Phonon anomalies in FeS

We present results from light scattering experiments on tetragonal FeS with the focus placed on lattice dynamics. We identify the Raman active $A_{1g}$ and $B_{1g}$ phonon modes, a second order scattering process involving two acoustic phonons, and contributions from potentially defect-induced scattering. The temperature dependence between 300 and 20 K of all observed phonon energies is governed by the lattice contraction. Below 20 K the phonon energies increase by 0.5–1 ${\mathrm{cm}}^{-1}$, thus indicating putative short range magnetic order. Along with the experiments we performed lattice-dynamical simulations and a symmetry analysis for the phonons and potential overtones and find good agreement with the experiments. In particular, we argue that the two-phonon excitation observed in a gap between the optical branches becomes observable due to significant electron-phonon interaction.

Figure 1

Raman spectra of FeS at $T=80$ K measured with light polarizations as indicated. The inset shows the crystal structure of FeS and the polarization directions with respect to the crystal orientation.

Figure 2

Raman spectra of FeS in $bb$ polarization projecting $A_{1g}+B_{1g}+E_g$ symmetries measured at temperatures given in the legend. The inset shows the light polarizations with respect to the crystal orientation.

Figure 3

Temperature dependence of energy and width of the four observed phonon modes in FeS. Black squares show the phonon energies $\omega$; open circles denote the phonon linewidths ${\mathrm{\Gamma }}_{\mathrm{L}}$. The red dashed and solid lines represent the temperature dependencies of the phonon linewidths and energies according to Eqs. (1) and (2), respectively. For better visualizing the low-temperature part, the data of this figure are plotted on a logarithmic temperature scale in Fig. 8 of Appendix pp4.

Figure 4

Phonon dispersion of t-FeS. (a) Brillouin zone with high symmetry points and lines [43]. (b) Phonon dispersion along the directions as indicated and phonon density of states (PDOS). The gray-shaded area marks the gap in the phonon dispersion. The dispersion shown here is derived using experimental lattice parameters. For this reason some of the acoustic phonons are unstable and do not have a linear dispersion around the $\mathrm{\Gamma }$ point. Upon relaxing the structure the acoustic dispersion becomes linear at $\mathrm{\Gamma }$, and the energies at the zone boundary decrease slightly. The energies of the optical branches, on the other hand, increase by some 10%. $M^{\prime }=(0.4,0.4,0.0)$ and $A^{\prime }=(0.4,0.4,0.5)$. The experimental energies of the four observed modes are shown as black lines.

Figure 5

Magnetization measurements of t-FeS at an applied field of $B=1\mathrm{mT}$ cooled to 2 K with (red curve) and without applied field (black curve).

Figure 6

Decomposition of the asymmetric phonon peak at 305 ${\mathrm{cm}}^{-1}$. Measured data are shown as black dots. The orange line shows the sum of two Voigt profiles shown as blue and green lines, respectively. The convolution of Fano and Gaussian (red line) deviates in the peak flanks and the nearby continuum.

Figure 7

Temperature dependence of $A_{1g}$ and $B_{1g}$ phonon modes in the temperature range between 80 K and 300 K. Black squares denote the phonon energies; open circles denote the phonon linewidths.

Figure 8

Temperature dependence of energy and width of the four observed phonon modes in FeS on a logarithmic scale. The data is identical to Fig. 3 of the main text. Black squares show the phonon energies $\omega$; open circles denote the phonon linewidths ${\mathrm{\Gamma }}_{\mathrm{L}}$. The red dashed and full lines represent the temperature dependence of the phonon linewidths and energies according to Eqs. (1) and (2), respectively. The region below 20 K is shaded light gray. Since the data for the volume are limited to the range above 10 K the theoretical curves for the phonon energies (full red lines) end at 10 K.

Figure 9

Raman spectra of a t-FeS sample from a different batch taken at $T=310$ K in polarizations as given in the legend. The inset shows magnetization measurements on a sample from this batch similar to Appendix pp1.