Introduction

Research on novel magnetic materials aims to understand the relationship between microscopic magnetic orders and macroscopic physical phenomena along with the development of desired functional properties. Magnetic oxides composed of metal cations and oxygen anions have been widely explored because of the abundance of constituents and stability of compounds. In particular, some of the materials exhibit multiferroicity1,2 and magnetoelectricity3,4, which are characterized by cross-couplings between electric and magnetic order parameters. Such intriguing aspects provide a beneficial foundation for technological applications such as magnetoelectric data storage and sensors5,6,7,8,9. Recently, rare-earth orthoferrites (RFeO3; R: rare-earth ions) have received tremendous attention for materials research due to ultrafast spin switching10,11,12, large magnetocaloric effect13,14, reversible magnetic exchange-bias15,16, and magnetism-driven ferroelectricity17,18. In DyFeO3, a giant magnetoelectric tensor component is observed, and electric polarization is found to be reversible by switching the direction of the magnetic field17. In GdFeO3, the multiferroicity emerges below TGd = 2.5 K due to the symmetric exchange striction between Gd3+ and Fe3+ magnetic moments18. In addition, magnetoelectric domains, in which ferroelectric and canted-antiferromagnetic domain walls are strongly fastened, lead to cross-controls of electric polarization and magnetization by applying magnetic and electric fields, respectively.

Although the potential multiferroicity was suggested in ErFeO3 (EFO)19, no direct evidence has yet been presented. In a polycrystalline EFO, the dielectric responses with strong frequency dependence were observed in a broad temperature range20. For example, thermally-activated dielectric relaxation, ascribed to the polaron relaxation arising from carrier hoppings between Fe2+ and Fe3+ ions, was found at approximately 200 K, and relaxor-like broad dielectric anomalies was also presented at approximately 550 K. The ferroelectricity arising below TGd = 2.5 K in GdFeO3 suggests that the additional ordering of rare-earth ions in orthoferrites would provide a substantial modification to the magnetic properties, which are imperative when determining new functional characteristics. In this study regarding magnetoelectric properties in single-crystalline EFO, we find an absence of ferroelectricity but reveal strongly anisotropic and nonlinear magneto-dielectric effects below TEr = 3.4 K. The magnetodielectric (MD) behavior exhibits versatile magnetic-field dependences, which are possibly ascribed to the spin-phonon couplings.

Results and discussion

Figure 1a and b depict the crystallographic structures of EFO viewed from the a- and c-axes, respectively. The corner-shared octahedral units of Fe3+ ions are strongly distorted due to the small radius of the Er3+ ions. A detailed structure was obtained from the Rietveld refinement using the Fullprof Suite program for the X-ray diffraction pattern of the ground EFO, measured at room temperature. In Fig. 1c, the observed and calculated patterns are shown as open circles and solid lines, respectively. As suggested from the result, the EFO crystallizes in a Pbnm orthorhombic structure with lattice parameters, a = 5.2611 Å, b = 5.5835 Å and c = 7.5915 Å, with an agreement factor of χ2 = 3.29 (see Supplementary Information S1 for details).

Figure 1
figure 1

Crystallographic structure of ErFeO3. (a) and (b) Schematics of the crystallographic structure of perovskite ErFeO3 (EFO) from the a and c axes, respectively. The pink, grey, and yellow spheres represent Er3+, Fe2+, and O2− ions, respectively. (c) Observed (open circles) and calculated (solid line) powder X-ray diffraction patterns for the ground EFO single crystals. The blue curve represents the intensity difference between the observed and calculated patterns. The green short lines denote the Bragg positions.

