Cuprates,^[1–8] first discovered in 1986, and iron-based superconductors (IBS),^[9–13] discovered in early 2008, are two classes of unconventional high temperature superconductors which share many common features. Both of them are quasi-two-dimensional and the phase diagrams are similar in which superconductivity develops after a magnetic order is suppressed.^[14–16] All the cuprates share a common structure element CuO₂ plane, where the Cu atoms form a square lattice.^[17] The IBS share a common Fe₂X₂ (X = As and Se) layered structure unit, which possesses an anti-PbO-type (anti-litharge-type) atom arrangement. The Fe₂X₂ layers consist of a square lattice sheet of Fe coordinated by X above and below the plane to form face sharing FeX₄ tetrahedra.^[18]

One of the major questions in the field of high T_c superconductors is whether cuprates and IBS share a common superconducting mechanism.^[16,19] The answer to this question may be obtained if we can integrate the characteristics of both superconductors into a single compound so that their relations can be exclusively addressed. As both structures are featured with layered square lattices with similar in-plane lattice constants, it is possible to design a compound containing both building blocks, Cu–O layers of cuprates and Fe–As(Se) layers of IBS. Similar material designs have been adopted^[20,21] and a target material Nd₄CuO₆Fe₂As₂ has been theoretically investigated.^[20]

Recently, two new materials Ba₂MO₂Ag₂Se₂ (M = Co,Mn)^[22] have been synthesized via a solid-state reaction in experiment. These two compounds, whose structures belong to the I4/mmm space group, consist of infinite MO₂ square planes and antifluorite-type Ag₂Se₂ layers separated by barium. In this paper, motivated by the fact that the MO₂ plane resembles the CuO₂ plane in cuprates and the Ag₂Se₂ layer resembles the Fe₂X₂ layer in IBS, we consider the substitution of Cu and Fe for M and Ag in the compounds respectively to obtain two possible new compounds, Ba₂CuO₂Fe₂As₂ and K₂CuO₂Fe₂Se₂. Similar structures have been realized in Sr₂CrO₂A₂As₂ (A = Fe, Cr).^[23,24] We perform density functional calculations to study the stability and basic electronic structures of these new compounds. We find that these materials integrate basic electronic characteristics of both high T_c superconductors. In Ba₂CuO₂Fe₂As₂, the CuO₂ layers are electron-doped while the Fe₂As₂ layers are hole-doped. The situation is reversed in K₂CuO₂Fe₂Se₂. Such doping configurations suggest that many possible mixed phases, in particular, magnetic and superconducting phases, may be realized in these materials by introducing additional carriers and applying an external pressure.

Our calculations are performed using density functional theory (DFT) employing the projector augmented wave (PAW) method encoded in the Vienna ab initio simulation package (VASP).^[25–27] Both the local density approximation (LDA) and the generalized-gradient approximation (GGA)^[28] for the exchange–correlation functional are used. Throughout the work, the cutoff energy is set to be 500 eV for expanding the wave functions into the plane-wave basis. In the calculations of magnetic properties, the LDA + U method is used with the effective on-site Coulomb U being 7 eV for Cu 3d states.^[29] In the calculations, the Brillouin zone is sampled in the k space within the Monkhorst–Pcak scheme.^[30] The number of these k points depends on the lattice: 15 × 15 × 3 for the general unit cell with 4 Fe atoms and 2 Cu atoms and 9 × 9 × 3 for the unit cell. We relax the lattice constants and internal atomic positions with both LDA and GGA, where the forces are minimized to less than 0.01 eV/Å. The phonon dispersions are calculated using the finite displacement method^[31] as implemented in the PHONOPY code.^[32,33]

