Magnetism I

Simon Greaves1

1Research Institute of Electrical CommunicationTohoku University, Japan

4/2019

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Magnetism I

Units and origins of magnetism

Types of magnetism

Exchange and anisotropy

Magnetostatics

Domains and domain walls

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Units and origins of

magnetism

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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

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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.

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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.

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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.

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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.

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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).

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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.

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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.

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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).

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Exchange and

anisotropy

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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.

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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.

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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.

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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.

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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.

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Magnetostatics

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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.

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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.

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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.

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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.

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Domains and domain

walls

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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.

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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.

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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.

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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.

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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.

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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.

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