Unique crystal field splitting and multiband RKKY interactions in Ni-doped EuRbFe4As4

The relationship between magnetism and superconductivity has been one of the most discussed topics in iron-based superconductors. Using first-principles calculations, we have studied the electronic structure of 1144-type iron-based superconductor EuRbFe4As4. We find the crystal field splitting of EuRbFe4As4 is unique, such that the dz2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{z^2}$$\end{document} orbitals are closer to the Fermi level ϵF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{\mathrm F}$$\end{document} than the dxy orbitals. The Ruderman−Kittel−Kasuya−Yosida (RKKY) interaction strength is approximately 0.12 meV in pristine EuRbFe4As4. Upon Ni-doping on the Fe site, the RKKY interaction strength is barely changed upon Ni-doping due to the highly anisotropic Fermi surfaces and multiband effect, despite the drastically reduced dzx(y) density of state at ϵF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon _{\mathrm F}$$\end{document}. Finally, in both pristine and doped compounds, the RKKY interaction is primarily mediated through bands due to Fe-dz2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{z^2}$$\end{document} orbitals. Our calculations suggest the RKKY interaction mediated by dz2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_{z^2}$$\end{document} orbital is probably responsible for the magnetism in EuRbFe4As4 and doesn’t change upon Ni-doping. The intriguing coexistence of superconductivity and magnetism is examined via first-principle calculations in an iron-based superconductor. By calculating the RKKY interaction and bared susceptibility, the authors explained the unchanged Curie temperature and largely suppressed superconducting temperature upon doping observed in experiment.

O ver the last decade, enormous efforts have been put on iron-based superconductors (FeSCs) since the first discovery of superconductivity in F-doped LaOFeAs in 2008 1 . To date the relationship between magnetism and superconductivity has been extensively investigated both experimentally and theoretically to pursue the essence of pairing mechanism. As a matter of fact, a multitude of the FeSCs are magnetic metals with quasi-2D electron and hole pockets. Usually the parent compounds of FeSCs are non-superconducting and undergo structural transition from tetragonal lattice to orthorhombic lattice followed by a spin density wave (SDW) transition when the temperature is decreased. Once the system is doped, however, both structural and SDW transitions are suppressed and superconductivity emerges 2,3 .
Recently a new type of FeSC structure, AeAFe 4 As 4 (AeA-1144; Ae = Ca,Sr; A = K,Rb,Cs), has been successfully synthesized by Iyo et al. 4 with superconducting transition temperature T c > 30 K without doping. Despite the same stoichiometric ratio, these 1144-type compounds are different from the 50% K-doped BaFe 2 As 2 compounds such that the Ae atoms are separated from the A atoms in different layers, and therefore the structure can be viewed as layer-by-layer stacked AeFe 2 As 2 (Ae-122) and AFe 2 As 2 (A-122). Independently, Liu et al. and Kawashima et al. 5,6 managed to make an intergrowth structure of RbFe 2 As 2 and EuFe 2 As 2 leading to another 1144-type compound EuRbFe 4 As 4 . Similar to the AeAFe 4 As 4 (Ae = Ca,Sr; A = K,Rb,Cs) compounds, EuR-bFe 4 As 4 exhibits superconductivity at 35 K without any doping. In addition, ferromagnetic (FM) behavior due to the Eu-layers was observed below 15 K without destroying the superconductivity, suggesting a robust coexistence of superconductivity and ferromagnetism 5 . The 4f electrons are nearly fully localized, yielding a large moment of 6.5 μ B /Eu. Interestingly, the Curie temperature was robust against the Ni-doping, while the superconducting T sc was very sensitive 7 . It was therefore conjectured that the d−f superexchange or Eu-As-Eu superexchange is likely to be the mechanism mediating the magnetic interactions.
