Superconductivity-driven ferromagnetism and spin manipulation using vortices in the magnetic superconductor EuRbFe₄As₄

Significance

The interplay between superconductivity and magnetism creates numerous exotic physical phenomena, while the two phenomena rarely coexist in one material because they compete with each other. The recently discovered iron-based magnetic superconductor EuRbFe₄As₄ shows high-critical-temperature superconductivity and helimagnetic order at the ground state. We revealed that ferromagnetic alignment of Eu²⁺ moments is induced by superconducting vortices in EuRbFe₄As₄ by combining neutron diffraction and magnetization experiments. The direction of the Eu²⁺ moments is dominated by the distribution of pinned vortices based on the critical state model, highlighting a unique interplay between magnetism and superconductivity. The capability of manipulating the spin texture by controlling the direction of superconducting vortices paves the way to novel devices using magnetic superconductors.

Abstract

Magnetic superconductors are specific materials exhibiting two antagonistic phenomena, superconductivity and magnetism, whose mutual interaction induces various emergent phenomena, such as the reentrant superconducting transition associated with the suppression of superconductivity around the magnetic transition temperature (T_m), highlighting the impact of magnetism on superconductivity. In this study, we report the experimental observation of the ferromagnetic order induced by superconducting vortices in the high-critical-temperature (high-T_c) magnetic superconductor EuRbFe₄As₄. Although the ground state of the Eu²⁺ moments in EuRbFe₄As₄ is helimagnetism below T_m, neutron diffraction and magnetization experiments show a ferromagnetic hysteresis of the Eu²⁺ spin alignment. We demonstrate that the direction of the Eu²⁺ moments is dominated by the distribution of pinned vortices based on the critical state model. Moreover, we demonstrate the manipulation of spin texture by controlling the direction of superconducting vortices, which can help realize spin manipulation devices using magnetic superconductors.

The coexistence of superconductivity and magnetism has been a long-standing issue in the field of superconductivity due to the antagonistic nature of these two ordered states. Therefore, magnetic superconductors, exhibiting both behaviors simultaneously, have played an important role in the study of their interaction. Various novel phenomena have been reported in magnetic superconductors containing rare earth elements (R) such as Ho and Er, for example, RRh₄B₄ (T_c ∼9 K and T_m ∼1 K) (1), RMo₆S₈ (T_c ∼2 K and T_m ∼0.2 to 1 K) (2), and RNi₂B₂C (T_c ∼5 to 10 K and T_m ∼2 to 6 K) (3, 4), including reentrant superconducting transition, anomalous temperature dependence of the upper critical field, and enhanced vortex pinning (or critical current density [J_c]) associated with the magnetic transition. All have contributed significantly to the development of superconductivity research.

Recently, iron-based superconductors containing Eu (Eu-IBSs) have attracted attention as a new class of magnetic superconductors (5). Eu-IBSs are characterized by higher T_c (up to 37 K) and T_m (∼15 to 20 K) compared to other magnetic superconductors. The coexistence of superconductivity and magnetism extends over a wider range of temperatures and magnetic fields, allowing us to conduct experiments using various probes. For example, in the case of EuFe₂(As,P)₂ (6) and Eu(Fe,Rh)₂As₂ (7), where the Eu²⁺ moments exhibit ferromagnetic ordering, observations of a spontaneous vortex state, the domain Meissner state, and a vortex-antivortex state were reported using magnetic force microscopy (8) and magnetization measurements (9). Furthermore, for EuRbFe₄As₄ (10, 11) with a Eu²⁺ helical order (12), it was proposed that the Eu-spin subsystem serves as an internal pump for the magnetic flux based on magneto-optical imaging results (13, 14). In addition, optical conductivity measurements (15) and scanning Hall microscopy (16) revealed that the temperature dependence of the superfluid density shows a dip feature around T_m, which was attributed to the weakening of superconductivity at the magnetic transition.

Most of these peculiar phenomena in magnetic superconductors are considered to reflect the impact of magnetism on superconductivity. On the contrary, an example of the influence of superconductivity on magnetism is the shrinkage of magnetic domains associated with the superconducting transition (17), which is favorable for superconductivity, as it reduces the internal magnetic field. Meanwhile, in a type-II superconductor, the magnetic flux penetrates the material, and the mixed state is generated. Once vortices are pinned, the superconductor can act as a magnet (known as a trapped-field magnet). In this situation, magnetism is likely influenced by the magnetic field generated by superconductivity. For example, the manipulation of local moments via superconducting vortices has been proposed using superconductor–magnet hybrids (18). To further investigate the impact of pinned vortices on the local moments, a superconductor with strong pinning properties is considered as a suitable target.

