Evidence for conventional superconductivity in Bi$_2$PdPt and prediction of topological superconductivity in disorder-free $\gamma$-BiPd

We present comprehensive investigations into the structural, superconducting, and topological properties of Bi$_2$PdPt. Magnetization and heat capacity measurements performed on polycrystalline Bi$_2$PdPt demonstrate a superconducting transition at $\approx$ 0.8 K. Moreover, muon spin relaxation/rotation ($\mu$SR) measurements present evidence for a time reversal symmetry preserving, isotropically gapped superconducting state in Bi$_2$PdPt. We have also performed density-functional theory (DFT) calculations on Bi$_2$PdPt alongside the more general isostructural systems, BiPd$_{x}$Pt$_{1-x}$, of which Bi$_2$PdPt and $\gamma$-BiPd are special cases for $x=0.5$ and $x=1$ respectively. We have calculated the $Z_2$ topological index from our DFT calculations for a range of substitution fractions, $x$, between $x=0$ and $x=1$ characterizing the topology of the band structure. We find a non-trivial topological state when $x>0.75$ and a trivial topological state when $x<0.75$. Therefore our results indicate that BiPd$_{x}$Pt$_{1-x}$ could be a topological superconductor for $x>0.75$.