  CaBi$_2$ is a recently discovered type-I superconductor with $T_c=2$~K and a
layered crystal structure. In this work electronic structure, lattice dynamics
and electron-phonon interaction are studied, with a special attention paid to
the influence of the spin-orbit coupling (SOC) on above-mentioned quantities.
We find, that in the scalar-relativistic case (without SOC), electronic
structure and electron-phonon interaction show the quasi-two dimensional
character. Strong Fermi surface nesting is present, which leads to appearance
of the Kohn anomaly in the phonon spectrum and enhanced electron-phonon
coupling for the phonons propagating in the Ca-Bi atomic layers. However,
strong spin-orbit coupling in this material changes the topology of the Fermi
surface, reduces the nesting and the electron-phonon coupling becomes weaker
and more isotropic. The electron-phonon coupling parameter $\lambda$ is reduced
by SOC almost twice, from 0.94 to 0.54, giving even stronger effect on the
superconducting critical temperature $T_c$, which drops from 5.2~K (without
SOC) to 1.3~K (with SOC). Relativistic values of $\lambda$ and $T_c$ remain in
a good agreement with experimental findings, confirming the general need for
including SOC in analysis of the electron-phonon interaction in materials
containing heavy elements.

Superconductivity;Materials Science