is a ferromagnetic perovskite with a Curie temperature of 5 K, and it crystallizes in the GdFeO
Pnma structure with large TiO
octahedra tilting (19
). The Ti-O-Ti angles are ∼144
plane and 140
along the b
axis. The distance between the Ti and apical O along the b
axis is the shortest (2.0167 Å), while the Ti-O distances in the
plane are alternatively short and long (2.0178 Å and 2.0754 Å) [21
]. Figure 1
shows the spin density isolines on the
plane passing through the Ti ions. The oxygen atoms directly bound to the Ti ions are located 0.44 Å above and below the plane.
The atomic charges and spin densities integrated in the Bader basins of the charge density are reported in Table 1
. The effect of the Hubbard U on the Ti ion is to concentrate both the charge and spin density on the metal ions. Interestingly, the ACBN0 method yielded U(Ti-3d) = 0.26 eV and a large value for oxygen, U(O-2p) = 8.31 eV.
The calculated atomic charges are far from the formal valence charges (Y:+3, Ti:+3, O:−2). The sum of the partial charges, integrated on a real space mesh, differs by 10
electrons from the total number of electrons per unit formula. The total spin magnetization per unit formula is 1
k with 80–86% of it located on the Ti ion. This situation corresponds to a Ti
ion in the
configuration. Note that the spin density around the oxygen atoms shows a negative shell near the ion, surrounded by a region of positive spin density. The size of this negative spin region is large when the U is applied to the Ti ion, whereas it is reduced with ACBN0. The ACBN0 atomic charges and spins are close to those obtained with the hybrid HSE06 and PBE0 functionals. The PBE0 calculations accompanying the experimental paper were performed with the gaussian basis set code CRYSTAL [34
], and the QTAIM analysis was performed with TOPOND [35
]. Note that the small magnetic moment on the O1 (apical) ion is initially larger that those of the O2 ions at U = 0, but the situation changes for U > 3 eV. The larger magnetic moments on the basal O2 ions are also suggested by the experimental data, but this is not the case for ACBN0.
The electronic density of states (DOS) is shown in Figure 2
. Upon increasing the Hubbard U on the Ti ion, the system undergoes a metal-to-insulator transition. The singly occupied
orbital in the spin-up channel separates from the empty state manifold and moves to a lower energy below the Fermi level. The situation is different for ACBN0: The small value of U on the Ti ion preserves the metallic character, and the large U value on oxygen has the effect of shifting the occupied manifold down by ∼2 eV (which has a dominant oxygen character) and increasing its band width. Our results are in good agreement with those of Refs. [36
], where it was found that a U(Ti) value of 3.7 eV yields a band gap of 2.20 eV. In the same paper, they also calculated a band gap of 2.07 eV with the HSE06 functional, and the manifold of occupied oxygen bands was found 6 eV below the bottom of the upper Hubbard band (UHB). Overall, the HSE DOS of Ref. [36
] is more similar to the case of U = 5 eV, except that the the oxygen states are −5 eV below the left shoulder of the UHB. In ACBN0, the distance between the oxygen states and the UHB is ∼6 eV, which is similar to HSE. Unfortunately, ACBN0 is not able to split the lower Hubbard band (LHB ) from the UHB.
The spin density and the VSCCs are reported in Figure 3
. The analysis of the laplacian of the charge density shows that the there are six VSCC points, which are located ∼0.37 Å from the Ti ion, corresponding to the outer
shell of Ti. When U(Ti) is small (less than 2 eV), the VSCCs are arranged as a nearly regular octahedron. The VSCC octahedron is rotated such that two VSCCs are in the O1-Ti-O2 plane, and the remaining four VSCCs are located in the plane tilted by ∼40
from the Ti-O2-O2’ plane. For small values of U(Ti), the six VSCCs correspond to the six lobes of the spin density, showing that the spin density is given by a superposition of the
orbitals. When the Hubbard U on Ti is larger than 2 eV and the system becomes insulating, the VSCC octahedron changes its shape and becomes highly distorted: Four VSCCs were found to correspond two the four spin density lobes, while the two remaining VSCCs were located close to the Ti-O1 bond. This is reflected by the values of the laplacian (curvature of the charge density): When U(Ti) is small, the laplacian at the VSCC ranges from −9.3 to −10.0 au
, meaning that the charge density is concentrated nearly equally at the VSCC. When U(Ti) is larger that 2 eV, the laplacian is more negative (−12.5 au
) at the four VSCCs corresponding to the spin density lobes, while it is less negative (−8.7 au
) at the two remaining VSCCs. Therefore, in YTiO
the curvature of the charge density distribution gives a direct indication of the shape of the spin density.
