Introduction to magnetism Part II Magnetization reversal Olivier - - PowerPoint PPT Presentation

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Institut Néel, Grenoble, France

Introduction to magnetism Part II – Magnetization reversal

Olivier Fruchart

Institut Néel (CNRS-UJF-INPG) Grenoble - France

http://neel.cnrs.fr

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Institut Néel, Grenoble, France

Sep.2011: next European School on Magnetism (ESM). Romania.

http://esm.neel.cnrs.fr

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Institut Néel, Grenoble, France

Introduction to magnetism – Magnetization reversal ToC →

• 1. Energies and length scales in magnetism
• 2. Single-domain magnetization reversal
• 3. Magnetostatics
• 4. Magnetization reversal in materials
• 5. Recent ways for reversing magnetization

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Hext M

Manipulation of magnetic materials:  Application of a magnetic field

s Z

H.M

µ − =

E

Zeeman energy: Spontaneous magnetization Ms Remanent magnetization Mr

Hext M

Losses

M H E d

ext

∫

= µ

Coercive field Hc

J s=0 M s

Another notation Spontaneous ≠ Saturation

• 1. ENERGIES AND LENGTH SCALES – Hysteresis and magnetic materials

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• 1. ENERGIES AND LENGTH SCALES – Soft and hard magnetic materials

Soft materials

Transformers Flux guides, sensors Magnetic shielding

Hard materials

Permanent magnets, motors Magnetic recording

Hext M

Hext M

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• 1. ENERGIES AND LENGTH SCALES – Origins of magnetic energy

2 2 1 2 , 1 Ech

) ( .

θ ∇ = − =

A J E S S ) ( sin 2

mc

θ

K E

=

H M .

S Z

µ − =

E

1 2 d S d

. 2 1 H M

µ − =

E

Zeeman energy (enthalpy) Magnetocrystalline anisotropy energy Dipolar energy Echange energy

Hext M

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• 1. ENERGIES AND LENGTH SCALES – Magnetic characteristic length scales

Typical length scale: Bloch wall width B

( )

θ θ

2 2

sin / K dx d A e

+ =

Exchange Anisotropy

J/m

3

J/m

Numerical values

K A/

B

π λ =

nm 3 2

B

− = λ

nm 100

B ≥

λ

Hard Soft

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• 1. ENERGIES AND LENGTH SCALES – Magnetic domains

Bulk material

Co(1000) crystal – SEMPA

• A. Hubert, Magnetic domains

Mesoscopic scale

Numerous and complex magnetic domains Small number of domains, simple shape

Microfabricated dots Kerr magnetic imaging

• A. Hubert, Magnetic domains

Nanometric scale

Magnetic single-domain

R.P. Cowburn, J.Phys.D:Appl.Phys.33, R1 (2000)

Nanomagnetism ~ mesoscopic magnetism

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Institut Néel, Grenoble, France

Introduction to magnetism – Magnetization reversal ToC →

• 1. Energies and length scales in magnetism
• 2. Single-domain magnetization reversal
• 3. Magnetostatics
• 4. Magnetization reversal in materials
• 5. Recent ways for reversing magnetization

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Framework

• 2. SINGLE-DOMAIN REVERSAL – Coherent rotation (1/4)
• L. Néel, Compte rendu Acad. Sciences 224, 1550 (1947)
• E. C. Stoner and E. P. Wohlfarth, Phil. Trans. Royal. Soc. London A240, 599 (1948)

IEEE Trans. Magn. 27(4), 3469 (1991) : reprint

[ ]

) cos( sin

H ext S 2 eff tot

θ θ µ θ − − =

H M K V E

θ H θ M

H

Approximation:

Cte

= = M

r m ) (

(strong!)

d mc eff

K K K

+ =

Uniform rotation / magnetization reversal Coherent rotation / magnetization reversal Macrospin etc. Names used

