Thermodynamic stability of Borophene, $\mathrm{B_2O_3}$ and other $\mathrm{B_{1-x}O_x}$ sheets

The recent discovery of borophene, a two-dimensional allotrope of boron, raises many questions about its structure and its chemical and physical properties. Boron has a high chemical affinity to oxygen but little is known about the oxidation behavior of borophene. Here we use first principles calculations to study the phase diagram of free-standing, two-dimensional $\mathrm{B_{1-x}O_x}$ for compositions ranging from $x=0$ to $x=0.6$, which correspond to borophene and $\mathrm{B_2O_3}$ sheets, respectively. Our results indicate that no stable compounds except borophene and $\mathrm{B_2O_3}$ sheets exist. Intermediate compositions are heterogeneous mixtures of borophene and $\mathrm{B_2O_3}$. Other hypothetical crystals such as $\mathrm{B_2O}$ are unstable and some of them were found to undergo spontaneous disproportionation into borophene and $\mathrm{B_2O_3}$. It is also shown that oxidizing borophene inside the flakes is thermodynamically unfavorable over forming $\mathrm{B_2O_3}$ at the edges. All findings can be rationalized by oxygen's preference of two-fold coordination which is incompatible with higher in-plane coordination numbers preferred by boron. These results agree well with recent experiments and pave the way to understand the process of oxidation of borophene and other two-dimensional materials.


Introduction
The investigation and development of two-dimensional (2D) materials is currently a focus of research worldwide [1,2]. This class of materials was recently extended by another representative when two research teams were able to grow a boron monolayer on a silver surface in ultra-high vacuum [3,4]. The experimental discovery of this "borophene" was anticipated by various theoretical predictions ranging back to the mid-1990s [5,6,7,8]. Analogous to the existence of many boron bulk phases, free-standing borophene exhibits a pronounced polymorphism [7,  ; however it is lifted in the vicinity of a metal surface which enables the growth of specific borophene crystals [10,11]. A prototypical borophene (the α -sheet) is shown in Fig. 1(a).
Boron is also known to have a high chemical affinity to oxygen and therefore boron-rich (nano)-structures can only be prepared under inert conditions or vacuum; so is borophene. For bulk phases the boron-oxygen system is well studied and several works on the phase diagram exist [12,13,14]. Thermodynamically stable are the pure "icosahredal" boron phases in their various forms [15], boron suboxide B 6 O [16] and boron trioxide B 2 O 3 that is a vitreous phase under ambient conditions [17] but can be crystalline when synthesized under pressure [18,19,14].
Sometimes boron monoxide B 2 O [20,21] is also considered but its stability and existence is strongly debated [22,23,12,13]. However, as the bonding in borophene is different from the bonding in the bulk phases, we can expect it to have different chemical properties, which are largely unexplored so far. Feng et al. exposed borophene flakes to different oxygen concentrations and found that they tend to oxidize from the edges, while boron atoms inside the flakes are relatively inert to oxidation [4]. We will come back to this point in the discussion below. In the literature a variety of 2D boron-oxygen structures were studied. Two-dimensional variants of B 2 O 3 were proposed by Ferlat et al. [17], with building blocks formed by planar BO 3 -units (see Fig. 1 [20] and multiple hypothetical structural models for monolayers were previously considerd [25,26]. One possible B 2 O monolayer model is shown in Fig. 1(c). Several authors theoretically studied the adsorption of oxygen on the buckled triangular borophene [27,28,29,30] or related nanoribbons [31]. However, structures related to the buckled triangular structure are not thermodynamically favorable neither as stand-alone system nor when placed on a metal surface [11] and angle-resolved photoemission spectroscopy of the experimentally realized borophenes (β 12 and χ 3 sheets) are also highly consistent with non-triangular borophenes [32,33]. A different theoretical study was performed by Luo et al. who considered oxygen adsorption, dissociation and diffusion on χ 3 borophene on Ag(111) [34]. They find that oxygen is not incorporated into the borophene layer (which mostly remains structurally intact) but it rather adsorbs on top of it. Sheets of varying composition, where oxygen is incorporated into the boron plane, were studied by Zhang et al. To systematize and unify the view on these various structures we study 149 boron-oxygen layers by first principles calculations and use them to construct the convex hull of the 2D boronoxygen system. The convex hull is related to the phase diagram and allows to decide which of these many systems are thermodynamically favorable and are likely to be realized experimentally.
This also helps to understand the process of oxidation of borophene. Our findings shed a new light on the boron-oxygen binary system, put the previous literature into context and agree well with experimental reports.

