## 3 Results and Discussion
Figure 4: Band structure for all Cr concentrations at the Ti site in the CaTi\({}_{1-x}\)Cr\({}_{x}\)O\({}_{3}\) material, considering (**a**) spin-up and (**b**) spin-down polarization. In this image, E\({}_{F}\) represents the Fermi level. Figure 5: (**a**) Bandgap behavior with Cr\({}^{4+}\) ratio at Ti\({}^{4+}\) sites for the two spin polarizations. (**b**) Band structure showing the behavior of the energy variation for the spin-up and spin-down polarizations. On the other hand, for the spin-down orientation, there is an initial decrease in the bandgap value for x = 0.25, increasing thereafter and remaining constant at E\({}^{\downarrow}\)g = 1.95 eV, which is 83% of the value obtained for CaTiO\({}_{3}\) (E\({}^{\downarrow}\)g = 2.34 eV). The asymmetry of the bands for the two spin polarizations, as well as the type of indirect bandgap and the difference in the bandgap value, are features that can be seen in Figure 5b. As mentioned above, the charge-bearing states in the valence band are mostly contributed by the 2p orbitals of oxygen, while the available states in the conduction band are due to the 3d orbitals of chromium. The difference between the bandgap values for the up- and down-spin polarizations allows us to infer that energetic excitations in the E\({}^{\uparrow}\)g < E < E\({}^{\downarrow}\)g regime could contribute to the generation of currents with a single spin orientation (spin polarized currents), giving rise to the possibility of transporting both electric charge and electronic spin. This behavior is characteristic of so-called magnetic semiconductors, which have a wide range of applications in the spintronics industry [50]. The variation in the effective magnetic moment in the unit cell as Cr\({}^{4+}\) is substituted at the Ti\({}^{4+}\) sites is shown in Figure 6, in which a linear increase in the magnetic moment in the unit cell is observed with increasing Cr\({}^{4+}\) ratio. In the calculation, the average over the whole material is considered, normalized with respect to the unit cell, since it is not possible to have only 1/4, 1/2, or 3/4 Cr or Ti cations in the unit cell. Only in the case of a 100% inclusion of Cr\({}^{4+}\) is there an integer number of these cations in the cell, for which it is expected that the two electron spins in the d orbitals contribute a magnetic moment \(\mu=\sqrt{n(n+1)}\)\(\mu\)B, where \(n\) represents the number of unpaired spins. Thus, with \(n\) = 2, considering zero contributions from Ti and O ions, the expected magnetic moment is \(\mu\) = 2.45 \(\mu\)B. However, this value is only an estimate. The value found by the DFT calculation of this work is close to 2.0 \(\mu\)B because spin-orbit interactions were not taken into account, which, in some materials, tend to decrease the value of the total effective magnetic moment [51]. The value of the effective magnetic moment in the cell then corresponds to the ratio of Cr. CaTiO\({}_{3}\)'s magnetic moment is zero, so the partial substitution of Ti\({}^{4+}\) by Cr\({}^{4+}\) is expected to be related to the proportion of this cation in the unit cell; thus, for x = 1/4, 1/2, 3/4, and 1 in the generic formula CaTi\({}_{1-x}\)Cr\({}_{x}\)O\({}_{3}\), effective magnetic moments of 0.5, 1.0, 1.5, and 2.0 \(\mu\)g/f.u would be expected, as shown in Figure 6.  
Figure 6: Dependence of the effective magnetic moment on the Cr\({}^{4+}\) content in the unit cell of the CaTi\({}_{1-x}\)Cr\({}_{x}\)O\({}_{3}\) perovskite.