Introduction to Magnetism (2) : Magnetism today or How performs - - PowerPoint PPT Presentation
Introduction to Magnetism (2) : Magnetism today or How performs the micromagnetic theory now? Andr Thiaville Laboratoire de Physique des Solides Universit Paris-Sud & CNRS, Orsay European School of Magnetism, 1 Constanta, 2005:
European School of Magnetism, Constanta, 2005: André THIAVILLE 1
Introduction to Magnetism (2) : Magnetism today
How performs the micromagnetic theory now?
André Thiaville
Laboratoire de Physique des Solides Université Paris-Sud & CNRS, Orsay
European School of Magnetism, Constanta, 2005: André THIAVILLE 2
1- Introduction 2 - Micromagnetic theory and applications statics domain walls dynamics 3 - Micromagnetics of nano-elements macrospin limit quasi-uniform structures dynamics 4 - Beyond micromagnetics molecular magnetism clusters of a few atoms spin-polarized transport
European School of Magnetism, Constanta, 2005: André THIAVILLE 3
Nanometric size structures are already used
European School of Magnetism, Constanta, 2005: André THIAVILLE 4
for 103 kbit/in 1 bit = 25 nm
European School of Magnetism, Constanta, 2005: André THIAVILLE 5
European School of Magnetism, Constanta, 2005: André THIAVILLE 6
Magnetic hysteresis in novel materials, G.C. Hadjipanayis Ed., Nato ASI E338 (Kluwer, Dordrecht, 1997)
European School of Magnetism, Constanta, 2005: André THIAVILLE 7
European School of Magnetism, Constanta, 2005: André THIAVILLE 8
The “micromagnetic” description of magnetism
y
0.2 0.4 0.6 0.8 1
m my mx position x
atomic spins continuous distribution x Assumes that structures to describe are large compared to atomic sizes
European School of Magnetism, Constanta, 2005: André THIAVILLE 9
i j
j i
S S J E ⋅ − =
Magnetic Interactions
H
European School of Magnetism, Constanta, 2005: André THIAVILLE 10
Micromagnetic equations
1 = m
no thermal fluctuations m T M M
s
) ( =
D s s
H m M H m M m KG m A E ⋅ − ⋅ − + ∇ =
2
2 1 ) ( ) ( µ µ
exchange anisotropy applied field demagnetizing field
Statics : minimise ∫E Brown equations
v v = m x H eff r r r = ∂ ∂ n m
+ boundary conditions effective field
exchange aniso demag applied eff
H H H H H + + + = m E M H
s eff
δ δ µ 0 1 − = m M A
s
∆ 2 µ
European School of Magnetism, Constanta, 2005: André THIAVILLE 11
Magnetostatics of matter
= B div r j H rot r r =
( )
M H B r r r + µ =
with and
M div H div
D
r r − = =
D
H rot =
ext
H div
ext ext
j H rot = r
+ boundary conditions
n M n H H
D ext D
r r r r r ⋅ = ⋅ − ) (
int
) (
int
= ⋅ − t H H
D ext D
r r r
demagnetizing field applied field
European School of Magnetism, Constanta, 2005: André THIAVILLE 12
Magnetostatic energy
2 1 2 1
3
2
≥ µ = ⋅ µ − =
R D V D D
H H M E r r r φ ∇ − = r r
D
H
proof : introduce the scalar potential
∂
− ⋅ π + − π − = φ
V V
r r n M r r M div ' 4 1 ' 4 1 r r r r r r r M div r = φ ∆
and transform by integration by parts both expressions into
φ µ − φ ⋅ µ
∂ V V
M div n M r r r 2 1 ) ( 2 1
European School of Magnetism, Constanta, 2005: André THIAVILLE 13
Characteristic lengths
K A = ∆ Bloch wall width parameter
A=10-11 J/m, K=102 – 105 J/m3 ∆= 1 - 100 nm
2
2
s
M A µ = Λ exchange length
Ms= 106 A/m Λ= some nm
2 2
2 ∆ Λ = µ =
s
M K Q
Quality factor Q > 1 hard material Q << 1 soft material
European School of Magnetism, Constanta, 2005: André THIAVILLE 14
x θ Easy axis z y m y x
= m div
No demag energy
= ) ( cos ) ( sin x θ x θ m r θ θ
2 2
sin K dx d A E + =
( ) ( )
π θ θ = ∞ + = ∞ − ,
European School of Magnetism, Constanta, 2005: André THIAVILLE 15
cos sin 2 2
2 2
= + − θ θ θ K dx d A
× dx dx d and θ
Energy minimization equation
sin
2 2
= = + −
st
C K dx d A θ θ
First integral
K A = ∆ ∆ ± = θ θ sin dx d
Bloch wall width parameter
1 2 3 4 5 0.25 0.5 0.75 1 1
( )
π θ + ∆ − = exp Atan 2 x x
Linear width :
∆ π
European School of Magnetism, Constanta, 2005: André THIAVILLE 16
AK A 2 2 = ∆
Integrated exchange energy =
AK K 2 = ∆
Integrated anisotropy energy = 2 Integrated hard axis component
∆ = ∆ = = π θ θ d dx dx my sin
∆ = ∆ = = 2 sin sin
2 2
θ θ θ d dx dx my
etc.