Computational methods
The first principles calculations of 2D B 1−x O x structures were performed with the density functional theory (DFT) code SIESTA, version 3.2 [38], using norm-conserving Troullier-Martins pseudopotentials [39] as provided on the SIESTA homepage as part of the Abinit's pseudo database. Electronic correlations were treated using the Perdew-Burke-Ernzerhof (PBE) exchangecorrelation functional [40] within the generalized gradient approximation. Calculations were carried out using a default DZP basis set and an energy cutoff of 250 Ry for the real-space grid. The unit cells were built with 20Å separation between replicas in the perpendicular direction to achieve negligible interactions. The k-space integrations were carried out using the Monkhorst-Pack scheme [41], employing a Fermi smearing of 300 K. The k-grids were converged for every considered structure such that energy changes were smaller than 1 meV/atom. Atomic coordinates as well as lattice parameters were relaxed using the conjugated gradient method

Results and Discussion
In order to construct the convex hull of free-standing, two-dimensional allotropes of boron and where E(a) is the DFT total energy per atom of structure a, E(B) = −79.64 eV/atom is the energy of the borophene α -sheet [7,8] and E(B 2 O 3 ) = −301, 33 eV/atom is the energy of a B 2 O 3 sheet [17]. The relative oxygen fraction is given by y = x/0.6; this definition implies that 4 our maximum composition is x = 0.6 for B 2 O 3 . The mixing energy E mix can be considered as Gibbs free formation energy G(x, p, T ) at p = 0 Pa and T = 0 K for the fictitious reaction The results are shown in the scatter plot in Fig. 2.
The borophene structures on the left-hand side of the plot (green triangles) are taken from various publications [4,7,6,42,8]. The rich polymorphism of boron is discernible by the large number of data points. In agreement with the literature we also find that (for the PBE functional) the α -sheet (see Fig. 1(a)) is the lowest energy borophene for free-standing systems [7,9]. It was therefore chosen as reference structure for calculating E mix .
On the right-hand side of the figure (red triangles) we find another set of very stable structures -the B 2 O 3 sheets proposed by Ferlat et al. [17]. Despite their different structure the two are nearly degenerate in energy. Figure 1(b) shows the B 2 O 3 sheet with planar BO 3 -units that we use as second reference to define E mix .
The hexagons for a composition of x = 1/3 in Fig. 2 Fig. 1(c). To our knowledge it was not reported before. However, all these honeycomb-like B 2 O systems still have rather high, positive mixing energies, which indicates that they are thermodynamically unfavorable. Besides the many B 2 O realizations on a honeycomb lattice we also find systems with rather irregular, non-hexagonal geometry (gray squares) and three structures with a particularly low energy (empty, gray diamonds). One of them is shown in Fig. 1 Fig. 2), some transformed into irregular structures (squares), but most of the systems spontaneously separated into borophene and B 2 O 3 (empty diamonds). An example of the last category is shown in Fig. 1(e), which corresponds to x = 0.37 and E mix = 0.18 eV/atom. To lower E mix further, structures that are already separated in their initial geometry were studied. The typical structural elements used for this approach were triangular units for boron and BO 3 -units and boroxo-rings for B 2 O 3 . These were connected to form areas of pure boron linked to areas of B 2 O 3 . An example from this set of structures is shown in Fig. 1(f). The energies are shown as crossed, gray diamonds in Fig. 2. It is obvious that E mix is significantly lower for such systems. The last approach to construct separated systems was by using boron and B 2 O 3 ribbons (for details see the Supplementary Material). An example for these structures is shown in Fig. 1(g). These structures show the overall lowest mixing energies of all studied sheets (filled, gray diamonds in Fig. 2). We also added the energy of the structures studied by Zhang et al. [35] and Lin et al. [36] to Fig. 2 (purple symbols). Their mixing energy is significantly higher than the ones of the heterogeneous mixtures, indicating that they are thermodynamically unfavorable.
All the described approaches yield the same result for the considered range of compositionsthe lowest energy structures are heterogeneous mixtures of borophene and B 2 O 3 .
To generate systems with with small oxygen fractions 50 structures were constructed by adding oxygen atoms to different borophene sheets (α, α 1 , β, β 1 , χ 3 following the notation by Wu et al. [8]). We found that for x < 0.05 (orange triangles in Fig. 2) the systems are mostly characterized by divalent oxygen atoms binding out-of-plane to sites near the hexagonal holes of the borophene sheets. The sheets themselves remain intact and are structurally modified only in the vicinity of the O atom. Figure 1(h) shows one example of such a system. These findings agree well with similar results by Luo et al. [34]. For x ≥ 0.05 oxygen tends to be incorporated in-plane and the structure is distorted (indicated by squares in Fig. 2). The data points in Fig. 2 [26,35,36]. The gray diamonds in Fig. 2 [4]. All findings can be rationalized by oxygen's preference of two-fold coordination which is incompatible with higher in-plane coordination numbers favored by boron.
These results are an important step forward to understand the oxidation behavior of 2D boron, which is crucial for its practical use in potential future electronic, optical or chemical applications. For future investigations it would be interesting to reconsider the compositional phase diagram on metal surfaces as in the experiment, which might be simplified by studying systems as a function of global charge transfer.