Merits and limits of the modified Becke-Johnson exchange potential

The modified Becke-Johnson exchange potential [F. Tran and P. Blaha, Phys. Rev. Lett. 102, 226401 (2009)] (TB-mBJ) is tested on various types of solids which are difficult to describe theoretically: nonmagnetic semiconducting transition-metal oxides and sulfides, metals (Fe, Co, Ni, and Cu), and (anti)ferromagnetic insulators (e.g., YBa${}_2$Cu${}_3$O${}_6$). The results for the band gap and other quantities such as the magnetic moment or electric field gradient are analyzed in detail, in particular to have a better understanding of the mechanism which leads to improved (or sometimes worse) results with the TB-mBJ potential compared to the standard local density and generalized gradient approximations.

Figure 1

Variation of the TB-mBJ band gap and Cu EFG in Cu${}_2$O as functions of $c$. The experimental values for the band gap (Ref. 43) and the EFG (Ref. 54),as well as the band gap and the Cu EFG from PBE, are also shown.

Figure 2

Electron density (in $e$/bohr${}^3$) of the VBM (a) and the CBM (b) of SrTiO${}_3$. The image shows a cross section through the TiO${}_6$ octahedron with Ti in the center and O on the top, bottom, left, and right.

Figure 3

(a) Difference between the TB-mBJ and the PBE XC potentials (in Ry) in SrTiO${}_3$. (b) BR term of TB-mBJ potential [Eq. (4)]. (c) $\sqrt{t/\rho }$-term of TB-mBJ potential [Eq. (4)].

Figure 4

Electron density (in $e$/bohr${}^3$) of Cu${}_2$O in the (011) plane of the VBM (a) and the CBM (b).

Figure 5

Comparison between PBE and TB-mBJ band structure with VLEED measurements for Cu (Ref. 55).

Figure 6

Spin-up and spin-down density of states (in states/spin/cell/eV) of bcc Fe for PBE and TB-mBJ potentials.

Figure 7

Comparison of the NiO band structure calculated with PBE and TB-mBJ. The gray regions indicate the states at the valence and conduction bands whose densities are plotted in Fig. 8.

Figure 8

(a) Spin-up electron density in NiO (001) of states in the indicated energy regions in Fig. 7. (a) VBM with PBE. (b) CBM with PBE. (c) VBM with TB-mBJ. (d) CBM with TB-mBJ. (e) Density scale (in $e$/bohr${}^3$). (f) Position of atoms for (a)–(d).

Figure 9

Difference between the TB-mBJ and PBE spin-up XC potentials (in Ry) in the (001) plane of NiO. The positions of atoms are indicated in Fig. 8f.

Figure 10

The two contributions to the TB-mBJ exchange potential (in Ry) in NiO in the (001) plane: (a) BR and (b) $\sqrt{t/\rho }$. The positions of atoms are indicated in Fig. 8f.

Figure 11

Band structures of the majority spin in EuO obtained by different methods.

Figure 12

Spin-up electron density according to PBE at the $X$-point for a state at $-0.06$ eV: (a) in the (011) plane (Eu atoms on top and bottom and O atoms at the center) and (b) 3D image with 0.03 $e$/bohr${}^3$ isosurface. Spin-up electron density for a state at $-1.3$ eV: (c) in the (011) plane and (d) 3D image with 0.0084 $e$/bohr${}^3$ isosurface.

Figure 13

Difference between the TB-mBJ XC potentials (in Ry) obtained with $c=1.38$ and $c=c_{\mathrm{opt}}=1.97$ in the (011) plane of EuO.