In-plane anisotropy of charge density wave fluctuations in 1$T$-TiSe$_2$

TL;DR

This work analyzes in-plane anisotropy and stacking effects of CDW fluctuations in 1T-TiSe$_2$ using X-ray diffuse scattering and high-energy XRD to reveal a hierarchy of energy scales and a quasi-2D character of the CDW state.Experimentally, CDW fluctuations above $T_{ ext{CDW}}$ are highly anisotropic in-plane and decoupled between layers, with three independent phase fluctuations corresponding to the three CDW components.A Ginzburg-Landau framework shows the ground state is triple-Q (3Q) and that melting is driven by phase fluctuations confined to domain walls, with a coarse-grained domain model and structure-factor analysis connecting domain statistics to observed diffuse scattering and rod-like $k_z$ features.The results underscore a short CDW coherence length and a decoupled, domain-wall–driven melting process, providing a coherent picture of how domain structure and stacking disorder shape the scattering signatures in TiSe$_2$.

Abstract

We report measurements of anisotropic triple-$q$ charge density wave (CDW) fluctuations in the transition metal dichalcogenide 1$T$-TiSe$_2$ over a large volume of reciprocal space with X-ray diffuse scattering. Above the transition temperature, $T_{\text{CDW}}$, the out-of-plane diffuse scattering is characterized by rod-like structures which indicate that the CDW fluctuations in neighboring layers are largely decoupled. In addition, the in-plane diffuse scattering is marked by ellipses which reveal that the in-plane fluctuations are anisotropic. Our analysis of the diffuse scattering line shapes and orientations suggests that the three charge density wave components contain independent phase fluctuations. At $T_{\text{CDW}}$, long range coherence is established in both the in-plane and out-of-plane directions, consistent with the large observed value of the CDW gap compared to $T_{\text{CDW}}$, and the predicted presence of a hierarchy of energy scales.

Figures (16)

• Figure S1: Thermal diffuse scattering of the CDW peak at (-3.0, 1.5, 0.5). (a) Line cuts along $K$ with the Bragg peak subtracted. The thermal diffuse scattering increases as the temperature rises to $T_{\text{CDW}}$ and then decreases as the sample is further heated. This trend is evident from the raw data points without requiring a fit. The lines represent Lorentzian fits. (b) The peak intensity of the Lorentzian functions shown in (a), with the intensity reaching a maximum at $T_{\text{CDW}}=195$ K.
• Figure S2: In-plane momentum map of the CDW at $L=0.5$ r.l.u. at (a) 140 K (well below $T_{\text{CDW}}$), (b)200 K (just above $T_{\text{CDW}}$) and (c) 240 K (above $T_{\text{CDW}}$). At the base temperature, the CDW peaks are resolution-limited. Above $T_{\text{CDW}}$, the CDW fluctuations exhibit in-plane anisotropy, with the peak elongated along the $q^{\mathit{/}\!\mathit{/}}_\text{CDW}$ direction. The hexagonal white lines represent the normal state Brillouin zone. The scale bar is in arbitrary units and follows a logarithmic scale.
• Figure S3: Out-of-plane momentum map of the CDW at $K=0.5$ r.l.u. at (a) 140 K (well below $T_{\text{CDW}}$), (b)200 K (just above $T_{\text{CDW}}$) and (c) 240 K (above $T_{\text{CDW}}$). Below $T_{\text{CDW}}$, the CDW peaks are resolution-limited with the tails arising from the mosaicity of the sample. The "raindrop" pattern observed above $T_{\text{CDW}}$ is the same as that shown in Fig. 2 of the main manuscript.
• Figure S4: Line cuts of the CDW peaks and their correlation lengths. (a) Momentum scans of the CDW peak at (-4.0, 1.5, -2.5) along the $H$-direction, both below and above $T_{\text{CDW}}$. The CDW peak is sharp at the base temperature, while the diffuse scattering is broad above $T_{\text{CDW}}$. (b)$L$-cuts of the pair of CDW peaks at (-3.0, 1.5, $\pm$0.5). At the base temperature, there is negligible weight between the half-integer CDW peaks. As the temperature increases, spectral weight begins to accumulate at $L=0$, and the overall thermal diffuse scattering weakens. (c) In-plane and out-of-plane correlation lengths of CDW diffuse scattering at (-4.0, 1.5, -2.5). The lines are power-law fits of $T-T_{\text{CDW}}$.
• Figure S5: In-plane momentum maps around the CDW regions of the vacancy-reduced sample. (a)-(c) Scattering peaks associated with CDW components of $q_1$ type (2.5, -6.0, -1.5), $q_2$ type (0.5, -6.5, -1.5) and $q_3$ type (-7.0, 0.5, -0.5) at 30 K. $q_i$ is defined in Fig. 1 of the main manuscript and $Q_x$ is parallel to $H$. $\delta Q_x$ and $\delta Q_y$ are the distances away from the CDW peak position. The CDW peaks are resolution limited. (d)-(f) The same momentum regions as (a)-(c) respectively, but at 210 K, which lies above $T_{\text{CDW}}$=208 K. Intensity scales have been normalized to the maximum intensity in the respective color plots.