  A new two-dimensional (2D) material, borophene (2D boron sheet), has been
grown successfully recently on single crystal Ag substrates by two parallel
experiments [Mannix \textit{et al., Science}, 2015, \textbf{350}, 1513] [Feng
\textit{et al., Nature Chemistry}, 2016, \textbf{advance online publication}].
Three main structures have been proposed ($\beta_{12}$, $\chi_3$ and striped
borophene). However, the stability of three structures is still in debate.
Using first principles calculations, we examine the dynamical, thermodynamical
and mechanical stability of $\beta_{12}$, $\chi_3$ and striped borophene.
Free-standing $\beta_{12}$ and $\chi_3$ borophene is dynamically,
thermodynamically, and mechanically stable, while striped borophene is
dynamically and thermodynamically unstable due to high stiffness along $a$
direction. The origin of high stiffness and high instability in striped
borophene along $a$ direction can both be attributed to strong directional
bonding. This work provides a benchmark for examining the relative stability of
different structures of borophene.