European School of Magnetism, Constanta, 2005: André THIAVILLE 17
The vortex
2D magnetization − = 0/ / r x r y m
= m div = ⋅ n m
( )
2 2
/ r A m A Eech = ∇ =
Divergence of the exchange energy
3D magnetization approximation
− = ) ( cos / ) ( sin / ) ( sin r r x r r y r m θ θ θ = m div ≠ ⋅ n m θ + θ =
2 2 2
/ sin dr d r A Eech
European School of Magnetism, Constanta, 2005: André THIAVILLE 18
− = ) ( cos / ) ( sin / ) ( sin r θ r x r θ r y r θ m r + =
2 2 2
sin r θ dr dθ A Eech θ M E
s dem 2 2
cos 2 µ =
2 sin 1 1 ) 2 ( 1 ) 2 (
2 2 2 2
= − Λ + + θ r dr θ d r dr θ d
2
2
s
M A µ = Λ
European School of Magnetism, Constanta, 2005: André THIAVILLE 19
A.Hubert et R. Schäfer Magnetic Domains (Springer, 1998) Normalized thickness D / Λ Normalized widths W / Λ
Variational calculation
European School of Magnetism, Constanta, 2005: André THIAVILLE 20
Walls in films with perpendicular anisotropy
Bloch Néel Néel Λ= 4 nm
European School of Magnetism, Constanta, 2005: André THIAVILLE 21
The Néel wall (1955)
Thin film without anisotropy, or small in-plane anisotropy Bloch wall Néel wall
European School of Magnetism, Constanta, 2005: André THIAVILLE 22
Approximate analytical model
x/D mx Q= 0.04, 1.88 D= 2.5 Λ Q= 2-2.5 10-4 D= 20, 2.5 et 1.45 Λ The wall has logarithmic tails
European School of Magnetism, Constanta, 2005: André THIAVILLE 23
Walls in soft thin films
European School of Magnetism, Constanta, 2005: André THIAVILLE 24
2D instability of the Néel wall : cross-tie
map of the magnetic charges
+ + + + + + electron holography image
European School of Magnetism, Constanta, 2005: André THIAVILLE 25
Magnetization dynamics
m e g g
B
2 = = h µ γ
γ gyromagnetic ratio (>0)
γ / M L − = H
Angular momentum dynamics
Γ = dt L d m H m M s × = Γ µ m H dt m d × = γ . . 10 2 . 2
5
I S ≈ = γ µ γ
Can be found directly from quantum mechanics
0.28 GHz/ mT
European School of Magnetism, Constanta, 2005: André THIAVILLE 26
Dynamics of a magnetization continuum
Effective field
exchange anisotropy demag applied eff
H H H H H + + + = m E M H
s eff
δ δ µ 0 1 − = m M A
s
∆ 2 µ dt m d m m H dt m d
eff
× + × = α γ 0
Landau-Lifshitz-Gilbert
α : Gilbert damping parameter
× × + × + = m H m m H
eff eff
α α γ
2
1
European School of Magnetism, Constanta, 2005: André THIAVILLE 27
another magnetization dynamics equation
