Download September 6th, 2007

November 17, 2009
Class 20
Ferromagnetic materials
Ferromagnetic materials have atoms grouped together is domains. Inside each domain
atoms have their magnetic moment aligned. Let’s consider that domains are misaligned with
other domain and thus the total magnetic moment is zero. If an external magnetic field is applied,
the domains will try to rotate to align to the magnetic field. Since the entire domain needs to be
rotated, there is some resistance for that to happen and the magnetization curve is not linear. As
the applied field increases, more and more domains align until
saturation is reached (all domains are align to each other).
The thermal effect is a lot less important than for
paramagnetic materials.
Thermal energy is able to rotate
individual atoms, but not entire domains, thus the magnetic
susceptibility (depends on the field) is in the order of thousands.
If after the material is completely magnetized the magnetic
field is decreased, domains do not completely rotate back to their random orientation but due to
mechanical resistance to this rotation, the demagnetization is not complete even when the applied
field is brought back to zero. The material is now permanently magnetized. Only if a magnetic
field in the opposite direction is applied, the magnetization can be brought back to zero.
However if further increased in the opposite direction, magnetization in the opposite direction
occurs. The cycle continues forming what is called and hysteresis curve.
The magnetic field necessary to push the magnetization back to zero is call the coercivity,
large coercivity materials are called hard magnets and are characterized by domains with large
resistance to rotation, have wider hysteresis cycles and as permanent magnets show larger
magnetic fields. Low coercivity materials are known as soft magnets, have little resistant to
magnetization and normally do not present large magnetic field.
The interaction between spins can be considered as follow. Consider a paramagnetic material
with an added interaction between magnetic moments. This is as if an internal “exchange”
magnetic field were trying to align the spins together.
The interaction energy between two spins, in quantum mechanical terms can be written as:
U=-2JSi.Sj
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Where J is the exchange integral (not to confuse with J the spin-orbital angular momentum),
Si/j is the spin of each of the interacting magnetic moments.
A material like that is called ferromagnet. In the Mean field approximation, the exchange
field can be consider proportional to the magnetization, BE=M, thus each moment feels an
average interaction of the other moments. Interactions like this cannot lead to a (linear behavior).
p, the susceptibility in the paramagnetic regime can be defined as
 o M=p (Ba+BE)
by using the Curie’s law to define the paramagnetic susceptibility, the susceptibility for a
ferromagnetic material can be written us
By replacing  p 
o M
B

oC
T
and BE=M in the equation above,
M
C
For TC=C this function have a singularity, below that, the equation does not

