Equilibrium shape of single-layer hexagonal boron nitride islands on iridium

Large, high-quality layers of hexagonal boron nitride (hBN) are a prerequisite for further advancement in scientific investigation and technological utilization of this exceptional 2D material. Here we address this demand by investigating chemical vapor deposition synthesis of hBN on an Ir(111) substrate, and focus on the substrate morphology, more specifically mono-atomic steps that are always present on all catalytic surfaces of practical use. From low-energy electron microscopy and atomic force microscopy data, we are able to set up an extended Wulff construction scheme and provide a clear elaboration of different interactions governing the equilibrium shapes of the growing hBN islands that deviate from the idealistic triangular form. Most importantly, intrinsic hBN edge energy and interaction with the iridium step edges are examined separately, revealing in such way the importance of substrate step morphology for the island structure and the overall quality of 2D materials.


S1. OVERVIEW OF HBN ISLANDS SHAPES IN A LARGE FIELD OF VIEW
shows a large area (25 µm in diameter) of the Ir(111) substrate with many hBN islands present on the surface. All islands can be categorized as triangular or trapezoidal in shape, with possible deviations from the ideal, regular forms having interior angles of 60°a nd 120°. Figure S1. LEEM image of a large number of triangular and trapezoidal hBN islands on Ir(111).

S2. DETAILS OF THE v ZZ (α) DATA FIT
The red data points in Fig. 3(b), corresponding to B-terminated ZZ edges of hBN, were fitted with a combination of linear and quadratic functions and the blue data points corresponding to N-terminated ZZ edges of hBN were fitted with having the same slope as the one fitting the red data points, only with different intercept.
The application of a particular fit model (linear or quadratic) to the two regions does not qualitatively affect the rest of the analysis nor it brings any novelties. The choice of the fitting functions as shown above provided a good fit to the data points (the adjusted R-Square value of 0.90 for 0 • ≤ α ≤ 90 • and 0.42 for 90 • ≤ α ≤ 180 • ) and was therefore used further in our study. After requiring continuity at α = 90 • and a minimum at α = 180 • , the fit parameters for v ZZ in nm/s and α in degrees are given in Table S1.
parameter value Table S1. Parameters of the v ZZ (α) fitting functions.
Error bars in Fig. 3(b) have been added to account for deviation ofŝ andn as the islands grow, finite pixel size and smearing of hBN island edges in LEEM images, and deceleration effects related to the vicinity of neighboring islands.

S3. EVALUATION OF THE DIFFUSION COEFFICIENTS
The surface diffusion coefficient D for an energy barrier E at a temperature T can be calculated with the aid of equation of magnitude than the product v ZZ L ≈ 3nm/s · 1µm. Therefore, in our experiments hBN islands grow near thermodynamic equilibrium, meaning that the edge advancement speed is directly proportional to the edge free energy and that thermodynamic Wulff construction can be used for the construction of the island shapes S4 .

S4. DETAILS OF HBN EDGE ENERGY
An analytic expression for epitaxial hBN island edge energy per unit length as a function of polar angle and chemical potential has the form S3,S5,S6 γ (χ, ∆µ) = |γ 0 | cos (χ + C), where

and the subscript
x is "N" for −30 • < χ < 0 • and it is "B" for 0 • < χ < 30 • . The intrinsic energies of B-and N-terminated zig-zag (γ ZB and γ ZN ) and armchair (γ A ) edges with the inclusion of binding to the flat metal substrate are It is important to note that the equations outlined up to now do not include interaction with the Ir steps, and can therefore be used for construction of equilibrium shape of hBN island on a flat (step-less) Ir substrate only, as shown in Fig. 4(a). hBN-Ir step interaction is included by making the substitution γ (χ, ∆µ) → γ (χ, ∆µ)·v ZZ (χ), resulting in an alteration of the hBN island shape, as depicted in Figs. 4(b)-(d). Such substitution is justified since v ZZ (α) provides direct information about the relative modification of the energy of different ZZ edges, and only these relative (not absolute) changes are needed for reconstruction of the experimental hBN forms.
N-terminated ZZ edges growing in the step-down direction of Ir were never found in our experiments, and blue data points do not exist for 90 • < α < 180 • in the v ZZ (α) plot in Fig. 3(b). In order to estimate γ (χ, ∆µ) · v ZZ (χ) for N-terminated ZZ edges in the entire range of angles, which is required for plotting all of the blue points in Figs. 4(b)-(d), we assume that the offset between v ZZ,B (α) and v ZZ,N (α) is the same for 0 • < α < 90 • and S4