The interplay of disorder and fluctuations creates complex phases like spin glasses. The Bragg glass is an elusive example: algebraically ordered, appearing crystalline but with Bragg peak intensities diverging as power laws. Experimental resolution obscures this divergence, hindering detection. While proposed for charge-density-wave (CDW) systems and vortex lattices, unambiguous evidence has been limited, mostly to vortex lattices using magnetic field-dependent rocking curve width. STM studies offer suggestive evidence in some CDW systems, but bulk evidence in incommensurate CDWs remains scarce. This study aims to address the open question of whether an algebraically ordered CDW phase can exist as a bulk phase in a CDW system, or if CDWs always exhibit short-range order in response to disorder.

Existing theoretical frameworks predict the possibility of quasi-long-range ordering in vortex lattices and CDWs in the presence of disorder, forming a Bragg glass phase. This contradicts the notion that order parameters linearly coupled to a disorder potential are short-ranged. The key difference lies in the compactness of the phase (defined within [0, 2π)). In systems with compact phases, like CDWs and vortex lattices, the disorder-averaged potential energy depends on the exponential of phase fluctuations, allowing for quasi-long-range order in three dimensions. Evidence for this would be a vanishing width of structure factor peaks. However, the presence of dislocations disrupts this; their absence is a necessary (but not sufficient) condition for a Bragg glass. Challenges in observing Bragg glass phenomena in CDWs include the need for adjustable disorder, large data volumes for reliable analysis (separating fluctuation effects from crystal imperfections and finite resolution), and advanced data analysis techniques.

This study uses PdxErTe3, a material system with adjustable disorder via Pd intercalation, to investigate the Bragg glass phase. Pristine ErTe3 exhibits unidirectional CDW order below Tc and orthogonal order below Ta (Ta < Tc). Pd intercalation introduces localized disorder. Comprehensive single-crystal X-ray scattering data were collected across a large reciprocal space volume (20,000 Brillouin zones) using high-throughput methods at the Advanced Photon Source. The raw data volume exceeded 100 GB. The X-ray diffraction temperature clustering (X-TEC) algorithm, a machine learning method, was used to analyze the data. X-TEC clusters temperature trajectories of X-ray intensity at each reciprocal space point. This allowed for the analysis of CDW peak characteristics – peak height, peak width (quantified using a novel 'peak spread' metric), and satellite peak asymmetry – to distinguish between long-range order, Bragg glass, and short-range order phases. The peak spread metric is model-independent, doesn't require high-resolution data, and integrates with X-TEC, offering scalability to large datasets. The momentum dependence of the peak width, fitted to a quadratic function, helps separate CDW fluctuation contributions from other broadening sources (crystal imperfections). The momentum-independent width is then analyzed to identify the Bragg glass transition temperature (T_BG).

X-TEC analysis of pristine ErTe3 accurately identified two CDW transitions (T_c1 ≈ 260 K and T_c2 ≈ 135 K), consistent with transport measurements. In intercalated samples, increasing Pd concentration suppresses CDW intensity and broadens the temperature distribution. Analysis of peak spread reveals a Bragg glass transition temperature (T_BG) for moderate intercalation levels (x ≤ 2%). Below T_BG, the peak spread becomes constant, indicating resolution-limited peak widths, signifying quasi-long-range order. This is in contrast to short-range order where peak spread increases with decreasing temperature. Extrapolation of the peak spread to zero width provided estimates for T_BG. The obtained T_BG values align remarkably well with the onset of transport anisotropy previously observed. The asymmetry in satellite diffuse scattering intensity, quantified by a ratio α, was only present in intercalated samples, indicating the effects of disorder pinning. A phase diagram shows the transition between the long-range ordered phase and the Bragg glass phase as a function of Pd concentration. For moderate intercalation levels, the phase space exhibiting in-plane resistance anisotropy is primarily covered by the Bragg glass phase, extending to significantly higher temperatures than expected. The 2.9% intercalated sample shows fluctuations suggestive of a Bragg glass but does not reach that phase. The finding of the Bragg glass phase supports the results of STM studies that suggest the absence of phase dislocations and hence the presence of the Bragg glass phase, even at high temperatures.

The findings provide the first X-ray scattering evidence for a Bragg glass phase in a disordered CDW system. The use of X-TEC and peak spread allowed separation of intrinsic CDW fluctuations from other broadening effects. The close agreement between T_BG and the onset of transport anisotropy suggests that the Bragg glass dominates the phase space where this anisotropy is observed. The presence of diffuse scattering asymmetry further supports the disorder-pinned nature of the CDW in the intercalated samples. The Bragg glass phase extends to remarkably high temperatures, even above the temperatures where the short range order prevails in similar systems. The small range of phase space for the competing short range order phase strengthens the argument that the prevalent phase is the Bragg glass. This research advances our understanding of the Bragg glass phase in disordered CDW systems, showing its stability over a wider range of temperatures and disorder than previously thought.

This work presents the first bulk evidence for a Bragg glass phase in PdxErTe3 using X-ray scattering. The combination of high-throughput data acquisition, the novel peak spread metric, and the X-TEC analysis technique enabled the identification of the Bragg glass phase despite experimental resolution limitations. The findings showcase the power of machine learning in analyzing large scattering datasets and provide insights into the interplay between disorder and fluctuations in complex materials. Future higher resolution studies could further confirm the presence of Bragg glass through observation of power-law tails in the peak broadening.

The study relies on an empirical extrapolation to determine T_BG. While the linear approximation is reasonable near the transition, it could introduce some uncertainty in the precise determination of T_BG. The X-ray experiments were performed at a single energy, which could potentially affect certain aspects of the scattering data analysis.