B T  C 
longer apply and the material is in the ferromagnetic region which above it is a paramagnet. TC
is the Curie temperature and it is the temperature where the ferro-para phase transition occurs.
Ferrimagnetism and antiferromagnetism
For some materials the saturation magnetization at T=0 does not correspond to all the spin
aligned in the same direction. These materials are characterized by a negative exchange integral,
thus the ground state consist on spins that are antiparallel. You can think of these materials as
consistent of two sublattices that prefer to align antiparallel to each other. These materials are
known as Ferrimagnets. An special case is one where the two sublattices have the same
saturation magnetization, in that case, for each magnetic moment that points up there is one that
points down, these materials are known as antiferromagnetic materials. Above a temperature
known as Neel temperature, the material is paramagnetic, at the Neel temperature; the
susceptibility does not diverge but has a sharp change in slope. Show Figure 12.20, page 342.
Ferromagnetic Domains
Despite the fact that a large positive exchange integral J tend to align individual spins together, a
ferromagnet does not spontaneously exhibit saturation magnetization, the reason for that lies in
the microscopic structure of these materials. They consist of groups of individual spins aligned
with each other forming a domain. A typical ferromagnet has several domains within which all
the spins are aligned, but each domain is not necessarily aligned with other domains.
When a magnetic field is applied, the material becomes magnetized in two ways, 1) at low values
of the magnetic field, those domains that are aligned with the field grow slightly at the expenses
of domains with spins pointing in other directions. 2) At larger values of the field, the domains
that are not aligned with the field rotate to align with the field. At saturation, all the domains
points in the direction of the field.
Anisotropy Energy
In all crystals there is a direction that is preferred for the spins; this is known as the easy axis
because the spins easily align in this direction. The energy is a minimum if the magnetization is
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parallel to that axis called the easy axis. Different crystallographic structures have different
expression for the anisotropy energy but to fix concepts let’s consider
Uan=K sin2Φ
Where Φ is the angle between the saturated magnetization and the anisotropy axis. Notice that
parallel or antiparallel is the same! And this is often the case.
If the applied field B is not parallel (or antiparallel) to the anisotropy axis, there will be
two competing effects, the anisotropy effect trying to align the magnetization to the easy axis,
and the field that is trying to align the magnet to the field. The interaction energy (due to the
interaction of the magnetization and the applied field) when B is perpendicular to the easy axis
and the magnetization is at an angle Φ with respect to the easy axis is:
EB=-MsB cos(90-Φ) =-MsB sin(Φ) where Ms is the saturation magnetization
The equilibrium is reached when the total energy is a minimum
Or
d(-MsB sin(Φ)+ K sin2Φ)/dΦ=cos(Φ)( -MsB + 2K sinΦ)= 0
Thus either cos(Φ)=0 or
M sB
2K
It is customary to define an H-field such that B=oH. In doing an analogy with M, H would be
the magnetization needed to create a magnetic field equal to B. In what next, we’ll focus on H.
M M M
Notice also that   o  o 
B
o H H
sin(  ) 
The component of the magnetization parallel to the field is:
M ||
MH
H
 sin( )  s o 
Ms
2K
H an
where Han is called the anisotropy field and it is given by (B=oH)
H an 
2K
M s o
If H is applied perpendicular to the easy axis, as it increases in intensity, magnetization parallel
to the field increases linearly until H=Han, then it does not increases anymore since the
magnetization is now parallel to H.
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Domain’s Walls
The domain walls are not infinitely thin, they depend on the material. Materials with large
anisotropic constant will tend to have narrow walls since in the wall the spins are largely oriented
away from the easy axis (not to convincing) but the anisotropy energy is approximately
proportional to the wall thickness, Thus σan =KNa where a is the distance between spins.
On the other hand the exchange energy changes with the cosine of the angle between neighbor
spins. U=-2JESi . Sj=-2JS2cosφ~-2JS2+JS2φ2 by expansion of the cosine for small angle, then the
change in the energy due to a small rotation is given by the second term. If the wall thickness
given in terms of lattice constant, is N, then the angle between consecutive spins (assuming a 180
rotation) is
/N then the exchange energy is JS2(/N)2 and thus, for a line of N atoms the exchange energy is
JS22/N
The above is the exchange energy per each line of atoms perpendicular to the wall and there are
1/a2 lines per unit area, thus
σex= JS22/Na2
The total energy is then
σex+ σan= JS22/Na2+KNa
Thus the thicker the wall the lower the exchange energy and the larger the anisotropy energy is.
So there are two effects competing, the anisotropy energy that tends to make the wall thinner and
the exchange that tries to make it thicker. The equilibrium thickness is

