Electronic and magnetic structures of ternary iron telluride KFe₂Te₂

Abstract

We examine the electronic and magnetic structures of iron telluride KFe₂Te₂ using first-principle calculations. We demonstrate that the ground state of this compound is in bicollinear antiferromagnetic order with Fe local moments (\sim 2.6 μ_B) that are ferromagnetically aligned along a diagonal direction and antiferromagnetically aligned along the other diagonal in the Fe-Fe square lattice, similar to the alignment discovered in the parent compound of superconductor α-FeTe. This bicollinear antiferromagnetic order results from the interplay among the nearest, next-nearest, and next-next-nearest neighbor exchange interactions, which are mediated by Te 5p orbitals. This finding may aid our understanding of the interplay between magnetism and superconductivity in the family of iron-based materials.

Graphical abstract

Keywords

iron-basel materials / electronic and magnetic structures

1 Introduction

The search for high-temperature superconductivity and a novel superconducting mechanism is one of the most challenging tasks undertaken by condensed-matter physicists and material scientists. Since the discovery of iron-based superconductors in 2008, such as LaOFeAs [1] (1111-type), BaFe₂As₂ [2] (122-type), and α-FeSe(Te)(11-type) [3], as well as iron selenide, K_xFe_2-ySe₂ [4], the pairing mechanism for these materials and the search for high-transition temperature superconductors have attracted substantial interest. These compounds exhibit superconductivity after doping or under high pressures. Like cuprates, they share the same robust tetrahedral layered structure. Given the magnetic orders of their parent compounds, these compounds are in spin-density-wave (SDW) order and have collinear [5, 6], bicollinear [7–9], and block antiferromagnetic (AFM) structures [10–13] when contained in iron pnictides, iron chalcogenide, and iron selenide K_xFe_2-ySe₂, respectively, at temperatures below a tetragonal-orthorhombic structural transition temperature.

In addition to the presence of phase separation in K_xFe_2-ySe₂ [14–20], it has generated considerable experimental and theoretical interest. In particular, the scanning tunneling microscopy (STM) measurements obtained from K_xFe_2-ySe₂ clearly demonstrate phase separation [17, 18], which suggests that it is phase separated into iron-vacancy ordered regions and iron-vacancy free regions. In this case, the former regions insulate with a \sqrt{5}\times \sqrt{5} vacancy ordered pattern, whereas the latter region is superconducting. Furthermore, neutron scattering experiments reveal that the \sqrt{5}\times \sqrt{5} vacancy ordered phase is also AFM ordered [10], which is affected by the superconducting pairing [21]. Understanding this observation is difficult in terms of phase separation. In addition to the \sqrt{5}\times \sqrt{5} vacancy ordered phase, an insulating phase with a \sqrt{2}\times \sqrt{2} ordered pattern on the Fe layer was also observed [15, 16, 22, 23], which indicates the existence of an additional symmetry breaking order in the SC phases [17, 18, 24] using both STM [17, 18] and angle-resolved photoemission spectroscopy (ARPES) [24].

Theoretically, one author proposed [13] that the ground state of the alkali-doped iron selenide without iron vacancies should approach a checkerboard phase, in which each of the four Fe sites combine into a tetragonal structure. However, the interplay between magnetism and superconductivity remains unclear in K_xFe_2-ySe₂. Chemically, K_xFe_2-ySe₂ appears to be the most heavily electron-doped iron-based superconductor. This significantly differs from other iron-based superconductors, which have both hole-like and electron-like Fermi surfaces [25–27] in the Brillouin zone. In order to examine the common aspects of the electronic structure and pairing mechanism of the iron-based systems that have the most electron-doped end, we considered the electronic and magnetic structures found in the isovalent-substituted compound KFe₂Te₂, using the framework of density functional theory, as the keys to an initial understanding of the possible superconducting mechanism.

