Electrons and phonons in the hexagonal layered superconducting alloy CaAl$_{2-x}$Si$_x$

We report a first-principles study of structural, electronic and vibrational properties of the superconducting C$_{32}$ phase of the ternary alloy CaAl$_{2-x}$Si$_x$, both in the experimental range of stability, $0.6 \leq x \leq 1.2$, and outside, in the limits of high Al and high Si concentration. We find that the dependence of the electronic bands on composition is well described by a rigid-band model, which breaks down outside the experimental range of stability. This breakdown, in the limit of high Al concentration, is connected to vibrational instabilities, and results in important differences between CaAl$_2$ and MgB$_2$. Unlike MgB$_2$, the interlayer band and the out-of-plane phonons play a major role on the stability and superconductivity of CaAlSi and related C$_{32}$ intermetallics.