Ba₂CuO₂Fe₂As₂ and K₂CuO₂Fe₂Se₂ crystallize in a body-centered tetragonal lattice, as shown in Fig. 1. The Ba (K) spacer layer separates the tetrahedral Fe₂As₂ (Fe₂Se₂) layers and the square CuO₂ layers. There are double Fe₂X₂ (X = As and Se) and CuO₂ layers in a unit cell, similar to BaFe₂As₂. In order to predict the structures of Ba₂CuO₂Fe₂As₂ and K₂CuO₂Fe₂Se₂, the lattices are fully optimized based on the experimental structural parameters of Ba₂CoO₂Ag₂Se₂. The optimized and experimental structural parameters are summarized in Table 1. Both of the calculated lattice constants using LDA and GGA are close to those of Ba₂CoO₂Ag₂Se₂ in experiment, which validates the adopted substitution. As the calculated lattice parameters for Ba₂CoO₂Ag₂Se₂ using GGA are closer to the experimental data, we perform the following calculations for the two materials using the lattice parameters obtained by relaxation with GGA.

Fig. 1.

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	Fig. 1. Crystal structure of Ba2CuO2Fe2As2 (K2CuO2Fe2Se2).

The binding energy is usually calculated to estimate the stability of new structures. Here, the binding energy per atom, E_b, is defined as E_b = (4E_Ba/K + 2E_Cu + 4E_O + 4E_Fe + 4E_As/Se − E_total)/18, where E_Ba, E_K, E_Cu, E_O, E_Fe, E_As, and E_Se are the respective energies per atom of elemental Ba, K, Cu, O, Fe, As, and Se in the states at room temperature and ambient pressure, and E_total is the calculated total energy of a unit cell of Ba₂CuO₂Fe₂As₂ (K₂CuO₂Fe₂Se₂). The obtained binding energies are 1.61 eV and 1.23 eV per atom for Ba₂CuO₂Fe₂As₂ and K₂CuO₂Fe₂Se₂, respectively. Both of the binding energies are very close to that of Ba₂CoO₂Ag₂Se₂ (1.318 eV), indicating that the two structures are energetically favorable in experiment and may be synthesized using similar methods. To further test the stability of these two structures, we calculate their phonon dispersions, as shown in Fig. 2. No imaginary frequencies are observed throughout the whole Brillouin zone in the phonon dispersions, confirming their dynamically structural stability.

Table 1.

Table 1.

Table 1. Optimized structural parameters of Ba2CuO2Fe2As2 and K2CuO2Fe2Se2 using LDA and GGA in the paramagnetic phase. The 6th column shows the experimental structural parameters of Ba2CoO2Ag2Se2.[22] . Ba2CuO2Fe2As2 K2CuO2Fe2Se2 Ba2CoO2Ag2Se2 LDA GGA LDA GGA Expt a/Å 3.902 4.026 3.769 3.839 4.223 c/Å 19.867 20.273 18.320 22.165 20.036

Table 1. Optimized structural parameters of Ba2CuO2Fe2As2 and K2CuO2Fe2Se2 using LDA and GGA in the paramagnetic phase. The 6th column shows the experimental structural parameters of Ba2CoO2Ag2Se2.[22] .

Fig. 2.

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	Fig. 2. Calculated GGA phonon dispersions of (a) Ba2CuO2Fe2As2 and (b) K2CuO2Fe2Se2 in the paramagnetic state.

The optimized parameter a of Ba₂CuO₂Fe₂As₂ is determined to be 4.026 Å, slightly larger than those of cuprates (3.78 Å for La_1.8Sr_0.2CuO₄^[3] and 3.859 Å for YBa₂Cu₃O_{6 + x}^[4]). Compared with three typical kinds of iron pnictides, LiFeAs,^[10] BaFe₂As₂,^[11] and LaFeAsO,^[9] as shown in Table 2, the parameter a is very close to those of BaFe₂As₂ and LaFeAsO but deviates a little from that of LiFeAs. The obtained As height above the Fe plane in Ba₂CuO₂Fe₂As₂ is smaller than those of iron pnictides in experiment. This underestimation has been noted in the study of IBS.^[34–36] The Fe₂As₂ inter-layer distance is much larger than the values in the three families of iron pnictides, indicating that it is more two-dimensional than the conventional iron pnictides. The separation between Fe₂As₂ and CuO₂ layers is about 5 Å, which suggests couplings between these layers.

