  Following the idea that hydrogen-rich compounds might be high-T$_c$
superconductors at high pressures, and the very recent breakthrough in
predicting and synthesizing hydrogen sulfide with record-high T$_c$ = 203 K, ab
initio evolutionary algorithm for crystal structure prediction was employed to
find stable germanium hydrides. In addition to the earlier structure of germane
with space group Ama2, we propose a new C2/m structure, which is energetically
more favorable at pressures above 278 GPa (with inclusion of zero point
energy). Our calculations indicate metallicity of the new C2/m phase of germane
with T$_c$ = 67 K at 280 GPa. Germane is found to exhibit thermodynamic
instability to decomposition to hydrogen and the new compound Ge$_3$H$_{11}$ at
pressures above 300 GPa. Ge$_3$H$_{11}$ with space group I$\bar{4}$m2 is found
to become stable at above 285 GPa with T$_c$ = 43 K. We find that the
pressure-induced phase stability of germanium hydrides is distinct from its
analogous isoelectronic systems, e.g., Si-hydrides and Sn-hydrides.
Superconductivity stems from large electron-phonon coupling associated with the
wagging, bending and stretching intermediate-frequency modes derived mainly
from hydrogen.

Superconductivity