⇦Chinese Physics Letters, 2023, Vol. 40, No. 9, Article code 097405 Role of Lanthanide in the Electronic Properties of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ ($Ln$ = Sm and Ho) Superconductors Yi-Na Huang (黄衣娜)1*, Zhao-Feng Ye (叶兆丰)1, Da-Yong Liu (刘大勇)2, and Hang-Qiang Qiu (邱航强)1 Affiliations 1Department of Physics, Zhejiang University of Science and Technology, Hangzhou 310023, China 2Department of Physics, School of Sciences, Nantong University, Nantong 226019, China Received 21 April 2023; accepted manuscript online 23 August 2023; published online 1 September 2023 *Corresponding author. Email: ynhuang@zust.edu.cn Citation Text: Huang Y N, Ye Z F, Liu D Y et al. 2023 Chin. Phys. Lett. 40 097405    Abstract We focus on the effect of ionic radius of lanthanides and the number of electrons in $4f$ orbitals on the superconducting temperature in 12442-type iron-based superconductors Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ (Ln = Sm and Ho). Electronic properties of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ with the largest differences of ionic radii and numbers of electrons in $4f$ orbital, and the largest difference of superconducting temperatures by using first-principles calculations. We predict that the ground state of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ is spin-density-wave-type in-plane striped antiferromagnet, and the magnetic moment around each Fe atom is about $2\mu_{\scriptscriptstyle{\rm B}}$. RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ has a great influence on the energy band near the $\varGamma$ point, and a Dirac-like dispersion energy band appears. This band is mainly contributed by the $d_{z^2}$ orbital of Fe, which proves that RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ has a stronger three-dimensionality. At the same time, this extra Fermi surface appears at the $\varGamma$ point, which also shows that Sm can effectively enhance the coupling strength within Fe$_{2}$As$_{2}$ bilayers. This is also confirmed by the charge density difference $\rho$(RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$)$\,-\rho$(RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$). It increases the internal coupling strength of the bilayer Fe$_{2}$As$_{2}$ layers, which in turn leads to a higher $T_{\rm c}$ of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ than RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$. Determining the details of their electronic structure, which may be closely related to superconductivity, is crucial to understanding the underlying mechanism. Such microscopic studies provide useful clues for our further research of other high-temperature superconductors.
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 DOI:10.1088/0256-307X/40/9/097405 © 2023 Chinese Physics Society Article Text It is well-known that superconducting properties of iron-based compounds strongly depend on tiny details of the structure, namely the height of the pnictogen atom (here As) above the FeAs plane (denoted as $h_{\rm As}$), and the distance between the Fe atom and the nearby As atom.[1-3] The angle formed by bonds, that is, the As–Fe–As bond angle, is usually denoted as $\alpha$. In particular, the maximum of the critical temperature is associated with a specific combination of these parameters, namely $h_{\rm As} = 1.38$ Å and $\alpha = 109.5^{\circ}$. However, the As ion height and bond angle of the 12442-system material Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ (Ln is a lanthanide element) are quite different from the superconducting transition temperature $T_{\rm c}$, which is inconsistent with previous research. Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ is derived from 1111 and 122 iron-based superconductors.[4,5] When substrate 1111 is LnFeAsO, Ln is usually Sm, and then the Fe sites are doped on SmFeAsO to increase the $T_{\rm c}$.[6,7] However, there are almost no 1111 materials that take Ho to replace the Ln position, and only a small amount of Ho replaces Sm to form a new material Sm$_{1-x}$Ho$_{x}$FeAsO, because HoFeAsO alone is unstable,[8] and HoFeAsO cannot be synthesized under ambient pressure alone. However, in the 12442 material, both KHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and CsHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ are realized, which means that the 12442 structure can stabilize the 1111 phase. Different rare-earth elements affect not only the magnetic structure of iron, but also the absolute value of the magnetic moment of iron, which indicates that rare-earth elements play an important role in superconductivity.[9,10] As is known, 12242 systems have a unique double Fe$_{2}$As$_{2}$ layer (named interlayer and intralayer), which comes from the corresponding Fe$_{2}$As$_{2}$ layer of the parent 1111 and 122 iron-based superconductors.