) cos( 2 ) ( sin2

H

h e

θ θ θ − − =         = = = S a a

/ 2 / / M K H H H h VK E e

µ

Dimensionless units:

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Stability Equilibrium states

• 2. SINGLE-DOMAIN REVERSAL – Coherent rotation (2/4)

) cos( 2 ) ( sin2

θ θ

h e

+ =

• 90°

0° 90° 180° 270°

H>0

h e

= ⇒ = ∂ ∂

) cos(

m θ θ

( )2

2 2 max

1 2 2 1 ) ( ) ( h h h h e e e

− = − + − = − = ∆ θ θ θ θ θ θ

cos 2 2 cos 4 cos 2 2 cos 2

2 2 2

h h e

− − = − = ∂ ∂

( )

h e

− = ∂ ∂ θ θ θ

cos sin 2

[ ]

π θ ≡

( )

° = 180 H θ

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

2 2 2 m 2 2 2 2

h e h e h e

+ = ∂ ∂ − = ∂ ∂ − = ∂ ∂ π θ θ θ θ

Energy barrier Switching

s a

/ 2 1 M K H H h

µ = = =

1−h

 with exponent 1.5 in general

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Institut Néel, Grenoble, France 30 60 90 120 150 180 210 240 270 300 330

Easy axis Easy axis Hard axis Hard axis

) (

Sw H

H

θ

‘Astroid’ curve

• J. C. Slonczewski, Research Memo RM

003.111.224, IBM Research Center (1956)

• 2. SINGLE-DOMAIN REVERSAL – Coherent rotation (3/4)
• 90°

0° 90° 180° 270°

H

• 90°

0° 90° 180° 270°

H H = 0.2 Ha H = 0.7 Ha H = Ha H = 0 EASY ~ HARD

( )

2 / 3 H 3 / 2 H 3 / 2 Sw

cos sin 1

θ θ + =

H

• HSw(θ) is a signature of

reversal modes

( )

2 / 3 H 3 / 2 H 3 / 2 Sw

cos sin 1

θ θ + =

H

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• 2. SINGLE-DOMAIN REVERSAL – Coherent rotation (4/4)
• 1

1

• 1.5
• 1
• 0.5

0.5 1 1.5

M h

0° 10° 30° 45° 70° 90°

1.0 0.8 0.6 0.4 0.2 0.0 Normalized field 180 135 90 45 Angle

Reversal field Coercive field Reversal field Coercive field Coerciv e field

Switching field = Reversal field

A value of field at which an irreversible (abrupt) jump of magnetization angle occurs. Can be measured only in single particles. The value of field at which M.H=0 (H±) A quantity that can be measured in real materials (large number of ‘particles’). May be or may not be a measure of the mean switching field at the microscopic level

Coercive field

Easy Easy Hard Hard

) 2 (sin Abs

2 1 c H

h

θ =

( )

2 / 3 H 3 / 2 H 3 / 2 Sw

cos sin 1

θ θ + =

h

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• 2. SINGLE-DOMAIN REVERSAL – Coherent rotation, experimental relevance (1/2)
• M. Jamet et al., Phys. Rev. Lett., 86, 4676 (2001)

Experimental evidence Extensions: 3D, arbitrary anisotropy etc.

• M. Jamet et al., PRB69,

024401 (2004)

• A. Thiaville et al.,

PRB61, 12221 (2000)

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

• 0.3
• 0.2
• 0.1

0.1 0.2 0.3

µ 0 Hz(T)

µ 0 Hy(T)

0.04K

First evidence: W. Wernsdorfer et al.,

• Phys. Rev. Lett. 78, 1791 (1997)

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• 2. SINGLE-DOMAIN REVERSAL – Experimental relevance (2/2)
• J. Z. Sun et al., Appl. Phys. Lett. 78 (25), 4004 (2001)