dt m d m m H dt m d
eff
× + × = α γ 0
Landau-Lifshitz-Gilbert (1955)
× × + × = m H m m H dt m d
eff eff L
λ γ
Landau-Lifshitz (1935)
are mathematically equivalent γ H x m dm/dt λ m x (H x m) dm/dt α m x dm/dt m γ H x m H LL LLG
European School of Magnetism, Constanta, 2005: André THIAVILLE 28
Properties of the magnetization dynamics
. 2 ) (
2
= = dt m d m dt m d r r r
1)
Conservation of the magnetization modulus
2
) / ( . . . − = − = − = − = dt m d M m x H dt m d M dt m d x m H M dt m d H M dt dE
s eff s eff s eff s
γ αµ αµ αµ µ r r
2) Decrease of the energy with time : the magnetic system is not isolated
European School of Magnetism, Constanta, 2005: André THIAVILLE 29
European School of Magnetism, Constanta, 2005: André THIAVILLE 30
Nanoparticles and small elements
L
2 3 1 ,
2 s dem ech
M E E µ ≈ ≈ ,
2
≈ π ≈
dem ech
E L A E Λ π < ⇔ π < µ 3 2 3 1
2 2
L L A M s
stable monodomain state for Demagnetising factor N With anisotropy, the transition size increases too
N L Λ π <
European School of Magnetism, Constanta, 2005: André THIAVILLE 31
Nanoparticles in the monodomain state
« macrospin » several Λ
( )
m H M m G K V E
s
⋅ − = µ 1 = m
s K
M K H 2 µ =
( ) ( )
m h m G m V U
H
⋅ − = = 2
European School of Magnetism, Constanta, 2005: André THIAVILLE 32
Uniaxial case of degree 2
(Stoner-Wohlfarth 1948, Slonczewski 1956)
axe difficile axe facile A B H
JC Slonczewski, IBM Research Memorandum RM 003.111.224,
« astroid »
EC Stoner, EP Wohlfarth Phil. Trans. Roy. Soc. London A240 599 (1948), reprinted IEEE Trans. Magn. 27 3475 (1991)
European School of Magnetism, Constanta, 2005: André THIAVILLE 33
Geometric solution
(inspired from J.C. Slonczewski, IBM report, 1956)
( ) ( )
m h m G m V U
H
⋅ − = = 2
Initial problem (statics) : given H , find m Dual problem : given m, find H
= ⋅ m d U d u
2 2
> ⋅ u m d U d u
equilibrium : stability :
m u ⊥ ∀
European School of Magnetism, Constanta, 2005: André THIAVILLE 34
3D Solution in spherical angles
x y θ ϕ
sin 2 = ⋅ −
ϕ ϕ
θ e h G 2 = ⋅ −
θ θ
e h G sin 2 1 2 1 = + ⋅ + ⋅ = m e G e G h λ θ
ϕ ϕ θ θ
λ
θθ θθ
2 + = G U ∂ ∂ = θ θ θ
ϕ θϕ
sin sin G U θ λ θ θ
θ ϕϕ ϕϕ 2
sin 2 cos sin + + = G G U > +
ϕϕ θθ
U U ) (
2 >
−
θϕ ϕϕ θθ
U U U
λ λ− λ+ z
equilibrium stability
European School of Magnetism, Constanta, 2005: André THIAVILLE 35
2 2
5 .
