Evolution of charge-density-wave soft phonon modes in $\mathrm{Pd}_x\mathrm{ErTe}_3$

TL;DR

This study investigates how quenched disorder from Pd intercalation affects charge-density-wave order in the quasi-two-dimensional ErTe$_3$ system by combining x-ray diffuse scattering and meV-resolution inelastic x-ray scattering with density-functional perturbation theory. The authors show that the pristine material hosts two unidirectional CDWs, and that Pd intercalation rapidly suppresses the secondary $a$-CDW while inducing diffuse scattering linked to competing soft phonons associated with the primary $c$-CDW. The results reveal two soft phonon modes at $q_{1}^{c}$ and $q_{2}^{a}/q_{1}^{a}$, whose temperature dependence tracks the CDW transitions, with incomplete softening at higher intercalation ($x=0.023$) possibly connected to a CDW Bragg-glass state. Overall, the findings demonstrate a competition between soft phonons along orthogonal in-plane axes that governs CDW evolution under disorder and warrant revising the phase diagram for Pd$_x$ErTe$_3$ to emphasize phonon-mediated suppression of the $a$-CDW and the role of the $c$-CDW–driven diffuse scattering.

Abstract

We investigated the lattice dynamics of quasi-two-dimensional Pd-intercalated $\mathrm{ErTe}_3$ in relation to its charge-density-wave (CDW) transitions by means of x-ray diffuse and meV-resolution inelastic x-ray scattering. In pristine $\mathrm{ErTe}_3$, CDW order develops at orthogonal in-plane wave vectors $\boldsymbol{\mathrm{q}}_{1}^{c} = (0, 0, 0.29)$ (the $c\text{-}\mathrm{CDW}$) and $\boldsymbol{\mathrm{q}}_{2}^{a} = (0.31, 0, 0)$ (the $a\text{-}\mathrm{CDW}$), with transition temperatures $T_{1}^{c} = 270$~K and $T_{2}^{a} = 160$~K, respectively. Remarkably, we observe diffuse x-ray scattering already near the higher transition temperature $T_{1}^{c}$ along $a\text{-}\mathrm{CDW}$ but at a slightly different wave vector $\boldsymbol{\mathrm{q}}_{1}^{a} = (0.29, 0, 0)$. Inelastic x-ray scattering for $\mathrm{Pd}_{0.01}\mathrm{ErTe}_3$ shows that a partial phonon softening at $\boldsymbol{\mathrm{q}}_{1}^{a}$, underscoring the strong competition between ordering tendencies along the nearly equivalent in-plane axes of the orthorhombic lattice. For intercalation levels $x \geq 0.02$, the $a\text{-}\mathrm{CDW}$ state is suppressed. Nevertheless, a similar correlation between phonon softening and diffuse scattering persists along the $[100]$ direction, again observed at $\boldsymbol{\mathrm{q}}_{1}^{a} = (0.29, 0, 0)$ and $T_{1}^{c}$. These findings confirm that the $a\text{-}\mathrm{CDW}$ is fully suppressed for $x \geq 0.02$, and that the residual diffuse scattering at $\boldsymbol{\mathrm{q}}_{1}^{a}$ originates from the partial phonon softening associated with the $c\text{-}\mathrm{CDW}$, reflected by the near equality of the absolute size of $\boldsymbol{\mathrm{q}}_{1}^{c}$ and $\boldsymbol{\mathrm{q}}_{1}^{a}$. In highly intercalated $\mathrm{Pd}_{0.023}\mathrm{ErTe}_3$, the phonon softening remains incomplete, possibly linked to the recently reported CDW Bragg glass state.

