Lecture
19
&
20:
Magnetic
Properties
Continue
Magnetic
Materials
Ferromagnetism:
Iron,
Nickel,
Cobalt
• Xm:
positive
and
large
• Strong
attraction
to
magnetic
field
• They
retain
their
magnetic
properties
after
the
magnetic
field
1
• Note:
It
is
microstructure
dependent
Atoms
have
parallel
aligned
magnetic
moments
The
strong
magnetic
properties
are
not
only
because
their
atoms
carry
a
magnetic
moment,
but
because
of
magnetic
domains.
In
magnetic
domains:
all
the
atomic
dipoles
are
coupled
together
in
a
preferred
direction
(a
large
number
of
atom’s
moments
(10!" → 10!" )
are
aligned
parallel.
If
there
is
no
external
field:
The
domains
are
randomly
organized
and
the
net
magnetic
field
is
zero.
2
Temperature
Dependence:
As
ferromagnetic
materials
are
heated,
the
thermal
agitation
of
the
atoms
takes
place
and
the
degree
of
alignment
of
atomic
magnetic
moments
decreases
→
The
saturation
magnetization
(Ms)
decreases
If
we
plot
Ms
vs.
temperature:
3
The
curie
temperature,
the
constant
e
are
very
close.
After
TC,
the
material
becomes
paramagnetic
and
obeys
Curie‐
Weiss
law.
TC
:
Ni: 358℃
Co: 1131℃
Fe: 770 ℃
Hysteresis
loop
of
a
ferromagnetic
material:
DRAWING
Mr:
Remanence
(remanent
magnetization)‐
If
applied
field
H
is
reduced
to
zero,
the
material
keeps
a
positive
magnetization
(Mr)
4
HC:
coercive
field‐
The
value
of
magnetic
field
strength
H
that
can
remove
the
remanent
magnetization.
Hard
magnetic
materials:
The
area
inside
the
loop
is
large
(large
Mr
&
large
HC)
Soft
Magnetic
Materials:
Small
area
(small
Mr
&
small
HC)
5
‐pizomagnetization:
which
means
that
magnetization
of
ferromagnetic
materials
is
stress
dependent.
Look
to
Ni
as
an
example
Magnetostriction:
A
change
in
dimensions
when
a
ferromagnetic
material
is
exposed
to
magnetic
field
Antiferromagnetism
Small
and
positive
Xm
6
Spontaneous
alignment
of
moments
below
a
critical
temperature
similar
to
ferromagnetic
materials
(but
align
in
antiparallel
directions)
and
the
magnetic
field
cancel
out
and
the
materials
seem
to
behave
as
a
paramagnetic
material
Atoms
have
parallel
and
antiparallel
aligned
magnetic
moments
7
Temperature
Dependence:
Antiferromagnetic
materials:
Cr
&
insulators
and
semiconductors
8
Ferrimagnetism:
A
complex
form
of
magnetic
ordering
due
to
the
crystal
structure.
Large
and
positive
Xm
They
behave
similar
to
ferromagnetics:
spontaneous
magnetization,
TC,
hysteresis
and
remanence.
It
occurs
only
in
compounds,
which
have
complex
crystal
structure.
The
structure
is
composed
of
two
magnetic
sublattices
separated
by
anions
(oxygen).
Each
sublattice
contains
ions
whose
spins
are
aligned
parallel
to
each
other,
but
each
sublattice
contains
different
number
of
ions.
→
Some
magnetic
moments
do
not
cancel
Ferrimagnetic
materials
are
very
important
in
applications
because
they
are
poor
electrical
conductors
(ceramic
materials)
and
a
large
resistivity
is
required
for
high
frequency
applications
to
avoid
eddy
currents.
Origin
of
magnetization
Diamagnetism
(classical
theory):
The
orbital
motion
of
an
electron
induces
a
magnetic
moment
𝜇! .
Compare
it
with
magnetic
moment
created
by
a
current
in
a
loop
9
μ! = I ∙ A where I = current, A = Area
e
= A where e = electron charge, t = time
t
e
=
A v = electron velocity
S
v
eν πr !
=
S = 2πr , r = radius of orbit
2πr
𝑒𝜈𝑟
2
The
external
magnetic
field
induces
an
electric
field
E‐
The
force
exerting
on
the
electron
eE
eE=ma
dr
m = eE
dt
dv eE
= (eq. 1)
dt
m
V
induced voltage
E = L
orbit length
But
V
is
related
to
the
flux
by
Lenz’s
law:
𝑑𝜙
𝑉!
𝑑𝑡
=
10
𝜙 = 𝐵𝐴
dϕ
𝑑𝐴
= μ! HA ⇒ = μ! 𝐴 dt
𝑑𝑡
in
eq
1:
𝑑𝜈 𝑒 𝑑𝜙
=
𝑑𝑡 𝐿𝑚 𝑑𝑡
=
𝑒
𝑑𝐻
𝜇! 𝐴
𝐴 = 𝜋𝑟 ! , 𝐿 = 2𝜋𝑟
𝐿𝑚
𝑑𝑡
𝑑v 𝑒𝜇! 𝜋𝑟 ! 𝑑𝐻
=
𝑑𝑡
2𝜋𝑟 𝑚 𝑑𝑡
𝑑v 𝑒𝜇! 𝑑𝐻
=
𝑑𝑡 2𝑚 𝑑𝑡
!!
!!
𝑒𝑟𝜇!
𝑑v =
2𝑚
Δv =
!
𝑑𝐻
!
𝑒𝑟𝜇! 𝐻
2𝑚
Δ:
Change
in
velocity
due
to
magnetic
field
!"#
𝜇! = !
Derived
early
in
lecture
∴ Δ𝜇! =
11
𝑒𝑟Δv
2
Δ𝜇!
