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Magnetism I
Simon Greaves1
1Research Institute of Electrical CommunicationTohoku University, Japan
4/2019
1 / 31
Magnetism I
Units and origins of magnetism
Types of magnetism
Exchange and anisotropy
Magnetostatics
Domains and domain walls
2 / 31
Units and origins of
magnetism
3 / 31
Units
Convention in data storage technologies has been to use units such as
ergs and cm, as opposed to SI units.
Parameter Customary units SI units Other units
Magnetic field, H Oe A/m Gauss, Tesla
Saturation
magnetisation, Ms emu/cm3 A/m
Uniaxial anisotropy, K erg/cm3 J/m3
Exchange stiffness, A erg/cm or erg/cm2 J/m or J/m2
1 Oersted = 79.58 A/m (1000 / 4π)
1 Oersted = 1 Gauss = 0.0001 Tesla
1 Joule = 107 erg
1 emu/cm3 = 1 kA/m
1 emu = 1 erg/G
4 / 31
Orbital angular momentum
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
r
Electroncharge = e
ω
Magnetic moment M = IS
S = πr2, I = −eω/2π
M = −eωr2/2
Angular momentum L = n~ = mωr2
M = −en~/2m = −eP/2m = nµB
µB = Bohr magneton, P = n~
Suppose an electron makes a circular orbit around an atomic
nucleus. The moving charge generates a magnetic field
perpendicular to the plane of the orbit, similar to a current flowing in
a coil.
5 / 31
The Stern-Gerlach experiment
Silver atoms are emitted from a hot oven and split into two beams.
Silver has a single outer electron with zero orbital angular
momentum. The spin of the outer electron is responsible for the
splitting.
6 / 31
Spin angular momentum
A measurement of S along any axis
will give S = ±~/2.
M = −eP/m, with P = ±~/2.
In general M = −geP/2m, where g
is the Landé g factor.
g = 2 for spin angular momentum
g = 1 for orbital angular momentum
Sub-atomic particles, such as electrons, neutrons and protons
have an intrinsic property known as spin.
For electrons the spin angular momentum S is
S =√
s(s + 1)~ =√
3~/2 , where s=1/2.
7 / 31
The gyromagnetic ratio
The gyromagnetic ratio is given by
γ =ge
2m
The electron gyromagnetic ratio is γ = 1.761×1011 rad/s·T.
The g factor for an electron is slightly more than 2, being
2.0023193... This value has been measured; the increase from the
expected value of 2 can be explained by quantum field theory.
In magnetic materials the magnetic moment can originate from
spin and/or orbital magnetic moments. g factors can be measured
by ferromagnetic resonance experiments.
8 / 31
Diamagnetism
Magnetic materials are classified according to their response to a
magnetic field.
We measure and plot the magnetisation M in the direction of the
applied field H . The plot is called a M-H loop, or M-H curve.
-10 -5 0 5 10Applied field (kA/m)
-0.2
-0.1
0
0.1
0.2
Mag
net
isat
ion (
A/m
)
∆H
∆M
∆M / ∆H = χ
The M-H curve of a diamagnet has
a negative slope. The susceptibility
χ = ∆M/∆H is very small, typically
-10−5.
Materials such as glass are
diamagnets. The magnetic field has
a similar effect to changing the
magnetic flux in a coil, where a
back emf is induced (Lenz’s law).
9 / 31
Paramagnetism
-10 -5 0 5 10Applied field (kA/m)
-0.2
-0.1
0
0.1
0.2
Mag
net
isat
ion (
A/m
)
Paramagnetism is a weak
magnetism found in materials with
non-interacting, or weakly
interacting magnetic atoms.
Susceptibilities typically range from
10−5 to 10−2.
Examples of paramagnets are
ferrofluids, dilute alloys etc.
At low fields the M-H curve of a paramagnet appears to be linear
and proportional to H.
Saturation can only be achieved using very high fields, or by
reducing the temperature.
10 / 31
Langevin function
0 20 40 60 80 100α ( = µH / kT )
0
0.2
0.4
0.6
0.8
1
L(α
)
The magnetisation of a paramagnet
is described by the Langevin
function M(H) = coth(α)− 1/αwhere α = µH/kT .
µ represents the magnetisation of
an atom or particle.
k is Boltzmann’s constant and T is
the temperature.
If µ is very small compared to the thermal energy kT a very large
field is required to saturate the paramagnet.
If T is reduced, saturation occurs at lower fields.
