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Recently, an alkaline earth hydride CaH$_6$ having a sodalitelike clathratestructure has been experimentally synthesized at megabar pressures with amaximum $T_c$ of 215 K, comparable to that of a rare earth hydride LaH$_{10}$.Here, based on first-principles calculations, we find that CaH$_6$ exhibits ahuge peak in the Eliashberg spectral function ${\alpha}^{2}F$ around thelow-frequency region of H-derived phonon modes, in contrast to LaH$_{10}$having a widely spreading spectrum of ${\alpha}^{2}F$ over the wholefrequencies of H-derived phonon modes. It is revealed that the huge peak of${\alpha}^{2}F$ in CaH$_6$ is associated with an effective electron-phononcoupling (EPC) between low-frequency optical phonons and hybridized H 1$s$ andCa 3$d$ states near the Fermi energy. As pressure increases, the strengthenedH$-$H covalent bonding not only induces a hardening of optical phonon modes butalso reduces the electron-phonon matrix elements related to the low-frequencyoptical modes, thereby leading to a lowering of the EPC constant. It is thusdemonstrated that H-derived low-frequency phonon modes play an important rolein the pressure-induced variation of $T_c$ in CaH$_6$. Furthermore, unlike thepresence of two distinct superconducting gaps in LaH$_{10}$, CaH$_6$ is foundto exhibit a single isotropic superconducting gap.

Electron-phonon coupling and superconductivity in an alkaline earth hydride CaH$_6$ at high pressures