In this paper, we present theoretical results concerning the electronic and magnetic structures of the ground state of KFe₂Te₂, which are obtained using intensive first-principle calculations. Similar calculations were partially provided in Ref. [28]. We reveal that weak spin fluctuation instability appears at momentum \overrightarrow{q}=(\pi ,\pi /2,0), indicating that the ground state for KFe₂Te₂ has a bicollinear AFM structure, which is confirmed using magnetic state calculations. Further calculations demonstrate that there is a strong competition among the nearest, next-nearest, and next-next-nearest neighbor exchange interactions between the Fe local moments that are mediated by Te 5p orbitals. The exchange coupling is also evaluated, yielding the same magnitude. This magnetic competition strongly affects the electronic structure of KFe₂Te₂, producing a small orthorhombic lattice distortion and a bicollinear AFM order of the Fe moments, which are similar to those found in α-FeTe [7] and KFe₂Se₂ [29].

2 Structure and method

We perform first-principle calculations on KFe₂Te₂ and demonstrate that the electronic structure of parent KFe₂Te₂ can be regarded as an electron over-doped 11-type system, rather than on BaFe₂As₂, which is structurally much more similar [2]. This can be seen in Fig. 1(a). In our calculations, we used the projected augmented wave method [30], which was implemented in the VASP code [31], and the Perdew–Burke–Ernzerhof (PBE) exchange correlation potential [32]. All atomic positions and lattice constants were simultaneously relaxed in order to determine the ground state with space group I4/mmm. A 500 eV cutoff in the plane wave expansion ensures that the calculations converge to {10}^{-5} eV. Moreover, the atomic positions and lattice constants were optimized until the largest force on any atom was 0.005 eV/Å. We used a 9\times 9\times 5 Monkhorst–Pack k-grid Brillouin zone sampling throughout our calculations. Furthermore, in the calculation of the electronic and magnetic structures of the nonmagnetic (NM), ferromagnetic (FM), and square AFM Néel states, the a\times a FeTe cell was taken to be the base cell, and the unit cells were doubled. The base cells were the \sqrt{2}a\times \sqrt{2}a and a\times 2a, as well as the 4\times 4 FeTe, cells for collinear, bicollinear, and block AFM states, respectively. The magnetic configuration of the bicollinear AFM state is shown in Fig. 1(b).

3 Results and discussion

First, we considered the electronic properties of KFe₂Te₂ in terms of their dependence on structural factors. Therefore, we performed a full structural optimization of KFe₂Te₂ for both the lattice parameters and atomic positions, including the internal coordinate, z_{\mathrm{Te}}, of the Te atom using energy minimization. These results are summarized in Table 1. Here, we calculated five possible magnetic configurations using FM, square Néel AFM, collinear AFM, and bicollinear AFM, as well as the block AFM orders combined with the nonmagnetic states. The lattice parameters that were optimized in our NM calculations exceeded those for KFe₂Se₂, which suggests that the atomic radii of Te are larger than those of Se. In addition, our results agreed well with previous studies that applied the full-potential method [28].

Second, we examined the NM state of KFe₂Te₂. In other words, the spin degrees were not included in the calculation. This work can provide a reference for studying magnetization states. Analyzing the density of states (DOS) at the Fermi level lets us determine whether the magnetic state is preferred. Figure 2(a) displays the calculated total DOS, as well as the projected DOS (PDOS) calculations performed on the Fe 3d and Te 5p orbitals of KFe₂Te₂. The total DOS and PDOS of KFe₂Te₂ resemble those of KFe₂Se₂ systems [29, 33] and exhibit the typical characteristics of layered structures, which can also be seen in the Fermi surface topology in Fig. 2(c). Here, the DOS can be divided into two parts: the lower part (3 eV below the Fermi energy), which consists of the bands formed through the bonding of the Fe 3d and Te 5p orbitals, and the upper part, which basically contains the Fe 3d orbital bands ranging from -2 eV to 2 eV that are centered at the Fermi energy. Further examination of the PDOS data reveals that more than 95% of the states from -2.0 eV to 0 eV are from Fe 3d orbitals and that the Fe 3d/Te 5p orbital substantially hybridizes, ranging from -6.0 eV to 2.0 eV. Thus, we expect Fe 3d electrons to play a dominant role in conduction, which is mediated by Te 5p orbitals.

