Revealing the Nature of Antiferroquadrupolar Ordering in Cerium Hexaboride: ${\mathrm{CeB}}_6$

The cerium hexaboride (${\mathrm{CeB}}_6$) $f$-electron compound displays a rich array of low-temperature magnetic phenomena, including a “magnetically hidden” order, identified as multipolar in origin via advanced x-ray scattering. From first-principles electronic-structure results, we find that the antiferroquadrupolar (AFQ) ordering in ${\mathrm{CeB}}_6$ arises from crystal-field splitting and yields a band structure in agreement with experiments. With interactions of $p$ electrons between Ce and ${\mathrm{B}}_6$ being small, the electronic state of ${\mathrm{CeB}}_6$ is suitably described as $\mathrm{Ce}(4f^1{)}^{3+}(e^{-})({\mathrm{B}}_6{)}^{2-}$. The AFQ state of orbital spins is caused by an exchange interaction induced through spin-orbit interaction, which also splits the $\mathit{J}=5/2$ state into a ${\mathrm{\Gamma }}_8$ ground state and a ${\mathrm{\Gamma }}_7$ excited state. Within the smallest antiferromagnetic (AFM) (111) configuration, an orbital-ordered AFQ state appears during charge self-consistency, and it supports the appearance of a “hidden” order. Hydrostatic pressure (either applied or chemically induced) stabilizes the AFM (AFQ) states over a ferromagnetic one, as observed at low temperatures.

Figure 1

For FM ${\mathrm{CeB}}_6$, (a) $Pm\overline{3}m$ crystal structure, (b) (100) Fermi surface, and (c) bands (with SOC) along $M\text{−}X\text{−}M$ and $X\text{−}\mathrm{\Gamma }\text{−}X$, with $p$, $d$, and $f$ states identified, and density of states (DOS). Inset: Brillouin zone and high-symmetry points.

Figure 2

For AFM ${\mathrm{CeB}}_6$, (a) $\mathrm{DFT}+\mathrm{U}$ dispersion along $M\text{−}X\text{−}M$ and $X\text{−}\mathrm{\Gamma }\text{−}X$. (b) Schematic of AFM state. (c) (100) constant energy surface plot at $-0.30\mathrm{eV}$ below Fermi-energy (${\mathrm{\Gamma }}_7$ and ${\mathrm{\Gamma }}_8$ splitting is observed in $-0.20$ to $-0.35\mathrm{eV}$ energy range). (d) $\mathrm{Ce}-f$ DOS (matches experiment) [28]. (e) Energy-level diagram with SOC and crystal-field splitting.

Figure 3

For AFM ${\mathrm{CeB}}_6$, we show (a) (111)-projected Ce1 ($f_{xyz}$) and Ce2 ($f_{z(5z^2-3r^2)}$) orbitals, (b) total charge density in (001) plane, and (c) schematic of AFM Ce configuration. Together these show distinct AFM and orbital arrangement at the Ce1 and Ce2 sites, indicating the underlying AFQ order.

Figure 4

For ${\mathrm{CeB}}_6$, FM-AFM energy difference vs pressure, which alters hybridization between $4f$ bands and dispersive $5d$ bands (Inset: AFM constant energy surface (hole states) at $-0.30\mathrm{eV}$ below ${\mathit{E}}_{\text{Fermi}}$ at 0 and 21 GPa. Holelike states appear at $\mathrm{\Gamma }$ and $X$ points (see Fig. S8) [28].