In this article, we present a systematic first-principles study of EuRbFe 4 As 4 compound. We compare EuRbFe 4 As 4 with the CaRbFe 4 As 4 as well as the traditional BaFe 2 As 2 compound. The band structure of EuRbFe 4 As 4 resembles that of CaRbFe 4 As 4 , and there are ten bands crossing the Fermi level in both compounds. However, the d z 2 -band in EuRbFe 4 As 4 is now elevated close to ϵ F around the Γ point; thus, the undoped EuRbFe 4 As 4 system is very close to a Lifshitz transition. As a result, the d z 2 -orbital is more relevant in the process of electrical transport and magnetism in the EuRbFe 4 As 4 compound. By decomposing the J RKKY into contributions from different Fermi surface (FS) sheets, we find this interaction is primarily mediated by the outmost hole pocket due to d z 2 orbital around the Γ point, suggesting that this orbital is important to the Eu-magnetism. Under nickel doping, the Ruderman−Kittel−Kasuya−Yosida (RKKY) exchange interaction is little affected, despite the drastically reduced density of state (DOS) from d xz and d yz orbitals. This may serve as a possible explanation why the magnetic transition T FM is relatively constant while the superconductivity T sc is quickly suppressed by Nidoping in this compound.

Results
Crystal structure and electronic structure. The crystal structure of EuRbFe 4 As 4 is tetragonal P4/mmm with lattice parameters a = 3.89 Å and c = 13.31 Å as illustrated in Fig. 1a. The crystal consists of FeAs-layers separated by alternating Rb-layers and Eulayers. Therefore, each unit cell contains two FeAs-layers that are mirror-symmetric with respect to either the Rb-layer or the Eulayer. In contrast to most traditional FeSCs, the two intercalating layers sandwiching each FeAs-layer are different in 1144-type FeSCs; thus the Fe atom is no longer located at the center of the tetrahedral formed by the four closest As atoms (Fig. 1b). Therefore, the S 4 local symmetry around each Fe is reduced to C 2 in 1144-type FeSCs.
The band structures of EuRbFe 4 As 4 (red line), CaRbFe 4 As 4 (black line), and folded BaFe 2 As 2 (blue line, Γ−X) in paramagnetic (PM) phase are shown in Fig. 2a, b. Both EuRbFe 4 As 4 and CaRbFe 4 As 4 exhibit the same number of hole pockets around Γ points and electronic pockets around M points, and the dispersions resemble each other. Nevertheless, the highest occupied state at Γ is extremely close to the Fermi level ϵ F in EuRbFe 4 As 4 (while in CaRbFe 4 As 4 it is more than 200 meV below ϵ F ), suggesting the undoped EuRbFe 4 As 4 system is on the edge to a Lifshitz transition. Projected band structure shows that the orbital character of this state is d z 2 . With slight hole-doping (0.5 hole per f.u.), this band will cross Fermi level and create a new Fermi surface sheet. In Ba 1−x Rb x Fe 2 As 2 , it was reported that a crossover from nodeless to nodal superconducting occurs at x = 0.65 as a result of a hole-doping-induced Lifshitz transition 8 . Considering the similarities between these systems, such transition may also be present in hole-doped EuRbFe 4 As 4 . Additionally, in contrast to BaFe 2 As 2 , the bands structures in 1144-system splits along X−M, owing to the S 4 symmetry breaking. Similar splitting is also present in CaKFe 4 As 4 9 . We have also calculated EuRbFe 4 As 4 (Fig. 2c) in the non-superconducting FM phase, where Eu moments align in parallel. The exchange spin splitting is less than 20 meV shown in Table 1. The projected band structure (Fig. 2d) indicates that the d z 2 orbital has sizable contribution in EuRbFe 4 As 4 , in addition to the d xz , d yz and d xy orbitals that universally dominate the electronic states near ϵ F in FeSCs 10,11 . The d x 2 Ày 2 orbital strongly hybridizes with As-4p, and is present beyond ϵ F ± 1 eV range. The DOS of EuRbFe 4 As 4 in FM phase are illustrated in Fig. 2e. Similar to all FeSCs, the DOS of FeSCs near ϵ F is dominated by Fe-3d orbitals, hybridized with As-4p orbitals. The total DOS at ϵ F is 9.41 eV −1 /f.u., equivalent to electronic specific-heat coefficient γ = 21.642 mJ/(K 2 mol). We note that gðϵ F Þ of CaRbFe 4 As 4 is 9.78 eV −1 /f.u. 12 , similar to what we obtained for EuRbFe 4 As 4 . As a comparison, the experimental value of gðϵ F Þ is 60 eV −1 /f.u. 5 , indicating a large electron mass renormalization factor of ≈6.4. From spin-resolved DOS (Fig. 2e) the majority-spin channel/minority-spin channel of 4f electrons are located around 2.7 eV (10 eV) below(above) ϵ F . By integrating the 4f contribution below ϵ F , each Eu atom has a local moment of Single particle Hamiltonian and crystal field. To analyze the crystal field effect in EuRbFe 4 As 4 , we have employed the maximally projected Wannier function method 13 to fit the density functional theory (DFT) band structure. In order to capture the main features of the electronic structures of EuRbFe 4 As 4 , we take into account not only Fe-3d orbitals and As-4p orbitals, but Eu-4d and Eu-4f orbitals as well. Using these 44 orbitals not only allows us to describe the low-energy physics of this system, but also elucidates the electronic hopping process between d-p as well as d-f orbitals, which are relevant in the superconducting and magnetic properties of this compound. In Fig. 3e-h we compare the crystal field splitting between BaFe 2 As 2 , CaRbFe 4 As 4 , and undoped EuRbFe 4 As 4 and the corresponding values are listed in Table 1. In general, the electronic bands of 3d-systems strongly depend on the crystal symmetry. Under perfect tetrahedral crystal field, the five dorbitals split into two groups: triply degenerate t 2g levels and doubly degenerate e g levels. In the family of iron-based superconductors, all the members share the similar FeAs layer with tetrahedral symmetry. Due to the combined effect of anion crystal fields and surrounding cations, the hybridization between the Fed orbitals and the pnictogen-/chalcogen-p orbitals appears. Therefore, both t 2g and e g orbitals further split, leaving only degenerate d xz and d yz , as exemplified in BaFe 2 As 2 (Fig. 3e). In addition, we find that in EuRbFe 4 As 4 the energy levels of d xy and d z 2 are reversed, consistent with the observation in the band structure results. Furthermore, the exchange spin splitting in EuRbFe 4 As 4 is one order of magnitude smaller compared to the crystal field splitting, and the formation of long-range magnetic order does not affect magnitude and order of crystal splitting.
In Table 2 we show the hopping terms related to low-energy excitation for BaFe 2 As 2 , CaRbFe 4 As 4 , and EuRbFe 4 As 4 (both PM state and FM state), respectively. In the BaFe 2 As 2 compound, the As atoms above (As I ) and below (As II ) the Fe-plane are equivalent (S 4 local symmetry); thus, the hoppings between Fe-3d and As-4p orbitals are the same. However, this symmetry is broken in 1144-type compounds; thus, the hoppings between Fe-3d and As I -4p and those between Fe-3d and As II -4p are no longer identical. For all these compounds the strongest hopping comes from Fe-3d x(y)z and As-4p y(x) , followed by the hopping between Electronic band structure and density of states. a Electronic band structure of EuRbFe 4 As 4 (red line) compared with CaRbFe 4 As 4 (black line).
b Electronic band structure of BaFe 2 As 2 (blue line) in paramagnetic state in the folded Brillouin zone. The green dotted line denotes the Fermi level, which is aligned at zero. The degenerate points at X point as indicated by red circles in BaFe 2 As 2 are removed in the 1144-system. c Spin-resolved band structure of EuRbFe 4 As 4 in ferromagnetic state. The red and blue lines correspond to the two spin channels respectively. d Projected band structure of EuRbFe 4 As 4 (ferromagnetic phase) for majority-spin channel around Fermi level. The size of the red, turquoise, blue, and gray circles is proportional to the contributions from the d x(y)z , d xy , d z 2 and d x 2 Ày 2 respectively. e Total electronic density of state and projected density of state of EuRbFe 4 As 4 .The black, red, blue, and turquoise lines represent the total density of state, the projected density of state for Fe-d, As-p, and Eu-f orbitals respectively. The Fermi level is aligned at zero  15 . Therefore, the superconductivity may coexist with the FM Eu-layer.