𝑒 ! 𝑟 ! 𝜇! 𝐻
=
4𝑚
In
our
derivation,
we
assumed
that
magnetic
field
is
perpendicular
to
the
plane
of
the
orbiting
electron.
However,
the
orbit
plane
direction
changes
and
its
direction
with
the
magnet
goes
from
parallel
to
perpendicular
if
we
take
the
average
of
Δ𝜇! we
will
find
that
𝑒 ! 𝑟 ! 𝜇! 𝐻
Δ𝜇! =
6𝑚
This
is
the
Langevin
Theory
that
explains
diamagnetism.
𝜇!
𝜇
𝑀=
𝑋 = v
𝐻
Paramagnetism:
The
theory
can
explain
paramagnetism
due
to
orbital
motion.
M=A
1
𝑋! = 𝐶 𝑇
⇒ function of temperature
I’ll
skip
the
derivation
of
Xpara
due
to
orbit
motion
based
on
Langevin
Theory
12
Quantum
Mechanical
Theory:
Magnetic
moment
of
the
spinning
electrons
are
the
dominant
contribution.
Any
state
may
be
occupied
by
2
electrons
If
we
apply
magnetic
field,
some
electrons
tend
to
change
their
directions.
This
can
only
happen
if
their
energy
is
higher
than
Ef
It
can
be
seen
that:
Δ𝐸 = 𝜇! 𝐻𝜇!" 𝜇!" : magnetic moment due to electron spin
Number
of
electrons
that
change
direction:
Δ𝑁 = Δ𝐸 𝑧 𝐸! z E! density of states
𝜇
The
magnetization
𝑀 = 𝑚 𝑉
μ!
M=
Δ𝑁 where V = volume
V
μ!
=
Δ𝐸 𝑧(𝐸! )
V
13
𝑀
𝜒= 𝐻
!
M= ! μ! 𝐻𝜇!" 𝑧(𝐸! )
!
μ!" ! μ! 𝐻 𝑧(𝐸! )
=
𝑉
𝑀 μ! ! μ! 𝑧(𝐸! )
Χ= =
𝐻
𝑉
Important:
only
electrons
close
to
Fermi
energy
are
capable
to
aligning
in
the
magnetic
field
direction.
Χ
calculated
from
the
above
equation
agrees
very
well
with
experiments.
Note:
not
all
valence
electrons
contribute.
Let’s
look
to
few
elements:
Be
(1s22s2):
However
in
crystal,
bands
overlap
2s
&
2p
electrons
populate
the
bottom
of
the
band,
density
of
states
at
EF
is
small
Xpara
is
small
Look
to
the
next
figure
14
Be→ diamagnetic
Cu:
Fermi
energy
is
close
to
the
band
edge,
Z(EF)
is
small,
Xpara
(due
to
spin)
is
small
Cu→ diamagnetic
Intrinsic
semiconductors:
Density
of
states
at
the
top
of
valence
band
is
zero.
Xpara
(due
to
spin)
is
zero
→ diamagnetic
High
doped
extrinsic
semiconductors:
15
There
are
number
of
electrons
in
the
conduction
band
at
high
temperature
Xpara
is
not
zero,
there
are
small
contributions
Dependence
of
Temperature:
According
to
that,
there
is
no
dependence
on
temperature
for
diamagnetic
or
paramagnetic
𝑒 ! 𝑍𝑟 ! 𝜇!
𝑋!"#$ =
6𝑚𝑉
𝜇!" ! 𝜇! 𝑧(𝐸! )
𝑋!"#" =
𝑉
In
fact,
most
paramagnetic
materials
do
not
obey
the
Curie‐Weiss
law.
Dilute
gas
and
rare
earth
metals
obey
because
of
the
contribution
of
magnetic
moment
of
the
orbiting
electrons.
Ferromagnetic
Materials:
Fe,
Co,
Ni
have
unified
d‐bands
The
d‐bands
overlap
the
next
higher
s‐band.
16
You
can
see
that
density
of
states
is
very
high
at
ferri
energy
for
Ni,
Co,
and
Fe
(Compare
it
with
Cu)
17
Thus,
for
these
elements,
a
small
amount
of
energy
can
transfer
a
large
number
of
electrons
to
spin
up
configuration.
In
paramagnetic
materials:
Applied
magnetic
field
is
needed
to
give
an
external
energy
to
achieve
spin
alignment.
In
ferromagnetic
materials:
There
is
no
need
for
external
energy
(because
of
exchange
energy)
In
the
next
four
slides,
I’ll
give
you
some
explanation
for
the
role
of
exchange
energy
in
spontaneous
alignment
of
adjacent
spins.
However,
this
part
is
not
required
(just
for
your
information).
The
exchange
energy
causes
spins
to
align
parallel
to
each
other.
(Transfers
electrons
to
high‐energy
state)
→
Leads
to
spontaneous
alignment
and
formation
of
magnetic
domains
(1‐100𝜇m).
In
ferromagnetic
materials,
the
spins
of
unfilled
d‐bands
spontaneously
align
parallel
to
each
other
within
small
domains.
18
Exchange
energy:
quantum
mechanical
energy
similar
to
the
pendulum
oscillates
𝜔! , 𝜔! 𝜔! − 𝜔!
2
However,
why
are
the
spontaneous
divisions
into
many
individual
domains?
I
mean
why
the
magnetic
domain
does
not
extend
over
the
whole
crystal?
Because
of
magnestatic
energy:
To
have
lower
energy,
antiparallel
spins
are
supported.
The
magnestatic
energy
will
be
halved
if
the
crystal
contains
two
domains
that
are
magnetized
in
opposite
directions.
19
Bloch
Wall:
is
the
region
between
individual
domains
in
which
the
spin
rotates
from
one
direction
to
another.
20