0.2x0.5 0.37x0.75 0.2x0.75 0.27x1.37

Size-dependent magnetization reversal

Size in micrometers Astroids of flat magnetic elements with increasing size

The simplest model Fails for most systems because they are too large: apply model with great care!.. Hc<<Ha for most large systems (thin films, bulk): do not use Hc to estimate K! Early known as Brown’s paradox Conclusion over coherent rotation

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Barrier height

• 2. SINGLE-DOMAIN REVERSAL – Thermal activation

h=0.2

( )

K H M h 2 /

S µ =

( )2

max

1 ) ( ) ( h e e e

− = − = ∆ θ

T k T k E

T k E B B

25 ~ ) / ln( exp

B

τ τ τ τ = ∆ ⇒       =

∆

        − = KV T k

M K H

B

25 S c

1 2

µ T Hc Blocking temperature

B b

25 / k KV T

≈

Notice, for magnetic recording : t ≈109s

T k KV

B B

60 40 −

≈

Lab measurement :  = 1s

Thermal activation

Brown, Phys.Rev.130, 1677 (1963)

Information about anisotropy density Information about total effective anisotropy

• E. F. Kneller, J. Wijn (ed.) Handbuch

der Physik XIII/2: Ferromagnetismus, Springer, 438 (1966)

• M. P. Sharrock, J. Appl. Phys. 76,

6413-6418 (1994)

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• 2. SINGLE-DOMAIN REVERSAL – Magnetic length scales
• E. F. Kneller & F. E. Luborsky,

Particle size dependence of coercivity and remanence of single-domain particles,

• J. Appl. Phys. 34, 656 (1963)

Towards suparamagnetism Towards nucleation-propagation and multidomain

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Institut Néel, Grenoble, France

Introduction to magnetism – Magnetization reversal ToC →

• 1. Energies and length scales in magnetism
• 2. Single-domain magnetization reversal
• 3. Magnetostatics
• 4. Magnetization reversal in materials
• 5. Recent ways for reversing magnetization

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Magnetization

• 3. MAGNETOSTATICS – Notations

          =           =

z y x z y x

m m m M M M M

s

M 1

2 2 2

= + +

z y x

m m m

Magnetization vector M Can vary in time and space. Mean-field approach possible: Ms=Ms(T) Modulus is constant (hypothesis in micromagnetism)

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• 3. MAGNETOSTATICS – Treatment of dipolar energy (1/3)

) ( ). ( ) (

d 2 1 d

r H r M r

µ − =

E

Density of dipolar energy

∫∫∫

− − − = space 3 3 s d

' d ' 4 ) ' )].( ' ( [ div ) ( r M r r r r r m r H

π

H curl

=

) (

d

) ( div ) ( div

d

M H

− =

By definition . As we have (analogy with electrostatics):

] ) ( div[

• )

(

s

r m r M

= ρ

is called the volume density of magnetic charges To lift the divergence that may arise at sample boundaries a volume integration around the boundaries yields:

        − − + − − − =

∫∫ ∫∫∫

sample 2 3 space 3 3 s d

' d ' 4 ) ' )].( ' ( ). ' ( [ ' d ' 4 ) ' )].( ' ( [ div ) ( r r M r r r r r n r m r r r r r m r H

π π

) ( . ) ( ) (

s

r n r m r M

= σ

is called the surface density of magnetic charges, where n(r) is the outgoing unit vector at boundaries Do not forget boundaries between samples with different Ms

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• 3. MAGNETOSTATICS – Treatment of dipolar energy (2/3)

Some ways to handle dipolar energy ∫∫∫

− =

sample d 2 1

d . V M.H

µ

E

∫∫∫ ∫∫∫

= − =

space 2 d 2 1 sample d 2 1

d . d . V V H M.H

µ µ

E

Notice: six-fold integral over space: non-linear, long-range, time-consuming. Bottle-neck of micromagnetic calculations Integrated dipolar energy: Usefull theorem for finite samples:

