Introduction

One of the ideas behind examining magnetic materials aims to develop desired functional properties utilized in a wide range of technologies, for example, energy storage1, memory devices2, medical appliances3, and environmental monitoring sensors4. In particular, magnetic oxides comprising metal cations and oxygen anions have been extensively investigated owing to the abundance of the constituents and stability of the compounds. A prominent example can be found in perovskite rare-earth manganites that have been the focus of research on magnetic materials over the last few decades. In mixed-valence manganites, the subtle balance between hopping and localization of charge carriers leads to the phase coexistence of ferromagnetic (FM) metallic and antiferromagnetic (AFM) insulating states via the kinetic arrest of the phase transition. The formation of mixed magnetic glass, which is susceptible to an external magnetic field (H), is essential to the origin of colossal magnetoresistance5,6. The variation of rare-earth ions in perovskite manganites also generates several types of multiferroic (MF) phases7,8,9,10. In medium-sized rare-earth ions, the spiral spin modulation can be stabilized, inducing ferroelectricity via antisymmetric exchange strictions with strong controllability of ferroelectric properties by external magnetic fields7. With a smaller radius, the crystallographic structure changes into a hexagonal structure, which represents a unique improper ferroelectricity due to structural trimerization9. However, the perovskite structure remains intact under high pressure and accompanies the E-type AFM phase that results in another type of the MF phase driven by symmetric exchange strictions10.

As an extension of studies on perovskite manganites, double perovskites of R2CoMnO6 (R = La, …, Lu, and Y) have recently been explored owing to their fascinating magnetic and functional properties, such as metamagnetism11,12,13, spin-glass state14,15,16, exchange bias effect17,18,19, magnetocaloric effect20,21,22, and multiferrocity23,24,25,26,27. By replacing half the Mn ions with Co ions in perovskite manganites, a double perovskite structure is formed with Co2+ (S = 3/2) and Mn4+ (S = 3/2) ions, alternatingly located in corner-shared octahedral environments. As the size of rare-earth ions decreases, the magnetic transition temperature (T) arising from the dominant Co2+ and Mn4+ superexchange interactions decreases from 204 K for La2CoMnO6 to 48 K for Lu2CoMnO628. In these compounds, the difficulty in attaining the impeccable alteration of Co2+ and Mn4+ ions naturally entails additional AFM clusters which involve another valence state of Co3+- Mn3+, and anti-sites of ionic disorders and/or antiphase boundaries leading to Co2+- Co2+ or Mn4+- Mn4+ pairs. The formation of anti-sites in addition to the dominant FM order27,29,30 of Co2+ and Mn4+ moments is known as the mechanism for the observed magnetic exchange bias in polycrystalline Y2CoMnO619. In Tm2CoMnO6 and Er2CoMnO6 (ECMO), the neutron diffraction studies confirm that the order of Co2+ and Mn4+ moments is FM and the order of Er3+/Tm3+ moments at lower temperature activates the additional ferrimagnetic (FIM) order between Er3+/Tm3+ and ferromagnetic Co2+/Mn4+ sublattices31,32,33. The FIM order exhibits an inversion of the magnetic hysteresis loop in polycrystalline ECMO34. In Yb2CoMnO6 and Lu2CoMnO6, the Co2+ and Mn4+ ions display the up-up-down-down (↑↑↓↓) spin configuration in which the ferroelectricity emerges perpendicular to the c-axis from the cooperative O2− displacements through the symmetric exchange striction23,24,25. Evidently, a scientific understanding of diverse magnetic phases and interactions is crucial for finding novel functional properties in double perovskites.

In this work, the magnetic and magnetoelectric properties of single crystals of double-perovskite ECMO were studied to reveal the characteristics corresponding to the mixed FIM and MF phases. The dominant FIM order between Er3+ and FM Co2+/Mn4+ sublattices was identified by compensated magnetization (M) occurring at TComp = 3.15 K. From our precise measurement of isothermal M in the low T regime, the inversion of the magnetic hysteresis loop was observed below TComp, which can be explained by the delicate balance between different magnetic moments, and qualitatively by an extended Stoner–Wohlfarth model35,36,37,38. The ferroelectric polarization (P) and dielectric constant (ε′) measurements demonstrated an additional inclusion of the MF phase as found in Yb2CoMnO6 and Lu2CoMnO623,24. Associated with the coexistence of FIM and MF phases, the disappearance of MF phase by an external H occurs simultaneously with the metamagnetic transition, revealing exclusive characteristics of the double perovskite.