y x
m m G + =
surface S- surface S+ z x y x : hard axis ; z : easy axis ; y : intermediate
European School of Magnetism, Constanta, 2005: André THIAVILLE 36
2 2 2 2 2 2 x z z y y x
m m m m m m G + + =
z x y surface S+ surface S- iron sphere nickel sphere
European School of Magnetism, Constanta, 2005: André THIAVILLE 37
Measurements on an isolated nanoparticle
(E. Bonet et coll., Phys. Rev. Lett. 83, 4188 (1999))
Zone axis for 18 cuts G= degree 2 +(degree 4 et 6, disoriented)
European School of Magnetism, Constanta, 2005: André THIAVILLE 38
The anisotropy energy of that nanoparticle
European School of Magnetism, Constanta, 2005: André THIAVILLE 39
3 nm cobalt cluster
Thèse M. Jamet, Lyon 2001
118 (2001)
European School of Magnetism, Constanta, 2005: André THIAVILLE 40
Surface anisotropy in ultrathin films
Ks Keff = Kv + 2 Ks/D D Kv Ks Keff < µ0 Ms
2/2
Keff > µ0 Ms
2/2
Ks : 10-3 J/m2
v s s c
K M K D − µ = 2 / 2
2
Transition thickness Dc : 1 nm
European School of Magnetism, Constanta, 2005: André THIAVILLE 41
Another 3 nm cobalt cluster
European School of Magnetism, Constanta, 2005: André THIAVILLE 42
A cube with uniaxial anisotropy
Edge size / Λ Quality factor Q
2
2
s
M K Q µ =
European School of Magnetism, Constanta, 2005: André THIAVILLE 43
A square platelet
10 Λ 20 Λ 5 Λ
permalloy Λ= 5 nm
flower leaf
R.P. Cowburn et al. APL 72 2041 (1998)
European School of Magnetism, Constanta, 2005: André THIAVILLE 44
Configuration anisotropy
permalloy : no anisotropy
R.P. Cowburn et al. APL 72 2041 (1998); Phys. Rev. B (1998)
European School of Magnetism, Constanta, 2005: André THIAVILLE 45
Phase diagram of domain walls in a soft nanostrip
permalloy Λ= 5 nm
t = 7.5 nm : VW t = 6.0 nm : ATW t = 3.5 nm : TW
20 40 60 80 100 1 2 3 4
w / Λ t / Λ Vortex Wall Symmetric Transverse Wall Asymmetric Transverse Wall t w
European School of Magnetism, Constanta, 2005: André THIAVILLE 46
Macrospin : magnetization reversal strategies
m(t=0) θ0 H dϕ/dt = γ0H dθ/dt = αγ0Hθ θ= αθ0 γ0Ht dθ/dt = γ0H θ= γ0Ht H Precessional switching θ0 : thermal fluctuations
European School of Magnetism, Constanta, 2005: André THIAVILLE 47
Precessional switching in a platelet
Field cutoff strategy: Magnetisation vector back to the film plane i ii iii i ii Field pulse profile i ii iii t Ha A basically 2 step process
Thèse G. Albuquerque, Orsay, 2002
European School of Magnetism, Constanta, 2005: André THIAVILLE 48
Macrospin magnetisation trajectories
Thèse G. Albuquerque, Orsay, 2002
European School of Magnetism, Constanta, 2005: André THIAVILLE 49
Macrospin precessional dynamics : switching phase diagram
Green: Static switching threshold Red: Dynamic switching threshold Blue: Ballistic trajectories Most favorable case: Transverse field Hy = Hk/2 Main Conclusion: Switching possible below the static threshold
Thèse G. Albuquerque, Orsay, 2002
European School of Magnetism, Constanta, 2005: André THIAVILLE 50
Precessional switching of a MRAM memory cell
H= 81 Oe H= 205 Oe T= 175 ps T= 240 ps
H.W. Schumacher et al.
European School of Magnetism, Constanta, 2005: André THIAVILLE 51 155 Oe 195 Oe 215 Oe @ 230 Oe
H.W. Schumacher et al.
European School of Magnetism, Constanta, 2005: André THIAVILLE 52
Precessional reversal of small elements
NiFe 500x 250x 5 nm, « S » state
Ha M - [M ×Ha] (a)
Hd M (b)
European School of Magnetism, Constanta, 2005: André THIAVILLE 53
(1) Initial phase : quasi-coherent reversal 250 ps
Spin Dynamics in confined structures I,
(Springer, 2002)
European School of Magnetism, Constanta, 2005: André THIAVILLE 54
(2) Breaking into magnetization waves with large out of plane components
European School of Magnetism, Constanta, 2005: André THIAVILLE 55