E is always positive Significance of (BHmax) for permanent magnets

∫∫∫ ∫∫∫ ∫∫∫

= − = + −

sample \ space 2 d 2 1 sample d 2 1 sample d d 2 1

d . d . d . ) ( V V V H B.H .H H M

µ µ µ

Energy available outside the sample, ie usefull for devices

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• 3. MAGNETOSTATICS – Treatment of dipolar energy (3/3)

+ + + +

• -

+ + + + + + + + + + + + + + + + + + + + + +

• + + +

x

Examples of magnetic charges

Notice: no charges and E=0 for infinite cylinder + + + + + + + + +

• Charges on

surfaces Surface and volume charges

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∫∫ ∫∫

− − = − − =

sample 2 3 s sample 2 3 s d

' d ' 4 ) ' ).( ' ( ' d ' 4 ) ' )].( ' ( . [ ) ( r n m M r M

i i

r r r r r r r r r r n m r H

π π

• 3. MAGNETOSTATICS – Demagnetizing coefficients (1/3)

( )

i i z y x

m M m m m M r u z y x M M

s s

) (

= + + = ≡

∫∫ ∫∫∫ ∫∫ ∫∫∫ ∫∫∫

− − − = − − − = − =

sample 2 3 sample 3 d sample 2 3 sample 3 2 s 2 1 sample 3 d 2 1 d

' d ' 4 ) ' ).( ' ( d ' d ' 4 )] ' .( ).[ ' ( d d . ). ( r r r r r r r r r r m r r r M r H

π π µ µ

j j i j i i i

r r n m m K n m M E m N m . .

d d d

V K m m VN K

j i ij = =

E

Assume uniform magnetization

See more detailed approach: M. Beleggia and M. De Graef, J. Magn. Magn. Mater. 263, L1-9 (2003)

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• 3. MAGNETOSTATICS – Demagnetizing coefficients (2/3)

m N m . .

d d d t j i ij

K m m N K

= =

E

N is a positive second-order tensor

          = z y x

N N N N m N r H . ) (

s d

M

− > = <

i

N M H

s i d,

) (

− > = <

r 1

= + +

z y x

N N N ) (

2 2 2 d d z z y y x x

m N m N m N K

+ + =

E

What with ellipsoids???

Self-consistency: the magnetization must be at equilibrium and therefore fulfill m//Heff Assuming Happlied and Ha are uniform, this requires Hd(r) is uniform. This is satisfied

• nly in volumes limited by polynomial surfaces of order 2 or less:

slabs, cylinders, ellisoids (+paraboloïds and hyperboloïds).

• J. C. Maxwell, Clarendon 2, 66-73 (1872)

…and can be defined and diagonalized for any sample shape

Valid along main axes only!

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• 3. MAGNETOSTATICS – Demagnetizing coefficients (3/3)

η η η η η

d ) )( )( ( ) (

1 2 2 2 2 2 1

− ∞

∫

      + + + + =

c b a a abc Nx

        −         − − − =

1 1 sinh A 1 1 1

2 2 2 2 α α α α α x

N

For prolate revolution ellipsoid: (a,c,c) with =c/a<1

                − − − − = α α α α α

1 sin A 1 1 1 1

2 2 2 2 x

N

For oblate revolution ellipsoid: (a,c,c) with =c/a>1

• J. A. Osborn, Phys. Rev. 67, 351 (1945).

General ellipsoid: main axes (a,c,c)

) 1 (

2 1 x z y

N N N

− = =

) /( ); /( ; c b b N c b c N N

z y x

+ = + = =

For a cylinder along x For prisms, see: More general forms, FFT approach:

• A. Aharoni, J. Appl. Phys. 83, 3432 (1998)
• M. Beleggia et al., J. Magn. Magn. Mater. 263, L1-9 (2003)

Ellipsoids Cylinders

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• 3. MAGNETOSTATICS – Dipolar energy and hard-axis loops (1/2)

