Magnetism and Magnetic MaterialsMagnetism and Magnetic Materials

DTU (10313) – 10 ECTSDTU (10313) – 10 ECTSKU – 7.5 ECTSKU – 7.5 ECTS

Module 6

18/02/2011

Micromagnetism I

Mesoscale – nm-mReference material:Blundell, section 6.7Coey, chapter 7These lecture notes

Intended Learning Outcomes (ILO)Intended Learning Outcomes (ILO)

(for today’s module)(for today’s module)

1. Explain why and how magnetic domains form2. Estimate the domain wall width3. Calculate demagnetizing fields in simple geometries4. Describe superparamagnetism in simple terms5. List Brown’s equation in micromagnetics6. Explain how hysteresis arises in a simple Stoner-Wolfharth model

FlashbackFlashback

€

M = M S BJ (y)

y =gJμ BJ(B + λM )

kBT

€

TC =gJμ B(J +1)λM S

3kB

=nλμ eff

2

3kB

Edge effects and consequencesEdge effects and consequences

This is a bit misleading

DipolesDipoles

Two interacting dipoles

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E =μ 0

4πμ1 ⋅μ2

r 3 − 3μ1 ⋅r( ) μ2 ⋅r( )

r5

⎡

⎣ ⎢ ⎤

⎦ ⎥

€

Hdip(r) =1

4π3

μ ⋅r( )r

r5 −μr 3

⎡

⎣ ⎢ ⎤

⎦ ⎥ Dipole field

Dipolar energy

€

EZ = −μ ⋅B

Torque

€

A(r) =μ 0

4πμ × r

r 3

⎛ ⎝ ⎜

⎞ ⎠ ⎟ Dipole vector potential

1

2

H12

H21

€

τ =×B

Zeeman energy

Energy of magnetized bodiesEnergy of magnetized bodies

This is to avoiddouble-counting

€

dE =12

μ 0

4πΜ(r) ⋅Μ(r')

r − r' 3 − 3Μ(r) ⋅ r − r'( )[ ] Μ(r') ⋅ r − r'( )[ ]

r − r' 5

⎧ ⎨ ⎩

⎫ ⎬ ⎭

drdr'

€

dμ = Μ(r)drd2

d1

€

E =12

μ 0

4πdrΜ(r) ⋅ dr'∫∫ Μ(r')

r − r' 3 − 3Μ(r') ⋅ r − r'( )[ ] r − r'( )

r − r' 5

⎧ ⎨ ⎩

⎫ ⎬ ⎭

Each dipole (magnetic moment) within a magnetized body interacts with each and every other. The sum of all that is the “self energy” of a magnetized body.

Recognize this?It’s the dipole field “density”.

€

Ed = −μ 0

2Μ(r) ⋅Hd (r)∫ dr

€

Hd (r) = hdip (r − r' )∫ dr' The demagnetization field

The demagnetization fieldThe demagnetization field

For spheres, ellipsoids, and a few other shapes, the demag field is uniform throughout the shape. In general, the demag field is highly non-uniform.

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A(r) =μ 0

4πΜ(r' )× (r − r' )

r − r' 3∫ dr'

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B(r) =∇× A(r) = μ 0 Μ(r)+Ηd (r)[ ]

= +

B M H

Demag field for uniformly magnetized objects Demag field for uniformly magnetized objects

Introducing the characteristic function D(r), with value 1 inside the object, and 0 outside, we disentangle shape effects and get a convenient expression for the demag field.

€

A(r) =μ 0M 0

4πm ×(r − r' )

r − r' 3∫ dr'

=μ 0M 0

4πm × D(r' )

r − r'

r − r' 3∫ dr'

=μ 0M 0

4πm × D(r)⊗

rr 3

⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

A(k) = −iμ 0M 0D(k)m × k

k2

B(k) = μ 0M 0D(k)k × m × k

k2

= μ 0M(k) −μ 0M 0D(k)m ⋅k

k2 k

Hd (r) = −M 0

8π 3 D(k)m ⋅k

k2 keik⋅r d∫ k Representation of the demag field for a uniformly magnetized tetrahedron

Demag energy and Demag energy and demag factorsdemag factors

€

Hd (r) = −M 0

8π 3 D(k)m ⋅k

k2 keik⋅r d∫ k = − ˆ N (r)M

€

Ed = −μ 0

2Μ⋅Hd (r)∫ dr =

μ 0

2M i

ˆ N ij (r)∫ dr[ ]M j

=12

μ 0M 02V N imi

2

i=x,y,z

∑ = KdV N xmx2 + N ymy

2 + N zmz2

( )

€

N i = ˆ N ii (r) =1V

ˆ N ii (r)∫ dr

€

H di (r) = − ˆ N ij (r)M j

Demag field as a result of a tensor operation on the magnetization

Demag factors

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ˆ N ij (r) = −1

8π 3 D(k)kik j

k2 eik⋅r d∫ k The demag tensor (a function of position)

The demag energy: a 2-form involving the three demag factors along main axes and the magnetization direction cosines

This is valid for any shape, provided its magnetization is uniform.