[11] It appears that the spacing between the bilayers increases with the increasing radius of the lanthanum ion, but in fact it is the lattice mismatch[12,13] that really affects the variation of the spacing in the bilayer. Since $T_{\rm c}$ basically increases with lattice mismatch, that is to say, the strength of the interaction between the bilayers directly affects the $T_{\rm c}$. The change in ionic radius will lead to the $a/c$ ratio changing, and $T_{\rm c}$ decreases almost linearly with $c/a$. Lanthanide contraction (chemical stress) systematically shortens the distance between Fe–As, Fe–Fe, Fe–Ln, and $\alpha_{_{\scriptstyle \rm Fe-As-Fe}}$. For the chemical pressure of lanthanide contraction, the contribution of $4f$ electrons should be considered at the same time, because it is well known that $4f$ electrons not only hybridize with anions, but also participate in magnetic coupling at a certain low temperature,[14] especially at the temperature at which superconductivity occurs.[15,16] Experimentally, it is impossible to separate these factors in the exploration of the superconducting mechanism. However, in this case, recent advances in DFT-based first-principles calculations have facilitated a breakthrough in the study of electronic states in strongly correlated electronic systems, which will help us to gain a clearer understanding of the intrinsic mechanism of superconductivity. The superconducting behavior is closely related to the pairing mechanism.[17,18] Determining the details of their electronic structure, which may be closely related to superconductivity, is crucial to understanding the underlying mechanism. Such microscopic studies will provide useful clues for our further study of other high-temperature superconductors. In this Letter, first-principles methods are described. Then, we apply this method to study the electronic structures and magnetic ground states of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$, and provide the main results. It is confirmed that the ground state of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ are spin-density-wave-type in-plane striped antiferromagnets, and the magnetic moment around each Fe atom is about $2\mu_{\scriptscriptstyle{\rm B}}$. Several other differences between RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ compounds are analyzed, including the effect of the chemical pressure caused by their introduction on the FeAs conductive layer. We investigate the role of differential charge transfer in the Fe$_{2}$As$_{2}$ bilayer, which is a structural peculiarity of the 12442 compound. Finally, we present a concise summary of our results. Crystal Structure. To perform calculations related to RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, their respective unit cells' experimental lattice parameters are required. The values for these lattice parameters are $a=3.92092$ Å and $c=31.3812$ Å for RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, and $a=3.86881$ Å and $c=31.24247$ Å for RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$. It is important to note that these values differ, dependent on the specific phase of each material, including the non-magnetic (NM) phase, ferromagnetic (FM) phase, and antiferromagnetic (AFM) phase. The crystal structures of NM, FM1, and FM2 are characterized by the I4/mmm (No. 139) space group, while AFM1, AFM2, and AFM3 exhibit the Cmma (No. 67), P42/mmc (No. 131), and I4/mm (No. 107) space groups, respectively. There are various types of antiferromagnetic arrangements in the Fe layer, including stripe antiferromagnetism (SAFM) and Néel antiferromagnetism (NAFM). SAFM (AFM1) involves each Fe atom being aligned antiparallel to its second nearest neighbors along the $a$ and $b$ axes, whereas NAFM (AFM2) occurs when the spin direction of the nearest neighbor atoms in the Fe layer is opposite. AFM3 represents a distinct interlayer antiferromagnet that differs from both AFM1 and AFM2. The material has a quasi-two-dimensional layered structure formed by stacking alternating layers of Fe$_{2}$As$_{2}$ and $Ln$O along the $c$-axis, within the space group denoted as P4/nmm. The arrangement of atoms in each layer adopts a collinear connection via four-coordinate tetrahedral structures. A shorter Fe-As bond length indicates higher metallicity, which implies greater capacity for electron loss and weaker control over outermost electrons by the metal. As a result, the interaction between cations and free electrons becomes weaker, leading to a decrease in electron correlation. The Fe$_{2}$As$_{2}$ layer features a two-dimensional square lattice of $3d$ transition ferromagnetic Fe atoms, with charges being transferred from the LnO layer to the Fe$_{2}$As$_{2}$ layer. This leads to a balanced chemical bond formation between the positively charged (LnO)$^{1+}$ layer and the negatively charged (FeAs)$^{1-}$ layer within the Fe$_{2}$As$_{2}$ layer.
Table 1. Total energy difference, for Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ (Ln = Sm and Ho), of seven magnetic phases including FM1, FM2, AFM1, AFM2, and AFM3 phases. The reference is the energy of the FM1 phase ($\Delta E=E_{\rm AFM}-E _{\rm FM1}$). The corresponding magnetic moment in the Fe sphere is given.