Magnetization loop of a macrospin along a hard axis

) sin( 2 ) ( sin2

θ θ

h e

− = d s a a

/ 2 . K N K M K H H h H

i = = = µ

θ H θ M

H

) cos( 2 ) ( sin2

H

h e

θ θ θ − − =

2 /

π θ = H

( )

h e

− = ∂ ∂ θ θ θ

sin cos 2

h

u m. ) cos( sin

= − = = H

h

θ θ θ

Hard axis: Dipolar energy: Equilibrium position

Ha

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• 3. MAGNETOSTATICS – Dipolar energy and hard-axis loops (2/2)

Case of a bulk soft magnetic material

Hypotheses:

• 1. Use an ellipsoid, cylinder or slab along a main direction

so that the demagnetizing field may be homogeneous.

• 2. Domains can be created to yield a uniform and effective magnetization Meff

ext eff 2 eff 2 1 Z d tot

H N E E E M M

µ µ − = + =

ext eff eff tot

H N E

µ µ − = ∂ ∂

M M

ext eff

1 H N

µ =

M

Density of energy: Minimization:

Ha

Susceptibility is constant and equal to 1/N

Conclusion for soft magnetic materials

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Easy axis, coercitive Ideally soft

• 3. MAGNETOSTATICS – Compensation of dipolar energy in loops (1/4)

Ha

N=0 (slab, infinite cylinder) N>0 (here N=1: slab, perpendicular) N=0 (slab, infinite cylinder) N>0 (here N=1: slab, perpendicular)

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• 3. MAGNETOSTATICS – Compensation of dipolar energy in loops (2/4)

1. Measure a hysteresis loop M1(Happl) 2. Internal field during loop: Hd=-Ni.M1 (must be corrected to access intrinsic properties) 3. Plot M1(Happl-NiM1) M2(Htot)

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.30

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• 3. MAGNETOSTATICS – Collective effects: range of interaction

∫

≤

3 d

d 2 ) ( r r r R

π

H

Position (a.u.) Average Real

Estimation of an upper range of dipolar field in a 2D system R Local dipole: 1/r3 Integration

R R / 1 Cte ) (

d

+ ≤

H

Convergence with finite radius (typically thickness)

Dipolar fields are weak and short-ranged in 2D or even lower-dimensionality systems Dipolar fields can be highly non-homogeneous in anisotropic systems like 2D Consequences on dot’s non-homogenous state, magnetization reversal, collective effects etc.

Upper bound for dipolar fields in 2D Non-homogeneity of dipolar fields in 2D

Example: flat stripe with thickness/height = 0.0125

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.31

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• 3. MAGNETOSTATICS – C and S states

’

 At least 8 nearly-equivalent ground-states for a rectangular dot  Issue for the reproductibility of magnetization reversal ‘C’ state ‘S’ state

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.32

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• 3. MAGNETOSTATICS – Domain walls in thin films
• L. Néel, Énergie des parois de Bloch dans les couches minces,
• C. R. Acad. Sci. 241, 533-536 (1955)

Thickness t Wall width W

W t

K E

2 d d = t W

K E

2 d d =

Bloch wall Néel wall

Bloch versus Néel wall

Crude model: wall is a uniformly-magnetized cylinder with an ellipsoid base

At low thickness (roughly t ≈W) Bloch domain walls are expected to turn their magnetization in-plane > Néel wall Model needs to be refined Domain walls not changed for films with perpendicular magnetization Conclusion

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.33

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Theory / Simulation

• 3. MAGNETOSTATICS – Magnetic vortices ('1D domain walls')

5 10 15 20 100 200 300 400 500

Thickness (nm)

Vortex state Single domain state

R.P. Cowburn, J.Phys.D:Appl.Phys.33, R1–R16 (2000)

Experiments

P.-O. Jubert & R. Allenspach, PRB 70, 144402/1-5 (2004)

 Vortex state (flux-closure) dominates at large thickness and/or diameter  The size limit for single-domain is much larger than the exchange length Experimentally the vortex may be difficult to reach close to the transition (hysteresis)

2 ex

20 .