The IBSs, especially the 122, 1111, and 1144 types, are known to show high critical current densities (J_c) of ∼1 MA/cm²; that is, they have a strong vortex pinning ability (19–26). Based on Bean’s critical state model (27), J_c is proportional to the gradient of the magnetic flux density (μ₀J_c = |dB/dx|). Then, when an IBS with a sample width (t) of 0.1 mm is magnetized in an external field, a large number of vortices are pinned and a strong magnetic field (H_sc = J_ct/2) of ∼5 kOe is estimated to be generated at the center of the sample. Therefore, in the case of Eu-IBSs, the superconductivity and the magnetic subsystem are both a source of strong magnetic fields; consequently, a nontrivial situation, where the magnetic fields generated by each state mutually affect each other, is considered. In this context, EuRbFe₄As₄ with the highest T_c among Eu-IBSs is an optimal material to investigate the influence of superconductivity on magnetism.

Fig. 1A shows the crystal structure of EuRbFe₄As₄. Since the Eu and Rb layers are alternately stacked without a solid solution, the Eu layer spacing is approximately twice that of EuFe₂As₂. Fig. 1B illustrates the temperature dependence of the magnetic susceptibility, showing a sharp superconducting transition at T_c = 37 K with perfect diamagnetism at zero-field cooling (ZFC). The kink structure at T_m = 15 K found below T_c corresponds to the magnetic ordering of the Eu²⁺ moments. The arrows in Fig. 1A indicate the direction of the Eu²⁺ moments. In the ordered state, the Eu²⁺ moments align ferromagnetically in the ab plane, and the orientation of the ferromagnetic alignment rotates by 90° in the next layer; that is, the ground state is a helical order (12). Meanwhile, magnetization measurements showed that the Eu²⁺ moments are reoriented to the ferromagnetic alignment at relatively low external fields (the saturation field [H_sat] is estimated to be ∼2 kOe) (28). In the ferromagnetically aligned state, the internal field (H_Eu) calculated from the Eu²⁺ moments (7 μ_B/Eu) is ∼4.5 kOe, which is approximately half of that in EuFe₂As₂, reflecting the Eu layer spacing.

Fig. 1.

Crystal structure and magnetic ordering of EuRbFe₄As₄. (A) Crystal structure of EuRbFe₄As₄. Blue arrows indicate the direction of the Eu²⁺ moments. The schematic diagram of the helical magnetic structure is shown on the Right. (B) Temperature dependence of the magnetization of a EuRbFe₄As₄ single crystal measured for ZFC and field-cooling processes. A magnetic field of 10 Oe was applied along the ab plane. The Inset shows a photograph of a EuRbFe₄As₄ single crystal with a black scale bar corresponding to 1 mm.

Thus, in EuRbFe₄As₄, the magnitudes of the characteristic magnetic fields related to superconductivity and magnetism, that is, H_sc, H_Eu, and H_sat, are comparable to each other, such that the Eu²⁺ moments cannot neglect the influence arising from superconductivity. In principle, it is possible to control the arrangement of local Eu²⁺ moments using H_sc. Moreover, because the internal field generated by superconducting currents has spatial distributions, the direction of the Eu²⁺ moments is expected to be locally manipulable by controlling the magnetic field profile. In this study, we demonstrate that 1) the orientation of the Eu²⁺ moments in magnetic fields is dominated by the direction of pinned vortices, and 2) the domain structure of the Eu²⁺ moments can be manipulated by the distribution of superconducting vortices.

Results and Discussion

Fig. 2A shows the magnetization hysteresis loop (MHL) of EuRbFe₄As₄. The shape of the MHL can be interpreted as a superposition of the ferromagnetic contribution of the Eu²⁺ moments on the hysteresis caused by vortex pinning of the superconductor. Assuming that the Eu²⁺ moments do not exhibit hysteresis in the increasing and decreasing field processes (Fig. 2B), as the simplest two-component model, the hysteretic part of the MHL (ΔM = M_- – M₊, Fig. 2C) depicts the contribution of the vortex pinning, and the average magnetization (M_ave = (M₊ + M_-)/2, Fig. 2D) depicts that of the Eu²⁺ moments (28, 29). In fact, the hysteresis of the Eu²⁺ moments in EuFe₂As₂ (a related nonsuperconducting material) is negligibly small (30), which likely justifies the assumption. However, this approach is too simplistic, as we show subsequently.