Results and Discussion

Figure 1(a) shows the X-ray powder diffraction pattern for the ground single crystals of double perovskite ECMO at room T. The crystallographic structure was refined as a monoclinic structure with the P21/n space group. The lattice constants were found to be a = 5.228 Å, b = 5.594 Å, and c = 7.477 Å with β = 90.244° with good agreement factors, χ2 = 1.74, Rp = 7.97, Rwp = 6.23, and Rexp = 4.72. The crystal structures viewed from the a- and c-axes are depicted in Fig. 1(b,c), respectively. Co2+ and Mn4+ ions are alternatingly located in corner-shared octahedral environments. The oxygen octahedral cages are strongly distorted due to the small radius of the Er3+ ion28.

Figure 1
figure 1

Crystallographic structure of Er2CoMnO6. (a) Observed (open circles) and calculated (solid line) powder X-ray diffraction patterns for ground Er2CoMnO6 (ECMO) single crystals. Yobs, Ycal, and Yobs−Ycal represent the intensities of the observed patterns, calculated patterns, and their difference, respectively. The green short lines denote the Bragg positions. (b,c) Views of the crystal structure of double perovskite ECMO from the a- and c-axes, respectively. The purple, pink, blue, and yellow spheres represent Er3+, Co2+, Mn4+, and O2− ions, respectively.

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To investigate intricate magnetic properties as anticipated in the double perovskite incorporating three different magnetic ions, the T-dependence of magnetic susceptibility (χ = M/H) was obtained. The anisotropic χ in H = 0.05 kOe along (H//c) and perpendicular (H⊥c) to the c-axis was measured upon warming in H after zero-field cooling (ZFC) and upon cooling at the same H (FC), as shown in Fig. 2(a). The overall T-dependence of χ’s for two different orientations exhibits strong magnetic anisotropy, which indicates that the spins are mainly aligned along the c-axis. The FM order relevant to the dominant Co2+ and Mn4+ superexchange interactions sets in at TC = 67 K, which can be determined by the sharp anomaly in the T derivative of χ in H//c. The T-dependence of heat capacity divided by the temperature (C/T) measured upon warming in zero H also exhibits the anomaly starting from TC, shown in Fig. 2(b). Upon further cooling, C/T shows an abrupt increase below TEr ≈ 10 K, which corresponds to the ordering of Er3+ moments. Below TEr, the reversal of χ was observed in both ZFC and FC measurements (Fig. 2(a)) as a characteristic signature of a ferrimagnet39,40,41,42,43,44,45,46.

Figure 2
figure 2

Temperature-dependent magnetic properties of Er2CoMnO6. (a) Temperature (T) dependence of the magnetic susceptibility (χ = M/H, 1 emu = 4π × 10−6 m3) of a double-perovskite ECMO single crystal along (H//c) and perpendicular (H⊥c) to the c-axis, measured upon warming in H = 0.05 kOe after zero-magnetic-field cooling (ZFC) and upon cooling in the same field (FC), shown up to 80 K. (b) T-dependence of specific heat divided by temperature (C/T) measured without magnetic field (H). (c) T-dependence of the thermoremanent magnetization (Mrem) of the ECMO crystal, measured in H//c warming from 2 K in the absence of H after cooling in 50 kOe. The vertical dashed lines indicate the ferromagnetic transition temperature (TC), the Er3+ spin ordering temperature (TEr), and the compensation temperature (TComp), respectively.

