**1.1. Nanomagnetism**

In addition to the changes related to the decrease in scale down to the nanometric range, magnetic materials provide another source of novel properties. Many of the critical parameters in magnetism are found in the 1 to 200-nm scale of length, see Table 1 [1]. Two of the most interesting changes in magnetic structure as a consequence of the reduction in size are the change from multidomain to single domain and superparamagnetism.

Bulk samples of ferro- and ferrimagnetic materials are divided into magnetic domains (separated by domain walls) in order to decrease the magnetostatic energy, i.e., the energy associated with the presence of magnetic flux just outside the sample surface. Inside the magnetic domains exchange energy is at a minimum (all spins are parallel coupled) as well as magnetocrystalline energy (all spins are oriented into easy axes), and the magnetization of domains is oriented to provide a continuous magnetic flux inside the sample, thus avoiding any external flux. This is why most magnetic materials do not manifest any attraction or repulsion force (in the absence of an applied magnetic field). Magnetostatic energy is therefore eliminated, except for a small contribution from domain walls, where spins rotate from the orientation of a domain toward the orientation of the neighboring domain. There is also a small contribution to both exchange and anisotropy energy due to domain walls, as spins cannot be strictly parallel, neither oriented into easy directions within the domain wall. These contribu‐ tions, however, are small as domain wall thickness is in the 10- to 100-nm range. Domain walls represent a very sensitive equilibrium in ferromagnetic materials. They can be displaced by very small applied, fields and their dynamics of propagation though defects (pinning sites) have a fundamental importance for soft magnetic materials [2].

As samples become smaller, there is a limit where it becomes more convenient, on an energetic basis, to eliminate the domain walls and form a single domain on the whole sample. This transition becomes feasible because magnetostatic energy is a function of the sample volume, while domain wall energy depends on their total surface. As a single domain, materials become magnetically harder; the coercive field typically exhibits an inverse relationship with the particle size going through a maximum for the transition between single domain and multi‐ domain structure. Single-domain ferromagnets become harder because in the absence of domain wall, the main magnetization process is spin rotation, i.e., the applied magnetic field has to rotate each spin against the anisotropy. This is very different than having domain walls; as they are in a delicate equilibrium, a small field can easily displace them with large variation in oriented domains associated with their passage.


**Table 1.** Critical lengths in magnetism. Adapted from Guimaraes [1].

The reduction in scale can also lead to a superparamagnetic (SPM) phase. As discussed with more detail in Section 3, in an SPM phase, the thermal energy dominates over anisotropy, and the magnetization is therefore subjected to random fluctuations. It is interesting to note that in the SPM phase, the exchange interaction maintains the spin coupling up to the Curie temperature, which is higher than the temperature of transition from ordered to SPM phase, also known as blocking temperature. The magnetic properties in the SPM phase are different to both those in the ordered phase and in the paramagnetic phase. SPM properties allow many novel applications, particularly in the field of biomedicine.
