Defining shapes of two-dimensional crystals with undefinable edge energies

Predicting equilibrium crystal shapes traditionally uses the Wulff construction based on direction-dependent surface or edge energies. In low-symmetry 2D crystals, edge energies cannot be uniquely defined because opposite or distinct edges are not symmetry-equivalent; ribbon or polygonal cutouts then only yield averages, not individual edge energies. This indeterminacy, rooted in gauge invariance noted by Cahn and colleagues, implies that absolute edge energies are not uniquely determinable for all directions in low-symmetry systems. The research question addressed here is how to predict unique equilibrium shapes of low-symmetry 2D crystals when their edge energies are fundamentally undefinable. The authors propose a computational framework that uses auxiliary closure relations combined with total energies of a minimal set of ribbons and triangles to recover a unique Wulff shape, despite the indeterminacy of individual edge energies.

The study builds on classical Wulff construction for equilibrium shapes from direction-dependent surface/edge energy, and on first-principles density functional theory (DFT) as a means to compute total energies of finite slabs or ribbons. Prior approaches could determine edge energies when sufficient symmetry allowed constructing ribbons or equilateral polygons with identical edges (e.g., GaAs polyhedra; 2D materials like hBN and metal chalcogenides). However, for many 2D materials lacking inversion or high rotational symmetry, edge energies are not uniquely definable due to gauge invariance, as highlighted by Cahn and co-workers, leaving a gap in predicting shapes without empirical input. The present work contributes a general resolution for such low-symmetry cases.

The authors define a master system of linear relations between basic edge energies and total energies of carefully chosen finite shapes whose total energies can be computed. For a given 2D lattice, a set of basic edge directions is selected (often along lattice vectors and diagonals). For each ribbon with opposite parallel edges of type i and i′ and for selected triangles combining three basic edges, they write linear equations relating unknown basic edge energies εi to the total energy per cell length/size of the computed structures, referenced to the bulk chemical potential. In low-symmetry cases, the number of unknown edge energies exceeds the number of independent equations from available ribbons and triangles, yielding an underdetermined system. To close the system, they introduce auxiliary closure equations—arbitrary constraints on linear combinations of edge energies (e.g., ε3 − ε1 = α), acknowledging that these auxiliaries do not alter the resulting Wulff shape due to gauge invariance but enable solving for a consistent set of εi. With the basic εi determined for any chosen auxiliary values, they obtain a continuous ε(α) for arbitrary edge orientation α using an interpolation ansatz: viewing a vicinal edge as a sequence of projections of basic edges, leading to ε(α) = ε0 cos(α + C), with amplitude and phase determined by lattice geometry and basic εi. For multicomponent materials, chemical potentials are included: total energies on the RHS are referenced to bulk phase chemical potentials, and imbalances (e.g., μ = ½(μSe − μSn) for SnSe) enter equations where structures contain excess of particular elements around perimeters. For ternary AgNO2, composition is treated effectively as Ag and NO2 with μAg + μNO2 fixed, introducing μAg into relevant RHS terms for ribbons and triangles containing extra Ag per cell. The equilibrium shape is then obtained by conventional Wulff construction from the interpolated ε(α). The authors also provide a practical workflow: identify if edge energies are undefinable by symmetry, choose basic edges with their lowest-energy reconstructions, compute total energies of a minimal set of independent ribbons/triangles (via DFT or suitable atomistic potentials), add one or two auxiliary closure relations as required, solve the linear system for εi, interpolate ε(α), and perform the Wulff construction.

- Established a general auxiliary-edge-energy framework that yields unique equilibrium Wulff shapes even when individual edge energies are fundamentally undefinable in low-symmetry 2D crystals. - For C2 symmetry (SnSe as exemplar), five non-equivalent basic edges lead to four independent equations from two ribbons and one triangle; introducing one auxiliary closure recovers a unique shape. Interpolation yields ε(α) whose gauge-induced shifts do not affect the Wulff envelope. - DFT-derived RHS values for SnSe (per unit length/cell): E11/l1 = 0.47 μ + 0.44, E22/l2 = 0.10, E10/l0 = 0.10, triangle E123 = 1.11. At μ = −0.67 (selenium-poor conditions), shapes are rhombi enclosed by ASn and ASe edges; as μ increases, shapes evolve to rectangles truncated at two corners, consistent with experimental observations of SnSe islands. - Demonstrated that only certain combinations of edge energies are μ-dependent and definite (e.g., εZSe + εZS = εASe + εASn), while individual εi vary with auxiliary parameters without changing the shape. - For no-symmetry C1 case (AgNO2), eight basic edges yield six independent equations (four ribbons, two triangles); two auxiliary closures complete the system. DFT RHS values used: ribbons 0.82, 0.01, 0.52, 0.64 eV; triangles ≈3.15 eV (with μAg subtracted where extra Ag appears). Predicted a highly elongated, asymmetric needle-like equilibrium shape with one slanted and one nearly straight tip, in good agreement with scarce experimental images. - Provided a ranking of definability: (1) trivial-definable (all edges from ribbons by inversion symmetry), (2) non-trivial-definable (regular polygons like hBN, MoS2, GaS), (3) partially definable (e.g., SnSe class), (4) no definability (e.g., AgNO2); the auxiliary approach resolves (3) and (4) without empirical input.

The work resolves the apparent paradox that Wulff construction requires known edge energies while low-symmetry crystals lack uniquely definable ε(α). By leveraging gauge invariance, the authors show that although individual edge energies are underdetermined, the Wulff envelope—the physical equilibrium shape—is invariant to auxiliary shifts. Thus, a minimal set of computable total energies for ribbons and triangles, supplemented by arbitrary but fixed auxiliary closures, suffices to reconstruct ε(α) up to a gauge and to determine a unique shape. Incorporating chemical potentials enables exploring how growth conditions (elemental chemical potentials) alter equilibrium shapes in multicomponent 2D materials. The approach restores predictive capability for a broad class of low-symmetry systems, with results validated against observed morphologies for SnSe and AgNO2.

The study introduces an auxiliary edge energy methodology and a master system of linear relations that, combined with a simple interpolation ansatz and Wulff construction, predicts unique equilibrium shapes of 2D crystals even when edge energies are fundamentally undefinable. Demonstrations on SnSe (C2) and AgNO2 (C1) match experimental morphologies, and a taxonomy of definability across 2D materials is provided. The framework is general: it can incorporate chemical potentials, extend to 3D crystals (with a larger linear system and three auxiliary parameters), and be adapted to finite temperatures by replacing energies with Gibbs free energies. Future directions include applications to substrate-supported layers, solvent/ligand effects, low-symmetry twisted bilayers, and biomolecular crystals, as well as systematic exploration of temperature-dependent rounding and edge reconstructions.

- The approach relies on accurate total energy calculations of ribbons and triangles and on correctly identified lowest-energy edge reconstructions; errors here propagate to ε(α) and shapes. - The interpolation ansatz for ε(α) assumes vicinal edges decompose into projections of basic edges; while the Wulff shape is robust to small ansatz inaccuracies, strongly reconstructed or complex edges might require refined interpolation or additional basic edges. - Chemical potentials must be specified within thermodynamic bounds and may be influenced by kinetics and environment; predicted equilibrium shapes may differ from kinetically controlled growth morphologies. - Finite-temperature effects are not explicitly included in the main calculations (though the method can incorporate Gibbs free energies); vertex rounding and entropy contributions could modify details. - Experimental validation for some systems (e.g., AgNO2) is limited, constraining comprehensive benchmarking.