Evolution of charge density wave soft phonon modes in Pd$_x$ErTe$_3$

We investigated the lattice dynamics of quasi-two-dimensional Pd-intercalated 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 ErTe$_3$, CDW order develops at orthogonal in-plane wave vectors q$^𝑐_1$=(0,0,0.29) (the 𝑐−CDW) and q$^𝑎_2$=(0.31,0,0) (the 𝑎−CDW), with transition temperatures 𝑇$^𝑐_1$=270K and 𝑇$^𝑎_2$=160K, respectively. Remarkably, we observe diffuse x-ray scattering already near the higher transition temperature 𝑇$^𝑐_1$ along 𝑎−CDW but at a slightly different wave vector q$^𝑎_1$=(0.29,0,0). Inelastic x-ray scattering for Pd$_{0.01}$⁢ErTe$_3$ shows that a partial phonon softening at q$^𝑎_1$, underscoring the strong competition between ordering tendencies along the nearly equivalent in-plane axes of the orthorhombic lattice. For intercalation levels 𝑥≥0.02, the 𝑎−CDW state is suppressed. Nevertheless, a similar correlation between phonon softening and diffuse scattering persists along the [100] direction, again observed at q$^𝑎_1$=(0.29,0,0) and 𝑇$^𝑐_1$. These findings suggest that the 𝑎−CDW is fully suppressed for 𝑥≥0.02, and that the residual diffuse scattering at q$^𝑎_1$ originates from the partial phonon softening associated with the 𝑐−CDW, reflected by the near equality of the absolute size of q$^𝑐_1$ and q$^𝑎_1$.