Screened hybrid functional applied to 3$d^0$$\to$3$d^8$ transition-metal perovskites La$M$O${}_3$ ($M$ = Sc–Cu): Influence of the exchange mixing parameter on the structural, electronic, and magnetic properties

We assess the performance of the Heyd-Scuseria-Ernzerhof (HSE) screened hybrid density functional scheme applied to the perovskite family La$M$O${}_3$ ($M$ = Sc–Cu) and discuss the role of the mixing parameter $\alpha$ [which determines the fraction of exact Hartree-Fock exchange included in the density functional theory (DFT) exchange-correlation functional] on the structural, electronic, and magnetic properties. The physical complexity of this class of compounds, manifested by the largely varying electronic characters (band/Mott-Hubbard/charge-transfer insulators and metals), magnetic orderings, structural distortions (cooperative Jahn-Teller–type instabilities), as well as by the strong competition between localization/delocalization effects associated with the gradual filling of the $t_{2g}$ and $e_g$ orbitals, symbolize a critical and challenging case for theory. Our results indicate that HSE is able to provide a consistent picture of the complex physical scenario encountered across the La$M$O${}_3$ series and significantly improve the standard DFT description. The only exceptions are the correlated paramagnetic metals LaNiO${}_3$ and LaCuO${}_3$, which are found to be treated better within DFT. By fitting the ground-state properties with respect to $\alpha$, we have constructed a set of “optimum” values of $\alpha$ from LaScO${}_3$ to LaCuO${}_3$: it is found that the optimum mixing parameter decreases with increasing filling of the $d$ manifold (LaScO${}_3$: 0.25; LaTiO${}_3$ and LaVO${}_3$: 0.10–0.15; LaCrO${}_3$, LaMnO${}_3$, and LaFeO${}_3$: 0.15; LaCoO${}_3$: 0.05; LaNiO${}_3$ and LaCuO${}_3$: 0). This trend can be nicely correlated with the modulation of the screening and dielectric properties across the La$M$O${}_3$ series, thus providing a physical justification to the empirical fitting procedure. Finally, we show that by using this set of optimum mixing parameter, HSE predict dielectric constants in very good agreement with the experimental ones.

Figure 1

Schematic representation of the typical magnetic orderings for the perovskites.

Figure 2

The structures of perovskite oxides studied in this paper. $P$${}_{nma}$ for LaScO${}_3$, LaTiO${}_3$, LaCrO${}_3$, LaMnO${}_3$, and LaFeO${}_3$, $P$${}_{2_1/b}$ for LaVO${}_3$, $R_{\overline{3}c}$ for LaCoO${}_3$ and LaNiO${}_3$, and $P$${}_{4/m}$ for LaCuO${}_3$, respectively. The large (green), medium-sized (blue), and small (red) balls denote La, $M$, and O atoms, respectively. Plot done using the vesta visualization program (Ref. 90).

Figure 3

Experimental trend of volume ($V$), average tilting angle ($\theta$), and JT distortions ($Q$${}_2$ and $Q$${}_3$) for the La$M$O${}_3$ series from $M$ $=$ Sc to Cu. The corresponding trend of the tolerance factor $t$ $=$ ($R_A$+$R_{\text{O}})/\sqrt{2}$($R_M$+$R_{\text{O}}$), and $R_M$ is also shown.

Figure 4

$l$-projected DOS of nonmagnetic LaScO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-35, HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 5

Band structure of LaScO${}_3$ computed at HSE level ($\alpha =0.25$) using the optimized structure.

Figure 6

$l$-projected DOS of AFM-G ordered LaTiO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-35, HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 7

Band structure of LaTiO${}_3$ computed at HSE level ($\alpha =0.15$) using the optimized structure.

Figure 8

$l$-projected DOS of AFM-C ordered LaVO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-35, HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 9

Band structure of LaVO${}_3$ computed at HSE level ($\alpha =0.15$) using the optimized structure.

Figure 10

$l$-projected DOS of AFM-G ordered LaCrO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-35, HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 11

Band structure of LaCrO${}_3$ computed at HSE level ($\alpha =0.15$) using the optimized structure.

Figure 12

$l$-projected DOS of AFM-G ordered LaMnO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-35, HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 13

Band structure of LaMnO${}_3$ computed at HSE level ($\alpha =0.15$) using the optimized structure.

Figure 14

$l$-projected DOS of AFM-G ordered LaFeO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-35, HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 15

Band structure of LaFeO${}_3$ computed at HSE level ($\alpha =0.15$) using the optimized structure.

Figure 16

$l$-projected DOS of nonmagnetic LaCoO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-35, HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 17

Band structure of LaCoO${}_3$ computed at HSE level ($\alpha =0.05$) using the optimized structure.

Figure 18

$l$-projected DOS of FM LaNiO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-35, HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 19

Band structure of LaNiO${}_3$ computed at PBE level using the optimized structure.

Figure 20

$l$-projected DOS of nonmagnetic LaCuO${}_3$ with experimental (left) and relaxed (right) structures based on PBE and HSE (HSE-25, HSE-15, HSE-10) functionals. The shadow area indicates the total DOS.

Figure 21

Band structure of LaCuO${}_3$ computed at PBE level using the optimized structure.

