implement local mesh refinement


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Design rules

Cell size smaller than the characteristic length of the problem (of the structure if statics only) In principle, results for decreasing size meshes should be extrapolated to zero mesh size to reach the continuous limit Warnings : - mesh friction - mesh orientation effects - Bloch points - Brown paradox : role of defects


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Time schemes

mHHmH

K

sDA

seff MM

A

00

12

effeff

mmmdtdm HH

00

21

TDtT

Heat diffusion equation :

Stability of explicit scheme for dt < (dx)2 / D

Small time steps, or implicit scheme


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Codes

1) OOMMF (http://math.nist.gov/oommf): since 1999 - public and free; finite differences - easy to use, versatile inputs (scripts) - requires expertise in C++ for modification 2) MuMax3 (http://mumax.github.io) : since 2011 - public and free; finite differences - extremely fast (runs on GPUs) 3) Nmag (http://nmag.soton.ac.uk/nmag) : since 2007 - public and free; finite elements Home-made programs Commercial codes


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http://math.nist.gov/oommf

http://mumax.github.io/



http://nmag.soton.ac.uk/nmag


Other energy terms and effective fields


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mMMHm

E

MH sth

s

eff

;

1

0

0

thH

H thi (t)H th

j (t' ) ij (t t' )

2kBT

0MsV

(Hthi )

2kBT

0MsV

System supposed to evolve "slowly"

Inclusion of some thermal fluctuations : Langevin model

W. F. Brown, Phys. Rev. 130 (1963) 1677 TkB


V: volume of a mesh cell


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Spin transfer torque (STT)

m m1

electrons

in

out

given

in out

Favors a parallel alignment of F to F1

m m1

in

out

given

in out

Favors an anti-parallel alignment of F to F1

electrons

mmmmmDeM

PJg

dtmd

s

B

transferspin

11

12

Slonczewski STT


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Spin transfer torque (CPP geometry)

mmmmmdt

mdmmH

dt

mdeff

110

1

m1

m

transferspindtmd

destabilizing stabilizing

Slonczewski field-like (damping-like)

Effective field for the Slonczewski term

10

erspintransfeff,

1 mmH

m

m1

q

No energy term to be associated with this spin transfer torque term !

transferspindtmd


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Torque by the spin Hall effect in an adjacent layer

Co 0.6 nm

Pt 3 nm electrons

CIP in Co + CPP

spin Hall in Pt with y polarized

(spin-orbit scattering) reference layer

)(1

ymmt

m

SHE

teM

gJ

teM

gJ

s

BHx

s

Bzspin

22

1 , q

Slonczewski

CPP-STT

First demonstration of the effect on domain walls : P.P.J. Haazen, E. Murè, J.H. Franken, R. Lavrijsen, H.J.M. Swagten, B. Koopmans Nat. Mater. 12, 299 (2013)

x

z

y


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Spin transfer torque (CIP geometry)

electrons

before after

CPP spin transfer between successive x slices

L. Berger, J. Appl. Phys. 49, 2156 (1978)

Adiabatic limit (walls are wide): carrier spins always along local magnetization -> angular momentum given per unit time in the slab dx

))()(( dxxsxsPe

J

dxx

mP

e

J

2dx

dt

mdM s

=

s

m


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mux

mu

dt

md

transferspin

)(_

s

B

Me

gPJu

2

Permalloy : Cm

Me

g

s

B /1072

311

1 x 1012 A/m2 & P = 0.5 u = 35 m/s

u : a velocity that expresses the spin transfer (spin drift velocity) (Zhang & Li : bJ)

Spin transfer torque in continuous form (CIP)

« adiabatic » term


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Full LLG equation under CIP-STT

mmumummmHm xxtefft

0

"adiabatic term" "non-adiabatic term"