Figures (5)

• Figure 1: (a) Orthorhombic unit cell of the crystalline structure of $\mathrm{ErTe}_3$ above $T_{1}^{c}$ [Er: large (chartreuse) spheres; Te: small (golden) spheres]. (b) Evolution of the ratio of the $a$ and $c$ in-plane lattice parameters of $\mathrm{Pd}_x\mathrm{ErTe}_3$ as deduced from the XDS data sets. Color-coded solid lines represent the smoothed ratios using spline interpolation. Error bars in (b) are derived from typical uncertainties determining lattice parameters of $2.5 \times 10^{-4}$. (c) Scheme of the reciprocal map indicating positions of the $c\text{-}\mathrm{CDW}$ superlattice peaks at $\boldsymbol{\mathrm{q}}_{1}^{c}$ (red circles) and $a\text{-}\mathrm{CDW}$ superlattice peaks at $\boldsymbol{\mathrm{q}}_{2}^{a}$ (blue circles). Bragg peaks are shown in black dots. Green dots denote the wave vector $\boldsymbol{\mathrm{q}}_{1}^{a}$ for which XDS is observed at high temperature ($x=0$) and in samples with $x \geq 0.02$. (d), (e) Different sections of the reconstructed $(h,-7,l)$ reciprocal plane, indicating the (d) $c\text{-}\mathrm{CDW}$ and (e) $a\text{-}\mathrm{CDW}$ superlattice peaks for pristine $\mathrm{ErTe}_3$ at 30 K. (f), (g) Temperature-dependent incommensurate wave vectors of the (f) $c\text{-}\mathrm{CDW}$ and (g) $a\text{-}\mathrm{CDW}$, extracted from the line cuts along $l$ and $h$ directions through the superlattice peaks at $\boldsymbol{\mathrm{Q}}=(9,-7,l)$ and $\boldsymbol{\mathrm{Q}}=(h,-7,9)$ [marked in white boxes in (d), (e), $x=0$], respectively. White arrows in (d), (e) indicate the in-plane binning range for extracting the line cuts from the reciprocal maps. Out-of-plane binning range was $\pm0.5$ r.l.u. along [010]. Horizontal dashed lines in (f), (g) denote $\boldsymbol{\mathrm{q}}_{1}^{c}$, $\boldsymbol{\mathrm{q}}_{2}^{a}$, and $\boldsymbol{\mathrm{q}}_{1}^{a}$ as discussed in the text. White (red) boxes in (d), (e) denote the binning area used to extract temperature dependences of CDW (background) intensities shown in Figs. \ref{['fig2']}(b)-(f).
• Figure 2: (a) XDS intensity along $\boldsymbol{\mathrm{Q}} = (-0.3, k, 9)$, $T = 30$ – 300 K obtained in $\mathrm{ErTe}_3$, reveal a considerable increase near $\boldsymbol{\mathrm{q}}_{2}^{a}$ around 260 K, i.e., close to $T_{1}^{c}$. (b)-(f) Integrated intensities of XDS (symbols) around $\boldsymbol{\mathrm{Q}} = (-0.3(\pm 0.1), -7(\pm0.5), 9(\pm0.1))$ (includes $\boldsymbol{\mathrm{q}}_{2}^{a}$ and $\boldsymbol{\mathrm{q}}_{1}^{a}$) for $x = 0$ (b), 0.005 (c), 0.020 (d), 0.026 (e), and 0.029 (f) and temperatures $T = 30$ - 300 K. Background scattering [$\boldsymbol{\mathrm{Q}}$ area indicated by red boxes in Fig. \ref{['fig1']}(e)] was subtracted. The XDS integrated intensities [$\boldsymbol{\mathrm{Q}}$ area indicated by white boxes in Fig. \ref{['fig1']}(e)] are normalized to a $[0-1]$ scale. Grey-shaded lines indicate corresponding results [$\boldsymbol{\mathrm{Q}}$ areas indicated by white and red boxes in Fig. \ref{['fig1']}(d)] at the orthogonal wave vectors $\boldsymbol{\mathrm{Q}} = (9(\pm0.1), -0.3(\pm 0.1), -7(\pm0.5))$ (including $\boldsymbol{\mathrm{q}}_{1}^{c}$). Red and blue vertical shades mark maxima of XDS intensity which coincide with the respective doping dependent values of $T_{1}^{c}$ and $T_{2}^{a}$, respectively [see Straquadine_prb_2019].