RKKY exchange interaction. The RKKY exchange interaction was reported to be inseparably related with the magnetic order in 122-type EuFe 2 (As,P) 2 16,17 and 1144-type EuRbFe 4 As 4 5 . In the weak coupling limit, the RKKY interaction can be approximatively J RKKY $ J 2 K gðϵ F Þ 18-20 with a bare Kondo coupling J K in the ordered phase and DOSs at the Fermi level gðϵ F Þ. Due to the large local moment of Eu atoms (6.5 μ B per Eu, close to half-occupied Eu-4f orbitals), the large-U f limit should be valid for EuRbFe 4 As 4 . Following the Schrieffer−Wolff transformation 21,22 , we calculate the Kondo coupling strength by In these expressions, n is the conduction band index, |nk〉 is the nth Bloch state of conduction electrons at k, |w α 〉 is the αth local Wannier state for the conduction electrons, ϵ nk and ϵ f are the eigen-energies of the conduction channel and local f states, g n ðϵ F Þ is the DOSs at the Fermi level of the nth conduction channel, and |w f 〉 denotes the local Wannier state of the f-electrons. The averaged Kondo coupling J K is roughly −0.73 meV, slightly smaller than EuFe 2 As 2 (−1.01 meV) 20 . Considering the multiband effect and highly anisotropic Fermi surfaces, the RKKY interaction is obtained by summing up the contributions from all conduction bands J RKKY ¼ P n J 2 0n g n ðϵ F Þ. The resulting J RKKY is around 0.12 meV, or equivalent to T FM ≈ 12.5 K, which is in good agreement with the experimental value 5 . It is worth noting that the J RKKY in our calculation is averaged in all directions. To tell the difference between the inter-layer coupling J ⊥ and intra-layer coupling J || , it In 1144-type, the hoppings between Fe-3d and As-4p will be different for As I (denoted with p) and As II (denoted with p′) due to the local S 4 symmetry broken PM paramagnetic ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-019-0112-1 is necessary to employ J RKKY ðrÞ ¼ J 2 K P μν χ μν ðrÞ (where 23 . Using the Wannier orbitalbased tight-binding (TB) Hamiltonian, we have estimated J ⊥ /J || ≈ 1.0. Such nearly isotropic behavior can be understood by calculating the band-decomposed contributions from each Fermi surface sheet (Fig. 3i), which suggests more than 60% of J RKKY is mediated by the outmost hole pocket around Γ point with prominent 3D feature. This Fermi surface sheet is mainly due to the Fe-d z 2 orbitals, and a similar mechanism was also previously proposed in EuFe 2 As 2 24 . It should be emphasized that the RKKY interaction in our calculation starts from the normal state. In superconducting state, the interplay between RKKY interaction and superconductivity is more complicated. It has been illustrated by Yao et al. that the antiferromagnetic contribution from indirect spin exchange will be enhanced due to the formation of Yu−Shiba−Rusinov (YSR) bound states, provided the binding energy of YSR state is close to the middle of superconducting gap Δ 25 . In EuRbFe 4 As 4 , however, this binding energy is estimated to be close to Δ, implying the weak hybridization between YSR state and superconducting condensate. Thus, the conventional RKKY interaction will still dominate in superconducting state.
Effect of nickel doping. In order to investigate the Ni-doping effects, we have also calculated EuRbFe 1−x Ni x As 4 with x = 6.25, 12.5 and 100% under the virtual crystal approximation (VCA). The DOS of these system are illustrated in Fig. 4a. With nickel doping, the Fermi level moves to higher energy, demonstrating the main effect induced by nickel is electron-doping. Comparing 6.25% Nidoping with the undoped EuRbFe 4 As 4 , the DOS of d zx(y) , d z 2 and d x 2 Ày 2 orbitals are suppressed by 17.6, 13.7 and 12.1% respectively while the DOS of d xy is enhanced by 4.0%. Note that the T sc for the superconductivity is usually very sensitive to the DOS at Fermi level gðϵ F Þ. Therefore, the large decreased gðϵ F Þ of d zx(y) is consistent with strongly suppressed superconducting temperature upon doping.
