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First-principles calculations of phase transition, low elastic modulus, and superconductivity for zirconium

The elasticity, dynamic properties, and superconductivity of $α$, $ω$, and $β$ Zr are investigated by using first-principles methods. Our calculated elastic constants, elastic moduli, and Debye temperatures of $α$ and $ω$ phases are in excellent agreement with experiments. Electron-phonon coupling constant $λ$ and electronic density of states at the Fermi level $N$(\emph{E}$_{\rm{F}}$) are found to increase with pressure for these two hexagonal structures. For cubic $β$ phase, the critical pressure for mechanical stability is predicted to be 3.13 GPa and at \emph{P}=4 GPa the low elastic modulus ($E$=31.97 GPa) can be obtained. Besides, the critical pressure for dynamic stability of $β$ phase is achieved by phonon dispersion calculations to be $\mathtt{\sim}$26 GPa. Over this pressure, $λ$ and $N$(\emph{E}$_{\rm{F}}$) of $β$ phase decrease upon further compression. Our calculations show that the large value of superconducting transition temperature $\emph{T}_{\rm{c}}$ at 30 GPa for $β$ Zr is mainly due to the TA1 soft mode. Under further compression, the soft vibrational mode will gradually fade away.