Bragg glass signatures in PdxErTe3 with X-ray diffraction temperature clustering

The study investigates whether a Bragg glass—an algebraically ordered, disorder-pinned phase with quasi–long-range order—can exist as a bulk phase in an incommensurate charge-density-wave (CDW) system. Prior wisdom suggested that order parameters linearly coupling to disorder should be short-ranged; however, for periodic states with compact phase (defined modulo 2π), theory predicts quasi–long-range order and power-law divergent Bragg peak intensities. Experimentally, detecting a Bragg glass in CDWs is difficult due to finite instrumental resolution, noise, and crystal imperfections that mask the vanishing intrinsic peak width. Existing unambiguous evidence has been largely limited to vortex lattices, leaving open whether CDWs exhibit a Bragg glass phase in the bulk or instead form a short-range vestigial nematic. The work addresses this by studying Pd-intercalated ErTe_3 (Pd_xErTe_3), a tunably disordered CDW system, using comprehensive single-crystal X-ray scattering combined with machine-learning-based analysis (X-TEC) to track peak widths and extract intrinsic correlation-length behavior across many Brillouin zones and temperatures.

The Bragg glass concept emerged from elastic theory of disordered periodic media, notably for vortex lattices and CDWs (Giamarchi, Le Doussal; Nattermann; Fisher, et al.). Unambiguous bulk evidence has been reported primarily in vortex lattices via field-independent rocking curve widths and observations of quasi–long-range order near Bragg-glass transitions. STM has observed Bragg-glass-like decay of translational order in vortex lattices of CDW defects in 1T-TaS2 and suggested Bragg glass behavior in NbSe2 and Pd_xErTe_3, but direct bulk evidence in incommensurate CDWs remained elusive. In rare-earth tritellurides (RTe_3), pristine samples show two orthogonal unidirectional CDWs with distinct transition temperatures and transport anisotropy; disorder via Pd intercalation suppresses long-range order and introduces pinning centers. Prior transport studies showed the onset temperature of in-plane anisotropy decreases with intercalation. Theoretical and X-ray studies established that disorder can yield asymmetric satellite scattering due to pinning (Ravy, Pouget) and that a stable Bragg glass requires three-dimensionality, with no Bragg glass in two dimensions (Zeng, Leath, Fisher; Le Doussal, Giamarchi).

Materials and synthesis: Single crystals of Pd_xErTe_3 (x = 0, 0.5%, 2.0%, 2.6%, 2.9%) were grown by Te self-flux with Pd intercalation controlled via melt composition. Intercalation levels were verified via resistivity benchmarks. X-ray scattering: Measurements were performed at Sector 6-ID-D, Advanced Photon Source. Samples were mounted on polyimide capillaries, measured in transmission geometry with 87 keV monochromatic X-rays, cooled by an Oxford N2/He cryostream (30–300 K). Each dataset comprises continuous 360° rotations at 1°/s with a PILATUS 2M CdTe detector (10 Hz acquisition), repeated three times with offsets to fill detector gaps and remove artifacts. The raw data (>100 GB per temperature series) were transformed into reciprocal-space meshes encompassing over 10,000 Brillouin zones (BZs) and ~5×10^6 q-bins; across temperatures this covered ~20,000 BZs total. Orientation consistency across temperatures was ensured during transformation. Machine-learning analysis (X-TEC): The X-TEC package (unsupervised Gaussian mixture model clustering) represents intensity–temperature trajectories at each reciprocal-space point q as vectors {I_q(T_i)} in a d-dimensional space (d = number of temperatures). After rescaling log-intensities, Gaussian mixture clustering identifies distinct temperature trajectories, enabling automatic segmentation of CDW satellite peaks and diffuse scattering. Benchmarking on pristine ErTe_3 recovered the two CDW transitions and selection-rule-consistent distributions of CDW-1 and CDW-2 peaks. High-throughput peak-width metric (peak spread): Traditional FWHM fitting on linecuts is not scalable to tens of thousands of peaks and is limited by coarse pixelation (peaks spanning only 2–3 pixels). The study defined a model-independent peak spread, Γ_q(T) = I_q^{int}(T) / I_q^{Max}(T), with units of pixels, computed within X-TEC–defined peak boundaries for every CDW peak. This measure correlates with conventional width estimates and enables high-throughput extraction of effective inverse correlation length without high-resolution fits. Separating contributions to peak width: To disentangle instrument resolution, finite correlation length, and crystal imperfections, Γ_q(T) was analyzed versus momentum across many BZs. At each T, Γ_q(H,K,L) was fitted to a quadratic form: Γ_q(T) = Γ_0(T) + γ_H(T) H^2 + γ_K(T) K^2 + γ_L(T) L^2. The momentum-independent term Γ_0(T) was taken as the intrinsic CDW-fluctuation-related width after removing BZ-dependent broadening from imperfections and resolution effects. Transition estimates: For pristine samples, Γ_0(T) drops to a resolution-limited plateau at T_c. For intercalated samples, Γ_0(T) exhibits a threshold temperature below which it is constant (resolution-limited), indicating divergent correlation length. To extrapolate to the vanishing intrinsic width, Γ_0(T) was fit near the threshold using an empirical linear form with a Heaviside step: Γ_0(T) = Γ + α (T − β) Θ[T − β], yielding T_BG = β − Γ/α. Peak intensity (order-parameter-like) trajectories and diffuse-scattering asymmetry around satellite peaks were also quantified; asymmetry was measured via α = (I_1 − I_2)/(I_1 + I_2) computed for congruent satellite diffuse regions and further analyzed by two-cluster X-TEC after masking Bragg and satellite peaks.