MFM of magnetic dots in a vortex state
First observation: T. Shinjo et al., Science 289 (2000) 930 Natural state After saturation under 1 T Sample : permalloy, 50 nm thick
European School of Magnetism, Constanta, 2005: André THIAVILLE 56
Vortex core switching : Experimental measurements
H
diameter : 200 nm 400 nm 1000 nm
European School of Magnetism, Constanta, 2005: André THIAVILLE 57
A Bloch point mediates the vortex core switching
B: from 331 to 332 mT at t=0 mesh: 4x4x5 nm damping α= 0.5 d=100 nm thickness=50 nm
European School of Magnetism, Constanta, 2005: André THIAVILLE 58
Bloch points at zero field
hedgehog circulating spiraling
The exchange energy density diverges at the center (singularity) r r m r r= It is lowest when up to a uniform rotation εA = (2A/r2) EA = 8π A R R: radius of the BP structure
European School of Magnetism, Constanta, 2005: André THIAVILLE 59
Bloch points at zero field : calculated structure
Vortex (diameter=200 nm, thickness=50 nm, meshing=2.5 nm; image size: 60nm) z= 0 nm z= 22 nm z= 24 nm z= 26 nm z= 28 nm z= 50 nm color code Vortex with a Bloch point in the middle z= 0 nm z= 22 nm z= 24 nm z= 26 nm z= 28 nm z= 50 nm
The BP is stabilized at zero H because of mesh friction; as soon as the BP is not perfectly centered it is expelled
European School of Magnetism, Constanta, 2005: André THIAVILLE 60
Cobalt 30 x 30 nm diamètre 32, 64 nm Permalloy Ni80Fe20 200 x 5 nm
European School of Magnetism, Constanta, 2005: André THIAVILLE 61
Confinement effect on the domain wall width
dx x S K dx d A E ) ( sin
2 2
∫
θ + θ =
S(x)
2 cos sin 2 = θ − θ θ S dx d A dx d S K
x
∫
= θ ) (x S dx C = θ ⇒ + = d x Arctg d x S S
2 0 1
d x x d x S S + π = θ ⇒ + = 2 1
2
wall width ∝ d
European School of Magnetism, Constanta, 2005: André THIAVILLE 62
20 40
0.5 1
x (nm) <mx>
0.1 0.2 0.3 0.4 0.5 0.6 0.7
<m >
∆x= 6.63 nm ∆y = ∆z = 5.33 nm
3 nm
30 nm x y z
Cobalt, 30x30 nm, maille 3 nm (167x 10x 10 points) ) / ( ch / 1 ) / tanh( ∆ ≈ ∆ − ≈ x m x m
y x
y
European School of Magnetism, Constanta, 2005: André THIAVILLE 63
Damping constant α= 0.1
1 2 3 4 200000 400000 600000 800000 1000000 1200000 1400000time (ns) Mt (A/m)
Hx = 50 Oe
Mx
1 2 3 4Mx (A/m) My Mz
Angle of the transverse magnetization
H = 50 Oe
1 2 3 4 200000 400000 600000 800000 1000000 1200000 1400000time (ns) Mt (A/m)
Hx = 100 Oe
Mx
1 2 3 4Mx (A/m) My Mz 1 2 3 4 1 2 3 4 5 6
time (ns) Mt angle in yz plane (rad.)
H = 100 Oe
1 2 3 4 200000 400000 600000 800000 1000000 1200000 1400000time (ns) Mx (A/m)
Hx = 150 Oe
Mx
1 2 3 4Mt (A/m) My Mz
H = 150 Oe
European School of Magnetism, Constanta, 2005: André THIAVILLE 64
Bloch point wall in a cobalt nanowire
Cobalt, 30x30 nm, mesh 3 nm (167x 10x 10 points)
Bloch point
European School of Magnetism, Constanta, 2005: André THIAVILLE 65
A permalloy nanostrip 200 nm thickness 5nm ≈ Λ = (2A/ µ0 Ms
2)1/2
H antivortex displacement direction
European School of Magnetism, Constanta, 2005: André THIAVILLE 66
Domain wall dynamics in a permalloy nanostrip (200 x 5 nm)
Nature Mater. 2, 521-523 (2003)
Perfect strip
European School of Magnetism, Constanta, 2005: André THIAVILLE 67
Effect of the roughness of strip edges
European School of Magnetism, Constanta, 2005: André THIAVILLE 68
Thermodynamics of a macrospin
dS = sinθ dθ dϕ /4π sinθ dθ / 2 E = K V sin2θ
45 90 135 180 0.2 0.4 0.6 0.8 1
θ (degrés) énergie / KV
0.2 0.4 0.6 0.8 1
Maxwell-Boltzmann statistics : p(E) = exp(-E/kBT) / Z
T k V K
B
parameter
densité d'états (u.a.)
European School of Magnetism, Constanta, 2005: André THIAVILLE 69
45 90 135 180 0.005 0.01 0.015 0.02 0.025
θ densité de probabilité (u.a.) kBT/KV = 0.1 0.2 0.3 0.5 1. 5.