Table 2.

Table 2.

Table 2. Optimized structural parameters of Ba2CuO2Fe2As2 using GGA in the paramagnetic phase and those of LiFeAs,[10] BaFe2As2,[11] and LaFeAsO.[9] The angle β is the Fe–As–Fe bonding angle of two next nearest Fe atoms. . Ba2CuO2Fe2As2 LiFeAs BaFe2As2 LaFeAsO a/Å 4.026 3.791 3.963 4.035 β/(°) 118.7 103.1 111.0 113.5 h(Fe–As)/° 1.19 1.51 1.36 1.32 h(Fe–Fe)/° 10.14 6.36 6.51 8.74

Table 2. Optimized structural parameters of Ba2CuO2Fe2As2 using GGA in the paramagnetic phase and those of LiFeAs,[10] BaFe2As2,[11] and LaFeAsO.[9] The angle β is the Fe–As–Fe bonding angle of two next nearest Fe atoms. .

The band structure and density of states (DOS) for Ba₂CuO₂Fe₂As₂ with the GGA optimized structural parameters in the paramagnetic states are shown in Fig. 3. The band structure near the Fermi level resembles those of both typical iron pnictides LaFeAsO^[37] and cuprates^[14] where Fe 3d states (d_xz, d_yz, d_x²−y² orbitals) and Cu 3d states (d_x²−y² orbitals) dominate the Fermi level. Similar to iron pnictides, it is a bad metal with a low carrier density. The 3d states of Fe are mainly located near the Fermi level from −2.5 eV to 3.0 eV and a pseudogap appears at an electron count of six. The As p states mainly lie 2.5 eV below the Fermi level but slightly mix with the Fe 3d states in the energy range from −1.0 eV to 1.0 eV. The Cu 3d contribution is concentrated between −3.0 eV and 1.5 eV. The O p states are strongly coupled with the 3d states of Cu in the total energy range. The spacer layer Ba 5d and 6s states, which donate electrons to the Fe₂As₂ and CuO₂ layers, are empty and lie 2 eV above the Fermi level. At the Fermi level, the total DOS shows the same negative slope with the minimum slightly above the Fermi level, similar to the conventional IBS within DFT. The DOS at the Fermi energy is about 7.81 eV⁻¹/f.u. for both spins. Considering that there are two Fe atoms in one formula unit for Ba₂CuO₂Fe₂As₂ but only one for LaFeAsO, this value (3.91 eV⁻¹/Fe) is larger than that of LaFeAsO (2.62 eV⁻¹/f.u.^[37]). The corresponding Pauli susceptibility and specific heat coefficient are χ₀ = 2.53 × 10⁻⁴ emu/mol and γ = 18.4 mJ/(K²·mol).

Fig. 3.

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	Fig. 3. (a) Band structures and (b) projected density of states of Ba2CuO2Fe2As2 using the GGA relaxed parameters in the paramagnetic state.

The calculated Fermi surfaces of Ba₂CuO₂Fe₂As₂ are given in Fig. 4, which are very similar to those of both iron pnictides and cuprates. From Fig. 3(a), we find that the Fe d_xz and d_yz states yield two hole cylinders at the zone center. There is an additional heavy three-dimensional (3D) hole pocket centered at the Γ point, which intersects and anticrosses with the hole cylinders. The 3D pocket is derived from Fe d_z² states which hybridize with As p_z states. At the zone corner, there are two 2D small electron pockets and one 2D large hole pocket. The electron pockets are mainly attributed to Fe d_xz, d_yz, and d_x²−y² orbitals (d_xy orbitals in the usual Fe lattice). The hole pocket around the M point, however, is mainly derived from Cu d_x²−y² and O p_x and p_y states, in accordance with the Fermi surfaces of cuprates.^[14] All of the Fermi surface sheets are double degenerate as there are two Fe₂As₂ and CuO₂ layers in one unit cell.

Fig. 4.

