Coexisting spin resonance and long-range magnetic order of Eu in EuRbFe$_4$As$_4$

TL;DR

This study reveals the coexistence of spin resonance linked to superconductivity and long-range magnetic order of Eu in EuRbFe$_4$As$_4$, supported by neutron scattering and theoretical calculations, indicating weak coupling between these phenomena.

Contribution

It provides the first detailed investigation of magnetic excitations and structure in EuRbFe$_4$As$_4$, demonstrating coexistence of superconductivity and Eu magnetic order with specific magnetic and resonance characteristics.

Findings

Neutron spin resonance observed at specific wave vectors below T_c

Eu magnetic order characterized by a propagation vector (0,0,0.25)

Weak coupling between superconductivity and Eu magnetic order

Abstract

Magnetic excitations and magnetic structure of EuRbFe$_4$As$_4$ were investigated by inelastic neutron scattering (INS), neutron diffraction, and random phase approximation (RPA) calculations. Below the superconducting transition temperature $T_\text{c}=36.5$~K, the INS spectra exhibit the neutron spin resonances at $Q_\text{res}=1.27(2)$~$\text{\AA}^{-1}$ and $1.79(3)$~$\text{\AA}^{-1}$. They correspond to the $\mathbf{Q}=(0.5,0.5,1)$ and $(0.5,0.5,3)$ nesting wave vectors, showing three dimensional nature of the band structure. The characteristic energy of the neutron spin resonance is $E_\text{res}=17.7(3)$~meV corresponding to $5.7(1)k_\text{B}T_\text{c}$. Observation of the neutron spin resonance mode and our RPA calculations in conjunction with the recent optical conductivity measurements are indicative of the $s_\pm$ superconducting pairing symmetry in EuRbFe$_4$As$_4$. In addition to the neutron spin resonance mode, upon decreasing temperature below the magnetic transition temperature $T_\text{N}=15$~K, the spin wave excitation originating in the long-range magnetic order of the Eu sublattice was observed in the low-energy inelastic channel. Single-crystal neutron diffraction measurements demonstrate that the magnetic propagation vector of the Eu sublattice is $\mathbf{k}=(0, 0, 0.25)$, representing the three-dimensional antiferromagnetic order. Linear spin wave calculations assuming the obtained magnetic structure with the intra- and inter-plane nearest neighbor exchange couplings of $J_1/k_\text{B}=-1.31$~K and $J_c/k_\text{B}=0.08$~K can reproduce quantitatively the observed spin wave excitation. Our results show that superconductivity and long-range magnetic order of Eu coexist in EuRbFe$_4$As$_4$ whereas the coupling between them is rather weak.

Full text

I Introduction

Recently, new family of iron-based superconductors $AeA$Fe4As4 with $Ae=$ Ca, Sr, Eu and $A=$ K, Rb, Cs (termed “1144”) were reported by Iyo et al CaKFe4As4_powder . EuRbFe4As4 EuRbFe4As4_1 ; EuRbFe4As4_2 can be regarded as an intergrowth of EuFe2As2 and RbFe2As2, both of which belong to so-called “122” family. Unlike solid solutions such as (Eu1-xNax)Fe2As2 EuNaFe2As2_1 ; EuNaFe2As2_2 and (Eu1-xLax)Fe2As2 EuLaFe2As2_1 , in EuRbFe4As4, Eu2+ and Rb+ ions occupy crystallographically inequivalent sites alternately between the FeAs layers along the $c$ axis because of the large difference between their ionic radii EuRbFe4As4_4 . Consequently, EuRbFe4As4 crystallizes in the tetragonal symmetry ($P4/mmm$), and no structural phase transition takes place down to 2 K EuRbFe4As4_4 . The unique structure of the 1144 compounds is that there are two different kinds of As sites in contrast to the single As site in the 122 compounds.

