Electronic properties of lanthanide oxides from the $GW$ perspective

A first-principles understanding of the electronic properties of $f$-electron systems is currently regarded as a great challenge in condensed-matter physics because of the difficulty in treating both localized and itinerant states on the same footing by the current theoretical approaches, most notably density-functional theory (DFT) in the local-density or generalized gradient approximation (LDA/GGA). Lanthanide sesquioxides (Ln${}_2$O${}_3$) are typical $f$-electron systems for which the highly localized $f$ states play an important role in determining their chemical and physical properties. In this paper, we present a systematic investigation of the performance of many-body perturbation theory in the $GW$ approach for the electronic structure of the whole Ln${}_2$O${}_3$ series. To overcome the major failure of LDA/GGA, the traditional starting point for $GW$, for $f$-electron systems, we base our $GW$ calculations on Hubbard $U$ corrected LDA calculations (LDA+$U$). The influence of the crystal structure, the magnetic ordering, and the existence of metastable states on the electronic band structures are studied at both the LDA+$U$ and the $GW$ level. The evolution of the band structure with increasing number of $f$ electrons is shown to be the origin for the characteristic structure of the band gap across the lanthanide sesquioxide series. A comparison is then made to dynamical mean-field theory (DMFT) combined with LDA or hybrid functionals to elucidate the pros and cons of these different approaches.

Figure 1

Crystal unit cells of hexagonal, cubic, and monoclinic structures taken by the Ln${}_2$O${}_3$ series.

Figure 2

Density of states of Eu${}_2$O${}_3$ in different crystal structures calculated by LDA+$U$. For the hexagonal phase, we use the structural parameters determined by the linear extrapolation of those of early Ln${}_2$O${}_3$ as explained in Sec. 3a. For the monoclinic and cubic bixbyite phases, we use experimental structures from Refs. 93 and 94, respectively.

Figure 3

Density of states of Ce${}_2$O${}_3$ from LDA+$U$ and $G{W}_0$@LDA+$U$ in the ferromagnetic and antiferromagnetic phases. The spin-up and -down DOS in the AFM phase are identical, and therefore only one is shown.

Figure 4

Density of states of Ce${}_2$O${}_3$ from $G{W}_0$@LDA+$U$ with $U=5.4$ eV in different metastable states. To see the differences more clearly, the data from M1 and M2 are shifted horizontally by 0.15 and 0.6 eV, respectively, so that the upper edges of their O-$2p$ valence band match that of M0.

Figure 5

Density of states of Ce${}_2$O${}_3$, Eu${}_2$O${}_3$, and Er${}_2$O${}_3$ from $G_0{W}_0$ and $G{W}_0$@LDA+$U$.

Figure 6

The band gaps of Ce${}_2$O${}_3$, Eu${}_2$O${}_3$, and Er${}_2$O${}_3$ from LDA+$U$ and $G{W}_0$@LDA+$U$ as a function of $U$.

Figure 7

Density of states of Ce${}_2$O${}_3$, Eu${}_2$O${}_3$, and Er${}_2$O${}_3$ from LDA+$U$ and $G{W}_0$@LDA+$U$ with different $U$ (in units of eV).

Figure 8

Band gaps of the whole Ln${}_2$O${}_3$ series from LDA+$U$ and $G{W}_0$@LDA+$U$ are compared to experimental band gaps obtained from optical absorption (Ref. 104). Experimental values of compounds that crystallize preferentially in the hexagonal structure are denoted by filled circles. The upper part of the figure illustrates the position of occupied and unoccupied $4f$ states with respect to the valence and conduction bands schematically. A more complete comparison between different theoretical approaches and experiment is presented in Table .

Figure 9

Density of states of the Ln${}_2$O${}_3$ series from LDA+$U$ (dashed lines) and $G{W}_0$@LDA+$U$ (solid lines). The shaded regions illustrate the Ln-$4f$ character in the $G{W}_0$ DOS.

Figure 10

Density of states of Ce${}_2$O${}_3$ in LDA+$U$, $G{W}_0$@LDA+$U$, LDA+DMFT with self-consistency in the charge density [LDA+DMFT(SC)] (Ref. 23), and the hybrid density functional plus DMFT approach (HYF+DMFT) (Ref. 22) as well as experimental spectral data from XPS+BIS (Ref. 111), all aligned at the upper edge of the O-$2p$ valence band. We note that the small peak $\sim$3 eV above $f^{\mathrm{occ}}$ in the XPS+BIS data is likely due to unoccupied $f$ states of CeO${}_2$ contamination in the Ce${}_2$O${}_3$ sample as evidenced by Fig. 3 in Ref. 111. The edge of the conduction band can therefore be estimated to appear at around 4 eV above the $f^{\mathrm{occ}}$ peak, which then gives an experimental $p$-$d$ gap of $\sim$6 eV.