Fig. 2.

Magnetization of EuRbFe₄As₄ and supposed response of the Eu²⁺ moments to external field. (A) MHL of EuRbFe₄As₄ measured at 2 K with a magnetic field parallel to the ab plane. The Inset shows the geometric alignment. Black arrows indicate the field sweep direction. (B) Response of the Eu²⁺ moments to external magnetic field assuming the absence of hysteresis. The blue dashed line indicates the fraction of the helical structure, and the black (red) dashed line indicates that of the ferromagnetic structure in a positive (negative) applied field. (C) Hysteretic part of MHL, ΔM = M_– – M₊, where M₊ (M_–) is the magnetization measured in the field-increasing (decreasing) process, derived from A. (D) Average magnetization M_ave = (M₊ + M_–)/2. The saturation field H_sat was determined by the crossing point of the extrapolation lines from the slope at the low-field region and the saturation moment M_sat. (E) Magnetic field dependence of J_c calculated using the formula J_c = 20ΔM/t(1 – t/3l), where t and l are the sample dimensions (t < l). Here, for simplicity, the anisotropy of J_c flowing along the ab plane (J_c^ab) and c axis (J_c^c) is not considered, that is, J_c = J_c^ab = J_c^c. (F) Schematic diagrams of the superconducting currents and the distribution of vortices in EuRbFe₄As₄. The Top Left drawing shows the superconducting currents flowing in the critical state at zero field after applying a large field. The Top Right graph shows the profile of the magnetic flux density along the red dashed line in the Top Left drawing. The magnitude of the magnetic field generated at the center of the sample is estimated to be H_sc = J_ct/2. The Bottom drawing shows a schematic diagram of the distribution of vortices. Each orange rod surrounded by a blue clockwise arrow indicates a vortex composed of circular supercurrents generating an upward magnetic field.

M_ave saturates around H_sat ∼4 kOe, as shown in Fig. 2D. This indicates the reorientation of the Eu²⁺ moments from the helical structure to a ferromagnetic arrangement [regarding the difference in the magnitude of H_sat from literature data (28), reference SI Appendix]. The saturation magnetization of ∼330 emu/cm³ corresponds to 6.9 μ_B/Eu, which is close to the expected full moment of 7 μ_B/Eu. Meanwhile, based on Bean’s critical state model (27), the magnetically determined J_c can be estimated from ΔM, as shown in Fig. 2E. The magnitude of J_c in the low-field region exceeds 1 MA/cm², which is comparable to those reported for 122- and 1144-type IBSs (19, 20, 24–26). This indicates that numerous vortices are pinned in the sample at zero external field after applying a large field. The magnitude of the trapped magnetic field at the center of the sample is estimated to be ∼5 kOe, which is larger than H_sat required for the ferromagnetic alignment of the Eu²⁺ moments. Subsequently, as shown schematically in Fig. 2F, the pinned vortices significantly influence the Eu²⁺ moments, that is, neglecting any hysteresis of the Eu²⁺ moments may be too simplistic.

In the present case, even if the Eu²⁺ moments exhibit a ferromagnetic hysteresis, it is difficult to extract the hysteresis from the results of magnetization measurements due to the large magnetization hysteresis arising from the vortex pinning of superconductivity below T_c. To directly observe the behavior of the Eu²⁺ moments in the superconducting state, we investigated single-crystal neutron diffraction patterns of EuRbFe₄As₄ under magnetic fields.

Fig. 3A shows the neutron diffraction patterns along the (00L) direction of EuRbFe₄As₄ at 2.5 K, well below T_m, with the field applied along the ab plane. The open (solid) symbol data were measured under an increasing (decreasing) field. First, in the ZFC condition (open blue), diffraction peaks are observed at (00L ± 0.25) positions in addition to (00L) positions. This (0, 0, 0.25) magnetic propagation vector indicates that the direction of the Eu²⁺ moments rotates by 90° between neighboring layers, that is, the helical structure, as described in Fig. 1A. With increasing magnetic field, at 1 kOe (open green) and 2 kOe (open orange), the intensity of the (00L ± 0.25) peaks is suppressed (the enlarged view around the (001.25) peak is shown in Fig. 3B), while that of the (00L) peaks is enhanced. This change in the peak intensity demonstrates the transition of the Eu²⁺ magnetic order from a helical structure to a ferromagnetic arrangement. With further increasing field, the (00L ± 0.25) peaks almost disappear at 4 kOe (open red), and the diffraction pattern becomes identical to that observed at a large field of 2 T (open black). Thus, the Eu²⁺ moments completely align ferromagnetically at ∼4 kOe, which is comparable to H_sat extracted from the MHL at 2 K.