11 / 31
Ferromagnetism
-20 -15 -10 -5 0 5 10 15 20Applied field, H
z (kOe)
-600
-400
-200
0
200
400
600
Mz (
emu/c
m3)
Ms
Hs
Mr
Hc
The M-H loop of a ferromagnet is
characterised by various
parameters.
Coercivity, Hc
Saturation field, Hs
Saturation magnetisation, Ms
Remanent magnetisation, Mr
Ferromagnets are materials which have high magnetisation, even
in zero field. Examples are Fe, Ni and Co.
A field exceeding Hc must be applied to reverse the magnetisation
of a ferromagnet. Materials with high coercivity can be used to
store information.12 / 31
Weiss explanation for ferromagnetism
0 5 10 15 20α ( = µH / kT )
-0.2
0
0.2
0.4
0.6
0.8
M
L (α)M = kTα / µw
M = kTα / µw - H / w
A
BWeiss explained ferromagnetism by
postulating the existence of an
internal “molecular field” which
aligned the magnetisation.
If the molecular field is wM then αbecomes α = µ(H + wM)/kT .
Rearranging: M = kTα/µw − H/w
If H = 0 the equilibrium value of α is when
coth(α)− 1/α = kTα/µw , indicated by point A in the figure.
As H increases, the equilibrium value of M also increases (point B
in the figure).
13 / 31
Exchange and
anisotropy
14 / 31
Exchange coupling
The source of Weiss’ molecular field is exchange coupling between
spins in a ferromagnet. Consider two electrons in close proximity
with wavefunctions ψa and ψb. Two spin configurations can exist:
ψsymmetric(a,b) = ψa(a)ψb(b) + ψa(b)ψb(a)
ψanti−symmetric(a,b) = ψa(a)ψb(b)− ψa(b)ψb(a)
In case (1) the spins are in the opposite direction and ψa and ψb
can be the same. For case (2) the spins point in the same direction
and therefore ψa 6= ψb, otherwise the Pauli exclusion principle will
be violated.
Configurations (1) and (2) have a difference of energy of
∆E = 2Jab, where J is the exchange constant. The energy
difference favours the alignment of adjacent spins, leading to the
formation of regions of aligned spins, known as domains.
15 / 31
Heisenberg exchange
The magnitude of the Heisenberg exchange energy is given by
wij = −2JSi · Sj , where Si and Sj are two adjacent spins. The
energy is a minimum when Si = Sj (if J > 0).
For small angular differences between spins wij = −2JS2cosθ or
wij = JS2θ2.
For many spins the total exchange energy is a sum over all pairs of
spins, i.e.
w =1
2
−2J∑
i ,j
Si · Sj
= −J∑
i ,j
Si · Sj
where J depends on the lattice structure of the magnet. Exchange
coupling is a short-range interaction and it is usually sufficient to
consider adjacent spins only.
16 / 31
Heisenberg exchange II
For cubic lattices with a lattice constant a, exchange stiffness
constants A are:
A =JS2
afor a simple cubic lattice
A =2JS2
afor a bcc lattice
A =4JS2
afor a fcc lattice
J is an energy (Joules or ergs) and A has units J/m or erg/cm.
17 / 31
Anisotropy
Anisotropy means that the magnetisation vector ~M prefers to lie
along a particular direction. The actual direction is often closely
related to the crystal structure.
For example, in hcp cobalt the easy (preferred) axis is
perpendicular to the hcp planes. The anisotropy energy is given by
Ea = Ku1sin2θ + Ku2sin4θ + ... where Ku is the anisotropy constant
and θ is the angle of the magnetisation from the easy axis.
For a cubic crystal, such as Fe, Ea = K1(α21α
22 + α2
1α23 + α2
2α23)
where α1, α2 and α3 form a unit vector along ~M (direction cosines).
The hysteresis loops of materials with anisotropy can differ
depending on the axis of measurement.
The anisotropy field of a material with uniaxial anisotropy is
Hk = 2K/Ms.
18 / 31
Easy and hard axes
-30 -20 -10 0 10 20 30Applied field (kOe)
-1
-0.5
0
0.5
1
M /
Ms
Easy axis
Hard axis
The figure shows M-H loops for a
single particle measured along the
easy and hard axes.
If a material has uniaxial anisotropy
we can define the easy axis as the
axis along which ~M prefers to lie.
The hard axis is an axis orthogonal
to the easy axis.