Furthermore, it should be noted that the DOS provided by the Fe 3d orbitals produces a large value at the Fermi level. The corresponding value of the DOS at the Fermi level is N_{\mathrm{Fe}}(E_f)=1.24 states per eV per Fe atom. According to the Stoner criterion [34, 35], while magnetism may occur with lower DOS values, it must occur within a band picture if the Stoner criterion, N(E_f)I>1, is met. Here, I represents the Stoner parameter, which assumes values between 0.7 and 0.9 eV for ions near the center of the 3d series. In this case, the NM state becomes unstable. Also, we should note that effective I can be reduced using hybridization. Therefore, the NM state is unstable against the magnetic states in the FeTe layer. On the other hand, the low energy band structure and its corresponding Fermi surface, which are shown in Figs. 1(b) and (c), respectively, demonstrate that KFe₂Te₂ is a highly electron-doped compound when compared with iron-pnictides [25, 26]. The result of this is a diminished hole pocket at the zone center and an enlarged electron pocket at the zone corner, which exhibits quite two dimensionality arisen from the nature of sandwiched FeTe layered structure.

Before exploring the ground state with magnetic order, we calculated the bare electron susceptibility, {\chi }_0(\overrightarrow{q}), which provides a signature with magnetic instability at momentum \overrightarrow{q}. Applying the tight-binding model, which is fitted using maximally localized Wannier functions [36, 37], the bare electron susceptibility is given by

\begin{array}{}& \begin{array}{cc}{\chi }_0(\overrightarrow{q})=\hfill & {\displaystyle \frac{1}{N_k}}{\displaystyle \sum _{\mu \nu k}}{\displaystyle \frac{{|⟨k+\overrightarrow{q},\mu |k,\nu ⟩|}^2}{E_{\mu ,k+\overrightarrow{q}}-E_{\nu ,k}+i{0}^{+}}}\hfill \\ \hfill & [f(E_{\nu ,k})-f(E_{\mu ,k+\overrightarrow{q}})],\hfill \end{array}\end{array}

where E_{\mu ,k} represents the band energy measured at Fermi level E_F, and f(E_{\mu ,k}) is the Fermi-Dirac distribution function for an eigenstate, |k,\mu ⟩. In addition, N_k denotes the number of k points used for the irreducible Brillouin zone integration. The calculated real part of {\chi }_0 is found in Fig. 1(d). Here, we see that the highly electron dopant induced small peak appears at momentum \overrightarrow{q}=(\pi ,\pi /2,0), which indicates the presence of a weak AFM spin fluctuation in the FeTe plane. In other words, the peak in bare electron susceptibility {\chi }_0(\overrightarrow{q}) suggests that the ground state of KFe₂Te₂ has a bicollinear AFM structure.

In order to analyze the magnetic instability along momentum \overrightarrow{q}=(\pi ,\pi /2,0), which was indicated by the bare electron susceptibility, {\chi }_0(\overrightarrow{q}), we calculated five additional possible magnetic states using FM, square Néel AFM, collinear AFM, bicollinear AFM, and block AFM orders. These are listed in Table 1. If the energy for the NM state is set to zero, the ground state of KFe₂Te₂ has a bicollinear AFM order, in which the Fe moments ferromagnetically align along a diagonal direction and antiferromagnetically align along the other diagonal direction on the Fe-Fe square, as is shown in Fig. 1(b). Furthermore, the Fe local moments (\sim 2.6{\mathrm{\mu }}_B), which are close to those of KFe₂Se₂ (with Fe local moments \sim 2.5{\mathrm{\mu }}_B) [29], are also similar to those of \alpha-FeTe [7]. The magnetic moment around each Fe atom is approximately 2.1{\mathrm{\mu }}_B–2.7{\mathrm{\mu }}_B, weakly varying in our five magnetically ordering states. Given the bicollinear AFM structure, our calculations indicate a small structure distortion, which diminishes along the spin-parallel alignment [y-axis in Fig. 1(a)] and expands along the spin-antiparallel alignment [x-axis in Fig. 1(a)], causing a structural transition from a tetragonal structure to an orthorhombic one. This readily implies that the direct exchange favors a shorter Fe-Fe separation, whereas the superexchange favors a longer Fe-Fe separation.

