  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.
