Download 1 Basics of magnetic materials Definitions in SI

Basics of magnetic materials
We will briefly review magnetism and magnetic materials as
necessary background for understanding magnetoelectronics,
particularly the data storage industry.
• Definitions in SI
• Types of magnetic materials
• Origins of magnetic properties
• Ferromagnetism
• Domains
• Hysteresis loops
Definitions in SI
Units are the worst part of dealing with magnetic systems!
B:
• magnetic induction (usually what physicists mean by
magnetic field).
• most physically significant quantity - what shows up in
Lorentz force law, in determining NMR frequencies, etc.
• Unit is the Tesla. Earth’s magnetic field = 6 x 10-5 T.
• More convenient cgs unit is the Gauss = 10-4 T.
• Important boundary condition:
∇ ⋅B = 0
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Definitions in SI
H:
• the magnetic field.
• Caused by currents of free charge.
• Unit is the Amp/m.
• Important relation:
∇×H = J
• With no magnetic materials around,
“permeability of free
space” = 4π x 10-7 Tm/A
B = µ0H
M:
• the magnetization.
• magnetic moment per unit volume of a material.
• Unit is the Amp/m.
• For a material with a permanent magnetization M0,
M = χH + M 0
magnetic susceptibility
Susceptibility and permeability
Combining effects of external currents and material response,
B = µ 0 (H + M)
We define the permeability by:
B = µH
So, for a material with a magnetic susceptibility χ,
B = µH = µ 0 (1 + χ )H
→ µ = µ 0 (1 + χ )
Note that for real materials χ and µ are tensorial.
Relative permeability is defined as µr = µ/µ0.
→ B = µ0µr H
Susceptibility is most useful when discussing diamagnetic (χ < 0) and
paramagnetic (χ > 0) materials, rather than systems with nonzero M0.
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Magnetic dipoles and magnetization
m
A magnetic dipole can be modeled as a
loop of current: m = IA n
Units: Am2 = J/T
eh
Convenient
= 9.27 × 10 − 24 J/T
µB ≡
number:
2m
A group of identical dipoles in a
plane can be replaced by one big
dipole with the same current
circulating around the perimeter.
A 3d stack of dipoles can be
replaced by thinking about a sheet
current running around the
perimeter.
Forces, torques, and fields
B
Torque on a magnetic dipole:
τ = m×B
m
Force on a magnetic dipole:
F = m ⋅ (∇B )
Energy of magnetic dipole:
U = −m ⋅ B
What is B at the surface of a long
uniformly magnetized body with
magnetization M in the absence of
other fields?
Answer: just µ0M, since normal
component of B must be continuous.
M
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Field from a dipole
B (r ) = µ 0
3(m ⋅ r )r − r 2m
4πr 5
Note that B(r) ~ r -3
Current loop only
approximates a dipole
field in far field limit (r
>> loop radius).
Image from Purcell.
Diamagnetism
Some materials develop a magnetization that is antialigned
with the applied external field H.
Such materials are diamagnetic, and have χ < 0.
Simple classical picture for diamagnetism: Lenz’s Law
Try ramping up B = µ0H.
B
m
Result is a circumferential electric
field that opposes the direction of
the current in the loop.
This would act to reduce the dipole
moment along H, and would be
diamagnetic.
Correct quantum treatment involves 2nd order perturbation
theory - can end up with either sign, depending on particulars of
atoms. Larmor diamagnetism or Van Vleck paramagnetism.
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Paramagnetism
Also common is paramagnetism, when χ > 0.
Two common origins of paramagnetism:
• Curie paramagnetism - localized moments free to flip.
• Pauli paramagnetism - requires “free” electrons in a metal.
Start w/ Curie case, spin J and gyromagnetic ratio g, in
magnetic flux density B.
1
k BT
Do statistical physics here. Alignment of spin with external
field lowers spin energy.
g ≡ (m / J ) / µ B
β≡
Can write down partition function and solve to find
equilibrium magnetization.
Curie paramagnetism
Result:
M = nµ B gJB J ( βµ B gJB)
2J +1
1
 x 
 2J +1 
x −
coth 
coth  
2J
 2J
 2J
 2J 
Bj is called the Brillouin function.
B J ( x) ≡
As long as µBB << kBT, can expand the above to get:
M ≈ ng 2 (µ B ) 2
µ 0 H J ( J + 1)
k BT
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where we’ve also assumed that M’s contribution to B is small,
valid for dilute systems of spins.
