Electronic Structure of InCo2As2 and KInCo4As4: LDA + DMFT

A comparative analysis of the electronic structure obtained in the DFT/LDA and LDA + DMFT approaches of the possible isostructural analogues of iron superconductors InCo2As2 and KInCo4As4 with the electronic structure of the parent high-temperature superconductor system BaFe2As2 is carried out. It is established that in spite of the rather large value of the electron-electron correlations (local Coulomb interaction on the Co-\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$3d$$\end{document} shell \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$U = 4.0$$\end{document} eV, the Hund exchange interaction \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J = 0.85$$\end{document} eV), in the considered systems a relatively small quasiparticle mass renormalization 1.2–1.35 at the Fermi level is observed. The correlation effects lead to the remarkable shift and compression of the spectrum below –0.8 eV. The band structure of InCo2As2 near the Fermi level is qualitatively similar to the previously studied BaCo2As2, and differs significantly from the band structure of BaFe2As2. In the KInCo4As4 system, the bands near the Fermi level resemble the band structure of BaFe2As2, and the Fermi surfaces have a similar topology. This indirectly points to the possibility of superconductivity in KInCo4As4. Also according to the results of LDA + DMFT calculations it is seen that with a rather small hole or electron doping in the KInCo4As4 system will experience topological Lifshitz transitions. We believe that the synthesis of the InCo2As2 and KInCo4As4 compounds considered in this paper is important for the study of superconductivity in this class of materials.


INTRODUCTION
The discovery of a family of iron-based HTSC (high-temperature superconductors) pnictides and chalcogenides (see reviews [1][2][3][4] and some other recent works [5][6][7][8]) has generated an intense search for new chemical and/or structural analogues of the systems of this class (see, e.g., [9,10]). Among the analogues of iron superconductors there are superconducting systems, but with a sufficiently low critical temperature of the superconducting transition . It should be noted that many of the newly obtained related systems are not superconductors. In particular, the complete replacement of Fe by Co in the compound BaCo 2 As 2 does not lead to superconductivity [11], even at pressures up to 8 GPa [12]. A possible reason for the lack of superconductivity may be a completely different electronic structure to those of typical iron-based superconductors, in which the spin-fluctuation mechanism of superconductivity cannot be realized [11].
Recently, not yet synthesized compounds InCo 2 As 2 and KInCo 4 As 4 [13] have been theoretically studied within the framework of DFT (Density Func-tional Theory). Compared with the parent compound BaFe 2 As 2 (the so-called class 122 iron-based superconductors) these systems have more valence electrons. InCo 2 As 2 has one more valence electron as compared to that BaCo 2 As 2 . KInCo 4 As 4 has effectively the same number of valence electrons as in BaCo 2 As 2 . However, the electronic bands of KInCo 4 As 4 are very different from BaCo 2 As 2 . Note that the KInCo 4 As 4 system belongs to the so-called 1144 class of iron-based superconductors, which were first synthesized in 2016 [14,15]. In [13], the band structure, densities of states of InCo 2 As 2 and KInCo 4 As 4 obtained by DFT were considered. However, the Fermi surface for KInCo 4 As 4 has not been studied. Also, it was previously shown that in the BaCo 2 As 2 system, the electronic correlations on the Co-shell are essential in describing the electronic struc-ture [16].
In view of the above, a systematic detailed study of the electronic structure of isostructural analogues of iron superconductors InCo 2 As 2 and KInCo 4 As 4 has been performed in this paper using DFT/LDA and

