  We present a combined experimental and theoretical study of the effects of
pressure on T_c of the hexagonal layered superconductors nH-CaAlSi (n = 1, 5,
6), where nH labels the different stacking variants that were recently
discovered. Experimentally, the pressure dependence of T_c has been
investigated by measuring the magnetic susceptibility of single crystals up to
10 kbar. In contrast to previous results on polycrystalline samples, single
crystals with different stacking sequences display different pressure
dependences of T_c. 1H-CaAlSi shows a decrease of T_c with pressure, whereas 5H
and 6H-CaAlSi exhibit an increase of T_c with pressure. Ab-initio calculations
for 1H, 5H and 6H -CaAlSi reveal that an ultrasoft phonon branch associated to
out-of-plane vibrations of the Al-Si layers softens with pressure, leading to a
structural instability at high pressures. For 1H-CaAlSi the softening is not
sufficient to cause an increase of T_c, which is consistent with the present
experiments, but adverse to previous reports. For 5H and 6H the softening
provides the mechanism to understand the observed increase of T_c with
pressure. Calculations for hypothetical 2H and 3H stacking variants reveal
qualitative and quantitative differences.

Superconductivity;Materials Science