Solid State Physics

MAGNETISM III As T decreases the straight line M gets less steep. Thus for lower T
there is a solution to
Lecture 29 M

gJ µBJλM

= BJ
Ms kB T
for finite M.
A.H. Harker
Physics and Astronomy
UCL

10.6.5 Mean field theory of ferromagnetism

Armed with the mean field picture, and a picture of the way M de-
pends on B through the Brillouin function, we have
 
M gJ µBJ(B + λM)
= BJ . (1)
Ms kB T
Assume for the moment that B = 0. Then we can plot the two sides
of equation as functions of M/T :
Furthermore the shape of BJ , a convex curve, shows that there is a
critical temperature TC above which the M line is too steep to inter-
sect the BJ curve except at M = 0.

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For small values of M/T we can use Curie’s law, Below TC the spontaneous magnetisation varies with temperature.
µ0ngJ2 µ2BJ(J + 1)
χ=
3kBT
and
M ngJ JµBBJ
χ= =
H H
to deduce  
gJ µBJB g µ (J + 1)B
BJ ≈ J B .
kB T 3kBT
In terms of x = M/T , the straight line is
M Tx
=
Ms Ms
and the approximation to the Brillouin function is (putting λM for
B)
g µ (J + 1) g µ (J + 1)
BJ ≈ λM J B =λ J B x.
3kBT 3kB
Equating the gradients with respect to x,
TC g µ (J + 1)
=λ J B ,
Ms 3kB
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or 10.6.6 Paramagnetic regime

g µ (J + 1)Ms
TC = λ J B Above the Curie temperature, if we apply a magnetic field, we have
3kB
2 2 M g µ (J + 1)
=
λngJ µBJ(J + 1)
. BJ = ≈ (B + λM) J B
3kB Ms 3kBT
The critical temperature TC is the Curie temperature – often denoted which can be rearranged to give
by θ. Some ferromagnetic materials MsBgJ (J+1)µB
3kB
Material TC (K) µB per formula unit M= ,
λMsgJ (J+1)µB
Fe 1043 2.22 T− 3kB
Co 1394 1.715 and with Ms = ngJ JµB
Ni 631 0.605
nBgJ2 J(J+1)µ2B
Gd 289 7.5
3kB
MnSb 587 3.5 M= ,
λngJ2 J(J+1)µ2B
EuO 70 6.9 T− 3kB
EuS 16.6 6.9 or
nBgJ2 J(J+1)µ2B
3kB
M= .
T − TC
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This gives a susceptibility 10.6.7 Effect of magnetic field on ferromagnet
1
χ∝ , At low temperatures, the magnetisation is nearly saturated, so a B
T − TC field has little effect:
the Curie-Weiss law.

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The Curie-Weiss law works quite well at high T As we increase the temperature, we reach a regime where the field
has a large effect on the magnetisation:

but breaks down near the Curie temperature TC or θ, where the mean
field approximation fails.
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At high temperature we are in the Curie-Weiss regime than we de- 10.6.8 Anisotropy in magnetic systems
scribed above:
The quenching of orbital angular momentum in a crystal is one effect
of the crystal field – the electrostatic potential variation in the solid.
But as spin-orbit coupling links the spins to the spatial variation of
the wavefunctions, the spins tend to align more readily along certain
directions in the crystal – the easy directions of magnetisation.

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Overall, then, the effect of a field is: 10.7 Magnetic domains

In general, a lump of ferromagnetic material will not have a nett
magnetic moment – despite the fact that internally the spins tend to
align parallel to one another.

10.7.1 Magnetic field energy

The total energy of a ferromagnetic material has two components –
the internal energy (including
R the exchange energy) tending to align
spins and the energy B.HdV in the field outside it.

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The external field energy can be decreased by dividing the material For small δθ, expanding the cosine,
into domains. 1
δE = 2JS 2 (δθ)2
2
and if we extend the change in spin direction (total angle change of
π) over N spins, δθ = π/N , and there are N such changes of energy
δE, so the total energy change is
π2
∆E = JS 2 .
N
This favours wide walls, but then there are more spins aligned away
from easy directions, providing a balance. Bloch walls are typically
about 100 atoms thick.
In very small particles, the reduction in field energy is too small to
balance the domain wall energy – small particles stay as single do-
mains – superparamagnets. Small magnetic particles are found in
some bacteria (magnetotactic bacteria) which use the angle of dip of
the Earth’s magnetic field to direct them to food.
The internal energy is increased because not all the spins are now
aligned parallel to one another.
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10.7.2 Domain walls 10.8 Other types of magnetic ordering

What is the structure of the region between two domains (called a do- The three easiest types of magentic ordering to visualise are ferro-
main wall or a Bloch wall? The spins do not suddenly flip: a gradual magnetic (all spins aligned parallel), antiferromagnetic (alternating
change of orientation costs less energy because if successive spins are spins of equal size), and ferrimagnetic (alternating spins of different
misaligned by δθ the change in energy is only size, leading to nett magnetic moment).
δE = 2JS 2(1 − cos(δθ)),
where J is the exchange integral.

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As the exchange integral J can have complicated dependence on di- Thus the number of extra electrons per unit volume with spin up will
rection, other orderings are possible, for example: be
1
∆n↑ = g(EF)µBB
2
and there is a corresponding change in the number with spin down,
1
∆n↓ = − g(EF)µBB.
2
The magnetisation is therefore
M = µB(n↑ − n↓) = g(EF)µ2BB,
giving a susceptibility
M 2 3nµ0µ2B
χ= = µ0µBg(EF) = .
H 2EF
This is a temperature-independent paramagnetism, typically of order
Helical ordering (spins parallel within planes, but direction changing 10−6. The free electrons also have a diamagnetic susceptibility, about
from plane to plane) – e.g. Dy between 90 and 180 K. − 13 of the paramagnetic χ.
Conical ordering – e.g. Eu below 50 K.
Polarised neutron scattering reveals these structures.
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10.9 Magnetic properties of metals 10.9.2 Ferromagnetic metals

10.9.1 Free electron paramagnetism If we look at the periodic table we find that the ferromagnetic ele-
ments are metals.
In a metal, the free electrons have spins, which can align in a field. As
the electrons form a degenerate Fermi gas, the Boltzmann statistics
we have used so far are inappropriate.

This causes some complication in the magnetic properties – treated

The field B will shift the energy levels by ±µBB. in a simplified way by Stoner theory.

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The exchange interaction splits the narrow d bands: the wide free-
electron-like s bands are relatively unaffected.

The Fermi surface is determined by the total number of electrons:

this can lead to apparently non-integer values of the magnetic mo-
ment per atom (e.g. 2.2 in Fe, 0.6 in Ni).

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