chapter 7 micromagnetism, domains and hysteresis...chapter 7 micromagnetism, domains and hysteresis...
TRANSCRIPT
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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