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A ferrimagnet is a substance that involves a portion of opposing magnetic moments as in antiferromagnetism, but generates a net M from unequal magnetic moments in the opposite directions, thus exhibiting distinct characteristics of magnetism. The FIM interaction between Er3+ and FM Co2+/Mn4+ sublattices generates the intriguing T-dependence of χ following the different sequence of measurement. In the FC measurement in H//c, χ increases smoothly below TC with the parallel alignment of Co2+ and Mn4+ moments. Upon cooling further below TEr, the Er3+ moments begin to align oppositely to the Co2+/Mn4+ moments, which leads to a gradual decrease in χ. At lower T, χ intersects the zero point owing to the large moment of Er3+ spin. On the other hand, ZFC χ shows a positive value at 2 K since the Er3+ moments tend to orient along the H direction. The decrease in the effective Er3+ moments upon increasing T results in the sign change of χ. Above TEr, the negatively magnetized Co2+/Mn4+ spins begin to flip along the applied H due to thermal fluctuation, which causes another sign change of χ at 48 K. To find the compensation T precisely, the thermoremanent magnetization (Mrem)47 was measured in H//c (Fig. 2(c)). At 2 K, H = 50 kOe was applied in H//c and then turned off, and Mrem was recorded in the absence of H upon warming from 2 K. The sign reversal of Mrem occurs at TComp = 3.15 K, which manifests the FIM feature of this double perovskite compound.

The anisotropic M in H//c and H⊥c was measured up to ±90 kOe at T = 2 K, shown in Fig. 3(a). For the hysteresis loop in H//c, solid and dashed lines denote sweeping H from +90 to −90 kOe and from −90 to +90 kOe, respectively. M in H//c is not saturated at +90 kOe with the magnetic moment of 17.2 μB/f.u., but it is much larger than the moment in H⊥c (8.71 μB/f.u.), indicating the magnetic easy c-axis. Upon decreasing H from +90 kOe, M decreases smoothly until it drops precipitously below 15 kOe. At low H, M intersects the zero point at 1 kOe and exhibits the negative remanent M (Mr) of −1 μB/f.u. (inset of Fig. 3(a)). Further decrease in H in the negative direction induces a sharp drop in M at HC = −26.5 kOe. The measurement of M in H//c in the opposite direction completes the inverted magnetic hysteresis loop. The inversion of the hysteresis loop in H//c can be analysed by an extended Stoner–Wohlfarth model within the frame of the FIM order between Er3+ and Co2+/Mn4+ sublattices with a different magnetic anisotropy35,36,37,38 (see Experimental section for detail). The experimental observation of inversed magnetic hysteresis loop in ECMO suggests the considerable difference of magnetic anisotropy energies between Er3+ and Co2+/Mn4+ moments. In our calculation, we assumed that the magnetocrystalline anisotropy energy of Co2+/Mn4+ moments is three times larger than that of Er3+ moments. With qualitative similarity, the magnetic hysteresis loop was attained from the model, as illustrated in Fig. 3(b). Based on the result, the evolution of the spin configuration for Er3+ and Co2+/Mn4+ ions during the sweeping of H from +90 to −90 kOe in H//c is schematically depicted in Fig. 3(b). The red and blue arrows indicate the effective moments of Er3+ and Co2+/Mn4+ ions, respectively. At high H, the Er3+ and Co2+/Mn4+ moments tend to be aligned in the same direction due to the dominant Zeeman energy. Upon decreasing H, the negative exchange coupling between Er3+ and Co2+/Mn4+ spins accompanied by a smaller magnetocrystalline anisotropy energy and larger moment of Er3+ ions leads to the progressive decrease in the net Er3+ moments, followed by zero net M even at a positive H and negative Mr. Decreasing H further in the negative direction induces an abrupt drop in M, where the Co2+/Mn4+ spins are fully reversed because the Zeeman energy of Co2+/Mn4+ sublattices overcomes the anisotropy energy. Since the change in magnitude of M caused by the reversal of Co2+/Mn4+ moment at the metamagnetic transition is found to be ~9 μB/f.u. (Fig. 3(a)), the net magnetic moment of Co2+/Mn4+ spins should be ~4.5 μB/f.u., which is smaller than the summation of Co2+ and Mn4+ moments (6 μB/f.u.). The smaller net magnetic moment of Co2+/Mn4+ spins is acceptable because a small portion of Co2+/Mn4+ spins is naturally reversed during the magnetization process from +90 kOe to −26.5 kOe and antiferromagnetic exchange couplings of Co2+- Co2+ or Mn4+- Mn4+ pairs are originally included from the presence of anti-sites of ionic disorders and/or antiphase boundaries.