For a small, isolated particle the coercivity measured along the
easy axis will be equal to Hk .
The hard axis loop has low coercivity and remanence. If no field is
applied ~M will rotate to lie along the easy axis.
19 / 31
Magnetostatics
20 / 31
Magnetostatics I
Magnetic materials are comprised of many magnetic atoms, each
of which generates a magnetic field.
The magnetic field from each atom can be treated as the field from
a magnetic dipole.
For a magnetic dipole ~m located at the origin, the magnetic field at
a point ~r is given by
~H(~r ) =
(
3~r(~m ·~r)r5
−~m
r3
)
This assumes the dipole is a point source.
21 / 31
Magnetostatics II
m
B
A magnetic dipole produces a
magnetic field in the surrounding
space.
The magnetostatic field is the sum
of all the dipole fields at each point.
A magnetic moment perpendicular to the surface of a material
creates a stray field outside the material. A demagnetising, or
magnetostatic field, is generated within the material.
For uniformly magnetised materials the magnetic moments on the
surfaces of the material are the main sources of magnetostatic
fields.
22 / 31
Demagnetising factors I
Consider a solid magnetic object, e.g. a cuboid. If the cuboid is
magnetised along the x axis, the magnetostatic field is given by
Hdx = −NxMs, where Nx is the demagnetising factor along the x
axis.
Demagnetising factors depend on the shape of an object, but
Nx + Ny + Nz = 1. For a cube with axes parallel to the sides of the
cube Nx = Ny = Nz = 1/3.
The geometry influences the demagnetising factors and can lead to
a shape anisotropy effect in which ~M prefers to lie along an axis
which minimises the magnetostatic field.
23 / 31
Demagnetising factors II
Consider a cuboid of length Lz along the z axis, with Lx = Ly .
As Lz → ∞, Nz → 0.
Therefore, Nx = Ny → 1/2.
For a thin film of thickness Lz along the z axis:
As Lz → 0, Nz → 1.
Therefore, Nx = Ny → 0.
24 / 31
Domains and domain
walls
25 / 31
Domains
Magnetic domains in a CoPt film.
White = magnetisation up, black =
magnetisation down.
When a magnetic field is applied to
a ferromagnet, domains with
magnetisation aligned along the
field direction grow and other
domains shrink.
A domain is a region of a ferromagnet in which the spins
(magnetisation) are aligned in the same direction.
Domains are separated by domain walls.
26 / 31
Domain walls IExchange coupling wants to
minimise the angle between
adjacent spins and favours a wide
domain wall (more spins, smaller
angle between spins).
Anisotropy wants ~M to lie along the
easy axis and favours a narrower
domain wall (fewer spins
misaligned with the easy axis.
Domains are separated by domain walls.
There are several types of domain walls (Néel, Bloch, cross-tie
etc). The width of a domain wall is determined by a competition
between the anisotropy and the exchange coupling.
27 / 31
Domain walls II
Néel walls are favoured in thin films
and Bloch walls in thicker films.
The cross-tie wall occurs for
intermediate thicknesses.
Different types of domain wall have different energies.
The type of domain wall can change depending on the thickness of
the magnetic material.
28 / 31
Domain wall width I
We consider a 180◦ domain wall in a material with a lattice constant
of a. The domain wall width is Na, where N is the number of spins.
The exchange energy is
Eex =N
a2wij =
JS2π2
Na2
where 1/a2 is the number of spins per unit area, N/a2 is the
number of spins per unit area of wall, wij = JS2θ2 and θ = π/N (the
angular change per spin).
The anisotropy energy is Ek = KNa, where K is the anisotropy
constant.
29 / 31
Domain wall width II
The total energy is
Etotal = Eex + Ek =JS2π2
Na2+ KNa
Minimising:
dE
dN=
−JS2π2
a2N2+ Ka = 0
So
N =
√
JS2π2
Ka3and Na = π
√
JS2
Ka= π
√
A
K
with A = JS2/a. So stronger exchange coupling broadens the wall
and larger anisotropy narrows it.
30 / 31
Domain wall width and single domain particles
For a magnetic object the magnetostatic energy can be reduced by
the formation of domains. However, creating a domain wall
increases the energy. Hence, there is an optimum domain width
which minimises the overall energy.
As the size of the magnetic object decreases, the number of
domains also decreases. Below a certain size the creation of
domain walls becomes energetically unfavourable and the object
becomes a single domain particle.
31 / 31
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