Basic concepts
on magnetization reversal (1)
Static properties : coherent reversal
and beyond
Stanislas ROHART
Laboratoire de Physique des Solides
Université Paris Sud and CNRS
Orsay, France
Introduction: Hysteresis loop
Manipulation of a magnetization :
Application of a magnetic field
Zeeman energy : Ez=-µ0 H.MS
M
Spontaneous Magnetization MS
Remanent Magnetization MR
Coercive field HC
H
S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties
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Introduction: Soft and Hard materials
Hard Materials
SoftMaterials
M
M
H
Applications : Permanent magnets,
motors, magnetic recording
Ex: Cobalt, NdFeB, CoSm, Garnets
H
Applications : Transformer, flux
guide (for electromagnets…),
magnetic shielding
Ex : Iron, FeCo, Permalloy (Fe20Ni80)
S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties
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Introduction: Energies in magnetic systems
Exchange energy
Magnetocrystalline anisotropy energy
≠
rr
Eex = − JS i S j
rr2
EMC = K (m.e )
= A(∇θ )
2
Zeeman energy
(simplest form, may be more complicated)
reflects the cristal symmetry
Dipolar energy
r r r
r
µ  3(mi .rij )rij mi  r
ED = − 0 
− 3 .m j
4π 
rij5
rij 
For practical use :
shape anisotropy
r r
EZ = − µ 0 M .H
r r
1
E D = − µ 0 M .H d
2
r
r
1
= − µ 0 M .[ N ]M
2
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Introduction: Micromagnetism: Typical Length Scales
Bloch wall
Exchange length
-> Anisotropy vs. Exchange
-> Dipolar coupling vs. Exchange
Ex : Magnetic vortex
(
E = A dθ
) + K sin θ
dx
2
2
Bloch wall parameter δ B = A
K
Bloch wall width d B = π A
K
Bloch wall energy
σ B = 4 AK
Typical value : 2-3 nm (hard)
-> 100-1000 nm (soft)
Λ = 2A
~ 2.6Λ
µ 0 M S2
Typical value : 5-10 nm
Quality factor
Q = 2K
µ0 M
2
S
( δ)
= Λ
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hard
Q << 1 soft
Q >1
5
Introduction: Magnetic Domains
Bulk materials
Mesoscopic scale
Complex magnetic paterns
Small number of possible
configuration.
Well defined states
Self organization of domains
Nanometric scale
Magnetic single domain
but non collinearities are
still possible
True collinear state at very
reduced dimensions
(< few Λ)
120 µm
130 µm
•Except at very small scales, dipolar energy plays an
essential role [competition between dipolar energy (long
range) and domain wall energy (local)].
•Single domain state is observed well below 1 µm or for
hard material .
(square dots –
500 nm)
Cowburn et al. PRL 81, 5414 (1998)
Cowburn J.Phys.D: Appl. Phys.
33, R1 (2000)
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Contents
I. Coherent reversal
II. Magnetization reversal in nanostructures
III. Domain nucleation and domain wall
propagation
IV. Conclusion
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Coherent reversal: Macrospin hypothesis
Hypothesis : m(r)=cte=M (strong approximation)
Exchange energy is constant
Dipolar energy equivalent to anisotropy energy
Easy axis
θ
M
r
r r
E = G ( M ) − µ 0 M .H
Simplest model : Stoner and Wohlfarth
E = K eff sin 2 θ − µ 0 M S H cos(θ + θ H )
K eff = K mc + k d
Anisotropy field :
θΗ
H
H K = 2 K eff / µ 0 M S
Dimensionless equation : e = sin 2 θ − 2h cos(θ + θ H )
Different names : Uniform rotation, coherent
rotation, macrospin, Stoner and Wohlfarth
model…
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Coherent reversal: Equilibrium states and switching
θ H = 0 (Field aligned with the anisotropy axis)
e = sin 2 θ − 2h cos θ
∂e