Magnetic structure FM1 FM2 AFM1 AFM2 AFM3
Relative energy (eV) RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ 0 1.05 $-0.93$ $-0.41$ 1.01
RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ 0 3.06 $-0.37$ $-0.18$ 2.19
Fe moment (${\mu_{\scriptscriptstyle{\rm B}}}$) RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ 0.73 0.83 1.94 1.94 1.21
RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ 0.83 0.91 1.87 2.05 0.94
Calculational Methods. The electronic structure calculations were performed using the Wien2K package,[19] which employs the full-potential linearized augmented plane wave method. In our density functional theory calculations, we employed the generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof[20] version. We used specific sphere radii for Rb, Sm, Ho, O, Fe, and As atoms in our electronic structure calculations. The values were 2.5 Bohr, 2.36 Bohr, 2.31 Bohr, 1.9 Bohr, 2.3 Bohr, and 2.18 Bohr, respectively. We set the basis set cut-off parameter as $R_{\rm mt} \cdot K_{\max} = 7.0$, which was found to be sufficient. To ensure adequate sampling of the small Fermi surface, we used a $k$-point number of 4500 for the tetragonal cell and 3000 for the supercell in each calculation. Figure 4 presents the shapes of the supercells and the distribution of spins that were investigated under various magnetic conditions. To account for the strong correlation of $4f$ electrons in Sm and Ho, we utilized a code with GGA + $U$ (where $U = 9$ eV) to conduct electronic structure calculations for RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, respectively. Since they are newly discovered materials, there is currently a lack of relevant experimental data as a reference basis. Therefore, we can only estimate and speculate on the $U$ value within a range based on existing research results[21,22] and empirical knowledge. We think that the $U$ value between 6 eV and 12 eV is a relatively reasonable range. We finally choose $U = 9$ eV as a specific example, which does not affect our overall physical conclusion. Iron-based superconductors belong to bad metals. Applying $U$ to Fe $3d$ electrons will change the renormalization effect of the energy band, but it does not affect the final physical conclusion. Therefore, we did not apply $U$ to Fe $3d$ electrons in this study, and it will not reduce the reliability of the calculation results. See the Supplemental Material for details. Analysis and Discussion. Initially, we analyzed the variation in total energy for different magnetic configurations, such as NM, FM, and several AFM arrangements, for RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, and Fe in each magnetic phase. From the magnetic moments in Table 1, we found that the magnetic moment around each Fe atom is about $2\mu_{\scriptscriptstyle{\rm B}}$, much smaller than the local density approximation value in LaFeAsO.[23] The in-layer SAFM configuration has the lowest total energy, so all ground states of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ are in-plane SAFM phases. In the following, we will discuss the fundamental characteristics of the electronic structures of the ground-state in-plane SAFM and NM phases in RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, and compare the differences between these two materials. The roles of lanthanides and Fe$_{2}$As$_{2}$ bilayer are discussed in detail. Nonmagnetic Phase. Figure 1 displays the structures of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ (Ln = Sm and Ho) in the NM phase. Based on our calculations of the electronic properties of the NM structure, we have determined that the As atoms in the FeAs tetrahedron exhibit differences in their crystallographic equivalence. The atomic coordinates for the structures are as follows: Rb located at (0, 0, 0); Ln positioned at (0.5, 0.5, $z$); Fe situated at (0.5, 0, $z$); As1 present at (0.5, 0.5, $z$); As2 located at (0, 0, $z$); O positioned at (0.5, 0, 0.25). It can be seen from the experimental results[8] that from Ho to Sm: the distance from the FeAs layer to the LnO layer becomes smaller and the distance to the Rb layer becomes smaller, because the lattice parameters $a$ and $c$ become smaller. The change of $T_{\rm c}$ with $c/a$ is almost linear, which supports the inference that the enhancement of the internal coupling of Fe$_{2}$As$_{2}$ bilayers can be the inference of the increase of $T_{\rm c}$.
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Fig. 1. The structures of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ (Ln = Sm and Ho) in NM phase. The white one in the picture is the Fe$_{2}$As$_{2}$ interlayer, and the blue one is the Fe$_{2}$As$_{2}$ intralayer. The two diagrams on the right show the structures the LnO and FeAs layers.