CONDENSED MATTER
LDA + DMFT [17,18] approaches. The corresponding electronic bands, densities of states and Fermi surfaces, as well as spectral functions, are obtained. The DFT/LDA and LDA + DMFT results for the considered systems were compared with each other as well as with the previous results for the parent system BaFe 2 As 2 . The essential role of electron-electron correlations in KInCo 4 As 4 is shown.
2. CRYSTAL STRUCTURE AND METHODS The band structure of the systems under study was obtained within the framework of density functional theory in the DFT/LDA local electron density approximation implemented in the wien2k [19] software package (full-potential method of linearized augmented plane waves (FP-LAPW)). The crystal structure of the Co-based compounds under consideration was obtained by optimizing of the lattice parameters and atom positions in DFT calculations. As a result, the spatial symmetry group was found for InCo 2 As 2 with the following values of lattice parameters and atomic positions: Å, Å, . For the compound KInCo 4 As 4spatial symmetry group , lattice parameters and atomic positions: , In (0, 0, 0), K (0.5, 0.5, 0.5). For the DFT/LDA calculations, the optimal spacing of the irreducible Brillouin zone in k-space was chosen as 16 × 16 × 16. For InCo 2 As 2 and KIn-Co 4 As 4 , a DFT analysis of the possible magnetic states was performed, which showed that all the considered magnetic configurations (FM, AFM-A, AFM-C and AFM-G) converge to a paramagnetic solution. To specify the kinetic part of the Hamiltonian for DMFT, the LDA Hamiltonian projected on the basis of Co(Fe)-, As-, In-Wannier functions was obtained using the Wannier90 [20] package. It has been checked that the bands built on the Wannier functions coincide well with the original LDA bands, which indicates sufficient quality of projecting performed.
The continuous-time Quantum Monte-Carlo method (CT-QMC) [21] implemented in the software package AMULET [22] was employed to solve the impurity problem within DMFT. DMFT(CT-QMC) calculations were performed at inverse temperature which approximately corresponds to 116 K. The number of averaged Monte-Carlo steps was chosen to be 10 6 .
In this paper we chose following values of the direct Coulomb interaction in the Hubbard model-eV and the value of the Hund exchange interaction-eV. The value of is taken to be slightly bigger than typical values for iron arsenides [23,24], since article [25] shows that the U value in the row of metals increases with increasing of its atomic numbers. The exchange interaction is taken the same value as for Fe-states in iron arsenides. In this paper a self-consistent calculation of the double counting correction ( ) in the FLL (fullylocalized limit) [26,27] approximation is performed, the following values of and for the Co-states are obtained: InCo 2 As 2 , eV; KInCo 4 As 4 , 27.84 eV.
The self-energy at real frequencies is obtained using the Pade approximants method [28]. Also to check the correctness of the performed analytical continuation of the self-energy, the Green functions obtained by integrating the Dyson equation with , were compared with the Green functions calculated within the maximum entropy method [29].
3. RESULTS Figure 1 shows the densities of states (total, Coand As-) for (a) InCo 2 As 2 and (b) KInCo 4 As 4 systems calculated in LDA (black dashed lines) and LDA + DMFT (red solid lines) methods compared to BaFe 2 As 2 LDA results (blue dot-dashed lines). In contrast to iron-based superconductors, the Co-based systems have a Fermi level closer to the upper edge of the Co-band due to the extra electron (shift relative to BaFe 2 As 2 Fermi level is about 0.7 eV).
The overall structure of the density of states in systems with Co and in BaFe 2 As 2 is quite similar (see the blue line in Fig. 1). However, as will be shown below, the band structures of these three systems are quite noticeably different. In the systems under investigation, the Fermi level is located slightly below the peak, which is formed by flat sections of zones in the direction (see Fig. 2), in contrast to BaCo 2 As 2 , where the Fermi level falls exactly on the [30] peak. Note that for systems with Co the contribution of As-states to the density of states near the Fermi level is substantially larger than in BaFe 2 As 2 .
Due to electron-electron correlation effects the peak in the density of states around -1 eV region shifts 0.25 eV higher in energy (see the red line on Fig. 1). However, the shape of the density of states at the Fermi level and above stays almost unchanged in the presence of electronic correlations. Figure 2a shows the band structure of InCo 2 As 2 obtained within LDA (black dashed lines) and LDA + DMFT (red solid lines). The band structure of InCo 2 As 2 resembles the band structure of BaCo 2 As 2 [30], but the Fermi level is shifted lower in energy by about 0.3-0.5 eV. Thus, the flat bands in the direction are shifted from the Fermi level in BaCo 2 As 2 higher by 0.5 eV in energy.
It is known that the zones crossing the Fermi level in BaFe 2 As 2 are mainly formed by Fe- [31] orbitals. Due to the extra electron in the Co systems considered here, the corresponding cobalt orbitals are now located below the Fermi level, and the main contribution at the Fermi level is now given by the Co-orbital. It can also be seen that just above the Fermi level in the direction, the Co-orbital forms a flat band.
The classical band structure for iron-based superconductors such as BaFe 2 As 2 is three hole pockets at the Γ-point and two electron pockets at the M-point (see Fig. 2b) [31]. In the case of InCo 2 As 2 there is only one rather small electronic pocket at the Γ-point and two electron pockets at the X-point (see Fig. 2a). For KInCo 4 As 4 (Fig. 2c), the structure of the bands crossing the Fermi level also differs from the classical form of BaFe 2 As 2 .
Accounting for electronic correlations (LDA + DMFT) does not change the bands at the Fermi level in the X-point region, but shifts the flat band in the direction lower in energy in both systems with Co. The calculated value of the correlation renormalized quasi-particle mass at the Fermi level for all Co-orbitals is almost the same and is 1.25 for InCo 2 As 2 . Thus, in InCo 2 As 2 there is practically no band structure compression at the Fermi level in contrast to BaFe 2 As 2 , where the renormalization is two to three times larger. However, when moving down from the Fermi level, the compression and band shift become noticeable below -0.8 eV.
Let us consider the band structure of the 1144 KIn-Co 4 As 4 system (Fig. 2c). Around the Γ-point one electron band crosses the Fermi level and a second electron band forms a hole pocket in the center of the ( ) direction. Three electron bands are Fe-3d Co-3d Co-3d As-4p As-4p As-4p KInCo 4 As 4 Co-3d Co-3d As-4p As-4p   Fig. 3). Among the Co-based compounds considered, the band structure of KInCo 4 As 4 is the most similar one to that of BaFe 2 As 2 .
The calculated LDA + DMFT value of the correlation renormalization of the quasi-particle mass at the Fermi level for KInCo 4 As 4 is 1.22. Just as in InCo 2 As 2 in KInCo 4 As 4 there is no band compression at the Fermi level, which becomes noticeable at energies below -0.5 eV. A small but significant, in our opinion, change in the band structure due to the electron-electron correlation effects is the displacement of bands near the Fermi level in the middle of the direction and at the M-point. As a consequence of this displacement, the upper and lower part of the bands falls directly on the Fermi level. Due to this, the Fermi surface topology (Lifshitz transitions) might be changed with rather small hole or electron doping. Almost all of the Fermi surface sheets for InCo 2 As 2 have an explicit -dependence. In the KInCo 4 As 4 system, where the K and In layers alternate in the crystal structure, the Fermi surface becomes almost quasi two-dimensional (Fig. 3b). This can also be seen in the band structure in the and directions near the Fermi level (Fig. 2c), in contrast InCo 2 As 2 (Fig. 2a). It is also obtained that the Fermi surface for KInCo 4 As 4 looks similar to the Fermi surface of ironbased superconductors, but the shape of the Fermi surface sheets near the Γ-point is closer to a rectangular prism shape than to a cylinder one. This shape of the Fermi surface sheets may contribute to the occurrence of nesting. At the same time, all Fermi surface sheets near the Γ-point are electronic, unlike BaFe 2 As 2 , where hole Fermi surface sheets are concentrated near the Γ-point. Thus, we can conclude that the experimental synthesis of KInCo 4 As 4 samples is worth trying to check realization of superconductivity there.
The electron correlations within LDA + DMFT calculations practically do not change the appearance of the Fermi surface in InCo 2 As 2 (Fig. 4a). In the case of KInCo 4 As 4 the LDA + DMFT result shows two sheets of the Fermi surface with the largest volume are changed by shifting the bands in the middle of the direction. As a result, these Fermi surface sheets touch each other (Fig. 4b). At the same time, the other sheets remain practically unchanged. The manifestation of electronic correlations is clearly visible in the spectral functions obtained in the LDA + DMFT calculations (Fig. 5). Starting from -0.8 eV and below in energy, the lifetime of quasiparticle states for InCo 2 As 2 and KInCo 4 As 4 decreases. At the same time, there are well-defined quasiparticle bands near the Fermi level. A similar correlation effects were observed for BaCo 2 As 2 below -0.8 eV [16]. Whereas for iron-based superconductors (including BaFe 2 As 2 ), the manifestation of electronic correlations leads to a broadening of all branches of the Fe-spectral function, including at the Fermi level [24]. rest of orbital states are concentrated near -1 eV, forming a peak in the Co-density of states (Fig. 1).