Figure 22

Summary of the MARE for the structural properties (top panel), band gap $\Delta$ (middle panel), and magnetic moment $m$ (lower panel), at PBE and HSE levels. For the band gap $\Delta$ and the magnetic moment $m$, the MARE are indicated by the numbers associated to each bar. A few specifications for the labels “OK” and “wrong”: (i) LaScO${}_3$, $m$: all methods correctly predict a nonmagnetic ground state; (ii) LaCoO${}_3$, $m$: all methods correctly predict a zero magnetic moment; (iii) LaNiO${}_3$, $\Delta$: PBE is the only approach which correctly finds a metallic solution; (iv) LaNiO${}_3$, $m$: all methods wrongly predict a magnetic ground state; (v) LaCuO${}_3$, $\Delta$: all methods correctly predict a metallic solution; (vi) LaCuO${}_3$, $m$: PBE and HSE (0.05, 0.15, and 0.25) correctly predict a zero magnetic moment, whereas HSE 0.35 wrongly stabilized a magnetic ground state.

Figure 23

Graphical interpretation of the optimum values of $\alpha$ listed in Table showing the correlation between the HSE fitted parameters (HSE fit), the inverse dielectric constant relation ($1/{\epsilon }_{\infty }$), and ${\alpha }_{\mathrm{opt}}^{\Delta }$ [Eq. (8)]. The light-gray squares represent the $1/{\epsilon }_{\infty }$ values shifted by 0.07. This shift roughly reflects the amount of screening incorporated in HSE via the screening factor $\mu$ as compared to fully unscreened PBE0 (see text).

Figure 24

Change of the band gap as a function of $\alpha$ (think lines) and optimum values of $\alpha$ (circles) obtained throughout the practical formula ${\alpha }_{\mathrm{opt}}^{\Delta }$=$\frac{{\Delta }^{\mathrm{Expt}}-{\Delta }^{\mathrm{semilocal}}}{k}$ [Eq. (8)].

Figure 25

Trend of selected structural (volume $V$, tilting angle $\theta$, and JT distortions $Q$${}_2$ and $Q$${}_3$), electronic (band gap $\Delta$), magnetic (magnetic moment $m$), and dielectric constant (${\epsilon }_{\infty }$) quantities along the La$R$O${}_3$ series from $M$ $=$ Sc to Cu. We also show the trend of the tolerance factor $t$ $=$ ($R_A$+$R_{\text{O}})/\sqrt{2}$($R_M$+$R_{\text{O}}$), where $R_A$, $R_M$, and $R_{\text{O}}$ indicate the ionic radius for La, $M$ $=$ Sc–Cu, and O, respectively, as well as $R_M$. For LaTiO${}_3$ we used $\alpha =0.1$. The character of the insulating gap is also indicated (BI $=$ band insulator, CT $=$ charge transfer, MH $=$ Mott-Hubbard, CT/MH $=$ mixed CT and MH character).

Figure 26

Summary of the HSE electronic dispersion relations showing the complete trend from the band insulator LaScO${}_3$ to metallic LaCuO${}_3$. The thick (red) lines demarcate the $d$ bands responsible for the observed orbital ordering in LaTiO${}_3$ ($t_{2g}$), LaVO${}_3$ ($t_{2g}$), and LaMnO${}_3$ ($e_g$).

Figure 27

Isosurface of the magnetic orbitally ordered charge density for LaTiO${}_3$, LaVO${}_3$, and LaMnO${}_3$ associated with the topmost occupied bands highlighted in the insets of Fig. 26. Light (yellow) and dark (blue) areas represent spin down and spin up, respectively, indicating the different types of spin orderings in LaTiO${}_3$ (G-type), LaVO${}_3$ (C-type), and LaMnO${}_3$ (A-type). Top panel: three-dimensional view; bottom panel: projection onto the $xy$ plane.

Figure 28

Isosurface of the nonorbitally ordered magnetic charge density for LaCrO${}_3$ (G-type) and LaFeO${}_3$ (A-type) associated with the topmost occupied bands (see Fig. 26). Light (yellow) and dark (blue) areas indicate spin down and spin up, respectively.

Figure 29

Comparison between experimental (blue squares) and calculated (red full lines) valence and conduction band spectra at the optimum value of the $\alpha$ parameter: (i) LaScO${}_3$: $\alpha =0.25$; (ii) LaTiO${}_3$: $\alpha =0.15$; (iii) LaVO${}_3$: $\alpha =0.15$; (iv) LaCrO${}_3$: $\alpha =0.15$; (v) LaMnO${}_3$: $\alpha =0.15$; (vi) LaFeO${}_3$: $\alpha =0.15$; (vii) LaCoO${}_3$: $\alpha =0.05$; (viii) LaNiO${}_3$: $\alpha =0$; (ix) LaCuO${}_3$: $\alpha =0$. The calculated and measured spectra have been aligned by overlapping the valence band maxima and conduction band minima. The experimental data are taken from the collection of spectra presented in Ref. 78, originally published in separate articles: (i) LaScO${}_3$: Ref. 7; (ii) LaTiO${}_3$: Ref. 210; (iii) LaVO${}_3$: Ref. 211; (iv) LaCrO${}_3$: Ref. 212; (v) LaMnO${}_3$: Ref. 144; (vi) LaFeO${}_3$: Ref. 190; (vii) LaCoO${}_3$: Ref. 145; (viii) LaNiO${}_3$: Ref. 213; and (ix) LaCuO${}_3$: Ref. 200.