Phase stability and large in-plane resistivity anisotropy in the 112-type iron-based superconductor ${\mathbf{Ca}}_{1-x}{\mathbf{La}}_x{\mathrm{FeAs}}_2$

The recently discovered high-$T_c$ superconductor ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2$ 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) zigzag As chains. We present a comprehensive first-principles theoretical study of the electronic and crystal structures of ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2$. After discussing the connection between the crystal structure of the 112 family, which ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2$ is a member of, with the other known structures of Fe pnictide superconductors, we check the thermodynamic phase stability of ${\mathrm{CaFeAs}}_2$, and similar hyphothetical compounds ${\mathrm{SrFeAs}}_2$ and ${\mathrm{BaFeAs}}_2$ which, we find, are slightly higher in energy. We calculate the optical conductivity of ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2$ 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 corresponds to coherent interband transitions within a fast-dispersing band arising from the zigzag As chains, which are unique to this compound. We also study the Landau free energy for ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2$ 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 ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2$.

Figure 1

Crystal structures related to iron 1111 and 112 phase systems. Crystal structures of (a) ZrCuSiAs ($P4/nmm$), (b) ${\mathrm{HfCuSi}}_2\left(P4/nmm\right)$, and (c) ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2\left(P{2}_1\right)$.

Figure 2

(a) Phonon dispersion curve for the hypothetical tetragonal structure with $P4/nmm$ of ${\mathrm{CaFeAs}}_2$. The gamma-point phonon instability is clearly shown, and the unstable phonon bands have the As square-net character. (b) The corresponding unstable phonon normal mode. The main lattice distortion occurs in the As square net. Black arrows represent the direction of the lattice distortion. The As square net is changed into the zigzag As chain.

Figure 3

The sketch of the three different distortions present in the structure of ${\mathrm{Ca}}_{1-x}{\mathrm{La}}_x{\mathrm{FeAs}}_2$ with respect to the reference ${\mathrm{HfCuSi}}_2$-type structure with space group No. 129. (a) The structure sketched with the Fe layer in the center. (b) The ${\mathrm{\Gamma }}_2^{+}$ irrep involves a rumpling of the Fe and the As (which is in the CaAs layer) layers. (c) The ${\mathrm{\Gamma }}_5^{-}$ irrep is the in-plane polar irrep that involves a displacement of all the ions along the same direction. (d) The ${\mathrm{\Gamma }}_5^{+}$ irrep involves pair of ions moving opposite to each other in the same plane. This irrep is responsible of the formation of the As chains.

Figure 4

Group table showing the relevant structural phase transitions between the tetragonal $P4/nmm$ and the monoclinic $P{2}_1$ structures. The space group numbers are also shown below the Hermann-Mauguin notation. Other possible structural phase transitions are also shown with the corresponding symmetry of the lattice distortion.

Figure 5

(a) Tetragonal crystal structure with $P4/nmm$ of ${\mathrm{CaFeAs}}_2$. (b) Monoclinic crystal structure with $P{2}_1/m$ of ${\mathrm{CaFeAs}}_2$. The black lines represent the unit cell in both (a) and (b). The As square net is clearly shown in (a), whereas, the zigzag As chain appears in (b). Both spacers, As square net and the zigzag As chain, are metallic.

Figure 6

Ternary phase diagram for (a) ${\mathrm{CaFeAs}}_2$, (b) ${\mathrm{SrFeAs}}_2$, and (c) ${\mathrm{BaFeAs}}_2$. Red and blue dots represent thermodynamically stable and unstable phases, respectively. ${\mathrm{CaFeAs}}_2$, ${\mathrm{SrFeAs}}_2$, and ${\mathrm{BaFeAs}}_2$ are put above the convex-hull, and the energies above hull are 13, 24, and 17 meV/atom, respectively.

Figure 7

Crystal structures of ${\mathrm{BaFeAs}}_2$ having the lowest energy based on (a) spin nonpolarized GGA, (b) spin-polarized GGA, and (c) GGA + $U$ ($U=2$ and 4 eV) schemes. All of them are orthorhombic and their space groups are $Imm2$ (No. 44), $Cmma$ (No. 67), and $Cmcm$ (No. 63), respectively.