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The anisotropic magnetic properties of EFO were measured by the T dependence of magnetic susceptibility (χ = M/H) at H = 0.01 T for a, b, and c axes upon warming after the zero-field-cooling process, as shown in Fig. 2a–c. The canted antiferromagnetic ordering of Fe3+ magnetic moments is known to occur at TN ≈ 640 K21,22, below which the net magnetic moment becomes aligned along the c axis, and starts to rotate into the a axis by 90° at TSR = 113 K, evidenced with a significant increase of χa (Fig. 2a) and a reduction of χc (Fig. 2c) below TSR. In Fig. 2c, after completing the spin reorientation at approximately 93 K, χc, upon further cooling increases smoothly. The sharp peak of χc at TEr = 3.4 K indicates the long-range antiferromagnetic ordering of Er3+ magnetic moments aligned along the c axis. According to previous studies, the Er3+ spins are also canted along the a axis and their larger net magnetic moment tends to align in the opposite direction of the net moment of Fe3+ spins19,23,24,25. In Fig. 2a, the net moment of the Er3+ sublattice at a low T regime follows the direction of the applied magnetic field while the smaller net moment of the Fe3+ sublattice is aligned in the opposite direction. As T is increased, the dominant ferrimagnetic ordering between the Er3+ and Fe3+ sublattices is identified by the compensated magnetization at TComp = 46 K. Further decreasing the net moment for the Er3+ sublattice induces a sudden reversal of total magnetization at approximately 61 K. In Fig. 2b, the overall behavior of χb reveals only weak T variation, which indicates that the Er3+ and Fe3+ spins do not tend to align along the b axis. It appears that the several different studies of magnetic properties on the single crystalline EFO reveal the sample dependence of magnetic transition temperatures15,25,26. In oxide compounds, oxygen contents vary in a broad range depending on the growth conditions and/or post annealing procedures in different gas environments, which would influence on the electronic and magnetic properties27,28,29,30. To verify the oxygen content of our EFO, we used an EPMA (Electronic Probe Micro-Analyzer, JEOL JXA-8530F). Each sample was measured at several points on the surface to confirm oxidation of the sample surface, which shows the oxygen content of the EFO crystals as 2.81. The oxygen deficiency on the crystal surface possibly results from the growth nature of flux method and may incorporate small amount of Fe2+ ions.

Figure 2
figure 2

Temperature- and magnetic-field-dependent magnetic properties of ErFeO3. (a)–(c) Temperature dependence of magnetic susceptibility, χ = M/H, shown for the EFO crystal for H = 0.01 T upon warming after zero-magnetic-field cooling along the a, b, and c axes, respectively. The vertical dotted lines indicate the spin-reorientation temperature (TSR = 113 K), compensation temperature (Tcomp = 46 K), and ordering temperature of Er3+ moments (TEr = 3.4 K). (d)–(f) Isothermal magnetization of the EFO crystal in H//a (d), H//b (e), and H//c (f), measured at T = 2 K up to 9 T. The inset of (d) shows the magnified view in the range of H =  ± 0.2 T of the hysteresis loop in H//a. The inset of (f) displays the enlarged view in the rage of H = 0–1 T in H//c.

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Isothermal magnetizations for the three different orientations were measured by applying H up to ± 9 T at T = 2 K. In the inset of Fig. 2d, Ma exhibits tiny magnetic hysteresis, consistent with the canted antiferromagnetism with the net magnetic moment along the a axis. The susceptible response of a small net magnetic moment leads to a narrow hysteresis loop with negligible amount of residual net magnetization and tiny coercive field as ~ 0.007 T, similar to the behavior of a soft ferromagnet. Upon further increasing H, Ma increases linearly up to approximately 1.3 T, whereas the slope slowly declines afterward. Ma at a maximum of H (9 T) is approximately 3.85 μB/f.u. (Fig. 2d). Mb exhibits a smooth increase up to 9 T with a magnetization value of 3.25 μB/f.u., without magnetic hysteresis (Fig. 2e). In Fig. 2f, Mc rises sharply at ~ 0.34 T, identified by a sharp peak in the H derivative of Mc, which indicates the spin flop transition of the Er3+ and Fe3+ magnetic moments. The inset shows the enlarged view of Mc, which indicates the absence of a hysteretic behavior at the spin flop transition. Mc increases gradually above 1.5 T and reaches the largest magnetic moment of 6.98 μB/f.u. at 9 T.

The T-dependence of ε′ is displayed in Fig. 3a, measured along the c axis (εc′) at f = 500 kHz for H = 0 T. εc′ decreases monotonously from a high T regime and exhibits an abrupt decrease below TEr = 3.4 K that corresponds to the long-range ordering of Er3+ magnetic moments. This suggests that the intrinsic magnetic ordering of the Er3+ moments would influence strongly on εc′ in the EFO. Additionally, the T-dependence of the heat capacity divided by the temperature (C/T) measured upon warming in zero H shows a sharp peak at TEr, (Fig. 3b). In contrast, no anomaly of C/T was observed at TComp in spite of the abrupt change of χa (Fig. 2a). It involves only a sign change of χa due to thermal fluctuation but is not relevant to an additional entropy change. In GdFeO318 and DyFeO317, multiferroicity and magnetic-field-induced ferroelectricity emerge along the c axis below the ordering temperatures of Gd3+ and Dy3+ ions, respectively, due to the symmetric exchange strictions. However, the clear anomaly of εc′ at TEr in the EFO does not involve a ferroelectric polarization in pyro- and magneto-electric current measurements, which indicates the absence of multiferroicity (see Supplementary Information S2 for details). A similar decreasing behavior in ε' was observed in Y2Cu2O5 below the T of antiferromagnetically ordered Cu2+ spins, TN = 13.2 K31.

