Establishing best practices to model the electronic structure of ${\mathrm{CuFeO}}_2$ from first principles

The cuprous delafossite, ${\mathrm{CuFeO}}_2$, has received significant attention in recent years as a potential photocathode material in photoelectrochemical water-splitting cells. Presented herein is an investigation of the electronic structure of ${\mathrm{CuFeO}}_2$ in the framework of density functional theory. We have benchmarked three of the most popular formulations for the treatment of the electron exchange and correlation interactions, highlighting their strengths and weaknesses in predicting electronic structures compatible with the available spectroscopic measurements. Although some features are correctly reproduced by the simplest approach, which is based on the generalized gradient approximation, this fails in describing the fundamental semiconducting character of the material. The introduction of the fully self-consistent Hubbard $U$ correction in the exchange correlation functional accounts explicitly for the on-site Coulomb interaction among localized $d$ electrons, thereby opening a gap in the band structure. However, our results indicate that the $U$ correction disrupts the crystal-field splitting of the $t_{2g}$ and $e_g$ states, resulting in an inaccurate description of the conduction-band edge. We provide a qualitative and quantitative analysis to explain why the $t_{2g}$ and $e_g$ states behave differently when the Hubbard correction is switched on. We find that best practice for accurate, yet computationally viable, simulations of CFO makes use of hybrid functionals, where the fraction of exact exchange is not arbitrarily selected but tuned according to the static dielectric constant of the material. In this case, theoretical predictions are found to be in excellent agreement with experimental results.

Figure 1

(a) Crystal structure of ${\mathrm{CuFeO}}_2$ and (b) rhombohedral unit cell. Cu, Fe, and O atoms are represented by blue, yellow, and red spheres, respectively.

Figure 2

Spin-polarized projected density of states of ${\mathrm{CuFeO}}_2$, computed with the PBE exchange-correlation functional. The zero of the energy is set to the Fermi energy.

Figure 3

Minority-spin band structure. The different colors represent the main character of the band, according to the PDOS: $\mathrm{O}s$ (gold), $\mathrm{O}p$ (red), $\mathrm{Cu}d$ (green), $\mathrm{Fe}{t}_{2g}$ (dark blue), $\mathrm{Fe}{e}_g$ (light blue), and $\mathrm{Cu}s$ (black).

Figure 4

Representative MLWFs for the $t_{2g}$ and the $e_g$ set inside the unit cell. In the row (a) the bands with $t_{2g}$ ($e_g$) are included separately in the summation. We then add in the summation the Kohn-Sham states with a prevalent $\mathrm{O}p$ (b), $\mathrm{Cu}d$ (c), and $\mathrm{O}s$ (d) character.

Figure 5

On-site energies of the MLWFs corresponding to the $t_{2g}-e_g$ states, obtained including different sets of Kohn-Sham states in the summation in Eq. (1).

Figure 6

Projected density of states at the $\mathrm{PBE}+U$ level, computed with the self-consistent value of $U$.

Figure 7

Projected density of states onto the $\mathrm{Fe}{t}_{2g}$ (red line) and $e_g$ (green line) $d$ states for different values of the applied Hubbard $U$. The green dots represent the experimental data of $\mathrm{Fe}L\alpha$ edge x-ray emission spectrum, taken from Ref. [23].

Figure 8

On-site energies of the MLWFs corresponding to the $t_{2g}$ and $e_g$ states for three different values of the Hubbard $U$.

Figure 9

Projected density of states for different values of the fraction of exact exchange, using the PBE0 (a) and the HSE hybrid functionals (b). The blue dashed line represents the experimental data of $\mathrm{Fe}L\alpha$ edge x-ray emission spectrum, taken from Ref. [23].

Figure 10

Band gap (upper panel) and crystal-field splitting (lower panel) in CFO, computed with hybrid functionals using a variable fraction of exact exchange.

Figure 11

Projected density of states computed using the meta-GGA SCAN functional.