Unsurprisingly, the small U(Ti)=0.26 eV found by the ACBN0 method yields a spin density with six lobes, and the VSCCs form a nearly regular octahedron. However, the ACBN0 charge density around the Ti atom is more concentrated with respect to the U = 0 case: The laplacian at the six VSCCs ranges from −9.4 to −11.2 au, and this is due to the fact that the ACBN0 Bader charge on Ti is ∼10% larger than that computed with U = 0. In the HSE06 case, the spin density shows four large lobes and two small ones that become visible with an isosurface of 0.2 au. This situation is intermediate between the small and large U value cases. The laplacians at the VSCCs are −8.9 and −12.5 au, comparable to the case of U(Ti) > 2 eV.
is a weakly interacting ferromagnetic metallic perovskite. SrRuO
crystallizes in the Pnma structure, but differently from YTiO
the octahedra in SrRuO
are less tilted and less distorted. The Sr-O distances are 1.986 Å(Sr-O1 apical), 1.986 Å and 1.987 Å (Sr-O2 basal). The Sr-O-Sr angle is ∼161
shows the spin density isolines on the
plane passing through the Ru ions. The plane containing the Sr and O atoms is almost parallel to the (010) plane. The atomic charges and spin densities integrated in the Bader basins of the charge density are reported in Table 2
. Similarly to YTiO
, the ACBN0 method yielded a relatively small value of U on the transition metal and a large value of U on the oxygen ions: U(Ru-4d) = 2.06 eV and U(O-2p) = 5.08 eV.
The calculated atomic charges are far from the formal valence charges (Sr:+2, Ru:+4, O:−2). The sum of the partial charges, integrated on a real space mesh, differs by 10 electrons from the total number of electrons per unit formula. The atomic charges of oxygen are smaller than those of YTiO. The ACBN0 and HSE06 functionals tend to concentrate both the charge and the spin on the Ru ion, making the system slightly more ionic in character. The total magnetization is close to 2 and is compatible with a Ru ion in the configuration. The atomic magnetic moments on the oxygen atoms are about 0.2 , one order of magnitude larger than those of YTiO. With respect to YTiO, the smaller Bader charge and the larger magnetic moment on oxygen can be explained by the larger overlap between the O-2p orbitals and the Ru-4d, which has a larger spatial extent than the Ti-3d orbital.
Differently from YTiO the spin density map, atomic charges, and spin populations are nearly insensitive to the value of U for the Ru ion. The spin density map shows only regions of positive spin density.
The electronic densities of states of SrRuO
are shown in Figure 5
. At U = 0, the system is a ferromagnetic metal, and upon increasing U(Ru), the system turns into a half-metal with a gap in the spin-up channel opening for U > 2 eV. Consequently, the total magnetic moment reaches the value of 2
in the half-metallic state. The system is also half-metal with the ACBN0 method (U(Ru) = 2.06 eV), but the bandwidth is larger than in the DFT+U calculations. The half-metallic character is consistent with previous reports [38
], as well as with quasi-particle self-consistent GW (sc-QSGW) calculations [39
]. The transition from the metal to the half-metal has important consequences for the structural parameters. In Figure 6
, we report the equilibrium volume of SrRuO
as a function of U(Ru-4d). For small values of U, the agreement with respect to experiments is rather good. However, as the system becomes half-metallic, the volume increases away from the experimental value. ACBN0 underestimates the equilibrium volume by 1.88%.
The spin density and the VSCC are reported in Figure 7
. The analysis of the laplacian of the charge density shows that the there are eight VSCC points located ∼0.44 Å from the Ru ion. The VSCCs are arranged as a cube whose vertexes point in the directions of the center of the faces of the SrO
octahedron. As already shown in Table 2
, the spin density depends weakly on the U values and on the choice of the functional. The value of the laplacian at the VSCC goes from −8.18 au
at U = 0 to −8.30 au
at U = 5 eV,
for ACBN0, and
for HSE. Thus, the charge and spin concentrations around the Ru ion depend rather weakly on the DFT methods that we have used.