Rare-earth–boron bonding and 4f state trends in $\mathcal{R}{\mathrm{B}}_4$ tetraborides

The B–B bonding, boron–rare earth coupling, and the changes in $4f$ states across the lanthanide series in $R{\mathrm{B}}_4$ ($R=\text{rare}$ earth) compounds are studied using the correlated band theory $\mathrm{LDA}+U$ method. A set of boron bonding bands that are well separated from the antibonding bands can be identified. Separately, the “dimer B” $2p_z$ orbital is nonbonding (viz., graphite and $\mathrm{Mg}{\mathrm{B}}_2$), but mixes strongly with the metal $4d$ or $5d$ states that form the conduction states. The bonding bands are not entirely filled even for the trivalent compounds (thus the cation $d$ bands have some filling), which accounts for the lack of stability of this structure when the cations are divalent (with more bonding states unfilled). The trends in the mean $4f$ level for both majority and minority, and occupied and unoccupied, states are presented and interpreted.

Figure 1

(Color online) Structure of $R{\mathrm{B}}_4$ viewed from along the $c$ direction. The large metal ion spheres (red) lie in $z=0$ plane. Apical B1 atoms (small black) lie in $z\simeq 0.2$ and $z\simeq 0.8$ planes. Lightly shaded (yellow) dimer B2 and equatorial B3 (dark, blue) atoms lie in $z=0.5$ plane. The sublattice of $R$ ions is such that each one is a member of two differently oriented $R_4$ squares, and of three $R_3$ triangles.

Figure 2

(Color online) Plot of experimental lattice constants of $R{\mathrm{B}}_4$ vs position in the Periodic Table (atomic number), showing a lanthanide contraction of about 5% for $a$ and 3% for $c$. The smooth lines show a quadratic fit to the data.

Figure 3

Band structure of $\mathrm{Y}{\mathrm{B}}_4$ (top panel) and $\mathrm{Ca}{\mathrm{B}}_4$ (lower panel) within $6\mathrm{eV}$ of the Fermi level along high symmetry directions, showing the gap that opens up around $E_F$ (taken as the zero of energy) throughout much of the top and bottom portions of the tetragonal Brillouin zone. Notice the lack of dispersion along the upper and lower zone edges $R\text{−}A\text{−}R$ ($k_z=\pi ∕c$, and either $k_x$ or $k_y$ is $\pi ∕a$). Note also that, due to the nonsymmorphic space group, bands stick together in pairs along $X\text{−}M$ (the zone “equator”) and along $R\text{−}A$ (top and bottom zone edges).

Figure 4

(Color online) Projected density of states per atom of each of the B atoms for $\mathrm{Y}{\mathrm{B}}_4$. The curves are shifted to enable easier identification of the differences. The $\mathrm{B}2p$ bonding-antibonding gap can be identified as roughly from $-1\mathrm{eV}\text{to}4–5\mathrm{eV}$.

Figure 5

(Color online) Enlargement of the partial densities of states of $\mathrm{Y}4d$ and $\mathrm{B}2p$ states (per atom) near the Fermi level. The states at the Fermi level, and even for almost $2\mathrm{eV}$ below, have strong $4d$ character. The apical B2 character is considerably larger than that of B1 or B3 in the two peaks below $E_F$, but is only marginally larger exactly at $E_F$.

Figure 6

(Color online) Fermi surfaces of $\mathrm{Y}{\mathrm{B}}_4$. Light (yellow) surfaces enclose holes, and dark (red) surfaces enclose electrons. The full tetragonal Brillouin zone is pictured, the $\Gamma$ point being in the center of each figure, the $R$ point is the midpoint along the horizontal edges, and the $A$ point lies at the corner (see Sec. 3 for specification of high symmetry points). The wide gap throughout the top and bottom edges of the zone accounts for the lack of Fermi surfaces except for the one “football” centered at the $Z$ point at the center of the upper and lower faces (lower left panel).

Figure 7

The full valence band structure of $\mathrm{Dy}{\mathrm{B}}_4$, and up to $5\mathrm{eV}$ in the conduction bands. This plot is for ferromagnetic alignment of the spin moments, with the solid bands being majority and the lighter, dashed lines showing the minority bands. The flatbands in the $-4.5\text{to}-11\mathrm{eV}$ are $4f$ eigenvalues as described by the $\mathrm{LDA}+U$ method.

Figure 8

Band structure of $\mathrm{Dy}{\mathrm{B}}_4$ on a fine scale around the Fermi energy, see Fig. 7. The exchange splitting (between solid and dashed bands) gives a direct measure of the coupling between the polarized Dy ion and the itinerant bands (see text).

Figure 9

(Color online) Calculated mean $4f$ eigenvalue position (symbols connected by lines) with respect to $E_F$, and the spread in eigenvalues, of $R{\mathrm{B}}_4$ compounds. The smooth behavior from Pr to Tm (except for Eu) reflects the common trivalent state of these ions. Eu and Yb are calculated to be divalent and deviate strongly from the trivalent trend. Ce has a higher valence than 3, accounting for its deviation from the trivalent trend.