In order to quantify the magnetic interactions, we assume that the energy differences between these magnetic orders are predominantly due to the local exchange interactions between the Fe moments with spin \overrightarrow{S}, which can be effectively modeled by the following frustrated Heisenberg model [38], with respective nearest, next-nearest, and next-next-nearest neighbor couplings J₁, J₂, and J₃:

\begin{array}{}\text{(1)}& \widehat{H}=J_1{\displaystyle \sum _{⟨i,j⟩}}{\overrightarrow{S}}_i\cdot {\overrightarrow{S}}_j+J_2{\displaystyle \sum _{⟨⟨i,j⟩⟩}}{\overrightarrow{S}}_i\cdot {\overrightarrow{S}}_j+J_3{\displaystyle \sum _{⟨⟨⟨i,j⟩⟩⟩}}{\overrightarrow{S}}_i\cdot {\overrightarrow{S}}_j.\end{array}

Here, ⟨i,j⟩, ⟨⟨i,j⟩⟩, and ⟨⟨⟨i,j⟩⟩⟩ denote the summation over the nearest, next-nearest, and next-next-nearest neighboring sites, respectively. This model may overlook some contributions from the itinerant electrons; however, we believe it captures the general physics of the magnetic structures. From the calculated energy data listed in Table 1, we see that [7, 39], for KFe₂Te₂, we have J_1=-8.1\mathrm{meV}/{\mathrm{S}}^2, J_2=-5.37\mathrm{meV}/{\mathrm{S}}^2, and J_3=7.93\mathrm{meV}/{\mathrm{S}}^2. It should be noted that these magnetic exchange constants have relatively smaller values than those found in other iron-based materials [23] because the magnetic ordering is gradually suppressed by the electron doped in pristine KFe₂Te₂, which results in weak magnetic fluctuations. This also agrees with the calculation of susceptibility {\chi }_0(\overrightarrow{q}) shown in Fig. 2(d). In addition, these values indicate that strong competing exchange interactions exist between the Fe spins. The bicollinear AFM has less energy than the other configurated AFM states for a frustrated J_1–J_2–J_3 Heisenberg model in terms of J_3>J_2/2 and J_2>J_1/2.

Finally, we calculated the low energy band structure and the total DOS, as well as the projected DOS, for Fe 3d and Te 5p, given the bicollinear AFM state in KFe₂Te₂, which can be seen in Fig. 3. When compared with the NM phase shown in Fig. 2(a), we find that most of the states around the Fermi level are gapped by the bicollinear AFM order. The corresponding electronic DOS at the Fermi level is N(E_f)=0.448 states per eV per Fe atom, which is significantly less than that of the NM state (1.24 states per eV per Fe atom), as is intuitively expected. In addition, the stability of the obtained bicollinear AFM state in KFe₂Te₂ can be understood given the nature of the Fe-Te-Fe bond angle in iron-based materials [35], which uniquely determines the tetrahedral crystal environment and the competing orders between the orbital and lattice degrees of freedom. In KFe₂Te₂, the Fe-Te-Fe bond angle decreases to a small value (approximately 94.57°) because of the large radii of the Te atom, which enhance the direct FM interactions between the nearest neighbor Fe spin and the third-nearest neighbor superexchange interaction mediated by the Te 5p orbitals. This behavior is consistent with our calculated exchange coupling constants. Furthermore, the weak peak appearing in the spin susceptibility at momentum \overrightarrow{q}=(\pi ,\pi /2,0) in Fig. 1(d) not only suggests the presence of SDW in the FeTe layer but also implies the tight relation between superconductivity and spin fluctuation. Clarifying the role of this relationship may provide insight concerning the mechanism of superconductivity that is driven by electron-electron correlation [40]. We aim to address this issue in future studies.