Figure 3
figure 3

Observed and calculated isothermal magnetization at 2 K and temperature evolution of inverted magnetic hysteresis. (a) Isothermal magnetization (M) of the ECMO crystal in H//c and H⊥c, measured at T = 2 K up to 90 kOe after ZFC. The inset shows the magnified view in the range of H = ±10 kOe of the hysteresis loop in H//c. For the hysteresis loop in H//c, the solid and dashed curves indicate the data obtained by sweeping H from +90 kOe to −90 kOe, and by sweeping H from −90 kOe to +90 kOe, respectively. (b) Calculated hysteresis loop in H//c by adopting the extended Stoner-Wohlfarth model. The schematic spin configurations depicted as net moments of Er3+ (light red arrows) and Co2+/Mn4+ (light blue arrows) spins are illustrated for the curve of sweeping H from +90 kOe to −90 kOe. (c,d) Magnified views in the range of H = ±5 kOe of isothermal M’s in H//c, measured at various T’s below TComp (T = 2, 2.25, 2.5, 2.75, and 3 K) and above TComp (T = 3.25, 3.5, 4, 4.5, and 5 K), respectively.

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The close relevance of Mr to TComp was cautiously examined by the T dependent evolution of Mr. The full hysteresis curves up to ±90 kOe were recorded in H//c at various T’s. The hysteresis loops below and above TComp are shown within the range of ±5 kOe in Fig. 3(c,d), respectively. Below TComp, all the curves present the inverted magnetic hysteresis. Upon increasing T, the inverted loop becomes narrow and the magnitude of negative Mr decreases linearly, resulting from the reduced net Er3+ moments by thermal fluctuation. By crossing TComp, the sign of Mr changes and it increases gradually with an increasing T.

Recently, new magnetism-driven ferroelectrics, i.e. type-II multiferroics, were found in double-perovskite Yb2CoMnO6 and Lu2CoMnO623,24. The initial polycrystalline analysis of neutron diffraction and bulk electric properties for Lu2CoMnO6 suggested that the ferroelectricity arises from the symmetric exchange striction of the ↑↑↓↓ spin chains with alternating Co2+ and Mn4+ charge valences48, consistent with the Ising spin chain magnet of Ca3CoMnO649. However, studies on the single crystals of Yb2CoMnO6 and Lu2CoMnO6 revealed that the ferroelectricity emerges perpendicular to the c-axis below TC = 52 and 48 K, respectively. Several theoretical works provided a plausible explanation for the ferroelecticity, in which the symmetric exchange strictions along the ↑↑↓↓ spin chain with alternatingly shifted O2− ions generate cooperative O2− displacements perpendicular to the c-axis50,51,52.

The possible formation of an additional MF phase in ECMO was examined by the H-dependence of P obtained by integrating magnetoelectric current density (J), measured perpendicular to the c-axis (E⊥c) at 2 K, shown in Fig. 4(a,b). After poling from 100 K to 2 K in H = 0 kOe and E = 5.7 kV/cm, the J in H//c exhibits a very sharp peak with peak height of ~0.76 μA/m2 at the metamagnetic transition, HC = 26.5 kOe. The corresponding P value at H = 0 kOe and 2 K was estimated as ~0.9 μC/m2, which is only two orders of magnitude smaller than the P observed in Lu2CoMnO6 and signifies the presence of a small amount of the MF phase. The tiny magnitude of P at 2 K implies that the exact magnetic configuration of MF phase could hardly be identified by the neutron diffraction experiment. Upon increasing H, the P shows the sharp step at HC and disappears above HC. The simultaneous transitions at HC for the suppression of the ferroelectricity and the reversal of Co2+/Mn4+ spins in the FIM state suggest that the small amount of the additional MF phase is strongly influenced by the dominant FIM phase. In analogy with the ferroelectricity in Lu2CoMnO6, the P emerged perpendicular to the c axis at H = 0 kOe in ECMO suggests that the most plausible spin configuration of the minor MF phase would be ↑↑↓↓. The disappearance of the P by applying H along the c axis can be explained by the change of spin configuration from the ↑↑↓↓ to ↑↑↑↑.