Fig. 3.

Field dependence of neutron diffraction patterns of EuRbFe₄As₄ and response of the Eu²⁺ moments to external field. (A) Neutron diffraction patterns of EuRbFe₄As₄ along (00L) at 2.5 K under magnetic fields along ab plane. The pattern on the Top (open blue circles) was measured after ZFC, and the magnetic field was increased to 20 kOe from the Top to Middle (open black circles). From the Middle to Bottom, the field was decreased to –4 kOe (pink solid circles). (B) Magnified view around (001.25) peak plotted in log scale. (C) Magnetic field dependence of peak intensity for (001) peak (red) and (001.25) peak (blue). The peak intensity is normalized at 20 kOe and 0 Oe (ZFC) for the (001) and (001.25) peaks, where the Eu²⁺ moments are completely helical and ferromagnetic, respectively. Arrows indicate the field sweep direction. (D) Fraction of the Eu²⁺ moments assuming helical (blue) and ferromagnetic structures along positive (black) and negative (red) field directions, estimated from the results shown in C. Here, it is assumed that the upward ferromagnetic moment is aligned to the downward one via the helical structure for the H < 0 region in the field-decreasing process. (E) Field dependence of magnetization arising from the Eu²⁺ moments (M_Eu), calculated based on D. The dashed line corresponding to M_Eu in the field-increasing process from the negative field is derived from the field-decreasing data assuming M_Eu(H) = –M_Eu(–H).

Next, the field was decreased from 2 T. The diffraction pattern at 2 kOe (solid orange) is found to be almost identical to that at 20 kOe. At zero field, although the peak at (001.25) position can be identified, the intensity is approximately one-tenth of that in the ZFC condition. The results clearly show that the Eu²⁺ moments do not return to the helical structure but remain in the ferromagnetic arrangement even at zero applied field. By further increasing the field in the negative direction, the (001.25) peak is slightly enhanced at –1 kOe (solid light green), and a similar diffraction pattern is observed at –2 kOe (solid brown). At –4 kOe (solid pink), the (001.25) peak is completely suppressed again, indicating the ferromagnetic arrangement of the Eu²⁺ moments in negative direction.

The magnetic field dependence of the intensity of the (001.25) peak (blue) and the (001) peak (red), representing the fraction of helical and ferromagnetic structure, respectively, is summarized in Fig. 3C. The intensities of the (001) and (001.25) peaks are inversely correlated. Based on these results, the field dependence of the fraction of helical and ferromagnetic Eu²⁺ moments is plotted in Fig. 3D. Here, for simplicity, no intermediate magnetic structure between the helical and the ferromagnetic structures is considered. This assumption is reasonable because the fraction of intermediate structure is estimated to be small (SI Appendix). Evidently, this behavior of the Eu²⁺ moments in Fig. 3D is distinct from the model shown in Fig. 2B, as only one part of the sample exhibits the helical structure in the field-decreasing process, and the region is not centered at zero but appears in the negative field region. Fig. 3E shows the magnetic field dependence of the magnetization arising from the Eu²⁺ moments (M_Eu), calculated based on Fig. 3D. The calculated M_Eu clearly shows a ferromagnetic hysteresis with the coercive field of ∼1 kOe, which is two orders of magnitude larger than that of EuFe₂As₂. Therefore, the magnetization hysteresis (ΔM) in the low-field region measured at low temperatures below T_m must be considered as the sum of the hysteresis arising from the vortex pinning (ΔM_sc) and that induced in the Eu²⁺ moments (ΔM_Eu).