4 Conclusions

We performed first-principle calculations for the study of the electronic and magnetic structures of KFe₂Te₂. Our numerical calculations suggest that spin fluctuation instability appears at momentum \overrightarrow{q}=(\pi ,\pi /2,0), which indicates that the ground state for KFe₂Te₂ has a bicollinear AFM structure. This is consistent with magnetic states calculations. Furthermore, we also determined a strong competition among nearest, next-nearest, and next-next-nearest neighbor exchange interactions between Fe local spins mediated by Te 5p orbitals, providing the bicollinear AFM order of Fe spins in the ground states that have a magnetic moment of \sim 2.6{\mathrm{\mu }}_B around each Fe atom. This discovery may provide some understanding of the interplay between magnetism and superconductivity in the family of iron-based materials.

References

[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono, Iron-based layered superconductor La[O_1-xF_x]FeAs (x=0.05–0.12) with Tc=26 K, J. Am. Chem. Soc. 130(11), 3296 (2008)

[2] M. Rotter, M. Tegel, D. Johrendt, I. Schellenberg, W. Hermes, and R. Pöttgen, Spin-density-wave anomaly at 140 K in the ternary iron arsenide BaFe₂As₂, Phys. Rev. B 78(2), 020503 (2008)

[3] F. C. Hsu, J. Y. Luo, K. W. Yeh, T. K. Chen, T. W. Huang, P. M. Wu, Y. C. Lee, Y. L. Huang, Y. Y. Chu, D. C. Yan, and M. K. Wu, Superconductivity in the PbO-type structure a-FeSe, Proc. Natl. Acad. Sci. USA 105(38), 14262 (2008)

[4] J. Guo, S. Jin, G. Wang, S. Wang, K. Zhu, T. Zhou, M. He, and X. Chen, Superconductivity in the iron selenide K_xFe₂Se₂ (0⩽x⩽1.0), Phys. Rev. B 82, 180520(R) (2010)

[5] C. de la Cruz, Q. Huang, J. W. Lynn, J. Li, W. R. Ii, J. L. Zarestky, H. A. Mook, G. F. Chen, J. L. Luo, N. L. Wang, and P. Dai, Magnetic order close to superconductivity in the iron-based layered LaO_1-xF_xFeAs systems, Nature 453 (7197), 899 (2008)

[6] J. Dong, H. J. Zhang, G. Xu, Z. Li, G. Li, W. Z. Hu, D. Wu, G. F. Chen, X. Dai, J. L. Luo, Z. Fang, and N. L. Wang, Competing orders and spin-density-wave instability in La(O_1-xF)xFeAs, Europhys. Lett. 83(2), 27006 (2008)

[7] F. Ma, W. Ji, J. Hu, Z. Y. Lu, and T. Xiang, First-principles calculations of the electronic structure of tetragonal α-FeTe and α-FeSe crystals: Evidence for a bicollinear antiferromagnetic order, Phys. Rev. Lett. 102(17), 177003 (2009)

[8] S. Li, C. de la Cruz, Q. Huang, Y. Chen, J. W. Lynn, J. Hu, Y. L. Huang, F. C. Hsu, K. W. Yeh, M. K. Wu, and P. Dai, First-order magnetic and structural phase transitions in Fe_1+ySe_xTe_1-x, Phys. Rev. B 79(5), 054503 (2009)

[9] W. Bao, Y. Qiu, Q. Huang, M. A. Green, P. Zajdel, M. R. Fitzsimmons, M. Zhernenkov, S. Chang, M. Fang, B. Qian, E. K. Vehstedt, J. Yang, H. M. Pham, L. Spinu, and Z. Q. Mao, Tunable (δ⁢π,δ⁢π)-type antiferromagnetic order in α-Fe(Te,Se) superconductors, Phys. Rev. Lett. 102(24), 247001 (2009)

