Magnetic Domains in Soft Ferromagnets
A ferromagnet consisting of a single domain creates a large external field,
which costs energy ( B2). This stray field is reduced by forming several
domains. It is like cutting up the magnet and letting the pieces orient themselves optimally (i.e., with North and South poles next to each other).
This occurs in pure and defect-free materials (soft magnets). An external
field is needed to magnetize them by aligning the domains (electromagnet).
Imaging Magnetic Domains with Iron Powder
Small iron particles are attracted to the high magnetic stray field at the
transition between domains. They scatter light in a dark field microscope.
The orientation of the domains is determined by adding an external field
and watching them grow or shrink.
Magnetic Domain Walls
 0.1 
m
Magnetic Hysteresis Curve: M(H)
M
Saturation
Magnetization
Ms
Remanence Mr
H
Coercivity Hc
Small Coercivity :
Soft Ferromagnets
Simple Magnetic Hysteresis in nm - Sized Particles and Layers
(Smaller than a Domain Wall  Single Domain)
Permanent Magnets
The strongest permanent magnets (hard magnets) are alloys of 3d transition
metals (Fe, Co) with rare earths. They exhibit the highest energy product B·H.
Examples are neodymium-iron-boron (Nd2Fe14B) and samarium-cobalt (SmCo5) .
Such magnets are used for electric motors and for generators in wind turbines,
both important ingredients in becoming more energy-efficient. A Prius contains
1 kg of neodymium. That has caused increasing demand for rare earths.
To prevent the formation
of domains in a permanent
magnet, one uses materials
with a high defect density.
Defects prevent domains
from forming and thereby
“pin” the magnetization.
Two neodymium-iron-boron
magnets that crunched a
fingertip.
B, H, and M in a Bar Magnet
M
The magnetization M is constant in a
permanent magnet (by definition).
The two other fields need to satisfy:
B = H + 4 M
inside
B=H
outside
H|| is continuous at a boundary
H
B is continuous at a boundary
The H-field inside the magnet is the
demagnetizing field Hd . It is related
to the stray field outside the magnet
by the continuity of H|| at the side of
the magnet. Since the outside field
lines bend 180o from the side to the
end and since B is continuous at the
end face, Hd opposes B inside. To construct the field lines, start with M,
then jump to B, and do H last.
B
Magnetostriction and Piezoelectricity
Change in B, E

Length Change
Magnetostriction has a large influence on magnetic nanostructures, such
as hard disk reading heads. Small structures tend to be strained by the
surrounding material, and the strain magnetizes them permanently. Their
magnetization cannot be switched anymore. In permalloy (Ni0.8Fe0.2) the
magnetostriction vanishes, which makes it a key material for spintronics.
Piezoelectricity makes it possible to position a STM tip with sub-atomic
precision. A piezoelectric ceramic changes its length proportional to an
applied voltage. A common design is a tube-scanner which can move in all
three directions (x,y,z).