Figure 8

(a) $A(\mathbf{k},\omega )$ of ${\mathrm{Ca}}_{0.8}{\mathrm{La}}_{0.2}{\mathrm{FeAs}}_2$ at $T=116\text{K}$ as computed by DFT+eDMFT. The Brillouin zone is shown in inset in the bottom of (c). (b) $A(\mathbf{k},\omega )$ of ${\mathrm{Ca}}_{0.7}{\mathrm{La}}_{0.3}{\mathrm{FeAs}}_2$ at $T=116\text{K}$ as computed by DFT+eDMFT. $(\mathrm{The}A(\mathbf{k},\omega )$ of the 30% doped compound is also presented in Ref. [19] with orbital projection onto the in-plane $p$ orbitals of As ions forming the chains.) (c) Optical conductivities within DFT+eDMFT method of ${\mathrm{CaFeAs}}_2$ (top), ${\mathrm{Ca}}_{0.8}{\mathrm{La}}_{0.2}{\mathrm{FeAs}}_2$ (middle), and ${\mathrm{Ca}}_{0.7}{\mathrm{La}}_{0.3}{\mathrm{FeAs}}_2$ (bottom). Optical conductivities (interband contributions only) within DFT method are also shown for comparison. (Intraband transitions within DFT give a delta-function-like contribution at the zero frequency.) The values of in-plane resistivity anisotropy ${\rho }_a/{\rho }_b$ correspond to the eDMFT results. The in-plane average conductivity is shown in inset for comparison between eDMFT, DFT, and experiment. We used 0.01 eV for the broadening of intraband contributions in DFT. Experiment data are for ${\mathrm{Ca}}_{0.77}{\mathrm{Nd}}_{0.23}{\mathrm{FeAs}}_2$ at $T=125\mathrm{K}$ and digitized from Ref. [63]. For ${\mathrm{Ca}}_{0.7}{\mathrm{La}}_{0.3}{\mathrm{FeAs}}_2$, there is a low-energy peak of 0.34 eV in the optical conductivity, which corresponds to the transition marked by the green arrow in (b).

Figure 9

DOS of ${\mathrm{Ca}}_{0.8}{\mathrm{La}}_{0.2}{\mathrm{FeAs}}_2$. Total and projected DOS of Fe $3d$ orbitals calculated by DFT+eDMFT at $T=116\mathrm{K}$. The inset shows the direction of local axes for the projected DOS of Fe $3d$ orbitals. Our choice of Cartesian axes are ${45}^{\circ }$ rotated with respect to the $\langle 100\rangle$ axes. The occupations of Fe $d_{xz}$ and $d_{yz}$ are same, giving the zero orbital polarization.

Figure 10

(a) The mass enhancement $m^{*}/m_e$ of the iron $3d$ orbitals upon doping calculated by the DFT+eDMFT. The experimental data (open diamond) are obtained from angle-resolved photoemission spectroscopy experiments in Refs. [19, 20]. (b) The orbital occupation of the iron $3d$ orbitals upon doping.

Figure 11

(a) The ferro- and (b) the antiferro-orbital orders.

Figure 12

Sketch of the Fe plane with (a) no strain, (b) shear strain, and (c) normal strain. While there are multiple components of possible shear and normal strains, we only show the relevant ones, the ${\eta }_{12}$ shear strain and ${\eta }_{11}$ normal strain. (d) In the nonstrained structure, there are four mirror planes that are not parallel to [001]. Normal strain breaks the mirror symmetry of only two of these planes (${\mathrm{m}}_{110}$ and ${\mathrm{m}}_{\overline{1}10}$, shown in blue) whereas the shear strain breaks only the other two (${\mathrm{m}}_{100}$ and ${\mathrm{m}}_{010}$, shown in red).

Figure 13

The stripe AFM order with its wave vector along the $\left[\overline{1}10\right]$ direction ($L_y$). Note that in our theory we do not take into account the spin orbit coupling and consider only collinear spin arrangements. While the relative orientations of the spin moments are meaningful, the direction that all the spins are parallel or antiparallel to is not. The stripe AFM order breaks the ${\mathbf{m}}_{100}$ and ${\mathbf{m}}_{010}$ mirror symmetries.

Figure 14

$T_c$ vs anion height in iron pnictide and chalcogenide superconductors. (Reproduced with permission from Ref. [73].) The three data points we added (black) for the 112 superconductors display a similar trend as the other iron based high-$T_c$ superconductors. This suggests that the metallic spacer layer in the 112 family does not influence superconductivity.