Suppression of charge density wave order by disorder in Pd-intercalated ${\mathrm{ErTe}}_3$

Disorder is generically anticipated to suppress long range charge density wave (CDW) order. We report transport, thermodynamic, and scattering experiments on ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3$, a model CDW system with disorder induced by intercalation. The pristine parent compound ($x=0$) shows two separate, mutually perpendicular, incommensurate unidirectional CDW phases setting in at 270 K and 165 K. In this work we present measurements on a finely-spaced series of single crystal samples, in which we track the suppression of signatures corresponding to these two parent transitions as the Pd concentration increases. At the largest values of $x$, we observe complete suppression of long range CDW order in favor of superconductivity. We also report evidence from electron and x-ray diffraction which suggests a tendency toward short-range ordering along both wave vectors which persists even well above the crossover temperature and comment on the origin and consequences of this effect. Based on this work, ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3$ appears to provide a promising model system for the study of the interrelation of charge order and superconductivity in the presence of quenched disorder, for pseudotetragonal materials.

Figure 1

(a) Crystal structure of the $x=0$ parent compound ${\mathrm{ErTe}}_3$, which consists of alternating ErTe slabs with bilayers of square Te nets. Unit cell is shown by black lines. (b) The out-of-plane lattice parameter $b$ determined by single-crystal x-ray measurements, as a function of Pd content $x$ (determined by microprobe analysis). Error bars are calculated as the variance in a simultaneous fit to the Cu $K{\alpha }_1$ and $K{\alpha }_2$ peaks at the (0,8,0) position and are only barely larger than the size of the plot symbols. The lattice expands perpendicular to the planes as $x$ increases, which is consistent with incorporation of Pd atoms between the planes. (c) The two in-plane lattice parameters $a$ and $c$ on the same set of crystals, extracted from the (1,9,0) and (1,7,1) peaks. There is no significant trend in $a$ or $c$ with increasing $x$.

Figure 2

Stoichiometric ratios determined from electron microprobe measurements. The $x$-axis value is used throughout this work to quantify the concentration of Pd atoms per formula unit. We found no crystals with Pd concentrations above $x=0.055$ with this growth method. For growths of the highest Pd concentrations, the formation of ${\mathrm{Pd}}_x{\mathrm{ErTe}}_3$ competes with the formation of ${\mathrm{PdTe}}_2$. For $x>0.035$, a significant deviation from the ideal ratio of 3 Te per Er is observed. This decrease is believed to be due to an increased population of Te vacancies within the bilayers.

Figure 3

(a) Temperature dependence of the in-plane resistivity between 4 K and 300 K, resolved into the two in-plane directions, normalized to the value at room temperature, and offset for clarity. Data is shown for representative compositions only. Solid and dashed lines correspond to the resistivity along the $a$ and $c$ axes, respectively. The inset describes the measurement geometry, in which a rectangular bar is cut with a scalpel at a ${45}^{\circ }$ angle to the crystal axes. In a coordinate frame rotated by the same amount (denoted here by $x$ and $y$) current passing in the $x$ direction generates a longitudinal voltage $V_x$ proportional to ${\rho }_a+{\rho }_c$, and a transverse voltage $V_y$ proportional to ${\rho }_c-{\rho }_a$. Simultaneous measurement of these voltages allows for both components of the resistivity to be measured simultaneously [29]. (b) Temperature derivative of ${\rho }_c$ for each composition shown in (a). The dips marked by arrows correspond to the first (black) and second (red) CDW transitions in the parent compound, which are shown here to be suppressed towards zero as $x$ increases. We define the crossover temperature $T_{CDW}$ as the temperature at which the derivative reaches a local minimum. Curves were smoothed using a Loess filter prior to taking the numerical derivative.

