### 4.3.7 Summary to: Ferromagnetism

| In ferromagnetic materials the magnetic moments of the atoms are "correlated" or lined-up, i.e. they are all pointing in the same direction |  |
|  | The physical reason for this is a quantum-mechanical spin-spin interaction that has no simple classical analogue. |
|  | However, exactly the same result - complete line-up - could be obtained, if the magnetic moments would feel a strong magnetic field. |
|  | In the "mean field" approach or the "Weiss" approach to ferromagnetism, we simply assume such a magnetic field HWeiss to be the cause for the line-up of the magnetic moments. This allows to treat ferromagnetism as a "special" case of paramagnetism, or more generally, "orientation polarization". |
| For the magnetization we obtain Þ |  |
J | = | N · m · µ0 · L(b) | = | N · m · µ0 · L | æ è | m · µ0 · (H + w · J) kT | ö ø
|  | The term w · J describes the Weiss field via Hloc = Hext + w · J; the Weiss factor w is the decisive (and unknown) parameter of this approach. |
|  | Unfortunately the resulting equation for J, the quantity we are after, cannot be analytically solved, i.e. written down in a closed way. |
| Graphical solutions are easy, however Þ |  |
- |  | From this, and with the usual approximation for the Langevin function for small arguments, we get all the major ferromagnetic properties, e.g. Saturation field strength. Curie temperature TC.
  TC | = | N · m 2 · µ02 · w 3k
  --- | --- | ---
Paramagnetic behavior above the Curie temperature. Strength of spin-spin interaction via determining w from TC. |
|  | As it turns out, the Weiss field would have to be far stronger than what is technically achievable - in other words, the spin-spin interaction can be exceedingly strong! |

| In single crystals it must be expected that the alignments of the magnetic moments of the atom has some preferred crystallographic direction, the "easy" direction. |  |
Easy directions: Fe (bcc) <100> Ni (fcc) <111> Co (hcp) <001> (c-direction)
| A single crystal of a ferromagnetic material with all magnetic moments aligned in its easy direction would carry a high energy because: |  |
|  | It would have a large external magnetic field, carrying field energy. |
| In order to reduce this field energy (and other energy terms not important here), magnetic domains are formed Þ. But the energy gained has to be "payed for" by: |
|  | Energy of the domain walls = planar "defects" in the magnetization structure. It follows: Many small domains —> optimal field reduction —> large domain wall energy "price". |
|  | In polycrystals the easy direction changes from grain to grain, the domain structure has to account for this. |
|  | In all ferromagnetic materials the effect of magnetostriction (elastic deformation tied to direction of magnetization) induces elastic energy, which has to be minimized by producing a optimal domain structure. |
| The domain structures observed thus follows simple principles but can be fantastically complicated in reality Þ. |
| For ferromagnetic materials in an external magnetic field, energy can be gained by increasing the total volume of domains with magnetization as parallel as possible to the external field - at the expense of unfavorably oriented domains. |  |
|  | Domain walls must move for this, but domain wall movement is hindered by defects because of the elastic interaction of magnetostriction with the strain field of defects. |
|  | Magnetization curves and hystereses curves result Þ, the shape of which can be tailored by "defect engineering". |
| Domain walls (mostly) come in two varieties: Bloch walls, usually found in bulk materials. Neél walls, usually found in thin films. |
| Depending on the shape of the hystereses curve (and described by the values of the remanence MR and the coercivity HC, we distinguish hard and soft magnets Þ. |  |
| Tailoring the properties of the hystereses curve is important because magnetic losses and the frequency behavior is also tied to the hystereses and the mechanisms behind it. |
|  | Magnetic losses contain the (trivial) eddy current losses (proportional to the conductivity and the square of the frequency) and the (not-so-trivial) losses proportional to the area contained in the hystereses loop times the frequency. |
|  | The latter loss mechanism simply occurs because it needs work to move domain walls. |
| It also needs time to move domain walls, the frequency response of ferromagnetic materials is therefore always rather bad - most materials will not respond anymore at frequencies far below GHz. |
Questionaire
Multiple Choice questions to all of 4.3

© H. Föll (Advanced Materials B, part 1 - script)