Phase stability and large in-plane resistivity in the 112-type iron-based superconductor Ca$_{1-x}$La$_{x}$FeAs$_{2}$

The recently discovered high-T$_c$ superconductor Ca$_{1-x}$La$_{x}$FeAs$_{2}$ is a unique compound not only because of its low symmetry crystal structure, but also because of its electronic structure which hosts Dirac-like metallic bands resulting from (spacer) zig-zag As chains. We present a comprehensive first principles theoretical study of the electronic and crystal structures of Ca$_{1-x}$La$_{x}$FeAs$_{2}$. After discussing the connection between the crystal structure of the 112 family, which Ca$_{1-x}$La$_{x}$FeAs$_{2}$ is a member of, with the other known structures of Fe pnictide superconductors, we check the thermodynamic phase stability of CaFeAs$_{2}$, and similar hyphothetical compounds SrFeAs$_{2}$ and BaFeAs$_{2}$ which, we find, are slightly higher in energy. We calculate the optical conductivity of Ca$_{1-x}$La$_{x}$FeAs$_{2}$ using the DFT + DMFT method, and predict a large in-plane resistivity anisotropy in the normal phase, which does not originate from electronic nematicity, but is enhanced by the electronic correlations. In particular, we predict a 0.34 eV peak in the $yy$ component of the optical conductivity of the 30\% La doped compound, which correponds to coherent interband transitions within a fast-dispersing band arising from the zig-zag As-chains which are unique to this compound. We also study the Landau free energy for Ca$_{1-x}$La$_{x}$FeAs$_{2}$ including the order parameter relevant for the nematic transition and find that the free energy does not have any extra terms that could induce ferro-orbital order. This explains why the presence of As chains does not broaden the nematic transition in Ca$_{1-x}$La$_{x}$FeAs$_{2}$.