Figure 4
figure 4

Isothermal ferroelectric polarization and dielectric constant at 2 K. (a) H-dependence of ferroelectric polarization (P) at 2 K, obtained by integrating the magnetoelectric current in b). (b) H-dependence of current density (J), measured with the H variation of 0.1 kOe/s in H//c after poling from 100 K to 2 K in E = 5.7 kV/cm perpendicular to the c axis. (c) H-dependence of ε′ in E⊥c, measured up to ±90 kOe in H//c at 2 K. (d) H-derivative of isothermal M (dM/dH) at 2 K, taken from the data in Fig. 3a).

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In Fig. 4(c), the H-dependence of ε′ in E⊥c is plotted, measured in H//c up to ±90 kOe at f = 100 kHz and T = 2 K. By sweeping H between +90 to −90 kOe, the whole variation of ε′ is only about 1% with strong hysteretic behaviour. The maximum values occur at H = ±6 kOe, followed by the sharp transitions at HC. The ε′ shows the rather complicated H dependence in comparison with the H dependence of P. In addition to the small portion of MF phase, the additional AFM clusters formed by anti-site disorders and antiphase boundaries in the ferromagnetic Co2+/Mn4+ sublattices would also affect the isothermal ε′. The complicated but tiny magnitude variation of isothermal ε′ may result from the intricate contributions from the small portions of MF phase and AFM clusters. For a comparison with ε′(H), the H derivative of isothermal M, dM/dH at 2 K is also plotted in Fig. 4(d). The dM/dH reveals the similar hysteretic variation of ε′. The isothermal M mainly reflects the response of the FIM order between the Er3+ and Co2+/Mn4+ moments to the external H, as illustrated in Fig. 3(b), but also affects strongly on the hysteretic behaviour of ε′(H).

The T-dependence of the dielectric constant (ε′) and tangential loss (tanδ) is displayed in Fig. 5(a,b), respectively, measured perpendicular to the c-axis (E⊥c) at f = 100 kHz in H//c with H = 0, 10, 20, and 30 kOe. At zero H, a small and broad peak of ε′ at TC = 67 K was observed in Fig. 5(a), which signifies the emergence of a small amount of MF phase. Compared to the peak height of ~15%, normalized by the value at TC = 48 K in Lu2CoMnO623, it can be estimated as only about 1% in ECMO. Despite a small portion of the MF phase in ECMO, TC is fairly enhanced. The broad peak of ε′ is gradually suppressed by applying H along the c-axis, ascribed to the change in the spin configuration from ↑↑↓↓ to ↑↑↑↑, similar to that in Lu2CoMnO623. Upon decreasing T, ε′ decreases linearly until it declines faster below 20 K. The overall T-dependence of ε′ and tanδ (Fig. 5(b)) below TC appears similar to those of Lu2CoMnO6.

Figure 5
figure 5

Temperature dependence of the dielectric properties of Er2CoMnO6. (a,b) T-dependences of dielectric constant (ε′) and dielectric tangential loss (tanδ), respectively, measured upon warming from 2 K to 100 K in an applied AC voltage of V = 1 V at f = 100 kHz perpendicular to the c-axis (E⊥c), and H = 0, 10, 20, and 30 kOe along the c-axis (H//c).