To explore the ferromagnetic hysteresis of the Eu²⁺ moments, we review the magnetic field dependence of M in detail. Here, we focus on the MHLs in the low-field region (±10 kOe), where the hysteretic behavior of the Eu²⁺ moments is expected. Fig. 4A shows representative MHLs measured below and above T_m. A clear hysteresis is observed at all temperatures, and the size of the MHL is largest for the lowest temperature (2 K, black) and decreases with increasing temperature. Below T_m, the magnitude of M_- (M₊) under a positive (negative) field is almost constant, while the MHLs above T_m are shrunk around zero field. In nonmagnetic superconductors, the hysteresis is dominated by vortex pinning, and the width ΔM corresponds to the magnitude of J_c. Fig. 4B shows the magnetic field dependence of ΔM at each temperature derived from Fig. 4A. At 2 K (black), ΔM shows a peak at zero field and decreases with increasing/decreasing field. This appears to be a typical magnetic field dependence of J_c. Furthermore, a small kink structure (indicated by a black arrow) reflecting a jump in the MHL is observed around ±4 kOe. By increasing the temperature to 5 K (blue), 10 K (green), and 15 K (orange), ΔM decreases, and the kink position shifts to lower fields. By further increasing the temperature, at 20 K (red) above T_m, the peak observed at zero field changes to a dip, and the kink structure disappears. ΔM shows a similar field dependence with a smaller magnitude at higher temperatures (25 K [purple] and 30 K [light green]). This field dependence of ΔM (J_c), having a dip at zero field, is not common; however, similar behavior was observed for the ΔM of CaKFe₄As₄ and associated with the defect structure specific to 1144-type materials (26). Fig. 4C shows the temperature and field dependence of ΔM plotted in a contour plot. ΔM peaks at zero field below T_m = 15 K (horizontal dashed line) and dips at higher temperatures. The change in the field dependence of ΔM across T_m was a strange behavior, supposing it reflects that of J_c, while it is reasonable considering that the hysteresis of the Eu²⁺ moments contributes to ΔM at low fields only below T_m. Fig. 4D shows the decomposition of ΔM at 2 K into the contribution of the Eu²⁺ moments (ΔM_Eu) and superconductivity (ΔM_sc) using the magnetic field dependence of M_Eu determined from neutron diffraction experiments (Fig. 3E). ΔM_sc (blue) has a dip at zero field similarly to those above T_m, indicating that the field dependence of J_c does not change across T_m. Moreover, ΔM_Eu (red) becomes zero around ±4 kOe, which approximately coincides with the magnetic field, where a kink appears in ΔM (open triangles in Fig. 4C), suggestive of an interaction between those two.

Fig. 4.

MHLs of EuRbFe₄As₄ at low-field region. (A) MHLs under low fields measured at several temperatures below and above T_m (2 to 30 K). Note that the field was applied along the ab plane up (down) to 30 (–30) kOe, whereas only data in the field range of ±10 kOe is shown. (B) Field dependence of ΔM derived from A. The same color code is used for each temperature as in A. Arrows indicate the kink structure, where M shows a sudden change. (C) Temperature and field dependence of ΔM plotted by a contour plot. The plot is produced using MHLs measured every 1 to 2 K. The hot (cold) colored region indicates that ΔM is large (small). The horizontal dashed line indicates T_m (= 15 K). The open triangles correspond to positions of the kink structure. (D) Decomposition of ΔM at 2 K (black) into contributions from vortex pinning (ΔM_sc, blue) and from the Eu²⁺ moments (ΔM_Eu, red). ΔM_Eu is derived from Fig. 3C and ΔM_sc = ΔM – ΔM_Eu is assumed to extract ΔM_sc.

Subsequently, we test whether these experimental results can be understood assuming that the state of the Eu²⁺ moments (upward, helical, or downward) is determined by the local magnetic flux density (B) in the sample. Here, the change in the distribution of B in the field-decreasing process from a sufficiently high field is considered. We separate the B profile into those created by vortex pinning and the Eu²⁺ moments and model it.

Based on Bean’s critical state model (27), the distribution of B pinned in a superconductor exhibits a “roof-top” shape, as shown in the Left drawing in Fig. 5A. For simplicity, the field dependence of J_c is not considered; that is, the slope of the B profile inside the sample (dB/dx) is taken as constant. The graphs in the Right Upper panel of Fig. 5A show the B profiles of the roof-top shape cut along the red dashed line in the Left drawing under different external fields. B inside the sample is described as B = μ₀H_ex + μ₀J_cx, where H_ex is the external field, and x is the distance from the sample surface. At the center of the sample, B becomes μ₀H_ex + μ₀H_sc, where H_sc = J_ct/2, and t is the sample size along the cut. Setting J_c = 0.8 MA/cm² (Fig. 2E) and t = 0.13 mm, H_sc is estimated to be 6.5 kOe. Since the field dependence of J_c is ignored, the B profile shifts downward without changing the shape with decreasing H_ex [from (i) to (vi)].

Fig. 5.