We see that Curie paramagnets have susceptibilities like:
χ ≈ ng 2 ( µ B ) 2
µ 0 J ( J + 1)
k BT
3
~
1
T
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Pauli paramagnetism
E
Starting from unpolarized electrons, applying a field B shifts the
Fermi level for spin-up and spin-down electrons oppositely!
ν (ε )
f (ε − µ B B )
f (ε + µ B B) N down = V ∫ dε
2
2
Taylor expanding to find the difference,
N up = V ∫ dε
M =
ν (ε )
µB
V
( N up − N down ) ≈ µ B2 ⋅υ ( E F ) µ 0 H
Only good for metals - can’t have gap at Fermi surface.
Paramagnetic and diamagnetic plots
MH
Curie paramagnet
Pauli or Van
Vleck paramagnet
H
Larmor diamagnet
gµ B µ 0 H
~1
k BT
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Some comments
• Most insulators are weakly diamagnetic.
• Metals can be either.
• Susceptibilities are usually quoted per molar volume rather
than in their dimensionless form, for experimental reasons.
• Strictly speaking, susceptibilities and permeabilities are
defined as derivatives.
The literature on all this stuff can be terribly confusing!
People use H even when there are no “free currents” around,
and often use a mishmash of SI and cgs units.
Some numbers
Image from Marder.
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Ferromagnetism - toy model
Start from Curie paramagnetism picture - local moments
(spins), but now allow them to respond to the local magnetic
field at their position.
Assume M = χ0 (H + Hm)
“molecular field”, = ηM
M (1 − ηχ 0 ) = χ 0 H
χ eff =
At high temperatures,
χ0
(1 − ηχ 0 )
Recall that χ0 ~ 1/T, so we find at high temperatures
χ eff ~
1
T − Tc
Curie-Weiss law
At Tc, the Curie temperature, the susceptibility diverges!
Spontaneous magnetization = ferromagnetism.
Ferromagnetism
• Preferred direction of M can
depend on crystallographic
properties of material and material
shape, stresses.
FM occurs because of the same
sort of exchange interaction that
leads to Hund’s rules.
Effective spin-spin interaction
because the spin-ordered state
tends to reduce total Coulomb
energy of system.
Result: In metals, different spin
directions have different
densities of states at the Fermi
energy.
Image from Coey, Trinity College, Dublin.
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Other types of magnetic order
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antiferromagnet
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canted antiferromagnet
canted ferromagnet
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ferrimagnet
Ferromagnetism and equilibrium configurations
One contribution (the total dipole moment of the system
interacting with the applied external field) to a system’s energy
will be decreased if M lies along H.
However, the field produced by the ferromagnet itself contributes
to the total energy.
Competition between “external field” contribution and self-field
contributions leads to complicated behavior.
As a result, ferromagnetic properties are often characterized by
sweeping H along a particular direction, and measuring M.
What is seen is hysteresis - the dynamics of M at a given applied
field H depend on the direction (history) of M.
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Ferromagnetism: M vs H hysteresis
MH
slope = max. χ
Ms saturation
magnetization
Mr remanent
magnetization
slope = initial χ
H
Hc coercive field
“Hard” ferromagnetic materials: large values of Hc, large Mr.
“Soft” ferromagnetic materials: small values of Hc, small Mr.
Crystalline anisotropy
Because of band structure origins of FM, there can be certain
crystallographic directions along which it’s energetically
favorable for M to lie.
Define m as the unit vector pointing along M; u as energy
density due to anisotropy.
Most common cases:
• cubic anisotropy
u = K c1 (m12 m22 + m12 m32 + m22 m32 ) + K c 2 (m12 m22 m32 )
• uniaxial anisotropy
u = K u1 sin 2 θ + K u 2 sin 4 θ
where θ is angle between m and
anisotropy axis.
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“Stray field” energies
Energy bookkeeping for FM calculations requires keeping track of
energy cost of the field produced by the FM itself.
Remember
∇ ⋅ B = µ 0∇ ⋅ (H + M ) = 0
Therefore, can write
∇ ⋅ H d = −∇ ⋅ M
Looks like magnetic
charge density.
Turns out that the energy associated with
these stray fields can be written as:
Ud =
1
1
µ 0 ∫ H 2d dV = − µ 0 ∫ H d ⋅ MdV
2 allspace
2 sample
For a particular sample geometry, one can compute − ∇ ⋅ M
to find a fictitious magnetic charge density. Then
one can find Hd everywhere, and compute the
energy contribution from these stray fields.