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	Fig. 4. Calculated Fermi surfaces of Ba2CuO2Fe2As2 using the GGA relaxed parameters in the paramagnetic state.

It is interesting to see that the hole pockets at the Γ point are large, which indicates heavy hole doping in the Fe₂As₂ layers. From the size of the pockets, we can estimate that the hole doping concentration with respect to BaFe₂As₂ is roughly 0.15 holes per Fe atom. Considering the hole doping level, the Fermi surfaces attributed to the Fe₂As₂ layers should be similar to those of Ba_0.7K_0.3Fe₂As₂ except that the pockets here are much more two-dimensional. This hole doping level is very close to that in optimally-hole-doped pnictide Ba_0.6K_0.4Fe₂As₂,^[38–40] where T_c can reach 37 K. Actually, the calculated Fermi surfaces are similar to those of (Ba, K)Fe₂As₂ obtained in the ARPES experiment.^[41,42] As required by the charge conservation, the heavy hole doping in the Fe₂As₂ layers suggests that the CuO₂ layers must be heavily electron-doped with the same doping concentration. This consistency is confirmed by the estimation that the size of the largest hole pocket at the M point attributed to d_x²−y² orbitals in CuO₂ layers is about 0.3 electrons per Cu atom from their half filling configuration. The doping level is a little far away from the narrow superconducting region for electron-doped cuprates in the phase diagram.^[14]

The superconductivity in both cuprates and IBS is most likely related to the magnetism. Therefore, we investigate the magnetic properties of this new compound. With such a large doping in the Fe₂As₂ and CuO₂ layers, it is expected that the checkboard antiferromagnetic (AFM) order in the CuO₂ layers and the E-type collinear AFM order in the Fe₂As₂ layers should be significantly weakened or completely suppressed even in the meanfield type of DFT calculations. To check this, we consider four possible magnetic states for Fe₂As₂ and CuO₂ layers: paramagnetic state, ferromagnetic state, checkerboard AFM state, and collinear AFM state. We calculate the total energies of these magnetic states using the LDA + U approach. The calculations show that there is no statically ordered moment in the CuO₂ layers while the E-type collinear AFM order in the Fe₂As₂ layers is still possible. We find that in the DFT result, the E-type state has an energy gain of 103.5 meV/f.u. relative to the paramagnetic state and a spin moment of 1.59 μ_B for each Fe. It is known that magnetic moments calculated for the known parent compounds of IBS, for example, BaFe₂As₂, are typically over 2.0 μ_B.^[36,43] This moment value is significantly lowered, which is consistent with our expectation as the Fe₂As₂ layers are overdoped by holes. Figures 5(a) and 5(b) show the calculated band structure and DOS in the E-type collinear AFM state (with supercell) with internal coordinates fixed to the values obtained by non-spin-polarized energy minimization. The E-type state is still metallic with a low carrier concentration and N(E_F) decreases severely. Among the considered magnetic states above, the checkerboard AFM state in Fe₂As₂ layers is found to be a metastable state relative to the E-type collinear AFM state. The energy gain is 81.5 meV/f.u. relative to the paramagnetic state and the magnetic moment is about 1.58 μ_B for each Fe atom.

Fig. 5.

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	Fig. 5. (a) Band structures and (b) projected density of states of Ba2CuO2Fe2As2 using the GGA relaxed parameters in the E-type collinear AFM state.