Model of flux density profile in EuRbFe₄As₄ and comparison of calculation and experimental results. (A) Change in B profile in field-decreasing process from a sufficiently high field. The Left drawing shows a typical roof-top B distribution (indicated by blue lines) in superconductors expected from Bean’s critical state model. Blue (light blue) arrows indicate the superconducting currents flowing along the ab plane (c axis). The magnetic field (H_sc) generated by the currents is indicated by the red arrow. The graphs in the Right Upper panel show B profiles cut along the red dashed line in the Left drawing under different external fields. The horizontal pink shaded region indicates the range of |B| < B₀ = 0.25 kG, where the Eu²⁺ moments assume the helical structure. Vertical orange shaded regions indicate the corresponding helical domains. The graphs in the Right Lower panel show the B profiles created by the Eu²⁺ moments. The magnitude of the internal field is assumed to be H_Eu = 4.5, 0, and –4.5 kOe for upward ferromagnetic, helical, and downward ferromagnetic structures, respectively. (B) Calculation results of magnetic field dependence of fraction of upward ferromagnetic (black), helical (blue), and downward ferromagnetic (red) Eu²⁺ moments (Left) and total magnetization M = M_Eu + M_sc (Right). The arrows indicate the field sweep direction. (C) Experimental data extracted from Fig. 3D (Left) and Fig. 4A (Right). The same color code is used as in B.

Next, we model the B profile arising from the Eu²⁺ moments, supposing that the B profile created by vortex pinning determines the arrangement of the Eu²⁺ moments. For simplicity, we assume that the Eu²⁺ moments order helically for |B| < B₀, where B₀ is a threshold field (the range shaded by pink in the Upper panel of Fig. 5A) and align ferromagnetically in the upward or downward direction according to the sign of B for |B| > B₀. Fig. 3D shows that the fraction of the helical structure reaches ∼10% in a decreasing field. Therefore, we assume 2B₀ = 0.5 kG (∼10% of H_sc), where the fraction of the helical structure is expected to be ∼10%. Thus, the B profiles of the Eu²⁺ moments at each H_ex are derived, as shown in the Lower panel of Fig. 5A. For μ₀H_ex > B₀ (i), the Eu²⁺ moments maintain an upward ferromagnetic arrangement, and the internal field (H_Eu) is 4.5 kOe at any position. With decreasing H_ex, when B₀ > μ₀H_ex > –B₀ (ii and iii), the Eu²⁺ moments maintain an upward ferromagnetic arrangement around the center of the sample, whereas the helical structure (H_Eu = 0) is adopted at the edge of the sample, where |B| < B₀. Subsequently, when μ₀H_ex < –B₀ (iv), the Eu²⁺ moments take a downward (upward) ferromagnetic arrangement with H_Eu = –4.5 kOe (4.5 kOe) at the sample edge (center). The helical structure appears at the region where |B| < B₀, which corresponds to the boundary between the upward and downward ferromagnetic domains. By further decreasing H_ex (v), the downward (upward) ferromagnetic domain becomes larger (smaller), while the change in helical area is small, as it appears at the domain boundary. When μ₀H_ex = B₀ – μ₀H_sc = –6.25 kG (vi), the upward ferromagnetic domain disappears. Consequently, the helical region sandwiched by the downward ferromagnetic domain only feels the negative B (< –B₀), and thus, the whole region will suddenly turn into the downward ferromagnetic state (vii). This sudden disappearance of the helical region gives rise to the discontinuous change of M.

Based on the model, the magnetic field dependence of the Eu²⁺ moment state is calculated, as shown in the Left of Fig. 5B. The features are as follows: 1) the helical component appears around 0 Oe and is present down to –6 kOe (asymmetric with respect to zero field), and 2) the volume fraction of the helical component is ∼10% across the entire region. Comparing this with the neutron diffraction results (Fig. 5 C, Left), the magnetic field dependence of the upward ferromagnetic (black), helical (blue), and downward ferromagnetic (red) components is mostly reproduced. The Right of Fig. 5B shows the calculated M = M_sc + M_Eu where M_sc is constant in this model, and M_Eu is derived from the Left of Fig. 5B. The M in the field-increasing process is calculated supposing M₊(H) = –M_-(–H). The calculated MHL is in good agreement with the measured MHL shown in the Right of Fig. 5C, including the magnitude and the kink structure. The slight deviation is considered to come from the simplification in the present model; that is, the field dependence of J_c and the intermediate state of Eu²⁺ between the helical and ferromagnetic structure are not taken into account (SI Appendix).