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While the intrinsic coupling phenomena between magnetic and ferroelectric states in single-phase type-II multiferroics were extensively explored, detailed properties of an MF phase mixed with another magnetic phase have scarcely been revealed. The T evolution of magnetoelectric effect in the mixed FIM and MF phases was examined by comparison between isothermal P and M at T’s below Tcomp. Figure 6(a,b) show the H-dependence of P’s and M’s, respectively, in E⊥c and H//c at T = 2, 2.25, 2.5, 2.75, and 3 K, indicating that both of P and M vary delicately to the change of T. The estimated P’s at 2.25 and 2.5 K were 0.79 and 0.47 μC/m2, respectively. As H is increased, the P’s are suppressed with steep steps at H = 28.0 and 28.7 kOe. The initial curve of M at 2, 2.25, and 2.5 K also shows the step at the same H as P, suggesting the strong intercorrelation between FIM and MF phases. At 2.75 and 3 K, P magnitudes at 0 kOe are reduced as 0.43 and 0.32 μC/m2, respectively. Upon increasing H, the P’s are gradually reduced and vanish above ~37 kOe, corresponding to the overall broad feature of M’s. Note that P above Tcomp could not hardly be obtained because of the almost suppressed magnitude of J with a broadened feature. Figure 6(c,d) display isothermal ε′ in E⊥c at f = 100 kHz and J in H//c, respectively, at T = 2, 2.25, 2.5, 2.75, and 3 K. The initial curve of ε′ at 2 and 2.25 K indicates both a sharp peak and step-like feature at the metamagnetic transition but the ε′ at 2.5 K shows only a step. The sharp peak of the J at 2 K shifts to higher H and the peak height is reduced upon slightly increasing the T. The weak anomaly was observed in the ε′ at 2.75 and 3 K, corresponding to the disappearance of P. As shown in the inset of Fig. 6(d), J’s at 2.75 and 3 K exhibit wide and small peaks around 35 kOe.

Figure 6
figure 6

Temperature evolution of ferroelectric polarization below Tcomp in comparison with that of magnetization and dielectric constant. (a) H-dependence of P at T = 2, 2.25, 2.5, 2.75 and 3 K shown in the range of 25–40 kOe. (b) Initial curves of isothermal M at T = 2, 2.25, 2.5, 2.75 and 3 K. (c) H-dependence of ε′ at T = 2, 2.25, 2.5, 2.75 and 3 K. (d) H-dependence of J, measured with the H variation of 0.1 kOe/s in H//c at T = 2, 2.25, 2.5, 2.75 and 3 K after poling in E⊥c. The inset shows the magnified view of J.

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The T evolution of the magnetodielectric effect in a wide range of T’s in the mixed FIM and MF phases was also investigated by comparison between isothermal ε′ and M. Figure 7 displays the isothermal ε′ in E⊥c at f = 100 kHz and M in H//c and H⊥c, at T = 5, 10, 20, 35, 50, and 65 K. At 5 K, a butterfly-like shape of ε′ was observed with a strong magnetic hysteresis, with the absence of the step-like metamagnetic transition (Fig. 7(a)). The broadened feature of ε′ is compatible with the modulation of M in H//c with the narrow magnetic hysteresis described as small values of Mr = 1.22 μB/f.u. and the coercive field of Hc = 2.10 kOe (Fig. 7(g)). At 10 K, the butterfly-like shape of ε′ is maintained (Fig. 7(b)), but the magnetic hysteresis is considerably reduced. The central part of the hysteresis loop in H//c is extended as Mr = 2.73 μB/f.u. and Hc = 7.24 kOe (Fig. 7(h)), indicative of the reduced strength of the Er3+ spin order. In addition, the slight and elongated hysteretic behaviour of M in H⊥c emerges. As T increases further, the magnetic hysteresis in both ε′ and M is progressively reduced. At 65 K, just below TC, the sharp cusp of ε′ occurs at zero H with the hysteresis loop in M vanishing.

Figure 7
figure 7

Temperature evolution of the isothermal dielectric constant. (a–f) Isothermal ε′ in E⊥c, measured up to ±90 kOe in H//c at T = 5, 10, 20, 35, 50, and 65 K, respectively. The light red and orange curves indicate the data obtained by sweeping H from +90 kOe to −90 kOe, and by sweeping H from −90 kOe to +90 kOe, respectively. (g–l) Isothermal M in both H//c and H⊥c, measured up to ±90 kOe at T = 5, 10, 20, 35, 50, and 65 K, respectively.

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In summary, we explored the magnetic and magnetoelectric properties of mixed ferrimagnetic and multiferroic phases of single-crystalline double-perovskite Er2CoMnO6. The dominant Co2+ and Mn4+ superexchange interactions lead to the ferromagnetic order below TC = 67 K, aligned mainly along the c-axis. The long-range order of Er3+ moments below TEr = 10 K induces the ferrimagnetic order and magnetization compensation at TComp = 3.15 K, delicately balanced with the ferromagnetic Co2+/Mn4+ sublattice. The extended Stoner–Wohlfarth model depicts qualitatively the inverted magnetic hysteresis loop observed below TComp. The observation of electric polarization at low temperature is indicative of the presence of a small portion of a multiferroic phase simultaneously with the ferrimagnetic phase. The strong magnetoelectric correlation at the metamagnetic transition in the phase coexistence reveals the unique characteristic of the double perovskite compound, which offers crucial clues for exploring suitable materials for magnetoelectric functional applications.