λ ≈

D t

Zero-field cross-over

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.34

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• 3. MAGNETOSTATICS – Magnetic length scales

Typical length scale: Bloch wall width λB

( )

θ θ

2 2

sin / K dx d A e

+ =

Exchange Anisotropy

J/m

3

J/m

Numerical values

K A/

B

π λ =

nm 3 2

B

− = λ

nm 100

B ≥

λ

Hard Soft

=A/K is often called

the Bloch wall parameter. Notice also that several definitions of Bloch wall width have been proposed, e.g. with  or 2 as prefactor

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.35

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• 3. MAGNETOSTATICS – Magnetic length scales

Typical length scale: Exchange length λex

( )

θ θ 2 d 2

sin / K dx d A e

+ =

Exchange Dipolar energy J/m

3

J/m

2 d ex

/ 2 /

s

M A K A

µ λ = =

nm 10 3

ex − = λ

Critical size relevant for nanoparticules made of soft magnetic material

ex c

3λ

π ≈

D

2 s c

) /( 6 M N A D

µ π ≈

Generalization for various shapes

Quality factor Q

θ θ 2 d 2

sin sin K K e

+ − =

m.c. Dipolar energy J/m

3

J/m

d

/K K Q = Relevant e.g. for stripe domains in thin films with perpendicular magnetocristalline anisotropy

Critical size for hard magnets

for hard magnetic materials

B d w c

5 . 2 / 6

λ

Q K E D

≈ ≈

AK E 4

w ≈

Notice: Other length scales: with field etc.

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.36

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Introduction to magnetism – Magnetization reversal ToC →

• 1. Energies and length scales in magnetism
• 2. Single-domain magnetization reversal
• 3. Magnetostatics
• 4. Magnetization reversal in materials
• 5. Recent ways for reversing magnetization

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.37

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Institut Néel, Grenoble, France

• 4. COERCIVITY IN MATERIALS – Nucleation and propagation (1/3)

Brown’s paradox

In most systems

s c

2 M K H

µ



Micromagnetic modeling

Exhibit analytic however realistic models for magnetization reversal

x K K0 d

• d

Propagation Nucleation

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.38

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• 4. COERCIVITY IN MATERIALS – Nucleation and propagation (2/3)

Nucleation-limited Propagation-limited Ex: Sm2Co17 Ex: SmCo5 First magnetization Use first-magnetization curves to determine the type of coercivity

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.39

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Institut Néel, Grenoble, France

• 4. COERCIVITY IN MATERIALS – Nucleation and propagation (3/3)

Activation volume

Also called: nucleation volume

100 200 300 400 500 2 4 6 8 10

Pr17Fe75B8

T(K) v/δ

3

Can be used for:

Estimating Hc(T) Estimating long-time relaxation Determination of dimensionality

Note: of the order of domain wall width 

• D. Givord et al., JMMM258, 1 (2003)

More detailed models:

1/cos law

Hypothesis:

Based on

nucleation volume

Hc<<Ha

T k H v M E

H B a s

25 ) cos(

+ − = θ µ

θ

H

M H

Energy barrier E0

• vercome by gain in

Zeeman energy plus thermal energy

• E. J. Kondorsky, J. Exp. Theor. Fiz. 10, 420 (1940)
• D. Givord et al., JMMM72, 247 (1988)

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.40

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Nucleation of new reversed domains

• 4. COERCIVITY IN MATERIALS – Thin films (1/3)

[ ]

) exp( 1 Rt N N

− − =

N: number of nucleated centers at time t N0: total number of possible nucleation centers R: rate of nucleation

• E. Fatuzzo, Phys. Rev. 127, 1999 (1962)

t R N N dN d ) (

0 − =

Radial expansion of existing domains

Rc: radius of critical nucleus T: total area of sample V0: speed of propagation of domain wall New nuclei Growth of existing nuclei Fatuzzo/Labrune/Raquet model