With the substitution of As by Se and Ba by K in Ba₂CuO₂Fe₂As₂, we obtain a new compound K₂CuO₂Fe₂Se₂. Compared with Ba₂CuO₂Fe₂As₂, the parameter a decreases to 3.839 Å but c increases to 22.165 Å, which is consistent with those of iron chalcogenides. Table 3 lists the calculated lattice parameters using GGA and those of KFe₂Se₂^[13] and FeSe.^[44] The calculated Se height and Se–Fe–Se bond angle are very close to those of FeSe, indicating that they may share some common electronic and magnetic properties. The Se heights are severely underestimated in the calculations compared with those in experiment.^[45] The band structure and DOS are shown in Fig. 6. The band structure is similar to that of Ba₂CuO₂Fe₂As₂. The DOS between −2.5 eV and 2.0 eV is dominated by Fe 3d states, and Se 3p states are mainly located at 3.5 eV below the Fermi level. The Cu 3d states and O 2p states are strongly coupled from −4.0 eV to 2.5 eV. The K atoms have no contribution to the bands near the Fermi level. To analyze the orbital characters near the Fermi level, we plot the fat band in Fig. 6(a). The DOS at the Fermi level is 4.54 eV⁻¹/f.u., which is significantly lower than that of Ba₂CuO₂Fe₂As₂. The corresponding Pauli susceptibility and specific heat coefficient are χ₀ = 1.47 × 10⁻⁴ emu/mol and γ = 10.7 mJ/(K²·mol). Figure 7 shows the calculated Fermi surfaces. The Fermi surfaces are two-dimensional cylinders. The Fe 3d states yield two hole pockets at the Γ point and two small electron pockets at the M point. The hole pockets are attributed to the d_xz and d_yz states while the electron pockets are derived from Fe d_xz, d_yz, and d_x²−y² states. The Fermi surfaces are similar to those in KFe₂Se₂ where hole pockets are absent.^[46] Besides the pockets from the Fe₂Se₂ layers, there are additional two hole pockets around the Brillouin zone corners. The bigger pocket is mainly contributed from the Cu d_x²−y² states, which are strongly coupled with the O p_x and p_y states, while the smaller one is derived from the O p_x and p_y states. All of the Fermi surface sheets are double degenerate just like those in Ba₂CuO₂Fe₂As₂.

Table 3.

Table 3.

Table 3. Optimized structural parameters of K2CuO2Fe2Se2 using GGA in the paramagnetic phases and those of KFe2Se2[13] and FeSe.[44] The angle β is the Fe–Se–Fe bonding angle of two next nearest Fe atoms. . K2CuO2Fe2Se2 KFe2Se2 FeSe a/Å 3.839 3.913 3.765 β/(°) 111.6 106.6 111.3 h(Fe–Se)/Å 1.31 1.46 1.29 h(Fe–Fe)/Å 11.08 7.02 5.52

Table 3. Optimized structural parameters of K2CuO2Fe2Se2 using GGA in the paramagnetic phases and those of KFe2Se2[13] and FeSe.[44] The angle β is the Fe–Se–Fe bonding angle of two next nearest Fe atoms. .

Fig. 6.

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	Fig. 6. (a) Band structures and (b) projected density of states of K2CuO2Fe2Se2 using the GGA relaxed parameters in the paramagnetic state.

Compared with the Fermi surfaces of Ba₂CuO₂Fe₂As₂, we find that the hole pockets at the Γ point of K₂CuO₂Fe₂Se₂ are much smaller while the electron pockets at the M point are much larger (see Fig. 7), which indicate heavy electron doping in the Fe₂Se₂ layer. From the size of the pockets, we can estimate that the electron doping concentration with respect to FeSe is roughly 0.16 electrons per Fe atom. Similarly, we can check the consistency required by the charge conservation. The size of the hole pockets at the M point from the CuO₂ layers is about 0.32 holes for each CuO₂ layer from their half filling configuration.

Fig. 7.

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	Fig. 7. Calculated Fermi surfaces of K2CuO2Fe2Se2 using the GGA relaxed parameters in the paramagnetic state.

Fig. 8.

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	Fig. 8. (a) Band structures and (b) projected density of states of K2CuO2Fe2Se2 using the GGA relaxed parameters in the AFM state.