These results support the model that the local B created by superconducting vortices determines the Eu²⁺ spin domain structure in EuRbFe₄As₄. If this is the case, it is expected that various spin textures can be formed by changing the B profile. To provide an example, we attempt to introduce a larger number of ferromagnetic domains in the present EuRbFe₄As₄ sample. Here, we consider a process where H_ex is switched from a decreasing to an increasing direction at some field (H_rev), as shown in Fig. 6A. Then, following Bean’s critical state model, the B distribution changes from the sample edge, while it remains unchanged around the sample center. When μ₀H_ex > B₀, a domain structure shown in Fig. 6C is expected, where the sample center and edge are upward ferromagnetic (green), the region between them is downward ferromagnetic (blue), and the domain boundaries (gray) are helical. The Left of Fig. 6B shows the calculated M – H curves based on the model with various H_rev values of 0, –2, –5, and –10 kOe, and the Right shows the measured M – H curves. The calculated results are in good agreement with the experiments, supporting the validity of the proposed model. For H_rev = 0 (blue), because the Eu²⁺ moments maintain the upward ferromagnetic arrangement, the change in M reflects the change in the B profile associated with vortex pinning (B profile is reversed). Meanwhile, for H_rev = –10 kOe (red), because the Eu²⁺ moments in the entire sample are aligned downward ferromagnetically and the B profile by vortex pinning is almost reversed around H_ex ∼0 in the field-increasing process, the situation is similar to Fig. 5A with an opposite sign. In contrast, in the case of H_rev = –2 and –5 kOe, which satisfy the condition of μ₀|H_rev| < μ₀H_sc – B₀, the upward ferromagnetic domain remains at the sample center, and thus, the domain structure shown in Fig. 6C is expected to be realized for the field range of B₀ < μ₀H_ex < μ₀|H_rev| – B₀.

Fig. 6.

Demonstration of ferromagnetic domain control via trapped vortices in EuRbFe₄As₄. (A) Change in B profile with varying external field. The external field was first decreased [from (i) to (ii)] and then increased [from (ii) to (v)]. (B) Calculated (Left) and measured (Right) magnetization curves for various H_rev = 0 (blue), 2 (green), 5 (orange), and 10 kOe (red). The arrows indicate the field sweeping direction. (C) Eu²⁺ domain structure expected for graph (v) in A. Vortices along positive (negative) field direction are colored orange (blue). Green, gray, and blue regions correspond to upward ferromagnetic, helical, and downward ferromagnetic domains, respectively.

According to theoretical works (31, 32), the helical order of the Eu²⁺ moments of EuRbFe₄As₄ (in zero field) appears due to the presence of superconductivity. It is proposed that the frustrating interlayer interactions caused by the interplay between the normal (ferromagnetic) and superconducting (antiferromagnetic) parts lead to the helical structure. When an external magnetic field is applied, each vortex generates a magnetic field decaying over a distance of the penetration depth (λ) from the core. For EuRbFe₄As₄, λ is estimated to be 100 to 200 nm at low temperatures (16), which is much larger than the Eu–Eu layer distance of 1.3 nm. Accordingly, a direct magnetic interaction between vortices and the Eu²⁺ moments would become dominant and thus the Eu²⁺ moments favor the ferromagnetic alignment in the presence of vortices. Moreover, the present results show that the Eu²⁺ moments must be ferromagnetically aligned by a much smaller field than that previously estimated from H_sat because the experimental data were reproduced by setting B₀ to 0.25 kG, which is one order of magnitude smaller than μ₀H_sat. Notably, the average intervortex distance (a₀) at B₀ is calculated to be ∼300 nm using the formula B = 2Φ₀/√3a₀² for a triangular flux lattice where Φ₀ is the magnetic flux quantum. The value of a₀ is close to 2λ, indicating that the Eu²⁺ moments align ferromagnetically when the whole region is covered by the field generated by vortices. Therefore, we consider B₀ to be the intrinsic saturation field of the Eu²⁺ moments in EuRbFe₄As₄, while H_sat extracted from M_ave – H is a sample dependent parameter (SI Appendix).

