Coexistence of ferromagnetism, antiferromagnetism, and superconductivity in magnetically anisotropic (Eu,La)FeAs2

Materials with exceptional magnetism and superconductivity usually conceive emergent physical phenomena. Here, we investigate the physical properties of the (Eu,La)FeAs2 system with double magnetic sublattices. The parent EuFeAs2 shows anisotropy-associated magnetic behaviors, such as Eu-related moment canting and exchange bias. Through La doping, the magnetic anisotropy is enhanced with ferromagnetism of Eu2+ realized in the overdoped region, and a special exchange bias of the superposed ferromagnetic/superconducting loop revealed in Eu0.8La0.2FeAs2. Meanwhile, the Fe-related antiferromagnetism shows unusual robustness against La doping. Theoretical calculation and 57Fe M\"ossbauer spectroscopy investigation reveal a doping-tunable dual itinerant/localized nature of the Fe-related antiferromagnetism. Coexistence of the Eu-related ferromagnetism, Fe-related robust antiferromagnetism, and superconductivity is further revealed in Eu0.8La0.2FeAs2, providing a platform for further exploration of potential applications and emergent physics. Finally, an electronic phase diagram is established for (Eu,La)FeAs2 with the whole superconducting dome adjacent to the Fe-related antiferromagnetic phase, which is of benefit for seeking underlying clues to high-temperature superconductivity.


INTRODUCTION
Magnetism is believed to play an important role in high-temperature superconducting pairing, e.g., the Fe-related antiferromagnetism (Fe-AFM) in the iron-based superconducting family [1][2][3] . The competition between superconductivity (SC) and Fe-AFM in charge-lightly-doping region has been widely revealed [4][5][6][7][8] , however, the systems with unusual phase diagrams are also worth our concern. Typically, in the 112-type (Ca,La)FeAs 2 9 , the Fe-AFM exhibits robustness and is abnormally enhanced by La doping in the overdoped region, with SC gradually suppressed 10 . Lately, a series of the homogenous (Eu,La)FeAs 2 compounds were discovered 11 . The transport and magnetic measurements suggested a structural transition (110 K), a Fe-related antiferromagnetic (Fe-AF) transition (98 K), and a Eu-AF transition (46 K) for single-crystalline EuFeAs 2 12 . A recent Mössbauer spectroscopy investigation on the polycrystalline EuFeAs 2 sample confirmed an incommensurate spin-density-wave-type (SDW-type) AFM ordering of Fe 2+ around 106 K 13 . The transport measurements on the underdoped Eu 1-x La x FeAs 2 (x = 0-0.15) suggest that the Fe-AFM exists in the studied doping region 11 . Hence, the unusual relationship between Fe-AFM and SC in (Eu,La)FeAs 2 is anticipated in a broader doping region, than that of (Ca,La)FeAs 2 with lightly-doped samples unavailable.
On the other hand, various Eu-related magnetic properties were revealed in polycrystalline EuFeAs 2 under a low magnetic field of 10 Oe, including a spin glass (SG) transition, reentrant magnetic modulation, and moment canting induced by transition metal doping in the Fe site 14,15 . The SG and the moment canting indicate a tunable competition and coexistence of the ferromagnetic and AF interactions between the Eu 2+ ions, which was proposed to mainly originate in the Ruderman-Kittel-Kasuya-Yosida (RKKY) indirect exchange.
More intriguing is that the coupling between the two magnetic sublattices (see the crystal structure 12 in Fig. 1a) would lead to anisotropic interaction between Eu 2+ and Fe 2+ in EuFeAs 2 . Magnetic systems with anisotropic interactions exhibits various magnetic properties, including sign-reversible exchange bias (EB) 16 , spin reorientation (SR) 17 , thermal magnetic hysteresis 18 , etc. The most studied EB effect is an exchange anisotropy with a shift of the magnetic hysteresis loop along the magnetic-field axis, which was first discovered in oxide-coated cobalt particles with moment compensation in the ferromagnetic/AF interface of Co/CoO 19 . Later, single-phase compounds with double magnetic sublattices have been found to exhibit EB effect due to the existence of anisotropic interactions 20,21 . One explanation is that when a net moment is induced in one of the sublattices by the anisotropic interaction, a circumstance analogous to the ferromagnetism (FM)/AFM interface generates with compensation in between. The moment compensation between FM and SDW-type AFM could also trigger EB effect in alloys or interfaces 22,23 . However, EB effect associated with SDW-type AFM in a stoichiometric compound system is rare.
(Eu,La)FeAs 2 with robust SDW-type AFM and doping-modifiable Eu-related magnetism is a suitable compound system for exploring the EB anisotropy. Furthermore, the interplays of exotic magnetism and SC have shown interesting physics and application prospects in layered or wire-like heterostructures [24][25][26][27][28][29] . Hence, the (Eu,La)FeAs 2 system is worth deeper investigation, not only for the unusual relationship between SDW and SC, but also for the underlying physics originating from the interplay between anisotropic magnetism and SC.
In this article, we first illuminate the magnetic anisotropy in the parent EuFeAs 2 . Then, the La-doping-induced magnetic evolution and the coupling between anisotropic magnetism and SC are studied. The nature of the robust Fe-AFM is discussed and examined in the superconducting state. Finally, a La-doping phase diagram on structure, magnetism, and SC is established.