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Fig. 2. The Fe Pbands, PDOS, and Fermi surfaces of [(a), (c)] RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, [(b), (d)] RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, and (e) LaOFeAs in NM phase.
Figures 2(a)–2(e) show the project of bands (Pbands) of the NM phases Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and LaOFeAs. Difference in density of states: RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ materials are similar to 12442 materials from other iron-nickel-based materials, and the Fe $3d$ state is the main contribution near the Fermi level, and other atoms (Ln, O, Rb) contribute very little, the Fe $3d$ and As $4p$ orbitals are hybridized, resulting in a small contribution from the As $4p$ state. Among the five Fe $3d$ orbital projected density of states, the main contributions to the Fermi level are Fe $3d_{xz}$, Fe $3d_{yz}$, and Fe $3d_{z^2}$, which are much higher than the contributions of the other two orbitals. This is obviously different from the five $3d$ orbitals of the Fe base in the parent material 1111 (the energy of the five $3d$ orbitals is relatively close, and the five $3d$ orbitals all contribute to the Fermi level). The two materials have slightly different contributions to the Fermi level: in RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, $3d_{z^2} > 3d_{xy}=3d_{xz} > 3d_{x^2-y^2}=3d_{yz}$; in RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, $3d_{xy}>3d_{xz} > 3d_{z^2} > 3d_{yz} > 3d_{x^2-y^2}$. Differences in energy bands: in superconducting materials, it is important to probe the Fermi surface to understand the carrier topology because superconductivity is caused by charge carriers on the Fermi surface. Figures 2(a) and 2(b) show the electronic structures of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ near the high symmetry point. The hole-type Fermi surface at the $\varGamma$ point near the Fermi level is mainly contributed by the $d_{z^2}$, $d_{x^2-y^2}$, $d_{xy}$ and $d_{yz}$ orbitals of Fe, and the electron-type Fermi surface near $X$ is mainly contributed by the $d_{x^2-y^2}$, $d_{xz}$, and $d_{yz}$ orbital contributions. We note that RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ have 11 and 10 bands across the Fermi level, respectively, of which 6 bands are in the center of the Brillouin zone, that is, at the $\varGamma$ point showing closely cylindrical shaped hole-like pockets. The others are 5 and 4 electron-like pockets with electron-like pockets at the corners of the Brillouin zone, respectively. RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ has a great influence on the energy band near the $\varGamma$ point, and a Dirac-like dispersion energy band appears. Due to some characteristics of Dirac dispersion, there may be nontrivial topological properties in this system and similar compounds.[24] This band is mainly contributed by the $d_{z^2}$ orbital of Fe, which proves that RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ has stronger three-dimensionality. At the same time, this extra Fermi surface appears at the $\varGamma$ point, which also shows that Sm can effectively enhance the coupling strength within Fe$_{2}$As$_{2}$ bilayers. The difference of the Fermi surface: all the Fermi surfaces that are different from the 1111-type iron-based superconductor are contributed by Sm or Ho. Whether the Fermi surface contributed by Fe is a hole type or an electron type, it is a columnar Fermi surface with good two-dimensionality. This means that the ground states of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ also tend to be SAFM unstable. The energy band structure of the holes along the $\varGamma$–$Z$ direction shows dispersion, which corresponds to the oblate Fermi surface at the center of $\varGamma$. This is also the main difference between these two materials, and presents the Fermi surface of the 1111-type iron-based superconductor. This orbital is mainly contributed by the $d_{z^2}$ orbital of Fe. Sm versus Ho. Considering that Fe-based compounds and chalcogenides occupy Fe orbitals and their relationship with the properties is the focus of attention. To further study the effects of Sm and Ho atoms on Fe orbitals, we calculated the charge densities of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ in the NM phase. To facilitate the comparison, we calculated the density difference $\rho$(RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$)$\,-\rho$(RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$). To obtain the charge density of the two materials, we calculated both from the lattice constant of the previous compound. Figure 3 is a color map of the charge density difference in the (100) plane for the two materials. A positive value indicates a region where the charge decreases when Sm is introduced. These changes are rather small, consistent with the neutral and nonpolar nature of the two isolated systems, but involve orbital regrouping on different Fe$_{2}$As$_{2}$ layers and lanthanide ions. The most important change is a strongly anisotropic rearrangement of the density in Fe: $p_{z}$ of O, $d_{xz}$ and $d_{yz}$ of Fe1, and $d_{xy}$ domains of Fe2 increase in density, while the $p_{x}$ ($p_{y}$) domains of O decrease in density. The charge is “transferred” to the $p_{z}$ orbital at the expense of the $p_{x}$ and $p_{y}$ orbitals. Proved the difference between Fe1As1 (Fe$_{2}$As$_{2}$ interlayer) and Fe2As2 (Fe$_{2}$As$_{2}$ intralayer). The influence that the $Ln$O layer plays from it. The charge density difference diagrams both also further prove that Sm is more conducive to the interlayer transfer of electrons through 3$d_{xz}$ and 3$d_{yz}$ of Fe1 to 2$p_{z}$ of O and then to Fe2 3$d_{xy}$ compared to Ho, which improves the double-layer internal coupling strength of Fe$_{2}$As$_{2}$. In turn, the $T_{\rm c}$ of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ is slightly higher than that of RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$.
