Electronic properties, low-energy Hamiltonian, and superconducting instabilities in ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$

We analyze the electronic properties of the recently discovered stoichiometric superconductor ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ by combining an ab initio approach and a projection of the band structure to a low-energy tight-binding Hamiltonian, based on the maximally localized Wannier orbitals of the $3d$ Fe states. We identify the key symmetries as well as differences and similarities in the electronic structure between ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ and the parent systems ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ and ${\mathrm{KFe}}_2{\mathrm{As}}_2$. In particular, we find ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ to have a significantly more quasi-two-dimensional electronic structure than the latter systems. Finally, we study the superconducting instabilities in ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ by employing the leading angular harmonics approximation and find two potential $A_{1g}$-symmetry representations of the superconducting gap to be the dominant instabilities in this system.

Figure 1

Atomic structure of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ (a) and the definition of the two different FeAs layers (layers $A/B)$ (b). In (c) and (d) the space symmetries of the atom positions of ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ and ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ are illustrated. For ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ the Fe atoms are in high-symmetry points at $(0,\frac{1}{2},\pm \frac{1}{4})$ and $(\frac{1}{2},0,\pm \frac{1}{4})$. The distance of the As atoms above and below the Fe atoms is equivalent. For ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ the FeAs layers are shifted away from the high-symmetry points; also the distance of the As atoms is different for the upper and the lower case.

Figure 2

Electronic dispersion for ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ (black), ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ (shaded red), and ${\mathrm{KFe}}_2{\mathrm{As}}_2$ (shaded blue) along the high-symmetry directions of the first Brillouin zone obtained from DFT using experimentally determined lattice parameters [5, 21, 22]. For better comparison, the electronic structures of ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ and ${\mathrm{KFe}}_2{\mathrm{As}}_2$ were shifted with a rigid band shift to acquire the same doping level as ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$.

Figure 3

Electronic dispersion of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ (black) compared to the mean of the band energies from ${\mathrm{CaFe}}_2{\mathrm{As}}_2$ and ${\mathrm{KFe}}_2{\mathrm{As}}_2$ (green).

Figure 4

Influence of the off-symmetry position of the FeAs layers on the electronic band structure of ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$. (a) shows the low-energy part of the electronic dispersion for ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$  using experimental values for the atomic positions, while (b) refers to the electronic dispersion obtained for the high-symmetry Fe positions at $(0,\frac{1}{2},\pm \frac{1}{4})$ and $(\frac{1}{2},0,\pm \frac{1}{4})$. In the latter case, the distance of the As atoms is equivalent in $\pm z$ direction, using $d_{\text{As-Fe}}=0.105\mathring{\text{A}}$ as the mean of the experimental values.

Figure 5

Comparison between DFT results (a) and the tight-binding (b) parametrization of the energy dispersion and resulting fermiology in ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$. Here, the wannier90 evaluation has been restricted to the Fe $3d$-orbital states.

Figure 6

Introduced symmetry properties of the TB Hamiltonian. The operation $R_1$ rotates the unit cell by ${90}^{\circ }$ along $k_z$ and $R_2$ rotates by ${180}^{\circ }$ around the axis, shown on (c). From (a) to (b) the Fe atom A2 is mapped on A1, from (b) to (c) A2 is mapped on B1, thus both ${\mathrm{Fe}}_2{\mathrm{As}}_2$ layers can be related by this symmetry operation. The arrows x and y refer to the basis vectors in the single Fe unit cell $(k_x,k_y$ in the reciprocal space), while ${\mathrm{x}}^{\prime }$ and ${\mathrm{y}}^{\prime }$ denote the basis vectors in the 2Fe unit cell $(k_1,k_2$ in the reciprocal space).

Figure 7

Fermi surface structure of the ${\mathrm{CaKFe}}_4{\mathrm{As}}_4$ for two different $k_z$ cuts with six hole and four electron Fermi surface pockets. The electronic states at the Fermi level are colored regarding their largest contribution from the iron $3d$ orbitals.

Figure 8

Multiorbital spin susceptibility within RPA approximation at zero temperature for $q_z=0$ (a), $q_z=\pi$ (b), Coulomb repulsion $U=1.5\text{eV}$, and Hund $J=0.1U$.

Figure 9

Sign structure of the leading solution in the $A_{1g}$ channel of the linearized BCS-type gap equations as a function of $U$ and the $J/U$ ratio. The superconducting gap at the six-hole and four-electron Fermi surface pockets can either have a positive or negative sign, which is denoted by the vector $[h_1\cdots h_6,e_1\cdots e_4]$. We find that spin fluctuations at an antiferromagnetic wave vector determine the sign structure of the superconducting gap and lead to a conventional $s^{\pm }$-wave state. In particular, this scenario is likely for $U\gg J$. The crosses refer to the particular ratio of the gaps presented in Fig. 10.

Figure 10

Mean values of the $s^{\pm }$-wave superconducting gap evaluated on the different Fermi surface pockets for two different values of $U=1.5$ eV: (a) the conventional $s^{\pm }$ state and $U=1.2$ eV and (b) the so-called $C$ state and $J/U=0.1$. The red points refer to experimental values [6].