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Chapter 7

Micromagnetism, domains and hysteresis

7.1 Micromagnetic energy

7.2 Domain theory

7.5 Reversal, pinning and nucleation

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The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to amagnetic field . It reflects the arrangement of the magnetization in ferromagnetic domains.The magnet cannot be in thermodynamic equilibrium anywhere around the open part ofthe curve! M and H have the same units (A m-1).

coercivity

spontaneous magnetization

remanence

major loop

virgin curveinitial susceptibility

The hysteresis loop

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Domains form to minimize the dipolar energy Ed

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Magnetostatics

Poisson’s equarion

Volume charge

Boundary condition

1. solid

2. air

M( r) ! H( r) BUT H( r) ! M( r)

Experimental information about the domain structure comes from observations at the surface.The interior is inscruatble.

en

M

+

++

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7.1 Micromagnetic energy

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1.1 Exchange

eM = M( r)/Ms (",#)

A = kTC/2a

A = 2JS2Zc/a0

A ~ 10 pJ m-1

Lex ~ 2 - 3 nm

Exchange energy of vortex

$Eex = JS2ln (R/a)

Exchange length

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EK = K1sin2" Bulk K1 ~ 102 - 107 J m-3

Surface Ksa ~ 0.1 - 1 mJ m-2.

Interface Kea ~ 1 mJ m-2.

Exchange and anisotropy governthe width of the domain wall.

1.2 Anisotropy

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Demagnetizing field governs the formation of the wall

(integral over all space) and B = µ0(H + M)

Hd is determined by the volume and surface charge distributions %.M and en.M

&m = qm/4'r; %2 &m= -(m H = - %&m

1.3 Demagnetizing field

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Magnetoelastic strain tensor

For isotropic material, uniaxial stress

Induced uniaxial anisotropy

1.4 Stress

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Local stresses can be created by the magnetostriction of the ferromagnet itself:

Usually this term is small < 1 kj m-3 , but it can influence the formation of closure domains.

Elastic tensor

Magnetostrictive stress

Deviation due to magnetostriction

1.5 Magnetosriction

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A guide to how nature minimizes the micromagnetic free energy is the charge avoidance principle.

Avoid forming bulk or surface chage, and keep charge of like sign as far apart as possible

e.g Keep magnetization parallel to the surface, wherever possible.

1.6 Charge Avoidance

Toroid

Picture frame

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Brown’s Micromagnetic equations

General statement of the micromagnetic problem:

No torque on the magnetization at any point.

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7.1 Domain Theory

A ~ 10-11 J m-1

K1 ~ 10 5 J m-3

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2.1 Bloch Wall

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2.1 Néel Wall

Neel walls form in thin films of soft materialthinner than ~ 6 nm

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2.3 Magnetization processes

There are two magnetization processes for a ferromagnet:

1) Domain-wall motion

2) Magnetization rotation

If the domain walls are perfectly free to move, they will do sountil H =0; H’ = 1/N

H’

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7.3 Nucleation, reversal and pinning

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Brown’s paradox

Brown’s theorem; for ahomogeneous, uniformly-magnetized ellipse

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A very small particle will be single-domain. Larger particles form domain walls to reduce demagnetizing energy

Single-domain particle size:

Cost of making two 90 degree walls is 2!R2(AK)1/2 should offset thegain in demagnetizing energy -(1/2)NMs

2

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3.2 The Stoner Wohlfarth model Assume coherent rotation of the magnetization. H makesan angle ) with the axis of the particle.

NB R < Rsd does not guarantee coherent rotation.

When ) = 0, Hc=2Ku/µ0Ms

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The energy landscape of a Stoner Wohlfarth particle

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Hc = 0.479

Mr = 0.5

Area = 0.99

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Preisach Model

Model hysteresis loops with a distribution of elementary square loops.

These are known as ‘hysterons’

M

H

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Other reversal modes

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3.3 Reversal in thin films and small elements

The Stoner Wohlfarth asteroid.Locus of points where a bifurcation of energy occursSwitching occurs on the surface, never within it.

Take components of H along easy and hard directions, andnormalize them by the anisotropy field 2Ku/Ms

Consider a thin film as a 2D S-W ‘particle’. The reversalis assumed to be coherent

dEtot/d"=0

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3.4 The two-hemisphere model

A sphere made up of two halves with different anisotropy K) and K*

Exchange + dipole interactions Anisotropy + Zeeman interactions

If K1 = K) and K* = 0, Independent reveral of the soft hemisphere occurs when H ≈ (1/8) Ms

Except if R < lex, when the soft hemisphere cannot reverse independently.

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Exchange stiffening operates on a length scale of up to ≈ 4lex ≈ 10 nm.

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3.4 Switching dynamics;

Torque on a magnetic moment in a field causes precessionat the Larmor precession frequency

i.e. 28 GHz/T when g=2 and + = -e/m

In the presence of unixial anisotropy:

Gilbert damping term

H

M

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3.5 Domain wall pinning

Domain wall velocity.

Barkhausen jumps

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3.6 Real hysteresis loops

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Kronmuller Equation

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Approach to saturation

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Time Dependence

Magnetic Viscosity M = M0 - S ln t

The hysteresis loop shows the irreversible, nonlinear response of a ferromagnet to amagnetic field . It reflects the arrangement of the magnetization in ferromagnetic domains.The magnet cannot be in thermodynamic equilibrium anywhere around the open part ofthe curve! M and H have the same units (A m-1).

coercivity

spontaneous magnetization

remanence

major loop

virgin curveinitial susceptibility