Hexatic Phase in Covalent Two-Dimensional Silver Iodide

TL;DR

We study graphene-encapsulated 2D AgI to observe covalent 2D melting using time- and temperature-resolved STEM and NBED. A CNN-based atom-detection workflow, combined with Voronoi analysis, extracts translational and orientational correlations, local density, and defect statistics to classify solid, hexatic, and liquid states. We observe a hexatic phase within a narrow window around 1125–1145 °C, with $G_k(r) \propto r^{-\eta_k}$ and $G_6(r) \propto r^{-\eta_6}$, crossing at $\eta_k \ge 1/3$ and $\eta_6 \le 1/4$, consistent with a mixed solid–hexatic–liquid melting pathway rather than a pure KTHNY scenario. Local density distributions become bimodal near the hexatic–liquid transition, and defect-cluster analysis shows grain-boundary-like structures, supporting the mixed-melting picture. The data also indicate a size-dependent 2D melting point, with $T_{m,2D\infty} \approx 1220 \pm 40$ °C for an infinite crystal, higher than bulk due to graphene confinement, thus highlighting a practical mechanism for tuning stability in atomically thin covalent materials.

Abstract

According to the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory, the transition from a solid to liquid in two dimensions proceeds through an orientationally ordered liquid-like hexatic phase. However, alternative mixed melting scenarios, in which melting proceeds through the hexatic phase with both continuous and discontinuous transitions, have also been observed in some two-dimensional systems. In this study, we imaged silver iodide embedded in multilayer graphene using time- and temperature-resolved in situ atomic-resolution scanning transmission electron microscopy and nanobeam electron diffraction. We observed the hexatic phase and provide evidence supporting a mixed melting scenario.

Figures (33)

• Figure 1: Annular dark-field imaging of the melting of 2D crystalline silver iodide.(A) Density functional theory (DFT) model of 2D AgI shown beside an atomically resolved ADF image of the room-temperature structure. (B) Development of the AgI structure during heating and cooling: crystalline state at 1000 ° C, phase one of the 2D melting process at 1115 ° C with interspersed disordered crystalline and molten areas, and the melting phase two at 1130 ° C. After cooling down to 25 ° C, the AgI crystal appears again but is rotated by 12°. The bright spots visible on and around the AgI crystal are iodine atoms stuck at the edges of graphitized contamination (see Figures \ref{['fig:I_adsorption']} and \ref{['fig:iodine_STEM']}). The FTs in the bottom row are from 100 rapidly acquired STEM images (see Methods). The contrast of the FTs has been optimized to emphasize the AgI peaks. Additional temperature datapoints and larger field of views can be found in figures \ref{['fig:2D_melting_extras_1']}--\ref{['fig:2D_melting_extras_4']}. (C) Temperature dependence of the azimuthal width of the first-order AgI peaks marked with arrows in the Fourier transforms (FTs) of the ADF images in panel B.
• Figure 2: Phase analysis based on nanobeam electron diffraction (NBED).(A) NBED map of an AgI crystal with its right side in a liquid state and its left side in a dynamic state fluctuating between solid and hexatic phases. Supplementary Movies S4 and S5 show time series of diffraction patterns recorded from the indicated positions. (B-D) Average diffraction patterns of the solid, hexatic, and liquid phases within the NBED map. The coloring in the NBED map is based on the approximate width of the diffraction spots: $\leq$ 0.20 rad $\rightarrow$ solid phase (blue); > 0.20 rad $\rightarrow$ hexatic phase (purple); isotropic rings $\rightarrow$ liquid phase (orange).
• Figure 3: Local density distribution functions at different temperatures.(A) Change of local density in the solid-hexatic transition temperature range with Gaussian and double Gaussian fits. (B) Change of local density in the hexatic-liquid transition temperature range with Gaussian and double Gaussian fits. The inset contains the deconvolution of the double Gaussian fits of the 1145 ° C and 1150 ° C datapoints, showing the bimodal nature of these distributions.
• Figure S1: Optical compound microscope images of a Protochips Fusion heating holder. The graphene encapsulated 2D AgI is suspended on an amorphous carbon film, which is supported on a silicon nitride/silicon carbide membrane. The carbon membrane is only visible in panel c, whereas the silicon nitride/silicon carbide membrane with a total of nine holes, is visible in the lower magnifications images in a-b as well.
• Figure S2: Time-averaged Fourier transforms Workflow for obtaining time-averaged Fourier transforms of STEM-ADF images and CNN-extracted atom positions.