  Low-energy rattling modes and their effects on superconductivity are studied
in the cage compound GaxV2Al20. A series of polycrystalline samples of 0 < x =<
0.6 are examined through resistivity, magnetic susceptibility, and heat
capacity measurements. A weak-coupling BCS superconductivity is observed below
Tc = 1.4-1.7 K for all the samples. For small Ga contents below 0.20,
approximately 30% of the cages are occupied by rattling Al atoms having an
Einstein temperature TE of 23 K, probably with most Ga atoms substituting for
the cage-forming Al atoms. For higher Ga contents, approximately 0.05 Ga and
0.25-0.35 Al atoms coexist statistically inside the cages and behave as
rattlers with TE ~ 8 and 23 K, respectively. A significant effect of Ga
rattling on the superconductivity is clearly evidenced by the observation of a
sharp rise in Tc by 8% at x = 0.20 when 0.05 Ga atoms are introduced into the
case. Probably, the electron-phonon interaction is significantly enhanced by an
additional contribution to the phonon density of states from the extremely low
energy rattling modes of Ga atoms. In addition, a large softening of the
acoustic modes is observed for x => 0.20, suggesting that the cage itself
becomes anomalously soft in the presence of low-energy Ga rattling modes.