Methods

Rod-shaped single crystals of ECMO with a typical size of 2 × 2 × 5 mm3 were grown by the conventional flux method with Bi2O3 flux in air. Er2O3, Co3O4, and MnO2 powders were mixed in the stoichiometric ratio for ECMO and ground in a mortar, followed by pelletizing and calcining at 1000 °C for 12 h in a box furnace. The calcined pellet was delicately reground and sintered at 1100 °C for 24 h. The same sintering procedure after regrinding was carried out at 1200 °C for 48 h. A mixture of pre-sintered polycrystalline powder and Bi2O3 flux with a ratio of 1:12 ratio was heated to 1300 °C in a Pt crucible. It was melted at the soaking T for 5 h, slowly cooled to 985 °C at a rate of 2 °C/h, and cooled to room T at a rate of 250 °C/h. The crystallographic structure and absence of a second phase were checked by the Rietveld refinement53 using the FullProf program54 for the power X-ray diffraction data. The data were obtained with a Rigaku D/Max 2500 powder X-ray diffractometer using Cu-Kα radiation.

The T and H dependences of DC M were examined by using a VSM magnetometer in a Quantum Design PPMS (Physical Properties Measurement System). The specific heat (C) was measured with the standard relaxation method in PPMS. The T and H dependences of ε′ were observed at f = 100 kHz using an LCR meter (E4980, Agilent). The H dependence of electric polarization (P) was obtained by the integration of magnetoelectric current measured with the H variation of 0.1 kOe/s after poling in a static electric field of E = 5.7 kV/cm.

In our extended Stoner–Wohlfarth model35,36,37,38, the magnetic energy density can be expressed as

$$\begin{array}{c}E=-\,{M}_{Er}H\,\cos \,{\theta }_{Er}-{M}_{Co/Mn}H\,\cos \,{\theta }_{Co/Mn}-J\,\cos ({\theta }_{Er}-{\theta }_{Co/Mn})\\ \,+\,{K}_{Er}{\sin }^{2}({\theta }_{Er}-{\phi }_{Er})+{K}_{Co/Mn}{\sin }^{2}({\theta }_{Co/Mn}-{\phi }_{Co/Mn})\end{array}$$
(1)

The first two terms represent the Zeeman energy density, where \({M}_{Er}\) and \({M}_{Co/Mn}\) are the effective magnetic moments of Er3+ and Co2+/Mn4+ ions, respectively, and \(\theta \) is the angle between the corresponding M and applied H. The third term denotes the exchange energy density, where the moments in the Er3+ and Co2+/Mn4+ sublattices tend to be ordered oppositely, and thus, the exchange coupling constant (\(J\)) is negative. The last two terms signify the densities of magnetocrystalline anisotropy energy (\({K}_{Er}\) and \({K}_{Co/Mn}\)) for Er3+ and Co2+/Mn4+ ions, respectively, where \(\phi \) is the angle between H and the magnetic easy axis. The energy density can be minimized to determine the direction of net M at an applied H by solving both \(\frac{\partial E}{\partial {\theta }_{Er}}=0\) and \(\frac{\partial E}{\partial {\theta }_{Co/Mn}}=0\). The estimated net M can be written as follows: \(M={M}_{Er}\,\cos \,{\theta }_{Er}+{M}_{Co/Mn}\,\cos \,{\theta }_{Co/Mn}\). The calculated hysteresis loop in Fig. 3(b) was obtained with \(J=-\,4.241\times {10}^{5}\,{\rm{J}}/{{\rm{m}}}^{3}\), \({K}_{Er}=8.482\times {10}^{4}\,{\rm{J}}/{{\rm{m}}}^{3}\), and \({K}_{Co/Mn}=2.545\times {10}^{5}\,{\rm{J}}/{{\rm{m}}}^{3}\).