Demagnetizing factors and fields
The resulting Hd is often called a
demagnetizing field.
Generally one can relate Hd to the
magnetization M by the
demagnetizing factor tensor:
ˆ ⋅M
H d = −N
One can calculate N for given
sample geometries.
Special case: for ellipsoids of
revolution, Hd || M.
Image from Bertram, Th. of Mag. Recording
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Demagnetizing factors and fields
Why the name?
Inside a magnetized object, Hd opposes Hext.
Total magnetic field inside sample is Htot = Hext - Hd.
N
S
S S
N
N
S
N
S
N
Hext
Geometric or shape anisotropy
The demagnetizing field energy contribution leads to
“geometric” or “shape” anisotropy: it costs less, energetically,
to have the magnetization pointed along certain directions
because that minimizes the stray fields.
Example: uniformly magnetized infinite (x-y) plate.
NNNNNNNNNNNNNNNNNNNNNNNNNN
angle btw M and z-axis
SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS
Effective surface magnetic charge density ~ Msat cos θ
µ0
M s2 cos 2 θ
2
So, we find it energetically unfavorable for a thin plate to have
its magnetization lying perpendicular to its surface.
Resulting Hd = - Msat cos θ;
ud =
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Other energetic considerations
• FM depends strongly on the crystal structure of the
materials in question.
• Particularly important in insulators, where exchange
interaction between local moments is not mediated by
conduction electrons.
• Result: strong coupling between deformation and
magnetization: magnetostriction.
• Can be used intentionally: built-in stresses can be used to
“force” easy directions to lie where desired, or to “pin”
magnetization.
Domains
So, if Tc for iron is 1073 K, and its µ0Ms = 0.17 T, why don’t
we see massive magnetic effects from iron bars all the time?
• FMs can lower their total energy by spontaneously
breaking up their magnetization into domains.
• Total energy must include contribution from stray fields
extending all over all space.
• Domains can greatly reduce these stray fields:
NNNNN
SSSSS
closure domains
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Domain walls
Boundary between two domains is called a domain wall.
• Because of the FM exchange interaction, it’s very energetically
costly for the direction of M to change very sharply.
• Result is that local magnetization spreads out the change over
some distance = domain wall thickness.
• Two types of domain walls:
Bloch wall
Neel wall
Domain wall thickness
What sets domain wall thickness (often called “width”)?
Competition between exchange interaction (wants to keep nearest
neighbor spins aligned) and crystalline (or shape) anisotropy
(wants to keep M oriented only along “easy” directions).
Exchange energy for pair of spins: U = (2 JS 2 ) − 2 JS1 ⋅ S 2
Energy is minimized ( = 0) when neighboring spins are aligned.
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For small change in angle φ between neighboring spins, U ≈ JS φ
For a total rotation of π spread out over a chain of N+1 spins,
the total energy cost is U ≈ NJS 2 (π / N ) 2
For lattice spacing a, energy density per domain wall area for N
atom thick wall ~ π 2 JS 2 /( Na 2 )
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Domain wall thickness
If anisotropy energy per unit volume is given by
K sin 2 θ
where θ is angle between M and easy axis, then averaging over
the thickness of an N atom thick domain wall gives an average
anisotropy energy density of KNa
Total energy density is then π 2 JS 2 /( Na 2 ) + KNa
Can find Nm that minimizes this, and wall thickness = Nma.
N m ≈ π 2 JS 2 /( Ka 3 )
W ≈ π 2 JS 2 /( Ka)
Plugging in numbers for iron, for example, gives W ~ 100 nm.
Note that it’s easy to make structures on size scales comparable
to that of domain wall thicknesses….
Domains and hysteresis
What happens to domains when an external field is swept?
Domains continuously
rearrange themselves to
minimize the total energy of
the whole system.
Relative sizes of different
domains can change (at
right).
Individual domains can also
reorient their magnetizations.
Domain walls cannot always
move freely!
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Domain wall motion
Often domain walls can be “pinned” by disorder, grain boundaries, etc.
At large enough fields, pinning energy can be overcome.
Also, kBT can push domain walls “over barriers”.
Effect of pinning: Barkhausen noise.
MH
H
Next time:
Nanoscale issues in magnetic materials
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