The DOS at the Fermi level attributed to Fe 3d orbitals is significantly decreased with such a large electron doping, which may suggest that the magnetic order in the Fe₂Se₂ layers is strongly suppressed. Compared with Ba₂CuO₂Fe₂As₂, we consider an additional bicollinear magnetic state, which is the ground state in FeTe.^[47] The ground state is paramagnetic on Fe₂Se₂ layers and checkboard AFM ordered for CuO₂ layers with an energy gain of 196.0 meV/f.u. and a spin moment of 0.59 μ_B per Cu. The band structure and DOS are shown in Fig. 8 and the system remains metallic. In this case, the Fe 3d, Cu 3d, and O 2p states dominate the Fermi level. The DOS at the Fermi level is 7.04 eV⁻¹/f.u. for the spin up channel.

It is well known that the pairing symmetry is d-wave for the cuprates and s-wave for the iron pnictides with a potential sign change between the hole and electron Fermi pockets. As the proposed new materials contain both CuO₂ layers and Fe₂X₂ (X = As, Se) layers and the electronic structures are similar to both, it can be expected that two different pairing symmetries may coexist in a single material. The proximity effect between two types of layers with different pairing symmetries can result in interesting novel phenomena. The time reversal symmetry may be broken with mixed s-wave and d-wave pairing symmetries.

Introducing additional carriers or applying an external pressure to these materials can maximize T_c. For Ba₂CuO₂Fe₂As₂, the doping of the Fe₂As₂ layer is 0.15 holes per Fe and that of CuO₂ is 0.3 electrons per Cu. The doping for the Fe₂As₂ layer is near the optimal level but that for the CuO₂ layer is unfortunately not in the superconducting region. To realize superconductivity for both layers, we can introduce hole doping with partial replacement of Ba²⁺ ions by K⁺ ions. From the phase diagrams of cuprates and (Ba,K)Fe₂As₂, the above case can be realized in Ba_2−xK_xCuO₂Fe₂As₂ (0.24 ≤ x ≤ 0.32) by assuming that the Fe₂As₂ and CuO₂ layers are equally hole doped. In this doping region, the T_c for the Fe₂As₂ layers shows little change. The x = 0.3 corresponds the optimal doping value for the electron-doped cuprates.^[14] For K₂CuO₂Fe₂Se₂, however, the Fe₂Se₂ layers are electron doped and the hole doping for the CuO₂ layers is not in the superconducting region but very close to it. The superconductivity for both layers can be realized in K_2−xBa_xCuO₂Fe₂Se₂ (0.10 ≤ x ≤ 0.54). This doping region is much wider than that in Ba₂CuO₂Fe₂As₂ due to the wide hole-doped superconducting region in cuprates. The optimal doping value for the CuO₂ layers is x = 0.34. The pressure is also an effective way to tune the superconductivity.

In conclusion, we identify two hypothetical compounds Ba₂CuO₂Fe₂As₂ and K₂CuO₂Fe₂Se₂ containing Fe₂X₂ layers and CuO₂ layers, which are the basic structural units of IBS and cuprates, respectively. The metallic spacer Ba (K) separates the basic units. The calculations of binding energies and phonon spectra indicate that they are dynamically stable, ensuring that they may be experimentally synthesized. The Fermi surfaces derived from Fe₂As₂(Fe₂Se₂) layers and CuO₂ layers are very similar to those of iron pnictides (iron chalcogenides) and cuprates, respectively. With heavy self-doping, the ground state of Ba₂CuO₂Fe₂As₂ is determined to be an E-type collinear AFM state in the Fe₂As₂ layer and a paramagnetic state in the CuO₂ layer, while K₂CuO₂Fe₂Se₂ favors a checkboard AFM state in the CuO₂ layers and a paramagnetic state in the Fe₂Se₂ layers. Without external doping, superconductivity can only be achieved in the Fe₂X₂ layers. However, with external doping through substitution, superconductivity for both Fe₂X₂ and CuO₂ layers can be simultaneously achieved in Ba_2−xK_xCuO₂Fe₂As₂ (0.24 ≤ x ≤ 0.32) and K_2−xBa_xCuO₂Fe₂Se₂ (0.10 ≤ x ≤ 0.54). The synthesis of these new compounds will not only provide us with a unique opportunity to explore exotic properties in cuprates and IBS simultaneously but also help us to understand the mechanism of high T_c superconductivity.