Unraveling the hardness of W$_{1-y}$B$_{3+x}$ $-$ the highest boride of tungsten

Two-dimensional systems have strengthened their position as one of the key materials for novel applications. Very recently, boron joined the distinguished group of elements that are confirmed to possess 2D allotropes, named borophenes. In this work, we explore the stability and hardness of the highest boride of tungsten, which is regarded as built of borophenes separated by metal atoms. We show that WB$_{3+x}$ has Vickers hardnesses approaching 40 GPa only for small values of $x$. The insertion of extra boron atoms is, in general, detrimental for WB$_{3}$ in terms of hardness since leads to the formation of quasi-planar boron sheets that are less tightly connected with the adjacent W layers. Very high concentrations of boron ($x \approx 1$), give rise to a soft (Vickers hardness of about 8 GPa) and unstable $hP$20-WB$_{4}$ structure that can be considered as built of quasi-planar boron $\alpha$-sheets separated by graphitic W layers. On the other hand, we show that the formation of tungsten vacancies brings on structures, e.g. W$_{0.75}$B$_{3+x}$, with Vickers hardnesses that are less sensitive to variations in the boron content and are close in value to the experimentally reported load-independent values above 20 GPa.