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Fig. 3. Contour plots of the density difference $\rho$(RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$)$\,-\rho$(RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$). The large red region in denotes where contours have been cut off due to the large and meaningless difference of the RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ densities in NM phase.
Ground State: In-Plane SAFM Phase. Next, we explore the magnetic energies and fundamental characteristics of the electronic structure of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$, and making comparisons with other pnictides. Our focus is on the electronic properties of the antiferromagnetic ground state, specifically the SAFM structure. We have also conducted calculations on two Ln atoms by breaking the NM structure's symmetry, which has confirmed that the in-plane SAFM is indeed the primary magnetic configuration for the other AFM structure. Each magnetic configuration has a significant energy difference compared to the NM state. Table 1 displays the energies of seven basic magnetic configurations relative to the FM1 phase, along with the Fe atomic sphere moments. The energy difference between the NM state and the magnetic state is too significant to make a meaningful comparison. To facilitate comparison, the FM1 ferromagnetic state is used as the reference because it has a more manageable energy level. Although the DFT methods[25] tend to overestimate the Fe moment and thus exaggerate the magnetic energy, the relative ordering of the energies is still significant. Table 1 indicates that Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ possesses an in-plane spin arrangement known as SAFM (AFM1), which is represented by the corresponding structures shown in Fig. 4. The magnetic moment of Fe in this compound is around $1.9\mu_{\scriptscriptstyle{\rm B}}$, a value similar to the previously calculated magnetic moment of CaFeAsH[26] ground state antiferromagnetism ($1.92\mu_{\scriptscriptstyle{\rm B}}$) using Wien2K, and also close to the magnetic moment of $2.11\mu_{\scriptscriptstyle{\rm B}}$ found in CaFeAs$_{2}$[27] ground state antiferromagnetic force. Neutron scattering experiments on underdoped samples of the 1111 and 122 systems have revealed a magnetic moment for Fe of approximately $2\mu_{\scriptscriptstyle{\rm B}}$, which is almost identical to the ordered magnetic moment, implying that the size of magnetic fluctuations throughout the crystal lattice is uniform. The magnetic structures of five phases, namely FM1, FM2, AFM1, AFM2, and AFM3, are illustrated in Fig. 4, arranged from left to right. The Fe1 and Fe2 atoms are located at (0, 0, $z$) and (0.5, 0.5, $z$), respectively. Similarly, the Ln1 and Ln2 atoms belong to different layers of Ln and have coordinates of (0.5, 0.5, $z$) and (0, 0, $z$), respectively.
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Fig. 4. The structures of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ (Ln = Sm and Ho) in FM1, FM2, AFM1, AFM2, and AFM3 phases.
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Fig. 5. Fe Pband and PDOS for (a1)–(a3) RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and (b1)–(b3) RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ in the ground state: [(a1), (b1)] spin up, [(a2), (b2)] spin down. Note the small DOS at the Fermi energy (zero of energy).
