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Topics in Magnetism
III. Hysteresis and Domains
Anne Reilly
Department of Physics
College of William and Mary
After reviewing this lecture, you should be familiar with:
1. General features of ferromagnetic hysteresis curves
2. Affects of anisotropy
3. Affects of domains
Material from this lecture is taken from Physics of Magnetism by Chikazumi, Chapters 15 - 18
In ferromagnetic materials, exchange interaction leads to an alignment
of atomic spins. When a magnetic field is applied, these spins are
reoriented, leading to hysteresis.
H
M
H
H
M=magnetization along direction of H
Features of Hysteresis Curve:
Saturation magnetization (Ms)
M
Remnant magnetization (Mr)
H
Coercivity (Hc)
M=magnetization along direction of H
What determines shape of hysteresis loop?
1. Coherent rotation determined mainly by Anisotropy
2. Domain formation and domain wall motion
Important principle:
Magnetization will lie in direction which is an energy minimum
Consider a simple example:
H
f
M
q
“easy axis”
(Stoner-Wohlfarth model)
Simple example:
H
f
M
q
“easy axis”
U  MH cos(f  q )  K1 sin (q )
2
Zeeman energy
Find M (q) by condition:
U
0
q
Uniaxial anisotropy
See: http://www.student.uni-kl.de/~mewes/magnet.e.html
Coherent rotation of magnetization considering only uniaxial anisotropy::
f=00 (along easy axis)
M
f=900 (along hard axis)
M
H
For 00: Hc=2K1/Ms
Note: Hysteresis shown above is the component
of M in the direction of H
H
H
M
Magnetic Anisotropy
• Anisotropy: preferred (easy axes) and unfavorable (hard axes) directions
of magnetization
• Due to coupling of electronic spins to electronic charge density
For this rotation, as long as spins remain parallel, exchange energy does
not change, but dipolar and LS coupling energy will change.
Magnetic Anisotropy
• Anisotropy: preferred (easy axes) and unfavorable (hard axes) directions
of magnetization
• Due to coupling of electronic spins to electronic charge density
Example: hcp Co
M
c-axis (hard)
easy
hard
(easy)
8000
H (G)
Magnetic Anisotropy
Two major types of anisotropy, written in terms of empirical anisotropy
coefficients:
Uniaxial:
U A  K1 sin 2 q  K2 sin 4 q
Cubic:
U A  K1 (1222  1232  2232 )  K2122232
(e.g., Co)
(e.g., Fe, Ni)
Note: cubic lattices can have several easy and hard axes
Domains
In ferromagnetic materials, exchange interaction leads
to an alignment of atomic spins
However, this leads to a large external and dipolar magnetic fields
which will tend to demagnetize the material. Domains are formed
to minimize this effect.
Domain wall
From http://www.aacg.bham.ac.uk/magnetic_materials
Domains
Domain size and wall size determined by energy cost,
dependent on material and geometry.
Ni thin film
Domain Walls
Energy is minimized by having a wall of finite width
N spins
Energy cost (exchange)
Energy cost (exchange + anisotropy)
N  
U   JS 2    KNa
a N
2
U   JS
(per unit area)
2
2
K = anisotropy constant
a = lattice constant
Domain Walls
Energy is minimized by having a wall of finite width
N spins over d
U
JS
0 N 
3
N
Ka
d  Na
2
For iron, J=2.16x10-21, S=1, K=4.2x104 and a=2.86x10-10
d=42 nm (150 lattice constants)
domain size will depend on sample geometry (see Chikazumi, Chp. 16)
Domain Walls
• Domains have different shapes and orientations
• Two examples of thin film domain walls:
Neel wall (rotation in plane)
Bloch wall (rotation out of plane)
Domains and Hysteresis
Domain formation and domain wall motion affects the
shape of hysteresis loop:
M
H
H
H
Domains and Hysteresis
Barkhausen noise: Tiny steps of domain walls
M
H
Domains and Hysteresis
Domain walls move across energy landscape
(determined by film morphology)
irreversible motion
Uw
reversible motion
x
Domains and Hysteresis
Coercivity can be increased over that for single domain system
because domain walls can become pinned (hard to move).
Pinning on lattice defects (dislocations, voids, etc.) , impurities.
Walls move between pinning points.
Defects and stress in thin film can increase number of
pinning sites and thus coercivity.