= 2 sin θ (cos θ + h)
∂θ
cos θ = −h : θ = θ m
∂e
=0⇒
sin θ = 0 : θ = 0 or π
∂θ
h=0
-2
2
1
-0.5
-1
Stability
4
2
e
∂ 2e
2
2
θ
θ + 2h cos θ
=
−
2
sin
+
2
cos
2
∂θ
= 4 cos 2 θ − 2 + 2h cos θ
0
-2
-4
2
−π/2
0
π/2
ϕ
θ
π
3π/2
1.0
0.5
m=M/MS
∂e
>0
<0
(
0
)
=
2
(
1
+
h
)
∂θ 2
∂ 2e
2
θ
(
)
=
2
(
h
− 1) > 0
m
2
∂θ
∂ 2e
(π ) = 2(1 − h) >0
2
∂θ
Square hysteresis
loop
Hswitch = HK
0.0
-0.5
-1.0
-2
-1
0
1
2
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Coherent reversal: Equilibrium states
e = sin 2 θ − 2h cos θ
1.0
•Square hysteresis loop
->Hswitch = HK
m=M/MS
Energy barrier
∆e = e(ϕ m ) − e(0)
(
2
0.0
-0.5
-1.0
)
= 1 − h + 2h − (− 2h )
2
0.5
-2
-1
0
1
2
h
= (1 + h )
2
Important for thermally
activated switching
For arbitrary angle :
•no analytical solution
•HK/2<Hswitch<HK
•∆e=(1-h)α with α=1.5
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Coherent reversal: Hysteresis Loops
e = sin 2 θ − 2h cos(θ + θ H )
θ=0
θΗ = π/3
1.0
1.0
0.5
0.0
0.0
m
-0.5
-0.5
-1.0
4
-1
Hswitch=HC
0
h
1
2
4
-1.0
2
2
-2.0 -1.5 -1.0 -0.5 0.0
0
h
-2
Hswitch
−π/2
0
π
Coercive field : M.H = 0
-> may not be equal to the switching field
1.0
1.5
2.0
0
-2
-4
3π/2
ϕ
−π/2
0
π/2
ϕ
π
3π/2
4
4
2
2
e
Switching field (or reversal field)
-> abrupt jump of magnetization angle
π/2
0
e
-4
0.5
e
-2
e
m
0.5
HC
-2
0
-2
-4
-4
−π/2
0
π/2
ϕ
π
3π/2
−π/2
0
S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties
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π/2
ϕ
π
3π/2
11
Coherent reversal: Switching field plot : astroids
Astroid curve :
Polar plot of Hswitch
H switch =
HK
(sin
2/3
θ H + cos 2 / 3 θ H )
3/ 2
0
1.0
330
300
0.5
0.0
0.5
1.0
30
60
270
90
240
120
210
150
180
H switch
HC =
1
sin 2θ H
2
 π π
if θ H ∈ − ;  @ [π ]
 4 4
 π 3π 
if θ H ∈  ;  @ [π ]
4 4 
J.C. Slonczewski (1956)
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Coherent
reversal:
Switching
field
plot
:
astroids
r
r
r
e = G (m) − 2h .m
r
m = (cos θ , sin θ )
Equilibrim condition
rr
r
de
v
= G ' (m) − 2h.e = 0 with e = (− sin θ , cos θ )
dθ
2 stable states
Hard axis
1.0
0.5
hy
-> For given m : Straight line in the field space, tangente
to the critial astroid curve, directed along m
Stability condition
Easy axis
0.0
r r
r
d ²e
= G" (m) + 2h .m > 0
dθ ²
->For given m : Only one part of the line is stable
-0.5
-1.0
-1.0
-0.5
1 stable state
0.0
0.5
1.0
hx
J.C. Slonczewski Research Memo RM 003.111.224, IBM Research Center (1956)
A. Thiaville JMMM 182, 5, (1998)
S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties
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Coherent reversal: Switching field plot : astroids
-> To go further :
•Complex type of anisotropy
•Three dimensionnal extension
G = sin ²θ cos ²θ
(Cubic anisotropy)
G = sin ²θ cos ²θ + sin ²(θ + π / 6)
(Cubic anisotropy + uniaxial)
Thiaville PRB 61, 12221 (2000)
Thiaville JMMM 182, 5, (1998)
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Coherent reversal: Experimental relevance
First observation (2D) : Co (D = 25 nm) cluster
Wernsdorder et al. PRL 78, 1791 (1997)
In 3 D: (same Co cluster) E. Bonnet et al PRL 83, 4188 (1999)
3 nm Co cluster : M. Jamet et al PRL 86, 4676 (2001)
S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties
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Coherent reversal: Experimental relevance
In technological application : memory cells
Measurements of the astroids of elliptical M-RAM cells
(Dimensions in µm)
Conclusion :
Good agreement with coherent
rotation only for smallest
elements.
