Structure and composition tunable superconductivity, band topology and elastic response of hard binary niobium nitrides Nb$_2$N, Nb$_4$N$_3$ and Nb$_4$N$_5$

We perform a systematic \textit{ab initio} density functional study of the superconductivity, electronic and phononic band structures, electron-phonon coupling and elastic constants of all four possible structures of niobium nitride $β$-Nb$_2$N as well as Nb-rich $γ$-Nb$_4$N$_3$ and N-rich $β^\prime$-Nb$_4$N$_5$. First of all, we find that all four structures of $β$-Nb$_2$N are superconductors with superconducting transition temperatures ($T_c$) ranging from 0.6 K to 6.1 K, depending on the structure. This explains why previous experiments reported contradicting $T_c$ values for $β$-Nb$_2$N. Furthermore, both $γ$-Nb$_4$N$_3$ and $β^\prime$-Nb$_4$N$_5$ are predicted to be superconductors with rather high $T_c$ of 8.5 K and 15.3 K, respectively. Second, the calculated elastic constants and phonon dispersion relations show that all the considered niobium nitride structures are mechanically and dynamically stable. Moreover, the calculated elastic moduli demonstrate that all the niobium nitrides are hard materials with bulk moduli and hardness being comparable to or larger than the well-known hard sapphire. Third, the calculated band structures reveal that the nitrides possess both type I and type II Dirac nodal points and are thus topological metals. Finally, the calculated electron-phonon coupling strength, superconductivity and mechanical property of the niobium nitrides are discussed in terms of their underlying electronic structures and also Debye temperatures. The present \textit{ab initio} study thus indicates that $β$-Nb$_2$N, $γ$-Nb$_4$N$_3$ and $β^\prime$-Nb$_4$N$_5$ are hard superconductors with nontrivial band topology and are promising materials for exploring exotic phenomena due to the interplay of hardness, superconductivity and nontrivial band topology.