As is well established, the non-superconducting parent compound EuFe2As2 undergoes a spin-density-wave (SDW) order in the Fe sublattice accompanied by a tetragonal-to-orthorhombic structural phase transition below 190 K EuFe2As2_1 ; EuFe2As2_2 . Hole doping by substitution of Rb+ for Eu2+ in EuFe2As2 gives rise to suppression of the SDW order in EuRbFe4As4, in favor of the occurrence of superconductivity below $T_{\text{c}}=36.5$ K EuRbFe4As4_1 ; EuRbFe4As4_2 . Bulk property measurements indicated that EuRbFe4As4 shows behaviors similar to optimally doped 122 superconductors EuRbFe4As4_1 ; EuRbFe4As4_7 . The normal state of the FeAs layer in EuRbFe4As4 is paramagnetic metal EuRbFe4As4_5 . On the other hand, Curie-Weiss fitting on the magnetic susceptibility measurement suggests that Eu2+ ion has a local magnetic moment with $S=7/2$ and the Curie-Weiss temperature $\theta$ is positive, suggesting that the dominant magnetic interaction between Eu2+ ions is ferromagnetic EuRbFe4As4_2 ; EuRbFe4As4_4 ; EuRbFe4As4_7 ; EuRbFe4As4_3 . Upon further cooling below $T_{\text{c}}$, the Eu sublattice shows a long-range magnetic order at $T_{\text{N}}=15$ K EuRbFe4As4_2 ; EuRbFe4As4_7 . The Eu sublattice of parent EuFe2As2 undergoes a so-called $A$-type antiferromagnetic order below $T_{\text{N}}=19$ K EuFe2As2_6 ; EuFe2As2_7 , and smaller $T_{\text{N}}$ in EuRbFe4As4 is due to the longer interlayer distance between the Eu layers compared to EuFe2As2. In fact, upon increasing hydrostatic pressure, the lattice constant $c$ (or interlayer distance of the Eu sublattice) decreases while $T_{\text{N}}$ increases in EuRbFe4As4 EuRbFe4As4_6 ; EuRbFe4As4_11 . It is noticeable that no reentrant superconductivity was observed in EuRbFe4As4 EuRbFe4As4_1 ; EuRbFe4As4_2 unlike observed in Eu(Fe0.89Co0.11)2As2 EuFeCo2As2_1 , Eu(Fe0.86Ir0.14)2As2 EuFeIr2As2_1 , Eu(Fe0.75Ru0.25)2As2 EuFeRu2As2_1 , Eu(Fe0.93Rh0.07)2As2 EuFeRh2As2_1 , and EuFe2(As0.7P0.3)2 EuFe2AsP2_1 .

Although $s$-wave superconductivity with a complete gap throughout the Fermi surface is reported by the optical conductivity measurements EuRbFe4As4_8 , the microscopic mechanism underlying the superconductivity in EuRbFe4As4 has not been completely elucidated yet. In particular, due to the large neutron absorption cross section of Eu, the neutron spin resonance Review_2 ; Review_3 ; Review_4 ; Resonance1 ; Resonance2 ; Resonance3 ; Resonance4 ; Resonance5 ; Resonance6 ; Resonance7 , which provides a strong constraint on the relative signs of the superconducting gaps on different portions of the Fermi surfaces spm_1 ; spm_2 ; cuprate_1 ; cuprate_2 ; cuprate_3 ; cuprate_4 ; cuprate_5 ; HeavyFermion_1 , has not been reported. In the meanwhile, the detailed magnetic structure of the Eu sublattice in EuRbFe4As4 is not determined yet using a microscopic technique such as neutron diffraction. Investigation of the magnetic structure by means of the microscopic technique is necessary because various types of the magnetic structures of the Eu sublattice are reported in the doped EuFe2As2 compounds EuFeCo2As2_5 ; EuFeIr2As2_3 ; EuFe2AsP2_11 ; EuFe2As2_6 ; EuFe2As2_7 . Moreover, there is no experimental report on the spin wave excitation from the long-range order of Eu2+ in EuRbFe4As4.

So far, there is no experimental evidence about the sign-changing superconductivity in Eu-containing iron-based superconductors, which can demonstrate that Eu-containing iron-based superconductors also belong to the same paradigm as other iron-based superconductors in spite of the local magnetic moments of Eu2+ between FeAs layers. Moreover, there is a longstanding issue about the possible interplay between the long-range magnetic order of Eu2+ local moments and the superconductivity of the FeAs layer in the Eu-containing iron-based superconductors EuFeIr2As2_1 ; EuFeIr2As2_3 ; coupling_1 ; coupling_2 ; coupling_3 ; coupling_4 ; coupling_5 ; coupling_6 ; coupling_7 ; coupling_8 ; coupling_9 ; coupling_10 . It is, therefore, important to report such coupling in the case of EuRbFe4As4 since EuRbFe4As4 contains no substitutional disorder. Measurements of magnetic excitations from the FeAs and Eu layers in $(Q,\hbar\omega)$ space in EuRbFe4As4 will give us the key ingredient to discuss the possible cooperative behaviors. In this paper, we investigated the neutron spin resonance by combination of inelastic neutron scattering (INS) measurements and random phase approximation (RPA) calculations. In addition, we also measured spin wave excitation and single-crystal neutron diffraction patterns to determine the spin Hamiltonian and the magnetic structure of the Eu sublattice. Finally, we will discuss the effect of the magnetic long-range order on the superconductivity in EuRbFe4As4.

