micromagnetics for spintronics
TRANSCRIPT
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,
A. Thiaville 3
International School on Spintronics & Spin-Orbitronics, Fukuoka 16-17 dec. 2016,
A. Thiaville 4
International School on Spintronics & Spin-Orbitronics, Fukuoka 16-17 dec. 2016,
A. Thiaville 5
International School on Spintronics & Spin-Orbitronics, Fukuoka 16-17 dec. 2016,
A. Thiaville 6
International School on Spintronics & Spin-Orbitronics, Fukuoka 16-17 dec. 2016,
A. Thiaville 7
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
q 0mdiv
0 nm
222
/sindrdrAE
ech
Ansatz
θM
E s
dem
2
2
0 cos2
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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
J. Miltat, Nano Spin School, Cargese 05-2005
- 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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J. Miltat, Nano Spin School, Cargese 05-2005
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
J. Miltat, Nano Spin School, Cargese 05-2005
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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J. Miltat, Nano Spin School, Cargese 05-2005
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
E
M s0
0sin
q
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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