RESULTS AND DISCUSSION
Magnetic anisotropy in EuFeAs 2 . The phase transitions of EuFeAs 2 are reexamined by heat capacity, high-field magnetization, and single-crystal X-ray diffraction (SXRD) analyses, detailed in Supplementary Figure 1. Based on the phase transitions, zero-field-cooling (ZFC), field-cooling (FC), and field-cooled-warming (FCW) magnetization measurements were performed on single-crystalline EuFeAs 2 under a low magnetic field of 100 Oe below 50 K.
The temperature dependent magnetization (M-T) curves, depicted in Fig. 1b are summarized in Fig. 1b with H//ab has also been conducted, while, the magnetization curve basically overlaps with the FC and ZFC curves due to the measurement error of PPMS.
In a word, the single-crystalline EuFeAs 2 shows various magnetic properties, mainly associated with the magnetic anisotropy. The EB behaviors related to SDW-type AFM in a stoichiometric compound system enrich the EB effect and the platforms for investigating the mechanism of EB anisotropy.  The anisotropic exchange between Eu 2+ and Fe 2+ is adjusted by introducing nonmagnetic La 3+ , which leads to the change of the ferromagnetic-AF competition, similar to the dilution effect in (Sm,La)FeO 3 33 ; 2) The nonmagnetic La 3+ will not participate the RKKY interaction, which results in the doubling of the interaction distance between the Eu 2+ moments beside the La 3+ ion and might change the proportion of the ferromagnetic term of the RKKY interaction.

La-doping effects in Eu
To further illuminate the La-doping effect on the exchange anisotropy, isothermal magnetization measurements were performed at different temperatures in the thermal-hysteresis interval, as shown in Fig Briefly, La doping greatly affects the competing balance between the ferromagnetic and AF interactions of the Temperature dependences of magnetization were measured for these overdoped samples, as depicted in Fig. 3b.
Eu-related ferromagnetic transition occurs in all the samples, different from the AF behavior of the compounds with x ≤ 0.15 11  in a large field interval of -6 to 6 T is enlarged in Fig. 3c.    Fig. 4d and e, respectively, where the FS nesting is gradually weakened by electron doping but always exists.
The SDW-type Fe-AFM of (Eu,La)FeAs 2 in the underdoping region can be explained by the FS nesting, despite the La-doping induced structural transformation 11,12 , as discussed in Supplementary Figure 10. The spectrum collected at 6 K (< T c ), as shown in Fig.   5a, is similar to that of the undoped EuFeAs 2 13 in the form of a broadened, asymmetric, six-line Zeeman pattern, which can be explained by the distribution of hyperfine magnetic field due to the SDW-type AF order.
To fit the spectrum of 6 K in the main text, we follow the procedure reported in Ref. 41 . In general, the hyperfine magnetic field of the spin-density-wave order can be expressed as where h 2n-1 denote the amplitudes of subsequent harmonics, q stands for the wave number of the SDW, and X denotes the relative position of the resonant nucleus along the propagation direction of the stationary SDW. The root-mean-square value of the hyperfine magnetic field ! can be obtained as which is proportional to the ordered magnetic moment !" carried by the Fe atoms. It is generally accepted that the magnetic moment is approximately proportional to the measured hyperfine magnetic field. The obtained hyperfine parameters are listed in Supplementary Table 1, and the resulting SDW shape and the corresponding hyperfine field distribution are shown in Fig. 5b and c, respectively. The magnetic moment is determined to be 0.84(1) ! by using the s ame proportionality constant of a = 63 kOe ! -1 as was used for the calculation of the magnetic moment of the parent compound EuFeAs 2 13 . The ordered magnetic moment is much larger than those of other iron-based superconducting samples with suppressed Fe-AFM [42][43][44] . Another interesting result is that the SDW shape is almost rectangular rather than quasi-triangular as found in most iron-based superconductors 13,42,43 . The rectangular SDW shape at a low temperature has been observed in some of the parent compounds with relatively large magnetic moment and less pronounced itinerant character 41 . Besides, the ratio of the third and first amplitudes h 3 /h 1 ~ 0.36, which outclasses the range of 10 -3 -10 -2 expected from the itinerant-electron model [45][46][47] , implies that the Fe-AFM in Eu 0.8 La 0.2 FeAs 2 cannot be accurately described merely by the itinerant picture. All these unusual Mössbauer spectroscopy results put our sample closer to the localized-AFM nature with the itinerant character of the magnetic order less prominent. Also, the magnetic moment is enhanced from that of the parent EuFeAs 2 13 , which is in agreement with the increasing prominence of the local superexchange interaction suggested by the DFT calculation.
On the other hand, the Mössbauer spectrum obtained at 6 K manifests a microscopic coexistence of the Fe-AFM   Under the dome, the superconducting order coexists with the Fe-and Eu-magnetic orders.
In summary, we systematically investigated the electrical and magnetic properties of the 112-type (Eu,La)FeAs 2 .
Due to the magnetic anisotropy, various exceptional magnetic phenomena are discovered in the parent EuFeAs 2 . Nonmagnetic

