Competing soft phonon modes at the charge-density-wave transitions in $\mathrm{DyT}{\mathrm{e}}_3$

The family of rare-earth ($R$) tritellurides $R\mathrm{T}{\mathrm{e}}_3$ features charge-density-wave (CDW) order related to strongly momentum-dependent electron-phonon coupling. Similar to other CDW compounds, superconductivity is observed when the CDW order is suppressed via hydrostatic pressure [J. J. Hamlin et al., Phys. Rev. Lett. 102, 177002 (2009)]. What sets the heavier members of the $R\mathrm{T}{\mathrm{e}}_3$ series apart is the observation of a second CDW transition at lower temperatures having an in-plane ordering wave vector $q_{\mathrm{CDW},2}\parallel \left[100\right]$ of almost the same magnitude but orthogonal to the ordering wave vector $q_{\mathrm{CDW},1}\parallel \left[001\right]$ observed at higher temperatures [N. Ru et al., Phys. Rev. B 77, 035114 (2008)]. Here, we report an inelastic x-ray scattering investigation of the lattice dynamics of $\mathrm{DyT}{\mathrm{e}}_3$ In particular, we show that there are several phonon modes along both in-plane directions, which respond to the onset of the CDW transition at $T_{\mathrm{CDW},1}=308\mathrm{K}$. Surprisingly, these soft modes close to $q_{\mathrm{CDW},2}=\left(0.68,0,0\right)$ show strong softening near $T_{\mathrm{CDW},1}$ but do not exhibit any response to the lower-temperature transition at $T_{\mathrm{CDW},2}=68\mathrm{K}$. Our results indicate that the low-temperature CDW order is not just the 90° rotated analogue of the one appearing at high temperatures.

Figure 1

(a) Orthorhombic unit cell of the crystalline structure of $R\mathrm{T}{\mathrm{e}}_3$ above $T_{\mathrm{CDW},1}$ [$R$: large (green) spheres; Te: small (yellow) spheres]. [(b) and (c)] Calculated dispersion for in-plane transversely polarized phonon modes along the (a) [001] and (b) [100] directions. The given $\mathit{Q}$ values denote the wave vectors at which the respective soft phonon modes (red line sections) have large structure factors, i.e., scattering intensities. Corresponding values for the second-lowest phonon modes at the respective wave vectors (blue line sections) are (b) $\mathit{Q}=\left(1,0,l\right)$ and (c) $\mathit{Q}=\left(h,0,4\right)$. The dashed vertical lines indicate the experimentally observed ordering wave vectors ${\mathit{q}}_{\mathrm{CDW},1}=\left(0,0,0.295\right)$ and ${\mathit{q}}_{\mathrm{CDW},2}=\left(0.32,0,0\right)$.

Figure 2

Color-coded contour plots of the calculated (a) minimum phonon energy, (b) energy-integrated phonon linewidth ${\mathrm{\Gamma }}_{\mathrm{sum}}$, and (c) electronic joint density of states (eJDOS) for wave vectors in the $\left(h,0,l\right)$ plane. The CDW ordering wave vectors are indicated in (a).

Figure 3

(a) Observed elastic intensities at the in-plane ordering wave vectors ${\mathit{q}}_{\mathrm{CDW},1}\approx \left(0,0,0.3\right)$ and ${\mathit{q}}_{\mathrm{CDW},2}=\left(0.68,0,0\right)$ of the two different CDW orders in $\mathrm{DyT}{\mathrm{e}}_3$. Dashed vertical lines indicate the deduced transition temperatures $T_{\mathrm{CDW},1}\approx 308\mathrm{K}$ and $T_{\mathrm{CDW},2}\approx 68\mathrm{K}$. (b) Scans for $E=0$ along $\mathit{q}=\left(1-h,7,3\right)$ revealing a clear shift of ${\mathit{q}}_{\mathrm{CDW},2}$ on cooling from $T=50$ to $10\mathrm{K}$.

