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We report a detailed first-principles study of the structural, electronic andvibrational properties of the superconducting C$_{32}$ phase of the ternaryalloy CaAl$_{2-x}$Si$_x$, both in the experimental range $0.6 \leq x \leq 1.2$,for which the alloy has been synthesised, and in the theoretical limits of highaluminium and high silicon concentration. Our results indicate that, in theexperimental range, the dependence of the electronic bands on composition iswell described by a rigid-band model, which breaks down outside this range.Such a breakdown, in the (theoretical) limit of high aluminium concentration,is connected to the appearance of vibrational instabilities, and results inimportant differences between CaAl$_2$ and MgB$_2$. Unlike MgB$_2$, theinterlayer band and the out-of-plane phonons play a major role on the stabilityand superconductivity of CaAlSi and related C$_{32}$ intermetallic compounds

physical review. b, condensed matter and materials physics

Electrons and phonons in the ternary alloy<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msub><mml:mi>CaAl</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Si</mml:mi><mml:mi>x</mml:mi></mml:msub></mml:mrow></mml:math>as a function of composition

Electrons and phonons in the ternary alloyCaAl2−xSixas a function of composition