Investigation of the phonon dispersion associated with superlattice reflections in the ${\mathrm{BiS}}_2$-based superconductor ${\mathrm{LaBiS}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$

The phonon dispersion of a $\mathrm{Bi}{\mathrm{S}}_2$-based superconductor $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$ is investigated by first-principles calculations and inelastic x-ray scattering experiments. The origin of superlattice (SL) reflections arising from transverse-type lattice modulation, which were recently reported in [J. Kajitani et al., J. Phys. Soc. Jpn. 90, 103601 (2021)], is discussed in terms of lattice dynamics. Our first-principles calculations of phonon dispersion and the Fermi surfaces (FSs) demonstrate that the phonon mode corresponding to the transverse-type lattice modulation is unstable, and the propagation vector corresponding to the SL reflections is close to the FS nesting vector, which suggests that the phonon softening originates from the FS nesting. Against these calculated expectations, measured phonon dispersion in $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$ along the Z-A direction, where the SL point is located, shows no remarkable temperature dependence, and there are no steeply declining branches accompanied with a softening around the SL point. Based on these results, we discuss the two possibilities for the transverse lattice modulation in $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$: the order-disorder-type structural transition and the displacive structural transition with an overdamped mode, for both of which the local structure distortion or the short-range correlation within the $\mathrm{Bi}{\mathrm{S}}_2$ plane would be essential.

Figure 1

Crystal structure of $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$. The rectangle indicates the unit cell. The image of the crystal structure is generated using the vesta software developed by Monma and Izumi [24].

Figure 2

(a) Schematic Brillouin zone of $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$ with high-symmetry points. SL shows a reciprocal lattice point where the superlattice reflection appears below $\sim 260$ K (see the main text for detail). For visibility, the $c$* axis is elongated. Calculated phonon dispersion of (b) $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$ and of (c) $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.8}{\mathrm{F}}_{0.2}$. In the calculation, a tetragonal unit cell ($P4/nmm$) is employed. Red curves in panels (b) and (c) are “modified” phonon dispersions for the soft modes to have very small real frequencies, to confirm whether the soft mode actually exists by comparing with the experimental results (see the main text and the Supplemental Material [25] for more detail).

Figure 3

Dynamical structure factors of $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{1-x}{\mathrm{F}}_x$ at room temperature for (a) longitudinal $\mathit{e}\parallel \left[110\right]$ geometry, (b) transverse $\mathit{e}\parallel \left[1\overline{1}0\right]$ geometry, and (c) transverse $\mathit{e}\parallel \left[001\right]$ geometry, respectively, where $\mathit{e}$ is a polarization vector of a vibration. Left panels: measured dynamical structure factor of $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$. White circles are peak positions of spectra obtained by the fitting analyses. Middle and right panels: calculated dynamical structure factors for $x=0.2$ and $x=0.5$, respectively. To compare with the experimental results, the calculated dynamical structure factors are modified. For clarity, the color gauge showing the strength of intensity is adjusted in each panel.

Figure 4

[(a)–(e)] Measured IXS spectra of $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$ along the Z-A direction at room temperature. Data were obtained in the transverse $\mathit{e}\parallel \left[1\overline{1}0\right]$ geometry, where $\mathit{e}$ is a polarization vector of a vibration. Black arrows indicate peak positions of each phonon excitation obtained by the fitting analyses. [(f)–(j)] Calculated IXS spectra of $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.8}{\mathrm{F}}_{0.2}$ for the same mode as in panels (a)–(e) at room temperature (we compare the experimental data of $x=0.5$ with the calculation results of our model). Note that the minus sides of IXS spectra indicate the anti-Stokes scatterings, which entirely differ from the minus side of Fig. 2, because the phonon modes on the minus side in Fig. 2 are not observable vibration modes. Black and orange arrows indicate the identical peaks measured by the experiments and the predicted peaks of the soft modes.

Figure 5

Temperature dependence of IXS spectra at (a) (3.207, 2.792, 0.501), (b) (3.207, 3.206, 0.499), and (c) (0.200, $-0.201$, 16.495). The inset in panel (a) shows the temperature dependence of a phonon peak position around 4.5 meV [indicated by a black arrow in panel (a)]. (d) Integrated intensities of the elastic components for the three polarizations as a function of temperature.

Figure 6

Measured phonon dispersion of $\mathrm{La}\mathrm{Bi}{\mathrm{S}}_2{\mathrm{O}}_{0.5}{\mathrm{F}}_{0.5}$ along the Z-A direction at selected temperatures. Solid lines are guides for the eyes.