Resonant enhancement of charge density wave diffraction in the rare-earth tritellurides

We performed resonant soft x-ray diffraction on known charge density wave (CDW) compounds, rare-earth tritellurides. Near the $M_5$ ($3d$-$4f$) absorption edge of rare-earth ions, an intense diffraction peak is detected at a wave vector identical to that of the CDW state hosted on Te${}_2$ planes, indicating a CDW-induced modulation on the rare-earth ions. Surprisingly, the temperature dependence of the diffraction peak intensity demonstrates an exponential increase at low temperatures, vastly different than that of the CDW order parameter. Assuming $4f$ multiplet splitting due to the CDW states, we present a model to calculate x-ray-absorption spectrum and resonant profile of the diffraction peak, agreeing well with experimental observations. Our results demonstrate a situation where the temperature dependence of resonant x-ray-diffraction peak intensity is not directly related to the intrinsic behavior of the order parameter associated with the electronic order, but is dominated by the thermal occupancy of the valence states.

Figure 1

(a) Crystal structure of the rare-earth tritelluride, $R$Te${}_3$ ($R=\text{Tb}$, Dy, etc.). (b) Scattering geometry. The polarization of the photons is set to be parallel to the scattering plane ($\pi$ geometry).

Figure 2

The absorption curve (black) near the Tb $M_5$ edge and the intensity of the ($0,4,q_{\mathrm{CDW}}$) peak as a function of the incident photon energy, where $q_{\mathrm{CDW}}\approx 0.296c^{*}$. Inset shows a representing $\theta$-2$\theta$ scan of the ($0,4,q_{\mathrm{CDW}}$) diffraction peak. Data were taken at a temperature of 30 K.

Figure 3

(a) Upper panel: the intensity (red) and the width (blue) of the diffraction peak as a function of temperatures taken at incident photon energy of 1239.5 eV. Lower panel: the square root of the diffraction peak intensity, for the CDW diffraction peak obtained from both resonant soft x-ray diffraction (this experiment, red markers) and nonresonant hard x-ray diffraction (Ref. 12, green). (b) Same as (a), but the data taken at lower temperatures are plotted. (c) XAS (upper) and CDW resonance profiles (lower) at three representing temperatures. The structure of XAS and CDW resonant profiles is essentially unchanged.

Figure 4

(a) the XAS (black) and the CDW resonant profile (red) of DyTe${}_3$ (left) and CeTe${}_3$ (right). Please note that the CDW diffraction peak for CeTe${}_3$ is ($0,2,q_{\mathrm{CDW}}$) due to the smaller momentum transfer of the photons at Ce $M_5$ edge. (b) The CDW diffraction peak intensity of TbTe${}_3$, DyTe${}_3$, and CeTe${}_3$ are plotted and superimposed. Inset demonstrates the width of the diffraction peaks of TbTe${}_3$ and DyTe${}_3$.

Figure 5

The calculated resonant profile (RP) of the CDW diffraction peak and x-ray-absorption spectrum (XAS) for TbTe${}_3$, DyTe${}_3$, and CeTe${}_3$ near the $M_5$ edge of rare-earth elements at two representative temperatures, 4 and 40 K. The calculated temperature dependence of the CDW diffraction peak intensity taken at the photon energy equal to the maximum of the RP is plotted in the insets.