8-10 Magnetism, Superconductivity

Ferromagnetic materials have areas with the same spin orientation (domains) that all together are non-magnetized. The boundaries where they meet are blurry, there, the electron’s spins gradually flip.

All materials have a tendency to align the spin and angular momenta of their atoms against an external B-field (since N is attracted to S and vice versa). This is called Diamagnetism. The magnetic susceptibility resulting from Diamagnetism is negative

How much a material magnetizes when it is put into an external field:

chi = M / H = \mu_0 * M / B

In isotropically bound substances, e.g. certain ionic lattices, the diamagnetic susceptibility is temperature independent.

In metals, the free electrons behave in a similar way, but we have to consider effective masses and density of states, leading to the Landau diamagnetism.

When the material responds to an external magnetic field by aligning its inner dipoles with it instead of against it.

Paramagnetism (if the material is paramagnetic). Diamagnetism is a phenomenon happening in all materials

To find the Ground state of an atom:

1. Spin is maximized

2. L (sum of m_L) is maximized under the condition of 1.

3. If the orbital (d, s, …) is more than half full, J=L+S, otherwise J=|L-S|

With a Brillouin function.

For low ratios B/T (room temp, weak magnets), the magnetization is approximately proportional to C/T with C being the Curie constant (Curie’s Law).

For high ratios B/T, the magnetization is approximately constant (saturation)

Because lattices in a crystal disturb the rotational invariance used to describe H-atom orbitals. This leads to irregularities in angular momenta.

For 4f electrons, this does not play a big role, since they are more inside of the atom and get shielded by the outer electrons.

In a metal, spin up and down electrons will have different energies when a magnetic field is applied. So, the Density of state functions will shift. Since the Fermi energy needs to be constant over the whole system, some electrons will switch from one spin to the other, aligning with the field.

Materials that can spontaneously align their magnetic moments and keep this alignment even without a magnetic field

To approximate all dipoles around the dipole of interest in a ferromagnetic material as an effective field.

It works a little bit like Pauli Paramagnetism, but the Energy levels are not only shifted because of the external B-field, but because of an imbalance in the spin up and the spin down electrons. This leads to a self-enhancing shift of energies until all electrons are aligned

It tells whether a material is ferromagnetic. The Susceptibility of Stoner Ferromagnetism is like The Pauli Susceptibility, but divided by a Term 1-(I D(E_F)/(2n)) . If this term is bigger than one, spontaneous magnetization can happen. If it is smaller than one, the material is not ferromagnetic.

A magnon is a spin orientation wave that is caused by short ranged interactions, it works a lot like phonons. Magnons happen for T higher than 0.

The disperion relation looks like a phonon dispersion relation.

Both materials have two sublattices that align their spins opposing to each other. For Ferrimagnets, the two sublattices have different magnetic moments, leading to an overall magnetization. Antiferromagnets do not have an overall magnetization.

When all spins are aligned in one direction, the changing magnetic field first needs to overcome the interaction energy of the particle spins. This is why the magnetization changes slowly at the beginning. Then, M is proportional to B as in paramagnets. And for high B fields, it will start to approach the saturation magnetization.

The Barkhausen effect describes sudden noisy jumps in the magnetization curve of ferromagnets due to the sudden alignment of ferromagnetic domains.

Below a certain sample size, ferromagnetic materials loose their ability to maintain their magnetization after the field is turned off.

This is due to the fact that there are less other particles to interact with, causing the energy barrier for demagnetization to be very low (lower than thermal energy).

Thus, below the curie and above the “blocking temperature”, these particles will be thermically demagnetized (magnons). The blocking temperature value is dependent on the particle size.

The energy barrier can be heightened by using a magnetically anisotrop material.

It physically limits the storage density of magnetic storage devices

The relaxation times of superparamagnetic beads can be used for contrast in MRIs

For Temperatures higher than the Neel Temperature, it behaves like a paramagnet and is proportional to the inverse of T.

Below the Neel Temperature, the susceptibility is dependend on the direction of the field. If it is perpendicular to the direction of spontaneous magnetization, it stays constant. Otherwise, it falls with shrinking T.

Ferromagnetic magnons go with

E(k)=4JS(1-cos(ka))

Whereas Antiferromagnetic magnons go with

E(k)=-4JS|sin(ka)|

In antiferromagnetic materials, the Brillouin zone size depends on the magnetization.

If there is no magnetization at all (higher T), it is bigger, whereas if the sublattices are magnetized in different directions, it is half as big as before.

This causes new “superstructural” peaks in crystal x-ray diffraction experiments.