II Experimental details and theoretical methods

The details of the synthesis of EuRbFe4As4 is described in elesewhere CaKFe4As4_powder ; EuRbFe4As4_1 , and a magnetic susceptibility measurement exhibits $T_{\text{c}}=36.5$ K and $T_{\text{N}}=15$ K in the current polycrystalline sample as shown in Supplemental Material Supplement . Polycrystalline EuRbFe4As4 with a mass of $\sim 4$ g was used for our INS measurements. We performed high- and low-energy INS measurements using two Fermi-chopper spectrometers ARCS at Spallation Neutron Source (SNS) ARCS and 4SEASONS at Japan Proton Accelerator Research Complex (J-PARC) 4SEASONS_1 ; 4SEASONS_2 . Neutron incident energies $E_{\text{i}}=70.0$ meV were used at ARCS while $E_{\text{i}}=8.5$ meV at 4SEASONS, and energy resolutions (full width at half maximum) at the elastic channel were 3.2 and 0.36 meV, respectively. INS data were analyzed by the software suites Utsusemi 4SEASONS_3 . To reduce the effect of neutron absorption by Eu, thin-plate cans whose effective thickness were 1 mm (ARCS) and 0.5 mm (4SEASONS) were used. For the absolute intensity scale in the present INS measurements, incoherent scattering from EuRbFe4As4 was used after correction for neutron absorption AbsoluteNormalization . Single-crystal neutron diffraction measurements on EuRbFe4As4 were also carried out using the time-of-flight (TOF) single-crystal neutron diffractometer SENJU SENJU in J-PARC. A single crystal with dimension about $1\times 1\times 0.1$ mm3 was grown for neutron diffraction measurements, and the obtained lattice constants at 20.1 K are $a=3.85923(2)$ and $c=13.18563(7)~{}\text{\AA}$ for the space group $P4/mmm$.

The first-principles calculation package employed is VASPKresse1 ; Kresse2 , which adopts the PAW methodBlochl1 ; Blochl2 and the GGA exchange-correlation energyPerdew . The number of ${\bf k}$ points is taken as $16\times 16\times 5$, and self-consistent loops are repeated until the energy deviation becomes less than $10^{-6}$ eV, with the cutoff energy being 500 eV. The density functional theory often fails to handle $f$ orbitals correctly due to self-interaction error of localized $f$-electrons. To avoid this failure, localized 4$f$ electrons in Eu atoms are treated as core electrons in our calculations. In the RPA calculations, for the effective model, maximally localized Wannier functions from twenty Fe $3d$ bands are constructedMostofi . We employ the orbital-interaction coefficient $U_{s}=aU^{rs}_{qt,M}$ obtained by first-principles calculation for the 122 systemMiyake and we put $a=1.7$. The number of ${\bf k}$ point is $96\times 96\times 8$ and the smearing factor is $\eta=\Delta_{0}/8$ where $\Delta_{0}$ denote superconducting gap. We take $\Delta_{0}$ ($-\Delta_{0}$) on the hole (electron) Fermi surfaces around $\Gamma$ ($M$) point. We set $\Delta_{0}=0.1$ eV to avoid numerical difficulty. We introduce a Gaussian cutoff for the gap $\Delta^{\nu}_{k}=\Delta^{\nu}\exp[-\epsilon_{k}^{\nu}/(4\Delta_{0})]$ with the normal-state energy dispersion $\epsilon_{k}^{\nu}$ on the $\nu$-th band NagaiKuroki .

Linear spin wave (LSW) calculations were performed using the SpinW software SpinW . The squared magnetic form factor of Eu2+ ion InternationalTalbes and $\hbar\omega$-dependent energy resolution for $E_{\text{i}}=8.5$ meV at 4SEASONS 4SEASONS_4 were included in the LSW calculations.