Solved form

mmumu

mHmmHm

xx

effefft

1

1

1002

Initial velocity for step current

A. Thiaville et al., Europhys. Lett. 69 990 (2005)

uV20

1

1

m

E

MH

s

eff

0

1 effective field of other micromagnetic terms


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The “non-adiabatic” term : many models

2

1/sf

sd

sf

sd

<<1 expected for 3d metals

3) Ab initio calculations : spin-orbit coupling for the carriers 4) Non-local effects

1) True non-adiabaticity for narrow walls ? Requires sub-nm thin domain walls 2) Spin accumulation and precession

I. Garate et al., PRB 79 104416 (2009)

G. Tatara et al., Phys. Rep. 468, 213 (2008)

J.Q. Xiao, A. Zangwill, M. Stiles

PRB 73, 054428 (2006)

S. Zhang and Z. Li, PRL 93, 127204 (2004)

A. Manchon et al., arXiv 1110.3487

D. Claudio Gonzalez et al., PRL 108 227208 (2012)

C. Petitjean et al., PRL 109, 117204 (2012)


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jijiij SMSE

ijijij AntisymSymM

(good base)

3

2

1

00

00

00

ij

ij

ij

ij

A

A

A

Sym

SDSAntisym ijij

jiij

ijji

antisym

ij

SSD

DSSE

« pseudodipolar »

« anisotropic exchange »

« antisymmetric »

Only if no inversion symmetry

Antisymmetric exchange in asymmetric structures (Dzyaloshinskii-Moriya interaction : DMI)


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)( jiij

antisym

ij SSDE

ijjijijiijji

antisym

ji DDSSDSSDE

)()(

The micromagnetic forms of DMI

What about the orientation of the DMI vector ?

Localized magnetism, single crystal : Moriya rules apply

T. Moriya, Phys. Rev. 120, 91-98 (1960)

Isotropic « bulk case » uDuD

)(

Isotropic « interface case » uzDuD

)(

Chiral spin spirals favored

Chiral spin cycloids favored

y

mm

y

mm

x

mm

x

mmDe

y

zz

yx

zz

x

DM


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Physics without fully solving the LLG equation:

Thiele equation

Collective coordinates model for domain wall dynamics


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Derivation of the Thiele equation (1)

LLG equation mmmHm tefft

0

`solved’ form mmmmH tteff

0/

ASSUME a magnetization structure in RIGID MOTION

))((),( 0 tRrmtrm

Force on the structure

i

effs

i

effs

i

ix

mHM

R

mHM

dR

dEF 0

00

j j

jtx

mVm 0

j ijj

js

ix

m

x

m

x

mmV

MF 000

0

0

0


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A.A. Thiele, Phys. Rev. Lett. 30, 230 (1973)

0

FFF dissipgyro

Gyrotropic force case of a film in x-y plane of thickness h

VGFg

Sk

ssz Nh

Mdxdydzm

y

m

x

mMG

4

0

00

00

0

0

Dissipation force

VDF

dxdydz

x

m

x

mMD

ji

sij

00

0

0 .

j j

j

ji

sj

ji

si V

x

m

x

mMVm

x

m

x

mMF 00

0

00

00

0

0

Derivation of the Thiele equation (2)


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A. Thiaville et al., Europhys. Lett. 69, 990 (2005)

Applications of the Thiele equation

0

FuVDuVG

Simple wall (Gz=0) 0

FVD

hM

dxdzx

mMD

T

ssxx

2

0

0

2

0

0

0

hHMF sx 02HV T

x

0

(both per unit length)

Thiele equation under CIP STT

Free structure (F=0), no gyrovector uV

/

Defines the Thiele domain wall width T

A.A. Thiele, J. Appl. Phys. 45, 377 (1974)


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SHE longitudinal force on a magnetic structure

LLG with SHE pmmmmmHm eff

10

Solve for Heff

(Thiele procedure) mpmmmmHeff

11

0

The forces are mHM

dX

dEF xeff

sx

0

0

pmmM

mpmM

F xs

xsSHE

x

0

0

0

0

DMI energy density (interfacial DMI) xyyx

y

zz

yx

zz

x

DM

mmmmD

y

mm

y

mm

x

mm

x

mmDe

)()(

SHE: for j//x one has p//y

SHE force along J according to Néel chirality !