- Benchmarking in pristine ErTe_3 recovered two sharp CDW transitions with cluster-averaged intensity trajectories: T_c1 ≈ 262 K (CDW-1) and T_c2 ≈ 135 K (CDW-2). The CDW-1 intensity follows a BCS-like power law with exponent β ≈ 0.54. - CDW peaks are three-dimensional and sharp, satisfying the dimensionality condition for a stable Bragg glass. - Peak intensity trajectories in intercalated samples exhibit broadened onsets with long high-temperature tails due to disorder-pinned fluctuations, making intensity alone insufficient to locate a transition. - The high-throughput peak spread Γ_q(T) enables extraction of the momentum-independent intrinsic width Γ_0(T) from thousands of peaks across ~20,000 BZs, removing BZ-dependent broadening from imperfections. - In pristine samples, Γ_0(T) drops and plateaus at T_c at the resolution limit, as expected for long-range order limited by instrumental resolution. - In intercalated samples (x = 0.5%, 2.0%, 2.6%), Γ_0(T) becomes T-independent below a threshold temperature (resolution-limited widths), implying divergent correlation length and a Bragg glass regime. Extrapolation using Γ_0(T) = Γ + α (T − β) Θ[T − β] yields positive Bragg glass transition temperatures: • x = 2.0%: T_BG ≈ 138 K (from ~3,688 peaks) • x = 2.6%: T_BG ≈ 19 K (from ~2,109 peaks) • x = 0.5%: positive T_BG (value from Supplementary, not specified here) • x = 2.9%: no positive T_BG; trends toward Bragg glass without entering the phase. - The resolution-limited in-plane correlation length is ~20 nm; the average Pd–Pd spacing at x = 2.9% is ~2 nm, consistent with a weak-pinning regime (correlation length much larger than disorder spacing). - Diffuse-scattering asymmetry across satellite pairs is negligible in pristine samples but pronounced in intercalated samples, forming asymmetric half-diamond patterns; quantified asymmetry α increases with intercalation and signals disorder pinning. - The phase diagram constructed from T_BG aligns closely with the onset of in-plane transport anisotropy reported previously, indicating that the anisotropic phase is predominantly a Bragg glass over a wide temperature and disorder range.

The central question—whether CDWs in a disordered crystal can host a bulk Bragg glass rather than only short-range vestigial order—is addressed by combining comprehensive XRD with machine-learning-enabled feature extraction. The vanishing intrinsic peak width (after removing momentum-dependent broadening) and its temperature evolution provide direct bulk evidence for quasi–long-range order characterized by diverging correlation length in intercalated Pd_xErTe_3. The persistence of resolution-limited widths below a threshold temperature and positive T_BG extracted via extrapolation are incompatible with a disorder-pinned, short-range CDW scenario that would yield increasing width at low T. The systematic asymmetry in satellite diffuse scattering only in intercalated samples confirms the presence of disorder pinning, consistent with a Bragg glass where disorder dictates phase shifts but does not introduce free dislocations; STM reports of absent free dislocations at moderate x corroborate this. The close agreement between T_BG and the transport anisotropy onset suggests that the electronic anisotropy arises within the Bragg glass regime, not from a separate short-range-ordered nematic phase. Methodologically, the peak spread metric and q-space quadratic fitting enable disentangling intrinsic fluctuations from imperfections and finite resolution across thousands of peaks, overcoming limitations of traditional linecut-based analyses.

This work provides the first bulk X-ray scattering signatures consistent with a Bragg glass phase in a disordered CDW system, Pd-intercalated ErTe_3, by leveraging comprehensive temperature-dependent diffraction over ~20,000 BZs and unsupervised clustering (X-TEC). A new high-throughput peak spread measure enables extraction of vanishing intrinsic widths and Bragg glass transition temperatures T_BG across intercalation levels, with T_BG aligning with transport anisotropy onsets. The results indicate that any finite intercalation (x > 0) destroys true long-range order and stabilizes a Bragg glass over most of the anisotropic phase space, up to high temperatures, collapsing near x ≈ 2.9%. The approach bridges bulk diffraction, STM observations, and Bragg-glass theory. Future work should employ higher-resolution, higher-brightness diffraction to directly resolve power-law tails of Bragg peaks expected in a Bragg glass and to refine T_BG estimates, as well as extend the methodology to other fluctuating ordered states.

- Finite instrumental resolution prevents direct observation of truly vanishing peak widths and power-law Bragg tails; intrinsic widths are inferred via extrapolation using an empirical linear model near the threshold, which provides a lower-bound estimate of T_BG. - Pixelation and limited peak span (2–3 pixels) limit traditional FWHM accuracy; while peak spread mitigates this at scale, it is an indirect width proxy. - Crystal imperfections and orientation stability across temperatures must be modeled and controlled; residual effects may influence quadratic q-dependence fits. - Highest intercalation (x = 2.9%) does not enter a clear Bragg glass regime within measured temperatures, limiting phase boundary precision at high disorder. - Quantitative T_BG for x = 0.5% is referenced to supplementary data and not specified in the main text.