III Results and Discussion

Figure 1(a) depicts the neutron scattering intensity ($I$) map from EuRbFe4As4 as a function of momentum ($Q$) and energy $(\hbar\omega)$ transfers. The data at 50 K was subtracted from that at 4 K. There is a well localized excitation in both $Q$ and $\hbar\omega$: $1.1\leq Q\leq 1.3~{}\text{\AA}^{-1}$ and $13\leq\hbar\omega\leq 20~{}\text{meV}$ which are comparable to the neutron spin resonance in isostructural CaKFe4As4 ($1.0\leq Q\leq 1.4~{}\text{\AA}^{-1}$ and $9\leq\hbar\omega\leq 16~{}\text{meV}$) CaKFe4As4_Neutron1 . Since the localized excitation increases in intensity at lower temperature, the observed excitation is magnetic in origin. To further classify the localized magnetic excitation in EuRbFe4As4, $Q$ and $\hbar\omega$ dependences of the neutron scattering intensities [$I(Q)$ and $I(\hbar\omega)$] are investigated.

The temperature difference of $I(Q)$ between 4 and 50 K is shown in Fig. 1(b). $I(Q)$ is averaged over the energy-transfer range of $15\leq\hbar\omega\leq 20$ meV. A Gaussian function was fitted to the $Q$ cut at 4 K, yielding the peak position of $Q=1.27(2)$ $\text{\AA}^{-1}$. As in CaKFe4As4 CaKFe4As4_Neutron1 ; CaKFe4As4_Neutron2 , the peak position in $Q$ is close to both the $\mathbf{Q}=(0.5,0.5)$ antiferromagnetic wave vector ($1.14~{}\text{\AA}^{-1}$) and the in-plane propagation vector in the parent compound EuFe2As2 ($1.13$ $\text{\AA}^{-1}$) EuFe2As2_2 . As discussed later, the $(0.5,0.5)$ antiferromagnetic vector is in good condition for the nesting between the electron and hole pockets at the Fermi surface [Fig. 2(b)] via spin fluctuations spm_1 ; spm_2 as other iron pnictide systems. Thus, the observed $Q$ dependence indicates that the dominant magnetic fluctuation of the FeAs layers in EuRbFe4As4 originates in the $(0.5,0.5)$ nesting. There is nonnegligible signal at higher $Q$, and the Gaussian fitting yields the peak center $Q=1.79(3)$ $\text{\AA}^{-1}$. Recent single-crystal INS measurements on CaKFe4As4 reported the strong $L$ dependences of the neutron spin resonance modes, representing three dimensional nature of the band structure CaKFe4As4_Neutron2 . The low-energy spin resonance modes in CaKFe4As4 shows the maximum intensity at $\mathbf{Q}=(0.5,0.5,L)$ with odd $L$. In the case of EuRbFe4As4, $(0.5,0.5,1)$ and $(0.5,0.5,3)$ correspond to 1.25 and 1.84 $\text{\AA}^{-1}$, and the $L$ dependence of the neutron spin resonance, therefore, naturally explains both the first and second peaks observed in the $Q$ cut of EuRbFe4As4.

The temperature difference of $I(\hbar\omega)$ between 4 and 50 K is plotted in Fig. 1(c). $I(\hbar\omega)$ is averaged over $1.1\leq Q\leq 1.4$ $\text{\AA}^{-1}$. The intensity at 4 K is enhanced over the energy range of $13\leq\hbar\omega\leq 23$ meV, and the spectral-weight gain is transferred by the depletion at lower energies ($\hbar\omega\leq 13$ meV). These are the typical behaviors of the neutron spin resonance observed in many iron-based superconductors CaKFe4As4_Neutron1 ; 1111_2 ; Ba122_4 ; Ba122_1 ; CaKFe4As4_Neutron2 . In addition, the scattering intensity of the observed localized mode in EuRbFe4As4 is almost the same as that of the neutron spin resonance in CaKFe4As4 CaKFe4As4_Neutron1 [see the solid line in Fig. 1(b)]. Therefore, the observed localized magnetic excitation can be assigned to be the neutron spin resonance mode. By fitting a Gaussian function to $I(\hbar\omega)$ at 4 K using the data between 13 and 24 meV, the characteristic energy of the neutron spin resonance is estimated to be $E_{\text{res}}=17.7(3)$ meV which is equivalent to $5.7(1)k_{\text{B}}T_{\text{c}}$, in agreement with the quantitative relationship between $E_{\text{res}}$ and $T_{\text{c}}$ widely reported in iron-based superconductors Resonance1 ; Resonance2 ; Resonance3 ; CaKFe4As4_Neutron1 ; 1111_2 ; 111_1 ; Ba122_4 ; 1111_3 ; Ba122_1 ; CaKFe4As4_Neutron2 ; LiODFeSe_1 . The neutron spin resonance in EuRbFe4As4 is also studied by RPA calculations as in the following text.

