# Unexpected Ferromagnetism--A Review
## 1 Conventional vs. \(d^{0}\) Magnetism
In these cases, the introduction of a dopant magnetic cation (M) between the non-magnetic cations (A) and oxygen anions will induce the formation of a defect (D, named \"donor defect\") in the host lattice formulated now as [A1-xMx][ODs]n (where \(\delta\) is the donor concentration and x is 1 or 2 corresponding to divalent or tetravalent non-magnetic cations), a defect which is associated with an electron (similar situation with the hole in p-type DMSs) which was treated similarly with a 1s electron in a hydrogenoid atom, having a specific radius and mass [3]. The orbitals associated with these uncoupled electrons tend to overlap, forming a so-called \"impurity band\" when the concentration of donor defects increases. At low concentrations, these electrons are strictly localized due to their narrow band fluctuations of potential, whereas at higher concentrations, when they reach a critical donor concentration, these impurity states lose their localization which implies metallic-type conduction. Here the BMPs are formed by coupling the orbits of these hydrogenoid defect-bounded electrons with 3d moments of magnetic ion dopants [3]. Thus, when the radius of this hydrogenoid electron orbital is large enough, their overlapping with the nearby magnetic cations will induce the apparition of a FM exchange coupling which is also influenced by the spin of 3d cations. The calculation showed that in a sphere with the radius of a hydrogenoid 1s electron there can be from \(\sim\)10 to \(\sim\)100 cationic sites [3]. For example, in the case of the d5 cations such as Mn\({}^{2+}\) or Fe\({}^{3+}\), the remaining unoccupied orbitals are available only for electrons with spin down. Because the donor electron will have also spin down (\(\downarrow\)), if the two magnetic impurities will fall in the same donor orbital due the parallel alignment of the spins, the resulting effective coupling will be ferromagnetic. The authors concluded that the coupling between the donor electron and the magnetic cation is ferromagnetic for the case when the 3d ionic shell is occupied less than half and antiferromagnetic if the 3d shell is half or over half occupied, whereas, for the case of two similar impurities interacting both within the same donor orbital, the coupling is ferromagnetic [3]. Additionally, the landscape of different magnetic interactions is shaped by both polaron and cation percolation threshold. If the number of cationic dopants sites within the sphere of hydrogenoid electron is high enough, the FM will occur when the concentration of donors is greater than those of donors percolations accompanied by a lower than specific percolation concentration of magnetic cations. Other behaviors, like antiferromagnetism or ferrimagnetism will manifest at concentrations exceeding those of magnetic cations percolation [3]. Regarding the Curie temperature (Tc) in these doped oxide systems, it can be an estimated function of many parameters: J\({}_{\text{sd}}\) (s-d exchange parameter), S (3d cation spins), s (donor electron spin), concentrations of cations (x), and donors (\(\delta\)), \(\omega_{\text{c}}\) (cation volume fraction, \(\epsilon_{\text{F}}\) (Fermi energy in the impurity band) according to the Equation (1):  
\[\text{T}_{\text{C}}=\text{S}(\text{S}+1)\text{s}^{2}\text{nx}\delta\text{J}_{ \text{sd}}{}^{2}\omega_{\text{c}}\ ^{2/3}/\text{k}_{\text{B}}\epsilon_{\text{F}} \tag{1}\]  
where k\({}_{\text{B}}\) is Boltzmann constant [3] and even this expression is much simplified, given lower than real values of Tc. The authors from [3] suggest that, for the cases where the donors' concentrations are sufficiently high, the model involving the impurity band does not apply anymore, due to the fact that the energy of the donor state is merging with the inferior part of the conduction band and here the RKKKY (Ruderman-Kittel-Kasuya-Yosida) exchange is applicable. If the separation between nearby magnetic cations is smaller than the localization length, the RKKY interaction will work even if the mobility edge is above the Fermi level.