Acoustic phonon softening and lattice instability driven by on-site $f$-$d$ hybridization in CeCoSi

Abstract

Soft phonon modes in tetragonal CeCoSi, which undergoes a structural transition at $T_0=12$ K followed by antiferromagnetic order at $T_{\text{N}}=9.5$ K, have been investigated using high-resolution inelastic x-ray scattering. Pronounced softening was detected in the transverse acoustic modes corresponding to the $(yz+zx)$-type monoclinic distortion, consistent with the experimentally determined triclinic structure. Remarkably, the softening persists up to the zone boundary along (0, 0, $q$), indicating a short correlation length of the lattice instability. This instability, characterized by a Curie-type strain susceptibility, is interpreted as a consequence of the on-site $4f$-$5d$ hybridization, which is intrinsic to this crystal structure due to the lack of inversion symmetry at the two Ce sites.

Figures (9)

• Figure 1: (a) Crystal structure of CeCoSi drawn using VESTA Momma11; $a$=4.057 Å, $c$=6.987 Å, $z$(Ce)=0.678, Ce site symmetry $4mm$Tanida19. The cross at the center of the unit cell indicates one of the inversion centers. (b) Schematic illustration of the triclinic lattice distortion below $T_0=12$ K.
• Figure 2: Phonon-dispersion relations of Ce acoustic and optical branches along $\Delta$: $(q, 0, 0)$ and $\Lambda$: $(0, 0, q)$ at 300 K (top) and 15 K (bottom). The solid lines are the results of first-principles calculations, assuming that Ce-$4f$ is localized. The dashed lines are the fits to the data (see text). The subscript of the mode index denotes the irreducible representation and the direction of atomic displacement SM. Arrows indicate the data points whose $T$-dependences are shown in Fig. \ref{['fig:SpecH0L']}(d,e).
• Figure 3: (a,b,c) Temperature dependence of the phonon excitation spectra at $\bm{Q}=(0.07, 0, 9)$, $(0, 0, 8.5)$, $(0.01, 0, 8.36)$, respectively. The solid lines are the fits to the data. The dashed lines represent the inelastic contributions. (d,e) Temperature dependence of the excitation energies for the transverse acoustic modes $\Delta_{3,z}$ and $\Lambda_{5,xy}$, respectively. The solid lines are fits assuming a Curie-Weiss temperature dependence of the strain susceptibility. (f) Intensity contour map of the elastic ($\Delta E=0$) component at 15 K and 100 K. Contour levels range from 0.08 to 0.5 in equal steps. The $+$ symbols indicate the centers of the analyzers.
• Figure 4: (a) Schematic illustration of the energy levels including local $4f$-$5d$ hybridization and monoclinic distortion. The splitting of the $f$ levels in the hybridized state is exaggerated for clarity. (b) Calculated charge distribution of the $4f$-$5d$ hybridized ground state involving Ce-1 and Ce-2.
• Figure S1: Six phonon modes of the Ce atoms for wave vectors along (a) $\Delta: (q, 0, 0)$, (b) $\Lambda: (0, 0, q)$, and (c) $\Sigma: (q, q, 0)$. The subscript in each mode index represents the irreducible representation and the direction of atomic displacement.