We calculate the band structure and the Fermi surfaces of EuRbFe4As4 as shown in Figs. 2(a) and 2(b). There are six two-dimensional cylindrical and one three-dimensional small hole Fermi surfaces around $\Gamma$ point and four cylindrical electron Fermi surfaces around $M$ point, which are similar to those of CaKFe4As4 CaKFe4As4_Neutron1 . We calculate the dynamical spin susceptibility $\chi^{\prime\prime}(\mathbf{Q},\hbar\omega)$ of EuRbFe4As4 on the basis of multiorbital RPANagaiKuroki with the use of the effective twenty-orbital three-dimensional tight-binding model at $T=0$. Figure 2(c) shows $\eta-\hbar\omega$ dependence of the dynamical spin susceptibility of EuRbFe4As4. Here, $\eta$ is the parameter which indicates $\mathbf{Q}=(\eta,\eta,0)$. The resonance peak due to the $s_{\pm}$-wave pairing states occurs around $\eta=0.5$ or $\mathbf{Q}=(0.5,0.5,0)$, which is similar to those in other iron-based superconductors. The broadness of the peaks in the region $0.4<\eta<0.6$ originates from the fact that there are many Fermi surfaces with different sizes. As mentioned above, the optical conductivity measurements EuRbFe4As4_8 reported the $s$-wave superconductivity in EuRbFe4As4. INS measurements and RPA calculations indicate the sign-changing superconducting gaps on different portions of the Fermi surfaces connected by the $(0.5,0.5)$ nesting vector INStheory_1 ; INStheory_2 . Combined with these results, we thus conclude that $s_{\pm}$ superconducting pairing symmetry is realized in EuRbFe4As4 as in other iron based superconductors.

In addition to the spin resonance excitation, EuRbFe4As4 shows strong dispersive signals in the low-energy inelastic channel ($\hbar\omega\leq 5$ meV) at 4 K as we can see in Fig. 1(a). To investigate in detail the excitation, low-energy $I(Q,\hbar\omega)$ maps at $T=10.5$ and 25.0 K are presented respectively in Figs. 3(a) and 3(b). At 25.0 K ($>T_{\text{N}}$), a quasielastic excitation was observed [Fig. 3(b)]. This is a common feature of the paramagnetic state of correlated localized spins Ferromagnetic1 . Indeed, the signature of the short-range correlation above $T_{\text{N}}$ is also reported by the heat capacity measurement EuRbFe4As4_4 ; EuRbFe4As4_7 . In contrast, spin wave dispersion develops at 10.5 K ($<T_{\text{N}}$) due to the long-range magnetic order of the Eu sublattice [Fig. 3(a)]. The magnetic excitation arises from $Q=0$ (or close to $Q=0$), indicating that the dominant magnetic interaction and the corresponding magnetic structure are ferromagnetic as in conventional ferromagnetic spin systems Ferromagnetic1 . To determine the peak position of the spin dispersion relation at low $Q$ and $\hbar\omega$, $I(Q)$ cuts at several energy windows are plotted in Supplemental Material Supplement . The Warren’s function Warren was fitted to each $Q$ cut, and the obtained powder-averaged dispersion relation whose shape is sinusoidal is plotted in Fig. 3(d).