Finally, the experiments in this work were performed with a magnetic field along the ab plane, which is the magnetic easy plane of EuRbFe₄As₄. When the magnetic field is applied along the c axis, the Eu²⁺ moments are expected to tilt continuously from the ab-plane direction without destroying the helical structure. Meanwhile, similarly to the case of H_ex // ab, the Eu²⁺ moments are easily aligned along the c axis (28), and the superconducting properties of EuRbFe₄As₄ are rather three dimensional (33). Then, the present model is expected to be applicable for H_ex // c by changing the parameters, J_c, t, and B₀, and modeling M_Eu for the intermediate conical structure instead of using B₀ will improve the model. However, the situation is more complicated for H_ex // c because the present sample is thin along the field direction, which will result in significant demagnetization effects. In this case, the B profile inside the sample no longer follows that of the Bean’s critical state model owing to a strong stray field, hence the model must be modified to describe the B profile including the stray field.

Conclusions

In summary, we revealed that the Eu²⁺ moments exhibit a ferromagnetic hysteresis in the magnetic superconductor EuRbFe₄As₄, which is characterized by a high T_c and T_m, as well as a strong vortex pinning ability. These experimental results can be explained by a model where the state of the Eu²⁺ moments is dominated by the local B distribution created by vortex pinning. Using the unique interplay between the Eu²⁺ moments and superconducting vortices, we demonstrated that the magnetic domain structure can be manipulated by controlling the B profile. We changed the B profile by switching the external field and expect that the construction of more complex domain structures is possible by controlling the local B. Thus far, devices to manipulate local spins via superconducting vortices have been proposed for artificial superconductor/magnet hybrids (18). The present results demonstrate the possibility that such a device may be realized by utilizing EuRbFe₄As₄, which is a natural superconductor/magnet hybrid.

Materials and Methods

Single-Crystal Growth.

Single crystals of EuRbFe₄As₄ were grown by the RbAs-flux method (34). EuAs, Fe₂As, and RbAs precursors were prepared from Eu and As, Fe and As, and Rb and As, respectively, which were mixed at appropriate molar ratios. The mixtures were sealed in evacuated quartz tubes (EuAs and Fe₂As) and a stainless-steel tube with an alumina crucible (RbAs) and heated at 750 °C (EuAs), 900 °C (Fe₂As), and 600 °C (RbAs) for 20 h. EuAs, Fe₂As, and RbAs were weighed at a ratio of 1:1:15 to yield a total amount of 9 g and placed in an alumina crucible, then sealed in a stainless-steel container (35, 36). The container was heated for 5 h to 700 °C and maintained at this temperature for 5 h. It was then heated to 970 °C within 5 h and maintained this temperature for 10 h. Subsequently, it was cooled to 620 °C for 350 h (1 °C/h). For single crystals used in this study, X-ray diffraction patterns were measured at room temperature using a diffractometer with Cu Kα radiation (Rigaku, Ultima IV) to verify the presence of 00l peaks (SI Appendix). The as-grown crystals show a trace of RbFe₂As₂ (34); hence, the surfaces of as-grown crystals were carefully removed by cleaving. The cleaved surfaces only show 00l peaks from EuRbFe₄As₄.

Magnetization Measurements.

The samples for magnetization measurements were cut into rectangular shapes. The dimensions of the EuRbFe₄As₄ crystal used in this study were l = 1.2 mm, w = 0.7 mm, and t = 0.13 mm, with the shortest edge along the c axis. Measurements were performed using a magnetic property measurement system (Quantum Design). The magnetic field was applied along the ab plane (along the w edge); therefore, two J_c components (in-plane J_c (J_c^ab) and interplane J_c (J_c^c)) contribute to M. For Fig. 2E, we used a simplified formula for the evaluation of J_c by taking J_c^ab = J_c^c, that is, J_c = 20ΔM/t(1 − t/3l) where ΔM is the hysteretic part of the MHL, and l and t are the sample dimensions in units of emu/cm³ and cm, respectively. For the calculation of M in Figs. 5 and 6, the anisotropy of J_c (J_c^ab/J_c^c) was taken into account based on the evaluation of J_c^ab and J_c^c using an extension of Bean’s critical state model for anisotropic J_c (24) (SI Appendix).

Neutron Diffraction Measurements.

Single-crystal neutron diffraction measurements on EuRbFe₄As₄ were carried out using the time-of-flight single-crystal neutron diffractometer SENJU (37) installed at the Materials and Life Science Experimental Facility, Japan Proton Accelerator Research Complex. The sample size was 1.8 × 0.9 × 0.08 mm³. A neutron wavelength of 0.4 to 4.4 Å was used. The magnetic field was applied along the [110] direction up to 20 kOe. The data were visualized by the software STARGazer (38).

Data Availability

All data are included in the manuscript and/or SI Appendix.