Nx

NyNz

Domain wallsDomain walls

Large dipolar energy, no exchange energy

Snaller dipolar energy, some exchange energy

Idem

Bloch walls: bulk, thick objects

Neel walls: thin films, thin objects

Cross-over between dipolar and domain wall energies for a sphere (idealized model)

Wall widthWall width

€

dE θ →θ + dθ( ) = JS2 dθ( )2

€

E = −2JS1 ⋅S2 = −2JS2 cosθ

€

E θ = 0 →θ =θDW( ) = NJS2 θDW

N

⎛ ⎝ ⎜

⎞ ⎠ ⎟2

= JS2 θDW2

N

€

εK θ = 0 →θ = π( ) =NKπ

sin2 θdθ =0

π

∫ δK2a

€

εK = K sin2 θ

€

σ DWπ = JS2 π 2

δa

€

σ Kπ =

δK2

€

min JS2 π 2

δa+

δK2

⎡

⎣ ⎢

⎤

⎦ ⎥→δ = πS

2JaK

= πAK

with A =2JS2

a

The strong commercial magnet NdFeB has K=4.3e6 J/m3, and A=7.3e-12 J/m. Estimate the domain wall width in this material.

The domain wall energy is proportional to the area

€

σ DWπ = π AK

Magnetocrystalline anisotropyMagnetocrystalline anisotropy

The crystal structure creates anisotropy: some directions are more responsive (“easier to magnetize”) to applied fields than others.

Consider a sphere of radius R magnetized along some easy axis u with anisotropy constant Ku=4.53e5 J/m3 (value for Co). If the magnetization flips to –u, the energy remains the same (up and down states are degenerate). But, in order to rotate from +u to –u, the magnetization has to go through a high energy state, i.e. when M points perpendicular to u. Suppose that the temperature is such that kBT is of the same order of the energy barrier separating the degenerate states. What happens?

€

εK = Ku sin2 θ + Ku2 sin4 θ

€

εK = K1 mx2my

2 + mx2mz

2 + my2mz

2( ) + K2mx

2my2mz

2

Uniaxial

Cubic

M

u

Stoner-WolfharthStoner-Wolfharth

€

E = KdV (N x cos2 θ M + N y sin2 θ M ) −μ 0M 0H cos(θ M −θ H )

€

ε =sin2 θ M − 2h cos(θ M −θ H )

€

h =1

(N y − N x )HM 0

x

y€

∂ε∂θM

= 0 →θ M (h,θ H )

MH

The direction of M at any given applied field

Single-domain hysteresis is a consequence of anisotropy (shape or magnetocrystalline).

Brown’s equationsBrown’s equations

€

Ed = −μ 0M 0

2m(r) ⋅Hd (r)∫ dr

=Kd

8π 3

m(k) ⋅k 2

k2∫ dk

€

EK = −Ku m(r) ⋅u[ ]∫ 2dr

€

Ex = A ∇mx( )2 + ∇my( )

2+ ∇mz( )

2

[ ]∫ d 3r

= −A m(r) ⋅∇2m(r)∫ d 3r

€

EZ = −μ 0M 0 m(r) ⋅Happ(r)∫ dr

The whole set of equations provides a full description of the energy landscape of a micromagnetic system (such as the one shown above) and drives its evolution towards the ground state of minimum energy

€

Μ(r) = M 0m(r)

Sneak peekSneak peek

Micromagnetic simulations

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∂m∂t

= −γ m × Heff +α m ×∂m∂t

LLG equation

Magnetodynamics and evolution

Searching for ground states

Wrapping upWrapping up

Next lecture: Tuesday February 22, 13:15, KU (A9)

Micromagnetism II (MB)

•Magnetic domains•Bloch and Neel walls, and wall widths•Dipolar/magnetostatic/demag energy•Demagnetization fields and factors•Stoner-Wolfharth hysteresis•Magnetocrystalline anisotropy•Brown’s equations

Please remember to:•Install OOMMF on your laptop•Familiarize a little bit with it•Bring your laptop to class on Tuesday, February 22