To understand the spin-wave excitations from the Eu sublattice, the magnetic structure of Eu should be determined. We now discuss the magnetic structure of the Eu sublattice in EuRbFe4As4. Easy-plane magnetic anisotropy of the Eu2+ moments in EuRbFe4As4 is reported by the single-crystal magnetic susceptibility EuRbFe4As4_7 and heat capacity EuRbFe4As4_9 measurements as in EuFe2As2 EuFe2As2_8 ; EuFe2As2_9 . Furthermore, the recent Mössbauer spectroscopy study on EuRbFe4As4 also reported that the Eu moments are ferromagnetically aligned within the $ab$ plane EuRbFe4As4_4 . Therefore, the remaining issue on the determination of the magnetic structure of the Eu sublattice in EuRbFe4As4 is how the two-dimensional in-plane ferromagnetic layers stack along the $c$ axis. Magnetic Bragg reflections along the $(00L)$ direction give crucial information to address the issue. As such, single-crystal neutron diffraction measurements along $(00L)$ below and above $T_{\text{N}}$ were performed. As shown in Fig. 4(a), at 4.3 K, several magnetic Bragg peaks were observed as satellite peaks beside the nuclear Bragg peaks with integer $L$, and the temperature dependence of the neutron scattering intensity of the magnetic Bragg peak at $\mathbf{Q}=(0,0,1.25)$ clearly represents the evolution of the long-rang magnetic order below $T_{\text{N}}=15$ K [Fig. 4(b)]. Magnetic Bragg reflections appear at $L=0.75$, 1.25, 1.75, 2.25, 2.75, and 3.25, yielding the magnetic propagation vector $\mathbf{k}=(0,0,0.25)$. As described by the solid line in Fig. 4(a), the neutron diffraction pattern along $(00L)$ at 4.3 K is well fitted by considering $\mathbf{k}=(0,0,0.25)$; the amplitude and width of the Gaussian function for each peak as well as background are varied. It should be noted that the neutron diffraction results along $(00L)$ can only determine the relative angle between the adjacent Eu layers but not the in-plane direction of the Eu moments. For determination of the in-plane anisotropy in EuRbFe4As4, the magnetization curves along $\langle 100\rangle$ and $\langle 110\rangle$ are measured at 2 K as shown in Supplemental Material Supplement . The $\langle 110\rangle$ direction seems to be the easy axis, and the easy axis in EuRbFe4As4 is the same as that in EuFe2As2 EuFe2As2_6 ; EuFe2As2_7 . The obtained magnetic structure is illustrated in Fig. 4(c). Three-dimensional antiferromagnetic order of the Eu sublattice can be seen, and the relative angle between Eu spins belonging to the nearest-neighbor layers is $90^{\circ}$.

Comparison between semi-classical LSW calculation and spin wave excitation provides us abundant information about the spin Hamiltonian Lovesey . Here, we consider the following spin Hamiltonian for EuRbFe4As4: $\mathcal{H}=J_{1}\sum_{i,j}\mathbf{S}_{i}\cdot\mathbf{S}_{j}+J_{c}\sum_{i,j}\mathbf{S}_{i}\cdot\mathbf{S}_{j}$ where $\mathbf{S}$ is the spin operator of Eu2+ ions ($S=7/2$). $J_{1}$ and $J_{c}$ are the intra- and inter-plane nearest neighbor exchange coupling constants between Eu2+ ions [Fig. 4(c)]. The magnetic structure determined by our single-crystal neutron diffraction measurements illustrated in Fig. 4(c) is considered. Since $J_{c}$ is expected to be much smaller than $J_{1}$, we used the fixed value for $J_{c}$ (0.08 K) determined by the single crystal magnetization measurements EuRbFe4As4_7 . Fitting calculated powder-averaged dispersion relation to the experimental result as shown in Fig. 3(d) yields the intra-plane ferromagnetic exchange constant $J_{1}/k_{\text{B}}=-1.31(1)$ K. Similar intra-plane exchange coupling ($J_{1}/k_{\text{B}}\sim-1$ K) was reported in EuFe2As2 based on the local spin density approximation with the Coulomb repulsion (LSDA+$U$) EuFe2As2_2 . Calculated LSW intensity map in Fig. 3(c) reproduces the experiment [Fig. 3(a)] well; we mention that the LSW spectra based on $J_{c}=0.08$ K and 0 are almost the same as shown in Figs. 3(c) and Supplemental Material Supplement . Mean field approximation theory can estimate the Curie-Weiss temperature by the equation $\theta=-(z_{1}J_{1}+z_{c}J_{c})S(S+1)/3k_{\text{B}}$. Here, the number of the nearest neighbor bonds $z_{1}$ and $z_{c}$ are 4 and 2 [Fig. 4(c)]. By using $J_{1}$ obtained by our LSW analysis and $J_{c}$ EuRbFe4As4_7 , we estimate $\theta=26.7$ K in good agreement with the reported value (23.6 K) estimated by the susceptibility measurements EuRbFe4As4_2 . This result further corroborates the validity of our LSW analysis. Important to be noted is that the further distance interaction along $c$ is required to account for the obtained magnetic structure [Fig. 4(c)] which is not the conventional $A$-type antiferromagnetic order. Since the further distance interaction along $c$ is expected to be very small, it is difficult to determine such interaction unless measuring a single crystal, which is left for future work.