European School of Magnetism, Constanta, 2005: André THIAVILLE 70
u d p = 1/τ p = 1/T p = 1/τ
Discrete orientation model (Néel-Brown)
u d
] / )) ( exp[( ] / )) ( exp[( kT d E E T kT u E E
m m
− = − = τ τ τ
European School of Magnetism, Constanta, 2005: André THIAVILLE 71
Calculation of τ0
) ( ' 1 / 1
2
col f M cc
s
µ γ α α τ + =
formulas of Brown, Coffey… ωwell τ0 ≈ qq. 10-10 s with τ0 = 0.1 ns τ = 1s 1min 1h 1 jour 1 an 10 ans
∆E/kT = 23 27 31 34 40 43 Superparamagnetism : when τ < τmeasurement
European School of Magnetism, Constanta, 2005: André THIAVILLE 72
Langevin field description of thermal fluctuations
th s eff
H m E M H + − = δ δ µ 0 1 dt m d m m H dt m d
eff
× + × = α γ 0 =
th
H ) ' ( ) ' ( ) ( t t t H t H
ij j th i th
− = δ δ µ
From the fluctuation-dissipation theorem, or by matching the final probability distribution to Maxwell-Boltzmann
dt V M kT H
s i th
2 ) ( γ α σ = V M kT
s
2 γ α µ =
N.B. supposes a slow evolution
kT qq. < ω h
k/h = 2 1010 Hz/K
European School of Magnetism, Constanta, 2005: André THIAVILLE 73
European School of Magnetism, Constanta, 2005: André THIAVILLE 74
Mn12-acetate : a molecular magnet
European School of Magnetism, Constanta, 2005: André THIAVILLE 75
European School of Magnetism, Constanta, 2005: André THIAVILLE 76
European School of Magnetism, Constanta, 2005: André THIAVILLE 77
Parity effect in tunneling
JMMM 200, 182 (1999)
European School of Magnetism, Constanta, 2005: André THIAVILLE 78
Magnetism of Fe : free atom vs bulk
Free atom : Z= 28 1s2 2s22p6 3s23p6 3d6 4s2 Hund’s rule : L=2 & S=2 : 6 µB (4 µB : spin + 2 µB : orbital) Bulk metal : 3d & 4sp bands : 2.1 µB (2 µB : spin + 0.1 µB : orbital) < Li > = 0 < Lz> ≠ 0
European School of Magnetism, Constanta, 2005: André THIAVILLE 79
The simple model of P. Bruno (simplified)
S L H
spin
r r ⋅ =
−
λ
for an atom S L
( )
// //
L L S E E − = −
⊥ ⊥
λ
λ ≈ 10 meV : ∆L =1 K= 10 meV/atom (bulk Co : 5 105 J/m3 = 25 µeV/atom)
European School of Magnetism, Constanta, 2005: André THIAVILLE 80
Magnetic anisotropy and orbital moment in Co clusters of a few atoms
Co / Pt(111) 0.01 plan atomique 8.5 x 8.5 nm2
Science 300, 1130 (2003)
European School of Magnetism, Constanta, 2005: André THIAVILLE 81
Magnetism and transport
EF
) ( 2 1
2 F B diff
E N T k V h π τ = m ne τ σ
2
=
Schematic model of the 3D magnetic metals : d electrons : localized, magnetism s electrons : delocalized, transport s d
European School of Magnetism, Constanta, 2005: André THIAVILLE 82
Interlayer exchange coupling (1986)
Oscillation periods depend on spacer material and crystalline orientation
European School of Magnetism, Constanta, 2005: André THIAVILLE 83
Calculations based on the electronic structure
M.D. Stiles, JMMM 200, 322 (1999)
European School of Magnetism, Constanta, 2005: André THIAVILLE 84
Giant magneto-resistance (1988)
European School of Magnetism, Constanta, 2005: André THIAVILLE 85
European School of Magnetism, Constanta, 2005: André THIAVILLE 86
Spin pumping
European School of Magnetism, Constanta, 2005: André THIAVILLE 87
Spin transfer effects
I (large) F2 F1 electrons Angular momentum transfer due to the reorientation of the spins of the conduction electrons m p F1 F2
Spin-polarized current switching of a Co thin film nanomagnet
School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853
Laboratory of Atomic and Stolid State Physics, Cornell University, Ithaca, New York 14853
APPLIED PHYSICS LETTERS VOLUME 77, NUMBER 23 4 DECEMBER 2000
European School of Magnetism, Constanta, 2005: André THIAVILLE 89
Micromagnetics