Figure 3
figure 3

Temperature-dependent dielectric constant and specific heat. (a) Temperature dependence of dielectric constant, measured along the c axis (εc′) at H = 0 T. Inset shows the low-temperature regime of εc′. (b) Temperature dependence of the ratio of specific heat and temperature, C/T, measured at H = 0 T. Inset shows the low-temperature regime of C/T.

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In Fig. 4, the MD effect, described by the variation of ε' by applying H and defined as MD (%) = \(\frac{{\varepsilon^{{\prime }} \left( H \right) - \varepsilon^{{\prime }} \left( {0 {\text{T}}} \right)}}{{\varepsilon^{{\prime }} \left( {0 {\text{T}}} \right)}} \times 100\) was measured at f = 500 kHz and T = 2 K along the a, b, and c axes (MDa (Fig. 4a–c), MDb (Fig. 4d–f), and MDc (Fig. 4g–i) at Ha, Hb and Hc, respectively, up to ± 9 T. The MD curves vary in shapes and signs depending on the relative orientations of ε' and H. The full MD curves appear to be symmetric because the direction of each MD is indistinguishable in the applied AC electric field for the ε' measurements. MDa varies slightly at Ha with the value of approximately 0.1% at 9 T (Fig. 4a), and it shows only a negligible Hb dependence (Fig. 4b). In Fig. 4c, a small peak in MDa was observed at low Hc, after which MDa starts decreasing to a negative value with the change in slope at Hc ≈ 2.0 T and reaches approximately − 0.25% at 9 T. Applying both Ha and Hb (Fig. 4d,e), the initial curve of MDb exhibits a small bending at low-H and tends to increase linearly by exhibiting a change in slope and maintaining a positive value throughout the range of H. MDb at 9 T was found to be 0.32 and 0.74%, respectively, for Ha and Hb. In contrast, the MDb (Fig. 4f) tends to behave similarly to MDa at Hc (Fig. 4c), with the maximum variation of − 0.65% at 9 T. Starting from the linear decrease upon increasing Ha, MDc changes in slope and shows a broad minimum at 3.7 T with − 0.4% variation (Fig. 4g). In Fig. 4h, MDc begins to increase linearly at Hb ≈ 1.9 T and maintains the plateau mostly above 4 T. In Fig. 4i, the initial curve of MDc increases with a slight curvature at low Hc regime and shows a kink at approximately 0.7 T, above which it reduces gradually, becomes almost linear above Hc = 2.2 T, and crosses zero at Hc ≈ 3.2 T. The maximum variation of MDc is found to be approximately − 0.47% at 9 T.

Figure 4
figure 4

Magnetodielectric effect of ErFeO3. Magnetodielectric (MD) effect defined as MD (%) = \(\frac{{\varepsilon^{{\prime }} \left( H \right) - \varepsilon^{{\prime }} \left( {0 {\text{T}}} \right)}}{{\varepsilon^{{\prime }} \left( {0 {\text{T}}} \right)}} \times 100\) for (a)–(c) a (MDa), (d)–(f) b (MDb), and (g)–(i) c (MDc) axes, respectively, at Ha, Hb , and Hc up to ± 9 T and T = 2 K.

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Among a variety of MD responses, as shown in Fig. 4, MDc at Hc appears to be strongly correlated to the isothermal Mc. The T evolution of MDc examined this intriguing aspect at Hc compared with the T dependence of dMc/dHc. Figure 5 displays the Hc-dependence of MDc (Fig. 5a–f) and dMc/dHc (Fig. 5g–l), measured up to ± 9 T at T = 2, 2.5, 3, 3.5, 5, and 10 K. Additionally, dMc/dHc at 2 K is plotted in Fig. 5g for a precise comparison of the Hc dependence of MDc at 2 K in Fig. 4a. dMc/dHc demonstrates two sharp peaks, which coincide with the spin-flop transitions and sharp features in MDc. At 2.5 K, the characteristics of both MDc and dMc/dHc at 2 K are almost maintained (Fig. 5b,h). At 3 K, the anomalies in MDc, shown in Fig. 5c, are considerably diminished as small kinks along with the reduction of dMc/dHc (Fig. 5i). In Fig. 5d, the kinks disappear at 3.5 K and a cusp occurs at Hc = 0 T. The value of MDc at 9 T decreases continuously from − 0.47% at 2 K to − 0.74% at 3.5 K. At 5 and 10 K above TEr, the linear regime of MDc is progressively curved with further suppression of dMc/dHc. The highly nonlinear Hc-dependence of MDc and its close correlation to dMc/dHc below TEr would be ascribed to a magnetostrictive effect. In EFO, the magnetostriction that results in the lattice contraction can occur due to the exchange couplings between the Er3+ and Fe3+ magnetic moments along the c axis below TEr. This may lead to a change in phonon energies, which are relevant to the displacement modes of Er3+ ions and consequently modifies εc' based on the Lydanne-Sachs-Teller (LST) relation32,33.