Finally, we discuss the coupling between the long-range magnetic order of the Eu sublattice and the superconductivity of the FeAs layers in EuRbFe4As4. $Q$ and $\hbar\omega$ dependences of the neutron spin resonance below and above $T_{\text{N}}$ are compared in Figs. 1(b) and 1(c). Onset of the long-range order of the Eu sublattice does not induce any noticeable change in both $Q$ and $\hbar\omega$ dependences of the neutron spin resonance mode within the accuracy of our INS measurements. It should be noted that $T_{\text{c}}$ in EuRbFe4As4 is the highest among the 1144 systems although other 1144 compounds contain no local magnetic moment CaKFe4As4_powder ; EuRbFe4As4_1 ; EuRbFe4As4_2 . EuRbFe4As4 do not exhibit the reentrant superconductivity below $T_{\text{N}}$ EuRbFe4As4_1 ; EuRbFe4As4_2 as mentioned above. In Ni-doped EuRb(Fe1-xNix)4As4, the magnetic transition temperature of the Eu sublattice is independent of the Ni content $x$ although $T_{\text{c}}$ decreases rapidly EuRbFe4As4_3 . Moreover, substitution of nonmagnetic Ca decreases $T_{\text{N}}$ while $T_{\text{c}}$ is independent in the (Eu1-xCa$x$)RbFe4As4 system EuRbFe4As4_12 . Therefore, we conclude that the long-range magnetic order in the Eu sublattice plays a minor role in the superconductivity of the FeAs layers in EuRbFe4As4. This may be reasonable because the Eu-$4f$ bands locate $2.7$ eV below the Fermi energy in EuRbFe4As4 EuRbFe4As4_10 as in doped EuFe2As2 systems EuFeIr2As2_1 ; EuFeIr2As2_3 ; EuFe2As2_2 ; EuKFe2As2_2 ; EuFe2AsP2_7 .

IV Conclusion

In this work, we investigate both the static and dynamic magnetism in EuRbFe4As4. On one hand, the spin resonance studied by INS measurements and PRA calculations in conjunction with the optical conductivity measurements EuRbFe4As4_8 presents the strong evidence in favor of the sign-reversed $s_{\pm}$ superconductivity. Our results show that EuRbFe4As4 belongs to the same paradigm carrying $s_{\pm}$ superconductivity as other iron pnictide systems although there is a static field from the long-range order of Eu2+. On the other hand, neutron diffraction measurements and LSW analysis determined the magnetic structure and the spin Hamiltonian of the Eu sublattice. No signature of coupling between the long-range magnetic order of the Eu sublattice and the superconductivity of the FeAs layers in EuRbFe4As4 was observed in the $(Q,\hbar\omega)$ region probed in the current measurements. To our best knowledge, this is the first INS study on Eu-containing iron-based superconductors for investigating the superconducting pairing symmetry and possible interplay between superconductivity and long-range magnetic order of Eu.

Acknowledgements

We thank Ryoichi Kajimoto for helpful discussion. Magnetic susceptibility measurements were performed at the CROSS user laboratories. This research at SNS was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, U.S. Department of Energy. Travel expenses for the ARCS experiment were provided by the General User Program for Neutron Scattering Experiments, Institute for Solid State Physics, The University of Tokyo (GPTAS:16913), at JRR-3, Japan Atomic Energy Agency (JAEA), Tokai, Japan. The experiments at 4SEASONS and SENJU in J-PARC were conducted under the user program with the proposal numbers 2017A0005 and 2018B0008, respectively. The calculations were performed by the supercomputing systems SGI ICE X at JAEA. The present work was partially supported by JSPS KAKENHI Grant Numbers JP15K17712, JP17K14349, and JP18K03552, and the Cooperative Research Program of “Network Joint Research Center for Materials and Devices” (20181072). ADC was partiality supported by the U.S. DOE, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division. DK and ME were supported by the Austrian Science Fund (FWF) I2814-N36.