We have also calculated the imaginary part of bare electron susceptibility (nesting function) under different doping levels ( Fig. 4b-e). For pristine EuRbFe 4 As 4 , the nesting function shows a sharp peak at M (π, 0), which signals large spin-fluctuation that considered to be related with the superconductivity [26][27][28] . This peak is quickly suppressed by Ni-doping, and is already absent with 12.5% Ni-doping, suggesting that the superconductivity can be quickly suppressed by Ni-doping. The imaginary part of the 100%-doped compound (or EuRbNi 4 As 4 ) is completely flat, meaning that the Ni-lattice is without spin-fluctuation. In fact, our DFT calculation confirms that Ni-lattice is completely nonmagnetic in EuRbNi 4 As 4 .
To evaluate the influence of doping on the crystal field splitting and the RKKY interaction, we naively calculate the 100% Nidoping situation. With all the irons substituted, the energy level of d xy orbital rises, substantially higher than d z 2 orbital and even nearly degenerate with d zx(y) orbitals (Fig. 3h). Considering the main effect of Ni-doping at small x is to shift the Fermi level upward, the averaged Kondo coupling J K and the RKKY interaction J RKKY are also calculated (Table 3) with the rigid band model (RBM). The averaged Kondo coupling J K slightly decreases upon doping within RBM while the variation in VCA is nonmonotonic. Both of these two methods, however, show the RKKY interaction is around 0.1 meV, insensitive to the Nidoping, which is in agreement with experimental observations. 7 .

Discussion
In conclusion, we have performed first-principles calculation on a typical 1144-type iron pnictide EuRbFe 4 As 4 . We analyzed the detailed electronic band structure, DOS and Fermi surface as well as the distinction of electronic properties between EuRbFe 4 As 4 , BaFe 2 As 2 , and CaRbFe 4 As 4 . The energy levels of d z 2 and d xy are  Note that in the experiment the Ni-doping is within 10% 7 . For doped system (0 < x < 1), the numbers inside (outside) the bracket were calculated using rigid band model (virtual crystal approximation), respectively reversed in EuRbFe 4 As 4 , and d z 2 orbital becomes closer to the Fermi level around Γ point; thus, the system is close to a Lifshitz transition. Upon Ni-doping, the DOS of d zx /d zy orbitals at Fermi level as well as the spin-fluctuation at (π, 0) will be substantially suppressed, and both effects are detrimental to superconductivity. Finally, the RKKY interaction between Eu-layers is little affected by Ni-doping, and is mostly mediated through the outmost hole FS pocket around the Γ point due to the d z 2 orbital.

Methods
The calculations were carried out based on DFT with Quantum ESPRESSO (QE) [29][30][31] . Throughout the calculations, the Perdew, Burke, and Ernzerhof parameterization of generalized gradient approximation to the exchange correlation functional was used 32 . The energy cutoff of plane-wave basis was chosen to be 122 Ry (1220 Ry for the augmentation charge), which was sufficient to converge the total energy to 1 meV/atom. A Γ-centered 12 × 12 × 3 Monkhorst-Pack 33 k-point mesh was chosen to sample the Brillouin zone in the calculations. For the calculations of PM phases, the Eu-4f states were considered to be fully localized core states and do not hybridize with any conduction electrons; while for the ferromagnetic state calculations, the Eu-4f states were explicitly considered as semi-core valence states, with an additional Hubbard-like effective interaction U = 7 eV [34][35][36] . In addition, for ferromagnetic state calculations, only the initial magnetic moments on Eu atoms were set to non-zero values.
We have employed the experimental crystal structure to perform all the calculations, since a full structural relaxation will yield a 5.1% shorter c and wrong Asheights. This problem was known in other FeSCs 37,38 , and was due to the fact that spin fluctuations cannot be accounted for in the static mean-field implementation of DFT 39 .
The DFT results were then fitted to a TB model Hamiltonian with maximally projected Wannier function method 13,40 . Although TB Hamiltonian constructed with five Fe-3d orbitals is sufficient to describe the low-energy excitations in FeSCs, one must include Eu-4f orbitals and other relevant ones in EuRbFe 4 As 4 in order to analyze the crystal field splitting and magnetic exchange coupling interactions. Therefore, we have employed 44 atomic orbitals including the Fe-3d, As-4p, Eu-5d, and Eu-4f orbitals in the fitting procedure.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.