• Figure 3: Comparison of measured (symbols) and calculated phonon dispersions (dashed lines) in $\mathrm{Pd}_x\mathrm{ErTe}_3$ with (a),(b) $x=0.01$, $T = 195$ K, and (c),(d) $x=0.023$, $T = 75$ K across $\boldsymbol{\mathrm{q}}_{1}^{c}$ (left panels) and $\boldsymbol{\mathrm{q}}_{2}^{a}$/$\boldsymbol{\mathrm{q}}_{1}^{a}$ (right panels). Raw IXS intensities are shown in color-code (log scale). Vertical lines denote $\boldsymbol{\mathrm{Q}}$ values of performed IXS scans. Symbols represent approximated phonons energies where exemplary fits are shown in Figs. \ref{['fig4']} and \ref{['fig5']}. Dashed lines indicate calculated phonon dispersions.
• Figure 4: Phonon spectroscopy in $\mathrm{Pd}_{0.01}\mathrm{ErTe}_3$. Constant-$\boldsymbol{\mathrm{Q}}$ scans (symbols) at wave vectors (a), (b) $\boldsymbol{\mathrm{Q}} = (3,7,0.29)$ and (c)-(e) $\boldsymbol{\mathrm{Q}} = (0.31,7,3)$ corresponding to $\boldsymbol{\mathrm{q}}_{1}^{c}$ and $\boldsymbol{\mathrm{q}}_{2}^{a}$/$\boldsymbol{\mathrm{q}}_{1}^{a}$ (see text), respectively. Solid lines in red represent fits to the data consisting of damped harmonic oscillator (DHO) functions convoluted with the experimental resolution for the soft phonon mode (blue dashed) and other phonon modes (green solid), a sloping background (too small to be visible) and a resolution-limited pseudo-Voigt function for the elastic line (black dashed). (f) Derived temperature dependence of the soft phonon mode at $\boldsymbol{\mathrm{q}}_{1}^{c}$ and $\boldsymbol{\mathrm{q}}_{2}^{a}$/$\boldsymbol{\mathrm{q}}_{1}^{a}$ for $\mathrm{Pd}_{0.01}\mathrm{ErTe}_3$. The inset shows high-resolution momentum scans at zero energy transfer across $\boldsymbol{\mathrm{q}}_{1}^{c}$ and $\boldsymbol{\mathrm{q}}_{2}^{a}$/$\boldsymbol{\mathrm{q}}_{1}^{a}$ at $T=60$ K. The vertical grey-shaded bar indicates the transition temperature $T_{1}^{c} = 195$ K reported for $\mathrm{Pd}_{0.01}\mathrm{ErTe}_3$Straquadine_prb_2019.
• Figure 5: Phonon spectroscopy in $\mathrm{Pd}_{0.023}\mathrm{ErTe}_3$. Constant-$\boldsymbol{\mathrm{Q}}$ scans (symbols) at wave vectors (a), (b) $\boldsymbol{\mathrm{Q}} = (3,7,0.29)$ and (c), (d) $\boldsymbol{\mathrm{Q}} = (0.29,7,3)$ corresponding to $\boldsymbol{\mathrm{q}}_{1}^{c}$ and $\boldsymbol{\mathrm{q}}_{2}^{a}$/$\boldsymbol{\mathrm{q}}_{1}^{a}$ (see text), respectively. Symbol/line code is the same as in Fig. \ref{['fig4']}. (e) Derived temperature dependence of the soft phonon modes at $\boldsymbol{\mathrm{q}}_{1}^{c}$ and $\boldsymbol{\mathrm{q}}_{2}^{a}$/$\boldsymbol{\mathrm{q}}_{1}^{a}$ for $\mathrm{Pd}_{0.023}\mathrm{ErTe}_3$. The inset shows high-resolution momentum scans at zero energy transfer across $\boldsymbol{\mathrm{q}}_{1}^{c}$ and $\boldsymbol{\mathrm{q}}_{2}^{a}$/$\boldsymbol{\mathrm{q}}_{1}^{a}$ at $T=50$ K. The vertical grey shaded bar indicates the temperature range reported for $T_{1}^{c} = 50$ - 100 K in $\mathrm{Pd}_{0.023}\mathrm{ErTe}_3$Straquadine_prb_2019.