Magnetic polaron percolation on a rutile lattice: A geometrical exploration in the limit of low density of magnetic impurities

The evolution of magnetic polaron clusters on a rutile lattice has been explored, and the results are used to discuss the origin of ferromagnetism in transition-metal (TM)-doped rutile ${\text{TiO}}_2$ oxides. It is shown that percolation on a rutile lattice is characterized by a threshold different from that found for the percolation of randomly distributed spheres, which has been so far assumed as a model to treat the percolation of magnetic polarons in diluted magnetic semiconductors. Furthermore, unlike previous investigations, we explicitly considered the condition of small density $n_i$ of TM impurities, i.e., a regime that is quite far from that usually regarded for polaron formation, dominated by a high density $n_i$ of TM ions. Assuming that ferromagnetic coupling arises between polarons that share common TM impurities, we show the effect of this constraint on the magnetic properties of the sample compared to those reported for the $n_i⪢n_h$ regime.

Figure 1

(Color online) Percolation radius dependence on hole density. Simulations were carried out using a supercell with a 45 u.c. edge where the apical sites are more likely occupied by oxygen vacancies (circles) and where vacancies are equally distributed among all sites (triangles). Squares refer to data obtained using a supercell with 60 u.c. edge where the apical sites are more likely occupied by oxygen vacancies. In all cases, the geometric clusterization algorithm was used.

Figure 2

(Color online) Top panel: percolation radius dependence on hole density obtained using a supercell with 45 u.c. edges. Different symbols represent data obtained with different fixed impurities density. Bottom panel: percolation radius dependence on impurities density obtained using a supercell with 45 u.c. edges. Different symbols represent data obtained with different fixed holes density. The algorithm clusterizes polarons with common impurities.

Figure 3

(Color online) Top panel: holes (black line) and impurities (red line) amount in the largest cluster in a supercell with 45 u.c. edges. The system was populated with 1.5% holes and 0.5% impurities. Dashed lines refer to data calculated using the geometrical superposition, while solid lines refer to data obtained using the common impurity method. Bottom panel: same as above, with 1.5% holes and 4.0% impurities.

Figure 4

(Color online) Calculated $M$ vs $T$ curves for samples with 1.5% holes and 0.5% impurities (red line) and 1.5% holes and 4.0% impurities (green line). The common impurity clusterization method is employed. For the TM ion a magnetic moment ${\text{S}}_i=3/2$ is assumed. The results are compared to those obtained for the geometric percolation scheme (dashed line) by considering only the contribution of ${\text{S}}_i=3/2$ to the percolating cluster. This curve has been computed on the basis of Eq. (6). In all three cases $a_B$ was set at $1\text{Å}$. Inset: ${\text{CoTiO}}_2$ $M$ vs $T$ curve (adapted from Ref. 12).