Origin of large electron-phonon coupling in the metallic hydride TiH$_2$

The recent discovery of large superconducting transition temperature of $T_c=190$ K in metallic H$_2$S under high pressures of 200 GPa, has renewed the interest in the superconducting properties of metal-hydrogen systems. These materials are expected to be electron-phonon superconductors and hydrogen with its low mass can contribute new optic phonons that may couple with the conduction electrons. Often, though not always, a large electron-phonon coupling parameter $\lambda$ (and consequently high $T_c$) can result from a high electronic density of states at the Fermi level ($N(E_F)$) and the presence of soft phonons. With the help of first-principles calculations within density functional theory, we studied the cubic TiH$_2$ which has a large $3d$ $N(E_F)=2.8$ states/eV/f.u. Our calculated phonon dispersions show that Ti modes active below frequencies of 10 THz whereas much lighter H modes are active between 32 and 40~THz. Electron-phonon coupling calculations reveal a $\lambda=0.98$ which corresponds to a $T_c=6.1$ K. However, the large $N(E_F)$ also leads to a tetragonal instability at low temperatures in TiH$_2$, which may be overcome by a uniaxial strain, potentially making it a candidate for electron-phonon superconductor.