CONCLUSIONS
In this paper we carried out a comparative analysis of the electronic structure of perspective isostructural analogues of iron superconductors InCo 2 As 2 and KIn-Co 4 As 4 within the framework of DFT/LDA and LDA + DMFT approaches together with similar HTSC results of the parent BaFe 2 As 2 system. The band structure of InCo 2 As 2 near the Fermi level differs significantly from the band structure of BaFe 2 As 2 , while being qualitatively similar to the previously studied BaCo 2 As 2 . For KInCo 4 As 4 , the bands near the Fermi level and the Fermi surface resemble BaFe 2 As 2 ones, which somehow points to the possible realization of superconductivity in KInCo 4 As 4 .
It is shown that the influence of electronic correlation effects (with sufficiently large values of the interaction parameters) at the Fermi level is minimal, and manifests itself when departing from the Fermi level. In this case, the correlation effects lead to an apprecia- ble shift and compression of the spectrum below -0.8 eV. The Fermi surface in KInCo 4 As 4 is rearranged in the direction of the Brillouin zone due to electronic correlations. It is also found that in the KInCo 4 As 4 system with relatively small hole or electron doping, Lifshitz transitions are possible. The synthesis of the InCo 2 As 2 and KInCo 4 As 4 compounds considered in this work will serve as an important step in the study of superconductivity in this class of materials.

CONFLICT OF INTEREST
The authors declare that they have no conflicts of interest.

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