[10] W. Bao, Q. Z. Huang, G. F. Chen, D. M. Wang, J. B. He, X. Q. Wang, and Y. Qiu, A novel large moment antiferromagnetic order in K_0.8Fe_1.6Se₂ superconductor, Chin. Phys. Lett. 28(8), 086104 (2011)

[11] C. Cao and J. Dai, Block spin ground state and three-dimensionality of (K,Tl)_yFe_1.6Se₂, Phys. Rev. Lett. 107(5), 056401 (2011)

[12] X. W. Yan, M. Gao, Z. Y. Lu, and T. Xiang, Ternary iron selenide K_0.8Fe_1.6Se₂ is an antiferromagnetic semiconductor, Phys. Rev. B 83(23), 233205 (2011)

[13] W. Li, S. Dong, C. Fang, and J. Hu, Block antiferromagnetism and checkerboard charge ordering in the alkali-doped iron selenides R_1-xFe_2-ySe₂, Phys. Rev. B 85, 100407(R) (2012)

[14] C. H. Li, B. Shen, F. Han, X. Zhu, and H. H. Wen, Transport properties and anisotropy of Rb_1-xFe_2-ySe₂ single crystals, Phys. Rev. B 83(18), 184521 (2011)

[15] Y. J. Yan, M. Zhang, A. F. Wang, J. J. Ying, Z. Y. Li, W. Qin, X. G. Luo, J. Q. Li, J. Hu, and X. H. Chen, Electronic and magnetic phase diagram in K_xFe_2-ySe₂ superconductors, Sci. Rep. 2, 212 (2012)

[16] F. Chen, M. Xu, Q. Q. Ge, Y. Zhang, Z. R. Ye, L. X. Yang, J. Jiang, B. P. Xie, R. C. Che, M. Zhang, A. F. Wang, X. H. Chen, D. W. Shen, M. H. Jiang, J. P. Hu, and D. L. Feng, Electronic identification of the parental phases and mesoscopic phase separation of K_xFe_2-ySe₂ superconductors, Phys. Rev. X 1(2), 021020 (2011)

[17] P. Cai, C. Ye, W. Ruan, X. Zhou, A. Wang, M. Zhang, X. Chen, and Y. Wang, Imaging the coexistence of a superconducting phase and a charge-density modulation in the K_0.73Fe_1.67Se₂ superconductor using a scanning tunneling microscope, Phys. Rev. B 85(9), 094512 (2012)

[18] W. Li, H. Ding, P. Deng, K. Chang, C. Song, K. He, L. Wang, X. Ma, J. P. Hu, X. Chen, and Q. K. Xue, Phase separation and magnetic order in K-doped iron selenide superconductor, Nat. Phys. 8(2), 126 (2011)

[19] R. H. Yuan, T. Dong, Y. J. Song, P. Zheng, G. F. Chen, J. P. Hu, J. Q. Li, and N. L. Wang, Nanoscale phase separation of antiferromagnetic order and superconductivity in K_0.75Fe_1.75Se₂, Sci. Rep. 2, 221 (2012)

[20] A. Zhang, T. Xia, K. Liu, W. Tong, Z. R. Yang, and Q. M. Zhang, Superconductivity at 44 K in K intercalated FeSe system with excess Fe, Sci. Rep. 3, 1216 (2013)

[21] W. Bao, G. N. Li, Q. Huang, G. F. Chen, J. B. He, M. A. Green, Y. Qiu, D. M. Wang, and J. L. Luo, Superconductivity tuned by the iron vacancy order in K_xFe_2-ySe₂, Chin. Phys. Lett. 30(2), 027402 (2013)

[22] Z. Wang, Y. J. Song, H. L. Shi, Z. W. Wang, Z. Chen, H. F. Tian, G. F. Chen, J. G. Guo, H. X. Yang, and J. Q. Li, Microstructure and ordering of iron vacancies in the superconductor system K_yFe_xSe₂ as seen via transmission electron microscopy, Phys. Rev. B 83(14), 140505 (2011)