⇒Apply coherent reversal with
great care!
⇒In most systems Hswitch<<HK
«Brown’s paradox»
S. ROHART
: Basic Concepts on Magnetization Reversal : Static Properties
Sun et al. APL 78, 4004
(2001)
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Magnetization reversal in nanoparticles
• Is the coherent reversal model applicable to
nanoparticles?
• Is a uniform magnetized ground state
sufficient for coherent magnetization reversal?
• Can we go further than coherent
magnetization reversal?
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Nanoparticles: Phase diagram of nanostructures
1. Single domain ground-state criterion
Non uniform state :
1 domain wall
Single domain state
EdSD
-Exchange energy is minimum
-Anisotropy energy is minimum
(magnetization along easy axis)
-Dipolar energy :
NV
1
4
=
µ 0 M S2 = µ0 M S2 × πR 3
2
6
3
RSD ⇔ EdSD = EdDW + E DW
RSD =
-Bloch domain wall at the
centre -> exchange and anisotropy
cost : EDW = πR 2σ BW = 4πR 2 AK
-Dipolar energy gain : magnetic flux
is closed in two domains: EdDW = EdSD / 2
!
Domain wall width is neglected
-> calculation adapted for high
anisotropy materials
36 AK
µ 0 M S2
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Nanoparticles: Phase diagram of nanostructures
2. Coherent/Incoherent rotation criterion
- Schematic calculation : coherent rotation vs. 2 domain rotation
θa
θb
θa
θb
Hypothesis : 2 localized moments
ma / b = M SV / 2 at xa / b = ± R / 2
E  A 1
1

2
=  2 − µ0 M S2 (θ a − θ b ) + (2 K − µ0 M S H ) θ 2 a + θ 2 b
V R
24
4

Stability analysis :
 ∂2E 
diagonalize the 2x2 matrix 
 and find H that changes the sign of one eigenvalue
∂
θ
∂
θ
 i j 
The two eigenmodes are : θ a = θ b ⇒ H coh = H K (coherent rotation)
M
8A
θ a = −θ b ⇒ H incoh = H K − S +
3
µ0 M S R 2
(
Hincoh< Hincoh
if
24A
R>
µ0MS2
)
Remarks :
Rcoh< RSD for real materials
anisotropy does not enter in the criterion
S. ROHART : Basic Concepts on Magnetization Reversal : Static Properties
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Nanoparticles: Phase diagram of nanostructures
2. Coherent rotation criterion
E  A 1
1

2
=  2 − µ0 M S2 (θ a − θ b ) + (2 K − µ0 M S H ) θ 2 a + θ 2 b
V R
24
4

(
)
Exchange : the two moments are distant of R and the angle variation is ∆=θa-θb. The
exchange energy density is A(dm/dx)²~A(θa-θb)²/R²
Dipolar energy : if θa=θb, the dipolar energy is µ0MS²/6. If the two angles are different,
the total magnetization is lower so that Ed decreases by
-[µ0MS²/6]cos(θa-θb)/2~ -[µ0MS²/24](θa-θb)²
Anisotropy energy : each hemisphere of volume V/2 has the anisotropy energy
K[sin²θ]*V/2~2Kθ²V/4
Zeeman energy : each hemisphere of volume V/2 has the Zeeman energy
-µ0MSH[cosθ]*V/2~ -µ0MSHθ²*V/4
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Nanoparticles: Phase diagram of nanostructures
2. Coherent/Incoherent rotation criterion
Real reversal mechanisms :
3.0
Incoherent Reversal
Coherent Reversal
2.5
Coherent
Curling,
Buckling
HSW/HK
2.0
1.5
1.0
0.5
For curling :
MS
8.67 A
H sw = H K −
+
3
µ0 M S R 2
26A
Rcrit =
µ0MS2
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
R/Rcoh
See: Aharoni Introduction to
the theory of Ferromagnetism (1996)
Skomski and Coey Permanent Magnetism (1999)
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Nanoparticles: Phase diagram of nanostructures
Phase diagram for a flat disk with surface anisotropy
Keff = KS/t
Skomski et al. PRB 58, 3223 (1998)
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Nanoparticles : Influence of nanostructure inhomogeneity
Dipolar field effect
-> Soft elements: flux closure domains faciliate magnetic reversal
H K = 5000Oe
t
H C = 3M S
+ cte
W
Shape anisotropy ~ MS L/W
but switching field intependant of L
Uhlig and Shi APL 84, 759 (2004)
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Nanoparticles : Influence of nanostructure inhomogeneity
Dipolar field effect
-> Soft elements
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Nanoparticles : Influence of nanostructure inhomogeneity
Edge and surface anisotropy in very small elements
Origin : enhanced magnetic anisotropy at low coordinated atoms
Co islands on Pt(111)
Nearly spherical Co cluster
Néel pair anisotropy model Ea =
rr 2
L
(
m
∑ .uij )
<i , j >
n .n .