Figure 4

Representative AC magnetic susceptibility measurement for an $x=0.043$ sample. Field was applied parallel to the plane of the sample in this measurement. The sample shows near-perfect diamagnetism below $T_c=2.7\text{K}$, suggesting that the superconducting state is bulk, rather than filamentary. $T_c$ is extracted as the peak in the imaginary part of the susceptibility. Inset: M(H) curves above and below $T_c$, showing the small $H_{c1}\approx 5\text{Oe}$. The normal state just above $T_c$ also shows significant paramagnetism, a consequence of the proximity to antiferromagnetic ordering of the rare earth moments.

Figure 5

Phase diagram constructed from transport and magnetometry measurements. The temperature axis is rescaled below 3.5 K to better display the trend in superconducting transition temperatures. The second CDW is suppressed quickly as $x$ increases and extrapolates to zero around $x=0.01$. A superconducting ground state emerges abruptly from below our base temperature of 1.8 K at $x=0.02$ and continues with an approximately constant transition temperature out to the solubility limit of Pd. The first CDW extrapolates to zero around $x=0.035$. It should be noted that both CDW features decrease in magnitude and become harder to distinguish as $x$ increases. Lines extend upward from each data point to the first onset of the CDW signature in resistivity, illustrating how disorder smears the crossover. Background shading approximates the magnitude of CDW correlations along the two in-plane axes: blue signifies the primary $c$-axis CDW and red signifies the $a$-axis. As disorder increases, the material begins to exhibit short range correlations along both directions (indicated by purple shading). True long-range-ordered CDW phases exist only on the $x=0$ axis.

Figure 6

Representative selected area diffraction patterns (SADPs) at various Pd concentrations and temperatures. (a) Parent compound (${\mathrm{ErTe}}_3$) at 95 K (i) and 274 K $\sim T_{CDW,1}$ (ii), showing the disappearance of the satellite peaks on either side of the center peak at high temperatures. The secondary CDW peaks show up weakly and are strongest near the four Bragg peaks in the corner rather than the peak centered here. This is simply due to the structure factor of the material. (b) SADP for an $x=0.023$ sample ($T_{CDW,1}\approx 97\text{K}$) at 95 K (i), clearly showing satellite peaks forming along both the $c^{*}$ and $a^{*}$ axes, despite the fact that the transport signature of the second CDW has been completely suppressed for this composition. This suggests that the introduction of disorder elicits a nearly fourfold symmetric response from the material, reflective of the underlying electronic susceptibility and phonon dispersion. A small square of diffuse streaks also begins to appear between these peaks, suggestive of short-range ordering. It should be noted, however, that the width of the satellite peaks sets a lower bound on the correlation length of 40 nm or about 94 unit cells. In (ii) remnants of the CDW peaks and the diffuse squares can be seen on this same sample even at 202 K, more than double the CDW crossover temperature. (c) Finally for a $x=0.05$, which shows no signatures of CDW in resistivity, no sharp satellite peaks are observed to the lowest temperature attainable, but the diffuse square remains visible at all temperatures. This is suggestive of a structural distortion with a similar $q$ vector, likely stemming from nucleation of CDW correlations by individual Pd atoms. Panel (d) shows a schematic describing the origin of the various peaks for the parent compound. The small upward triangles correspond to higher order CDW peaks which are only observable in (a).

Figure 7

Measurement of the elastic CDW superlattice peak intensity observed in x-ray diffraction at zero energy transfer as a function of temperature for a sample with $x=0.023$. Data was taken at the HERIX beamline 30-ID at the Advanced Photon Source, with photon energy 23.72 keV and energy resolution of 1.5 meV. Above the transition temperature (95 K from resistivity) the amplitude of CDW correlations begin to grow identically, but the intensity of the primary CDW grows further at lower temperatures. Inset: FWHM measurements of longitudinal scans through $q_{CDW,1}$ and $q_{CDW,2}$ also show similar behavior above the crossover. Below the crossover, the correlation length of the secondary CDW along $a$ reaches a maximum around 90 K, while the correlation length of the primary continues to grow at least to the resolution limit. We estimate a reciprocal-space resolution for this beamline of 0.0112 r.l.u.