[23] D. X. Mou, L. Zhao, and X. J. Zhou, Structural, magnetic and electronic properties of the ironCchalcogenide A_xFe_2-ySe₂ (A = K, Cs, Rb, and Tl, etc.), Front. Phys. 6(4), 410 (2011)

[24] D. Mou, S. Liu, X. Jia, J. He, Y. Peng, L. Zhao, L. Yu, G. Liu, S. He, X. Dong, J. Zhang, H. Wang, C. Dong, M. Fang, X. Wang, Q. Peng, Z. Wang, S. Zhang, F. Yang, Z. Xu, C. Chen, and X. J. Zhou, Distinct Fermi surface topology and nodeless superconducting gap in a (Tl_0.58Rb)0.42Fe_1.72Se₂ superconductor, Phys. Rev. Lett. 106(10), 107001 (2011)

[25] I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, Unconventional superconductivity with a sign reversal in the order parameter of LaFeAsO_1-xF_x, Phys. Rev. Lett. 101(5), 057003 (2008)

[26] K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and H. Aoki, Unconventional pairing originating from the disconnected Fermi surfaces of superconducting LaFeAsO_1-xF_x, Phys. Rev. Lett. 101(8), 087004 (2008)

[27] W. Li, J. Li, J. X. Zhu, Y. Chen, and C. S. Ting, Pairing symmetry in the iron-pnictide superconductor KFe₂As₂, Europhys. Lett. 99(5), 57006 (2012)

[28] I. R. Shein and A. L. Ivanovskii, Structural, electronic properties and Fermi surface of ThCr₂Si₂-type tetragonal KFe₂S₂, KFe₂Se₂, and KFe₂Te₂ phases as parent systems of new ternary iron-chalcogenide superconductors, arXiv: 1102.4173 (2011)

[29] X. W. Yan, M. Gao, Z. Y. Lu, and T. Xiang, Electronic and magnetic structures of the ternary iron selenides AFe₂Se₂ (A = Cs, Rb, K, or Tl), Phys. Rev. B 84(5), 054502 (2011)

[30] P. E. Blöchl, Projector augmented-wave method, Phys. Rev. B 50(24), 17953 (1994)

[31] G. Kresse and J. Furthmuller, Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set, Phys. Rev. B 54(16), 11169 (1996)

[32] J. P. Perdew, K. Burke, and M. Erznerhof, Generalized gradient approximation made simple, Phys. Rev. Lett. 77(18), 3865 (1996)

[33] C. Cao and J. Dai, Electronic structure of KFe₂Se₂ from first-principles calculations, Chin. Phys. Lett. 28(5), 057402 (2011)

[34] D. J. Singh, Electronic structure and doping in BaFe₂As₂ and LiFeAs: Density functional calculations, Phys. Rev. B 78(9), 094511 (2008)

[35] W. Li, J. X. Zhu, Y. Chen, and C. S. Ting, First-principles calculations of the electronic structure of iron-pnictide EuFe₂(As,P)₂ superconductors: Evidence for antiferromagnetic spin order, Phys. Rev. B 86(15), 155119 (2012)

[36] N. Marzari and D. Vanderbilt, Maximally localized generalized Wannier functions for composite energy bands, Phys. Rev. B 56(20), 12847 (1997)

[37] A. A. Mostofi, J. R. Yates, Y. S. Lee, I. Souza, D. Vanderbilt, and N. Marzari, Wannier90: A tool for obtaining maximally-localised Wannier functions, Comput. Phys. Commun. 178(9), 685 (2008)

[38] W. Li, C. Setty, X. H. Chen, and J. Hu, Electronic and magnetic structures of chain structured iron selenide compounds, Front. Phys. 9(4), 465 (2014)

[39] F. Ma, Z. Y. Lu, and T. Xiang, Arsenic-bridged antiferromagnetic superexchange interactions in LaFeAsO, Phys. Rev. B 78(22), 224517 (2008)

[40] Y. Liang, X. Wu, W. F. Tsai, and J. P. Hu, Pairing symmetry in layered BiS₂ compounds driven by electron-electron correlation, Front. Phys. 9(2), 194 (2014)

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