## 3 Results and Discussion
On the other hand, for spin-down polarization, there are no marked systematics in the bandgap changes, but there is an increase in the DOS due to the presence of 3d-Cr orbitals. These changes can be best observed in the band structure for the two spin polarizations that are exemplified in Figure 4.  
A decrease in the bandgap value can be expected with the inclusion of Cr\({}^{4+}\) in the crystallographic sites of Ti\({}^{4+}\), since the former has two electrons in the 3d orbitals while Ti\({}^{4+}\) does not. The presence of these two electrons is represented in the band structure of Figure 4 by the appearance of new orbitals in the conduction band, which tend to decrease the total bandgap value according to the proportion of Cr\({}_{4+}\) in the unit cell. For the pristine material CaTiO\({}_{3}\), both for spin-up and spin-down polarization, it is evident in Figure 4a,b that the material behaves as a direct bandgap semiconductor. The inclusion of Cr\({}^{4+}\) in all ratios modifies this characteristic of the compound, making it an indirect bandgap for both spin orientations. Meanwhile, for spin-up orientation, a new band appears for spin-up polarization, decreasing the bandgap when 25% Cr is introduced into the crystal cell. More states appear in this band as the Cr\({}^{4+}\) concentration is increased,  
Figure 2: Variation in total energy of the perovskite CaTi\({}_{1-x}\)Cr\({}_{x}\)O\({}_{3}\) materials as a function of %Cr. as shown in Figure 4a. This phenomenon takes place thanks to the magnetic character of Cr\({}^{4+}\), whose [Ar]4s\({}^{0}\)3d\({}^{2}\) electronic configuration allows for the occurrence of two unpaired spins in the \(d\) orbitals. The behavior of the bandgap variation for the spin-up and spin-down polarizations is exhibited in Figure 5, in which it is clear the rapid decrease in the bandgap for the spin-up orientation is due to the appearance of a transport channel due to the 3d\({}^{2}\)-Cr orbitals, which are distributed with the same polarization, following [PERSON]'s rules. For higher Cr\({}^{4+}\) concentrations, although more orbitals appear in the band structure, the bandgap decreases slowly with a tendency towards saturation, being the lowest for 100% substitution, reaching a value E\({}^{\uparrow}_{\text{g}}\) = 0.65 eV corresponding to 28% of the bandgap value of the pristine compound (E\({}^{\uparrow}_{\text{g}}\) = 2.34 eV). This behavior is characteristic of magnetic semiconductor materials, in which each spin orientation corresponds to a distinct charge transport channel. Such materials are particularly valuable in spintronic devices [50], in which the ability to generate and control spin-polarized currents is crucial. The tunability of the bandgap and magnetic properties observed in Cr-doped CaTiO\({}_{3}\) suggests that these materials could be effectively integrated into spintronic applications, such as non-volatile memory, spin transistors, and magnetic sensors. The potential to engineer specific electronic and magnetic configurations in these perovskites opens up new avenues for developing advanced components that can operate with higher levels of efficiency and reliability in spintronic technologies. Figure 3: Density of total (**a**,**d**,**g**,**j**,**m**) and partial 3d-Ti and 3d-Cr (**b**,**e**,**h**,**k**,**n**) and 2p-O (**c**,**f**,**i**,**l**,**o**) states calculated for the CaTi\({}_{1-\text{x}}\)Cr\({}_{\text{x}}\)O\({}_{3}\) perovskite-like system. The colors in the plots represent contributions from different orbitals and spin polarizations: black and red lines, in the first row, correspond to the total density of states for the spin-up and spin-down configurations, respectively. In the second row, the blue and orange lines represent the spin-up and spin-down contributions from the 3d-Cr orbitals. The 2p-O states are shown in green and brown for the spin-up and spin-down configurations in the third row.