Gambardella et al. Science 300, 1130 (2003)
Jamet et al. PRL 86, 4676 (2001)
On spherical clusters, atom anisotropy axis are perpendicular
to the local surface
-> Hedgehog like orientation of anisotropy axis
Kachkachi and Dimian PRB 66,174419 (2002)
Néel J. Phys. Radium 15, 225 (1954)
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Nanoparticles : Influence of nanostructure inhomogeneity
Edge and surface anisotropy in very small elements
J >> Ksurface = Kvolume
mz
Consequence of surface anisotropy axis distribution
h
Non colinear magnetic configuration
Atomic simulation : L/J=2
(unphysically strong surface anisotropy!)
mz
1 particle
1 macrospin
J ~ Ksurface >> Kvolume
Kachkachi and Dimian PRB 66,174419 (2002)
Garanin and Kachkachi PRL 90 065504 (2003)
1 particle
h
assembly of many coupled spins
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Nanoparticles : Influence of nanostructure inhomogeneity
Edge and surface anisotropy in very small elements
Effective anisotropy model
1.0
Example: flat disk with edge anisotropy
Shape anisotropy Edge anisotropy
(out of plane)
(in plane)
m
HZ/HK
Exchange
0.5
0.0
Z
0.8535
-0.5
0.8190
0.7845
0.7500
Y (u.a.)
0.7155
-1.0
-1.0
-0.5
0.0
0.5
1.0
HX/HK
Typical astroid with a 4th order anisotropy
X (u.a.)
Twisted inhomogeneous configuration
Rohart et al. PRB 76, 104401 (2007)
Garanin and Kachkachi PRL 90 065504 (2003)
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Nanoparticles: Vortices
⇒Strongly inhomogeneous magnetic state
Wachowiak et al. Science 298 577 (2002)
Cowburn et al. PRL 83, 1042 (1999)
Four degenerated states :
-vortex cirulation (clock wise vs. Counter clock wise)
-> difficult to manipulate with magnetic field
-Vortex core (up vs. Down) -> easily coupled to magnetic field
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Nanoparticles: Vortices
Stable vortex state
Stable uniform state
Hysteresis loop with IN PLANE magnetic field
The energy cost to expell the vortex core
out of the disk creates an hysteresis
Cowburn et al. PRL 83, 1042 (1999)
Guslienko and Metlov PRB 63, 100403R (2001)
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Nanoparticles: Vortices
Vortex core magnetization reversal
Magnetic field is coupled to the vortex core only
but coherent reversal of vortex core magnetization
is topologically impossible
Shinjo et al. Science 289, 930 (2000)Normal vortex state : magnetization is
perpendicular to minimize exchange
energy
Eex ( r ) = 2 A / r
EBP = ∫∫∫ Eex (r )r 2 sin θdrdθdϕ =8πAR
Vortex with Bloch point: magnetic
moment are all almost in plane, mean
magnetization is zero at the core
Beyond micromagnetism
Thiaville et al. PRB 67, 094410 (2003)
Images from R. Dittrich http://magnet.atp.tuwien.ac.at/gallery/bloch_point/index.html
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Nanoparticles: Vortices
Vortex core magnetization reversal
Experiments : Okuno et al. JMMM 240, 1 (2006)
Thiaville et al. PRB 67, 094410 (2003)
Images from R. Dittrich http://magnet.atp.tuwien.ac.at/gallery/bloch_point/index.html
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Magnetization reversal in thin films
Is the macrospin model relevant for thin films ?
-Lateral dimension >> Micromagnetic lengthes
-Experimental observation : HC<<HK in most systems
Brown’s paradox
⇒Coherent reversal is unreallistic : need for micromagnetic modelling.