d
d  JS 2 2
JS 2 2


 KNa    2 2  Ka  0
dN dN  Na2
N a

Thus the equilibrium thickness (in terms of the number of atoms in a line is)
N
 2 JS 2
Ka3
By replacing N back into the wall energy per unit of area and rearranging
KJ
 w  2S
a
J is roughly equal for most materials, so the energy associated with the wall increases if the
anisotropy constant increases and decreases otherwise. But the width of the walls depend on the
relative value of the constants, materials with relatively large anisotropy constant will have
thinner walls and high surface energy thus making it difficult for the domains to align. These are
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good for hard magnets. On the contrary, materials with relatively low anisotropy constant will
have wide walls and low energy thus they make good soft magnets.
Superconductivity
It is a property that some materials have to reduce their resistance to zero; this effect was first
observed Heike Kamerling Onnes in 1911 when he observed that conductivity in Hg below 4.2
o
K went to infinity. If a current is induced in a superconductor it’ll stay forever without
attenuation even without a source.
Onnes wanted to do a super electromagnet in order to generate a magnetic field of 10 Teslas, but
he discovered that superconductiviy is lost when the current goes above a critical value. If the
temperature, current, or an applied magnetic field is increased above critical values,
superconductivity is lost.
For a superconductor
  T 2 
H c (T )  H o 1    
  Tc  
where Ho is the critical field extrapolated to the absolute zero. That is, Ho is the maximum value
of magnetic field for which the material maintains superconductive properties if the temperature
is 0K. Hc(T) is the critical field at temperature T, thus if T increases Hc decreases.
In addition, a current on a loop of radius r produces a magnetic field of
I
I
B o
or H 
2r
2r
The critical current is limited to that producing a field of Hc
Ic=2rHc
If some external magnetic field H is applied, then the critical current will be:
Ic=2r(Hc-H)
Conversely, if there is a current in the superconductor, the field generated by the current ads to
the applied field, then the maximum field is equal to the critical field minus the field generated
by the current. Also if there is no external field, the maximum temperature is that for which the
critical field is equal to the field produced by the current.
Cooper pair
The reason for superconductivity is a very unusual phenomenon, electrons couple in pairs, now
the behave as bosons. A fermion is a particle with semi-integer spin and it is subject to Pauli
exclusion principle, a boson instead, is a particle with an integer spin and it is not subject to Pauli
principle. In a superconductor, electron pairs “condense” to the ground state and increase their
mobility to infinity.
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Electron pairs are possible due to electron-phonon interaction, one of the electrons attract the
ionic network which attract the other electron. The attraction is only of a few meV but enough to
form the pair at high temperature
 Check: http://hyperphysics.phy-astr.gsu.edu/hbase/solids/coop.html#c4
2. Superdiamagnetism
B
a time-varying magnetic field produces an electric field that
t
will produce a current in the superconductor
E
 o J the current (and a time-varying electric
From Ampere-Maxwell equation x B   o o
t
field, will generate a magnetic field). Due to the asymmetry in the equations, (Faraday has a
minus sign) the magnetic field generated by the current will oppose to the applied magnetic field
thus decreasing it. In regular metals, the current is dumped by the resistance and the magnetic
field created by the current is less than the apply magnetic field. In superconductors, the current
induced is large enough to create a magnetic field that exactly compensate for the applied field,
thus the internal field is zero. In a superconductor, the super-current is not attenuated and the
field is completely compensated.
B=onI+oM=o(H+M)= o(1+)H=0
From the Faraday law x E  
So, the magnetic susceptibility is -1 and the material is superdiamagnetic. M=-H
Above the critical field, the material recover normal properties, the magnetization and magnetic
susceptibility goes to zero, because superconducting materials are very week diamagnets at high
temperature and B=oH
3. Type I superconductor
If the field goes above the critical field, the system become normal and the field will
penetrate, Onnes was expecting that if the field is decreased below the critical limit again,
super currents will trap the field inside, but such a thing does not happen, the super conductor
repel the magnetic field. This is known as Meissmer Effect and its existence indicate that the
diamagnetism of superconductors is an intrinsic property and not just caused just by the
supercurrents.
Expected behavior
Observed behavior
In simple terms, the reason is that the current in a superconductor is proportional to the vector
potential A, J=A, (which is for the magnetic field what the electrical potential is for electric
field).
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Since
B  A
then
  J  B which is the London equation
According to the Maxwell’s equation (when the electric field does not change in time)
xB  o J
Taking then the curl on the late equation
  xB  2 B  o  J  oB
B=constant is NOT a solution of the above equation, since the left-hand side would be zero but
the right hand side is not unless B=0
This relation between magnetic field and current density does apply to a perfect conductor.
Type II superconductor
Type II superconductors are mostly alloys, they have two critical fields, below Hc1, the system is
totally superconductor, above Hc1, it is in a mix state and above Hc2, it is normal. Increasing
doping level reduces the first critical field and increases the second.
In the mix state, between the two critical fields, there are some domains, like thin
columns, were the material is normal called fluxoids. Those domains are surrounded by strong
super-currents that confine the field inside. Some magnetic field can penetrate reducing the high
energy that keeps the field from penetrating.
Fluxoids can move around, each of them carries a given amount of magnetic flux
o=h/2e=2.07x10-15 webers (Tesla-m2), thus it is a quantum effect. The total magnetic field
inside is B=No. Perpendicular to a magnetic field (the one generated by the superconductor)
Lorenz forces appear that affects the fluxoids, If the fluxoids are free to move, current will hardly
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go through, is like an expert skier going through a forest there trees are moving around., If
defects are added (grain boundaries, dislocations, chemical inhomogeneities, etc) to prevent them
from moving, then current can find a superconducting path and will get through without
resistance.
High Tc superconductors
Up to 1986, the highest critical T was 20 K, In 1986 La-Ba-Cu-O was discovered to have a Tc
above 30 K. In 1987 Y-Ba-Cu-O, was found to have Tc above 90 K. When superconductors with
Tc above 77 K were discovered, the path to new applications opens up, above 77 K liquid
nitrogen can be used which is cheaper than liquid He.
A problem with high Tc superconductors is that the materials are very difficult to deal
with and they have a low Ic, apparently, it is not that easy to pin fluxoids at high T.
Also, the formation of the cooper pair is due to a process which is not exactly known and much
controversy is still around.
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