Hydrogen is potentially a key light element in the Earth's core. Determining the stability of iron hydride is essential for Earth's core mineralogy applications. We investigated the thermal stabilities of a range of Fe-H binaries at core P-T conditions. It is concluded that face-centered cubic phase FeH is stable in the Earth's inner core. The high mobility of hydrogen in the Fe lattice suggests that hydrogen is transferred to a superionic state under the inner core conditions, where the superionic state transfer temperature of H in Fe fcc lattice is similar to 500 K higher than that in hcp Fe system. The H concentration in the inner core is estimated to be similar to 0.92 wt% to explain its density deficit, this value was further constrained to similar to 0.21 wt% by matching the density jump at the inner-core boundary. H alongside other light elements are required to account for the geophysical observations of the Earth's inner core.