Figure 4

(a) Comparison between calculated (solid lines) and observed (squares) energies of transversely polarized phonon modes propagating along the [001] direction in $\mathrm{DyT}{\mathrm{e}}_3$. Measurements were performed at $\mathit{Q}=\left(3,1,l\right), l=0.1\text{–}0.5$, and $T=315\mathrm{K}$. Circles represent results from previous IXS measurements in ${\mathrm{TbTe}}_3$ at $T=330\mathrm{K}$ taken from Ref. [20]. Red and blue squares mark the energies of the phonon modes for which the corresponding linewidths $\mathrm{\Gamma }$ are displayed in panel (b). Crossed black squares denote additional phonon modes of minor intensities (corresponding $\mathrm{\Gamma }$ not shown). (b) Wave-vector dependences of linewidths $\mathrm{\Gamma }$ of the soft phonon mode (red squares) and optical mode (blue squares) at $T=315\mathrm{K}$. The solid line represents a DFPT calculation of the linewidth of the soft mode. For comparison it has an offset (dashed blue line) and is scaled (see text). Green circles again represent published results for ${\mathrm{TbTe}}_3$ [20]. Dashed vertical lines denote the ordering wave vector ${\mathit{q}}_{\mathrm{CDW},1}$ of the high-temperature transition. [(c) and (d)] Analogue plots for the transversely polarized phonon mode propagating along the orthogonal [100] direction. Measurements were performed at $\mathit{Q}=\left(1-h,7,3\right), h=0.5\text{–}0.85$, and $T=310\mathrm{K}$. There are no corresponding results for other $\mathrm{RT}{\mathrm{e}}_3$ in the literature for this direction.

Figure 5

Energy scans taken at [(a) and (b)] $\mathit{Q}=\left(1,1,0.3\right)$ and [(c) and (d)] $\left(1,0,0.3\right)$ for (top) $T=400\mathrm{K}$ and (bottom) $310\mathrm{K}$, i.e., well-above and very close to $T_{\mathrm{CDW},1}=308\mathrm{K}$. Solid (red) lines are fits consisting of damped harmonic oscillators (inelastic, blue dashed lines), estimated background (straight dotted line) and a pseudo-Voigt function for the elastic peak in (c) (black dash-dotted line). The observed intensities in (b) are scaled by a factor of 0.5 in order to plot the data at the same temperature on the same scale.

Figure 6

Energy scans taken at $\mathit{Q}=\left(0.32,7,3\right)$ and different temperatures (a) $T=400$, (b) 310, (c) $110,$ and (d) $60\mathrm{K}$. Solid (red) lines are fits consisting of damped harmonic oscillators (inelastic, blue dashed lines), estimated background (straight dotted line) and a pseudo-Voigt function for the elastic peak in (c) and (d) (black dotted line). The inset in (d) shows the data taken at $T=60\mathrm{K}$ but on a larger vertical scale in order to include the maximum intensities at zero energy transfer.

Figure 7

Energy scans taken at $\mathit{Q}=\left(0.68,0,4\right)$ and different temperatures (a) $T=400$, (b) 300, (c) $110,$ and (d) $60\mathrm{K}$. Solid (red) lines are fits consisting of damped harmonic oscillators (inelastic, blue dashed lines), estimated background (straight dotted line) and a pseudo-Voigt function for the elastic peak in (c) and (d) (black dotted line).

Figure 8

Temperature dependent [(a) and (b)] energies and [(c) and (d)] linewidths of the predicted soft phonon modes at the CDW ordering wave vectors [(a) and (c)] ${\mathit{q}}_{\mathrm{CDW},1}\approx \left(0,0,0.3\right)$ and [(b) and (d)] ${\mathit{q}}_{\mathrm{CDW},2}=\left(0.68,0,0\right)$. Dashed vertical lines denote the respective CDW transition temperatures. Lines are guides to the eye. Absolute wave vectors for the measurements are given in the legends.

Figure 9

Comparison of results from time-resolved pump-probe reflectivity measurements (squares) [41] and inelastic x-ray scattering (dots/circles). Horizontal thick bars denote the calculated phonon energies having the same symmetry as the soft mode. Negative values correspond to imaginary phonon energies [see Fig. 5]. The thin solid (red) line is the soft mode temperature dependence at $T>T_{\mathrm{CDW},1}$. The thin dashed (blue) line is a guide to the eye for the likely amplitude/soft mode temperature dependence for $T<T_{\mathrm{CDW},1}$. The vertical dashed line indicates $T_{\mathrm{CDW},1}=308\mathrm{K}$.