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Collective coordinates models of domain wall dynamics

J.C. Slonczewski, Intern. J. Magnetism 2, 85 (1972) … A.P. Malozemoff, J.C. Slonczewski, Magnetic domain walls in bubble materials (Academic Press, 1979)

Slonczewski equations for DW field dynamics in bubble materials (films with large PMA)

N.L. Schryer, L.R. Walker, J. Appl. Phys. 45, 5406 (1974)

1D field dynamics of a Bloch wall

1D DW dynamics in nanowires of soft magnetic materials

A. Thiaville et al., J. Magn. Magn. Mater. 242, 1061 (2002) D. Porter & M. Donahue, J. Appl. Phys. 95, 6729 (2004) A. Thiaville & Y. Nakatani, in Spin Dynamics in Confined Magnetic Structures III (Springer, 2006) A. Thiaville et al., Europhys. Lett. 69, 990 (2005) A. Thiaville & Y. Nakatani, in "Nanomagnetism and

Spintronics", (Elsevier, 2009, 2013)

Field

STT


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Construction of the model(s)

appHq

0

cossin0 KHq

q

qq

E

M s0

0sin

q

qq

E

M s sinsin

0

0

LLG in (q, )

variables (here with only conservative terms)

)(

)(expAtan2),(

t

tqxtxq )(),( ttx Assumption for the structure

(here, 1D assumption)

Case of DW driven by easy axis field

J.C. Slonczewski, Intern. J. Magnetism 2, 85 (1972)


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Walker field sKW

MKHH0

/2/

Explicit solution for angle (t) for field applied at t=0

2

2

2

2

1Atan1

1tan1tan

t

H app

app

W

H

H

Wapp HH 1

1Atanh1

1tanh1tan

22

2

2 / tH app Wapp HH 1

The Walker solution (1D)

N.L. Schryer, L.R. Walker, J. Appl. Phys. 45, 5406 (1974)


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appHqvWall velocity

Linear mobility

Velocity loss due to precession

Precessional regime, constant

Wapp HH

2

22

1

Wapp

app

HHHq

Limit case without transverse anisotropy

0WH

tH app

21

tHq app21

Uniform precession

« hard wall » mobility


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0 1 2 3 40

1

2

3

4

u / vW

v / v W

10

3

1.5

1

0.5

0.25

0

/

1D model in the case of STT

For stationary motion

uv /

stC

Wvuv

Döring limit of stationary motion

uq

uH

qK cossin0

A. Thiaville et al., Europhys. Lett. 69, 990 (2005)


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What Micromagnetics does not (fully) describe: The Bloch point

r

rm

r

Aeexc

2

2for + rotations

RAEexc 8R radius where BP profile applies

Exchange Energy density : Total energy:

hedgehog > circulating > spiraling

Demag energy

W. Döring, J. Appl. Phys. 39, 57 (1968)

Diverges ! Finite !

E. Feldtkeller, Z. angew. Phys. 19, 530-536 (1965) [17, 121-130 (1964)]


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Conclusions & perspectives

Versatile micromagnetic framework One (a few) phenomenological parameter(s) for each torque term Atomic-scale Micromagnetics also exists In most cases, a numerical calculation is required But many physical insights can be obtained analytically Limit: magnetization not fixed (ultrafast, large temperatures) ./..


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References

A. Hubert, R. Schäfer, Magnetic Domains (Springer, 1998)

Handbook of Magnetism and Advanced Magnetic Materials, volume 2, H. Kronmüller and S. Parkin Eds. (Wiley, 2007)


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Documents

Micromagnetics for Spintronics