Sample preparation
Single crystals of EuFeAs 2 and Eu 0.79 La 0.21 FeAs 2 were grown from a CsCl flux. A mixture of elementary Eu/La, Fe, and As in ratio of 1 : 1 : 4 (or 2 : 1 : 6) with 10-to 20-fold of dehydrated CsCl was sealed in a vacuum quartz tube, heated slowly to 800 °C, and held for 2 weeks before quenching. Polycrystalline Eu 1-x La x FeAs 2 (x = 0.2, 0.25, and 0.3) samples were synthesized following our previous work 11 . The reaction temperature in the last step was modulated to 850 °C to improve the La-doping homogeneity in the overdoped samples.

Phase and Property characterization
The SXRD experiments were carried out on a Single-crystal X-ray Diffractometer (Bruker). The PXRD patterns were collected on a Powder X-ray Diffractometer (PAN-analytical). The EDXS experiment was performed using a Scanning Electron Microscope (SEM) equipped with an Energy Dispersive X-ray Spectrometer (ZEISS). Electrical transport, heat capacity, and magnetic measurements were conducted on a Physical Property Measurement System (PPMS) and a Magnetic Property Measurement System (MPMS) (Quantum Design).
Transmission 57 Fe Mössbauer spectra were recorded by using a conventional spectrometer working in constant acceleration mode. A 50 mCi of 57 Co embedded in a Rh matrix moving at room temperature was used as the γ-ray source. The absorber was prepared with a surface density of ~8 mg cm -2 natural iron. The drive velocity was calibrated with sodium nitroprusside at room temperature and all the isomer shifts quoted in this work are relative to that of the α-Fe.

Theoretical calculations
Theoretical calculations were performed using the DFT as implemented in the Vienna ab initio simulation package (VASP) code [48][49][50] . The generalized-gradient approximation (GGA) for the exchange correlation functional was used.
The cutoff energy was set to be 400 eV for expanding the wave functions into plane-wave basis. In the calculation, the BZ was sampled in the k space within Monkhorst-Pack scheme 51 .

DATA AVAILABILITY
The data that support the findings of this study are available from the corresponding authors upon reasonable request.
Heat capacity measurement from 150 K to 2 K is performed on single crystalline EuFeAs 2 to confirm the phase transitions, as depicted in Supplementary Figure 1a. Two associated slight transitions around 106 K and 93 K, and one significant transition around 45 K are observed, which agrees with the previous electrical transport and magnetization results 1,2 .
The significant transition around 45 K is in consistent with the canceling of the large moment of Eu 2+ from the magnetic measurement. Focusing on the two associated transitions around 100 K, we conducted an SXRD study on EuFeAs 2 at different temperatures. The crystal structures for different temperatures share the same space group Im2m. We carefully checked the configuration of the FeAs layer, and found that a distortion occurs between 120 K and 100 K. The lengths of the Fe-As bonds in the bc plane undergoes a clear change, as displayed in Supplementary Figure 1b. Thus, the phase transition around 106 K is associated with the FeAs-layer-related structural distortion.
The temperature dependence of magnetization was measured under a magnetic field of 5 T parallel to the ab plane. The reciprocal of the magnetization is scrutinized around 100 K, see Supplementary Figure 1c. According to the Curie-Weiss law, the 1/M-T curve should be linear with a slope related to the effective moment in the paramagnetic region. We found that the experimental data deviate from the linear fitting at low temperatures.
After subtracting the fit, the data exhibit clearly a reduce of slope at 93 K, see Supplementary Figure 1d Our previous studies revealed a monoclinic structure with space groups of P2 1 /m for Eu 0.87 La 0.13 FeAs 2 by SXRD analysis, and the PXRD patterns of the polycrystalline Eu 1-x La x FeAs 2 (x = 0-0.15) were indexed based on this monoclinic structure 1 . Later, we found that the structure of the undoped EuFeAs 2 belongs to space group To support the procedure of using the FS nesting to explain the evolution of the Fe-AFM with the existence of the La-doping-induced structural transformation, we calculated the band structures of undoped EuFeAs 2 using the