⇒Defect (even at very low density) may drive the switching field if
domain wall propagation is involved
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Thin Films: Hard axis hysteresis loops
In some cases, coherent reversal can be applied to hard axis hysteresis loops
-In hard axis loop, domain nucleation is
M
prevented (the two orientation energies are
the same) and coherent rotation is possible.
H
M
MS
H
2K/µ0MS
Example2: « Anisometry »
Ga(Mn)As thin film
(with perpendicular magnetization)
•Polar Kerr effect measurment
with quasi in plane field (α)
H=
sin 2θK eff
M S cos(θ + α )
; M = M s cos θ
=> Application: determination of magnetic
anisotropy and MS
Example1: Soft magnetic film
Permalloy thin film
-no magnetocristalline anisotropy
-in plane magnetization due to the shape
anisotropy Ed = 12 µ0 M S2
=> Loop with perpendicular field yields a
saturation field of MS
MS -> Keff = 272 mT
J.P. Adam PhD. Thesis 2008
MScos(α)
Principle/Application:
Grolier et al. J. Appl. Phys. 73,
5939 (1993)
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Thin Films: Brown’s paradox
Origin of the lower coercivity : magnetic defects, temperature
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Thin Films: Nucleation vs. Propagation
First magnetization curve indicates the type of coercivity
M
Nucleation limited reversal
-> Need few nucleation event
followed by domain wall
propagation
-> Provides generally square
loops
H
Pinning
center
Magnetization reversal is controlled by the
micro/nanostructure
Propagation limited reversal
->Soft inclusion, misoriented grains…
-> Need many nucleation events
create nucleation centers
-> Provides generally rounded loops
->Hard inclusion,crystalline defect…
Ex : Recording media
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create pinning
centers
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Thin Films: Nucleation – « 1/cos θΗ law »
Nucleation of a reversed domain on defect -> Nucleation volume
• v ≈ δ 3 for bulk hard magnets Givord et al. JMMM 258, 1 (2003)
• Droplet theory for thin films Barbara JMMM 129, 79 (1994)
Nucleation cost (domain wall, anisotropy energy ) E0
is compensated by Zeeman energy gain (plus thermal energy)
M
If H c << H K : M is aligned with easy axis ∆E z = −2 µ 0 M S vH cos θ H
Magnetization reversal if ∆E z = E0 − 25kT
⇒ H C (θ H ) =
H
θH
H C ,θ H =0 Kondorsky, J. Exp. Theor. Fiz.
cos θ H
10, 420 (1940)
See also : Givord JMMM 72,
247 (1988)
Switching field of a Co(1 nm)
film grown on Au(233)
-> In plane magnetization with
well defined in plane
anisotropy (step edges)
G. Baudot PhD. Thesis,
unpublished (2003)
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Thin Films: Nucleation vs. DW propagation
Importance of dynamics
Key ingredient :
-> Nucleation rate
-> Domain wall propagation dynamics
Pt/Co/Pt
(From J. Vogel)
Coercive field may not be an intrinsic property
(ie. only linked to micromagnetic parameters)
-> depend on temperature
-> depend on sweeping rate
See second part of the lecture
on temperature activation and
slow dynamics
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Conclusion
Take home messages
•Coherent (uniform) magnetization reversal only takes place in very small particles
•Micromagnetism is powerfull to determine the switching mode in nanoparticles
•In thin films: Importance of micro/nano structure of magnetic films to determine the
reversal mechanism
•Coercive field may be ambiguous (different from switching field, not intrinsic…)
To go further
•Temperature influence and slow dynamics
•Precessionnal magnetization reversal (ns time scale, need loweer magnetic fields)
•Ultra fast magnetization reversal (beyond micromagnetism hypothesis M=cte)
•New driving forces for magnetization reversal (spin polarized currents, electrical field…)
Some readings
•Skomski and Coey Permanent magnetism (Taylor & Francis Group 1999)
•Hubert and Schäfer Magnetic domains (Springer 1999)
•Aharoni Introduction to the theory of ferromagnetism (Oxford 1996)
•Skomski Nanomagnetics J. Phys. Cond. Mat. 15, R841 (2003)
•Fiorani Surface effects in Magnetic Nanoparticles (Springer 2005)
•Fruchart and Thiaville Magnetism in reduced dimensions CR Physique 6 921 (2005)
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