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.41

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Model: fraction area not yet reversed

• 4. COERCIVITY IN MATERIALS – Thin films (2/3)

) /(

c

Rr v k = k is a measure of the importance of wall propagation versus nucleation events

• M. Labrune et al.,
• J. Magn. Magn. Mater. 80, 211 (1989)
• E. Fatuzzo, Phys. Rev. 127, 1999 (1962)

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.42

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• 4. COERCIVITY IN MATERIALS – Thin films (3/3)
• M. Labrune et al.,
• J. Magn. Magn. Mater. 80, 211 (1989)
• J. Camarero et al., PRB64, 172402 (2001)

Depending on structural defects Depending on measurement dynamics

Cross-over Note also for fast propagation of domain walls: breakdown of propagation speed (Walker)

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.43

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[1] Magnetic domains, A. Hubert, R. Schäfer, Springer (1999, reed. 2001) [2] R. Skomski, Simple models of Magnetism, Oxford (2008). [3] R. Skomski, Nanomagnetics, J. Phys.: Cond. Mat. 15, R841–896 (2003). [4] O. Fruchart, A. Thiaville, Magnetism in reduced dimensions,

• C. R. Physique 6, 921 (2005) [Topical issue, Spintronics].

[5] O. Fruchart, Couches minces et nanostructures magnétiques, Techniques de l’Ingénieur, E2-150-151 (2007) [6] Lecture notes from undergraduate lectures, plus various slides: http://perso.neel.cnrs.fr/olivier.fruchart/slides/ [7] G. Chaboussant, Nanostructures magnétiques, Techniques de l’Ingénieur, revue 10-9 (RE51) (2005) [8] D. Givord, Q. Lu, M. F. Rossignol, P. Tenaud, T. Viadieu, Experimental approach to coercivity analysis in hard magnetic materials, J. Magn. Magn. Mater. 83, 183-188 (1990). [9] D. Givord, M. Rossignol, V. M. T. S. Barthem, The physics of coercivity, J. Magn. Magn. Mater. 258, 1 (2003). [10] J.I. Martin et coll., Ordered magnetic nanostructures: fabrication and properties,

• J. Magn. Magn. Mater. 256, 449-501 (2003).

SOME READING

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.44

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Institut Néel, Grenoble, France

Introduction to magnetism – Magnetization reversal ToC →

• 1. Energies and length scales in magnetism
• 2. Single-domain magnetization reversal
• 3. Magnetostatics
• 4. Magnetization reversal in materials
• 5. Recent ways for reversing magnetization

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.45

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Institut Néel, Grenoble, France

• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Precessional switching (1/5)

Basics of precessional switching

Magnetization dynamics: Landau-Lifshitz-Gilbert equation:

[ ]

      × + × − =

dt d M dt d

s eff

M M H M M

α γ 0 Gyromagnetic factor

γ 0

H e f f

α

Démonstration: 1999

• C. Back et al., Science 285, 864 (1999)

M

∂ ∂ − = mag eff

E H

µ γ µ γ 0 =

GHz/T 28 2 /

= π γ

m gq 2

= γ

Effective field (including applied) Damping coefficient (10-3 > 10-1)

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.46

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• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Precessional switching (2/5)

y x z z

Hm M Km m N M E

s 2 2 2 s 2 1 µ µ − − = y x K z z

hm m h m N e

− − = 2 2 2 1

1

2 2 2 = + + z y x

m m m 1

= x

m

Precessional trajectories using energy conservation

In-plane uniaxial anisotropy (1) (2) (1) Using (2)

2 2

2 1

y K z z y K z x

m h N N m h N h m

+ − + − =

( ) ) ( 1 / 1 /

2 2 2 K z z z K z y x

h N N h N h N h m m

+ + = + + +

Starting condition: Can be rewritten: Using (2)

2 2

2

y K z K y K z z

m h N h m h N h m

+ − + =

Can be rewritten:

( )

2 2 2         =         − +

+

K K y h N h z

h h h h m m

K z K

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.47

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( ) ) ( 1 / 1 /

2 2 2 K z z z K z y x

h N N h N h N h m m

+ + = + + +

( )

2 2 2         =         − +

+

K K y h N h z

h h h h m m

K z K

• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Precessional switching (3/5)

mx my

• 1

1

• h

mz my h=0.01 h<0.5hK h>0.5hK h=0.5hK 0.5 1

Magnetization trajectories

mx mz

• 1

1 h=0.5hK h>hK hK>h>0.5hK h<0.5hK

2 / ) ( 847 .

s K

H H M

− ≈ γ ω

• M. Bauer et al., PRB61, 3410 (2000)

Field at 135°

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.48

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• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Precessional switching (4/5)

Stoner-Wohlfarth versus precessional switching

2.0 1.5 1.0 0.5 0.0 Energy (normalized) 360 315 270 225 180 135 90 45 In-plane angle 

h=0 h=0.5

0.1 0.2 0.3 0.4

Stoner-Wohlfarth model: describes processes where the system follows quasistatically energy minima, e.g. with slow field variation Precessional switching: occurs at short time scales, e.g. when the field is varied rapidly Applied field

Relevant time scales

ps.T 35 / 2

= γ π

ps 500 25 − Precession period Precession damping ) 2 /( 1

π α

per period ) 5 . 01 . (

− = α

Magnetization reversal allowed for h>0.5hK (more efficient than classical reversal

Notice

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.49

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• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Precessional switching (5/5)
• M. Bauer et al., PRB61, 3410 (2000)

Switched Non-switched Initial moment Stoner-Wohlfarth astroid (quasistatic limit) Reversal below the quasistatic value h=1 for hy, and even for m.h>0!

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.50

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Analytical models

• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Precessional switching - Overview
• T. Devolder et aL, Precessional switching of thin nanomagnets: analytical study, Eur. Phys. J. B 36,

57–64 (2003)

• T. Devolder et aL, Spectral analysis of the precessional switching of the magnetization in an isotropic

thin film, Sol. State Com. 129, 97 (2004)

• G. Bertotti et al., Comparison of analytical solutions of Landau–Lifshitz equation for ‘‘damping’’ and

‘‘precessional’’ switchings, J. APpl. Phys. 93, 6811 (2003)

• C. Serpico et al., Analytical solutions of Landau–Lifshitz equation for precessional switching, J. Appl.
• Phys. 93, 6909 (2003)

Most efficient for field applied perpendicular to the easy axis Analytical or near-analytical descriptions Beyond the simple example given here: field pulse in one or several directions, finite damping, spin-valves etc.

Conclusion on precessional switching

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.51

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• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Current-induced

Basics

Can be viewed as the GMR-reversed effect Conventionnal hysteresis loop Current-induced magnetization reversal

Group Myers et Ralph, Cornell University (2000)

Simplified architectures (MRAMs etc.) Fully electronic read/write Devices making use of domain wall motion (memory, logic) Devices using stationnary GHz oscillators

Motivations

• J. C. Slonczewski (1996)
• L. Berger (1996)

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.52

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• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Electrical control
• M. Weisheit et al., Science 315, 349 (2007)

Electric modification of intrinsic properties

See also: magnetic semiconductors, multiferroics etc.

Olivier Fruchart – School of GdR nanoalloys – Fréjus, June 2010 – p.53

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• 5. NEW WAYS FOR MAGNETIZATION REVERSAL – Polarized light

Gd22Fe74.6Co3.4

• C. D. Stanciu et al.,
• Phys. Rev. Lett. 99, 047601 (2007)

Principle

Combined heating + inverse Faraday effect Magneto-optical

• material. Tc=500K

Preliminary: one shot with large power

Demagnetized Magnetization reversed

Local reversal with controlled power

Ti:S laser: =800nm; =40fs.