A small number of states may be seen at first glance to undergo an $e_{\rm g}$–$t_{\rm 2g}$ split, but the reality is much more complex. Apparently, SAFM order substantially affects the iron configuration, based on the loss of tetrahedral symmetry due to magnetic re-establishment of electronic bonds. Since the Fe atoms are coordinated by the As tetrahedron, the crystal field usually splits the five $d$ orbitals into the two lower $e_{\rm g}$ states and the three higher $t_{\rm 2g}$ states. However, the As tetrahedron is actually distorted from its normal shape. This distortion will further split the $e_{\rm g}$ and $t_{\rm 2g}$, complicating the final orbital distribution. Figures 5(a1), 5(a2), 5(b1), and 5(b2) show the Pbands of Fe in Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ under SAFM phase. Comparing the energy bands of the two materials, we can find that the bands passing through the Fermi level surface near the $\varGamma$ and $Z$ points are mainly contributed by $d_{z^2}$, $d_{x^2-y^2}$, and $d_{yz}$ orbitals. The difference between them is that the energy bands of RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ near these two points have shifted upward compared to those of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$. In addition, the original Dirac band near the $\varGamma$ point in the NM phase has moved downward in the $\varGamma$–$Z$ path under the SAFM phase, and the Dirac band has disappeared. The energy bands of RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ have changed more than those of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$. All of these indicate the different effects of Sm and Ho atoms on the electronic properties of the materials. From the PDOS in Figs. 5(a3) and 5(b3), it can be seen that the material has very low density of states near the Fermi level. In the spin-up orbital density, the five orbitals are filled below the Fermi level, while the spin-down orbital density seems to be evenly distributed. The orbital densities at the Fermi level for the two materials are as follows: RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, $d_{xz} >d_{z^2} =d_{x^2-y^2} >d_{xy} >d_{yz}$; RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, $d_{xz} >d_{z^2} >d_{x^2-y^2} >d_{yz} >d_{xy}$. It can be seen that the difference between these two materials is that the contribution from the $d_{xy}$ orbital in the Fe PDOS of RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ is larger than that from the $d_{yz}$ orbital, so RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ material is more conducive to interlayer charge transfer, which is consistent with the conclusion drawn earlier. In summary, we have studied the high-temperature nonmagnetic state and the low-temperature magnetic ground state of two 12442 materials, RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$, and made a detailed study of the effects of lanthanides and bilayer Fe$_{2}$As$_{2}$. It is found that the two 12442 materials RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ have the same antiferromagnetic ground state. The ground state of Rb$Ln_{2}$Fe$_{4}$As$_{4}$O$_{2}$ is spin-density-wave-type in-plane striped antiferromagnets, and the magnetic moment around each Fe atom is about $2\mu_{\scriptscriptstyle{\rm B}}$. In the 12442 iron-based superconductor, the ionic radius of Sm is larger than that of Ho. However, the 3$d_{z^2}$ contribution of Fe in RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ causes the interlayer interaction of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ is stronger than that of RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$. Stretching and compression also lead to changes in the distance between the Fe$_{2}$As$_{2}$ bilayers. Stretching leads to a larger interlayer distance and a higher superconducting temperature. We find that the quasi-two-dimensional nature of the Fermi surface of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ is relatively strong, which means that the ground states of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ and RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ also have a tendency of SAFM instability. Also, RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ has a great influence on the energy band near the $\varGamma$ point, and a Dirac-like dispersion energy band appears. This band is mainly contributed by the $d_{z^2}$ orbital of Fe, which proves that RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ has stronger three-dimensionality. At the same time, this extra Fermi surface appears at the $\varGamma$ point, which also shows that Sm can effectively enhance the coupling strength within Fe$_{2}$As$_{2}$ bilayers. This is also confirmed by the charge density difference $\rho$(RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$)$\,-\rho$(RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$). The transfer path is: The electrons transfer to the 3$d_{xy}$ of intralayer Fe, through 3$d_{xz}$, $3d_{yz}$ of interlayer Fe and $2p_{z}$ of O. This increases the internal coupling strength of the bilayer Fe$_{2}$As$_{2}$ layers, which in turn leads to the fact that the $T_{\rm c}$ of RbSm$_{2}$Fe$_{4}$As$_{4}$O$_{2}$ is higher than that of RbHo$_{2}$Fe$_{4}$As$_{4}$O$_{2}$. We hope that these findings will help to study such materials and enable us to understand their relationship with superconducting properties, thus providing theoretical support to search for superconducting materials with higher $T_{\rm c}$. Acknowledgments. This research was supported by the National Natural Science Foundation of China (Grant Nos. 11904319 and 11974354), and the Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ18A040002). 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