Figure 5
figure 5

Temperature evolution of magnetodielectric effect along the c axis. (a)–(f) Hc dependence of MDc at T = 2, 2.5, 3, 3.5, 5 and 10 K. (g)–(l) Hc derivative of Mc and dMc/dHc measured at T = 2, 2.5, 3, 3.5, 5 and 10 K.

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Although a positive and negative MD has been previously presented34, it is unconventional that both the MD effects arise in a single-phase material such as EFO, depending on relative orientations between the electric and magnetic fields, as shown in Fig. 4. In addition to the possible magnetostrictive effect along the c axis, the spin-phonon couplings would be the possible cause for the versatile field dependences of the MD effects in EFO. As long-wavelength optical phonons are relevant for the frequency of f = 500 kHz used for the dielectric permittivity measurement, the spin-phonon coupling would be a plausible origin for the MD effects. The shift of optical phonon frequencies can be induced by the spin–spin correlation function as a result of the relation \(\Delta {\upomega } \approx {\uplambda }\left\langle {S_{i} \cdot S_{j} } \right\rangle\)35,36, where λ is the spin-phonon coupling constant and its typical value is known as a few cm−1 for the optical phonons in oxide materials37,38,39,40. Further, the spin–spin correlation function can be related to the magnetic contribution of heat capacity (Cm) obtained after subtracting the phonon contribution proportional to T3:

$$\left\langle {S_{i} \cdot S_{j} } \right\rangle = \frac{1}{{8N_{A} J_{1} + 3N_{A} J_{2} }}\int {C_{m} \left( T \right)dT} ,$$

where 8NA and J1 denote the number of bonds per mole and exchange constant for the Er-Fe pairs, respectively, and 3 NA and J2 do the number of bonds per mole by considering the double counting and exchange constant for the Er-Er pairs, respectively37. Recent experiment of low-T Raman spectroscopy for the GdFeO3 reveals that Raman shift for the mode relevant to the motion of Gd3+ ions occurs below the TGd and is found to be ∆ω ≈ 1 cm−138. In a theoretical work on the GdFeO3, J1 = 0.03 meV and J2 = 0.05 meV were also calculated41. Assuming similar results are expected in the EFO, the coupling constant based on the heat capacity data shown in Fig. 3b for the EFO was estimated as λ ≈ 3.2 cm−1. Moreover, close correlation between magnetic anisotropy and phonon spectra can be found in an example of Sr4Ru3O1040. Both increase and decrease of Raman shifts upon increasing magnetic fields were observed for the different field orientations. This implies that the spin correlations would be susceptible to the magnetic anisotropy and thus Raman shifts result in the positive or negative variations in a dielectric constant based on the LST relation. Our results motivate further optical experiments and theoretical studies to reveal the underlying mechanism for strongly anisotropic and nonlinear MD behaviors in EFO.

Conclusion

In summary, we have synthesized single crystals of orthoferrite ErFeO3 and explored their magnetic and magnetodielectric properties along different crystallographic orientations. We demonstrate highly nonlinear magnetodielectric responses with both positive and negative effects, which is rare as well as significant in a single-phase material, depending on the relative orientations between the electric and magnetic fields. Furthermore, the simultaneous anomalies of the dielectric constant and magnetic-field derivative of magnetization, corresponding to the spin-flop transition, were observed with the electric and magnetic fields along the c axis below TEr = 3.4 K. The symmetric exchange strictions, which act as mechanisms for multiferroicity and magnetoelectricity in GdFeO3 and DyFeO3, respectively, would be responsible for the magnetodielectric effect. The results of the intricate magnetodielectric properties demonstrated by ErFeO3 will encourage fundamental and applied research on magnetodielectric materials.

Methods

Single crystals of EFO were grown by the flux method utilizing PbO, PbF2, PbO2, and B2O3 fluxes in a high-temperature furnace. Er2O3 and Fe2O3 powders were prepared in a stoichiometric ratio and mixed with the flux compound. The mixture was heated to 1,290 °C in a platinum crucible for 16 h until it was completely dissolved. Then, it was cooled slowly to 850 °C at a rate of 2 °C/h, and further cooled to room temperature at a rate of 100 °C/h. Large EFO crystals with a cuboid shape and a length up to approximately 1 cm on one side were obtained. The crystallographic structure of the EFO crystals was confirmed using an X-ray diffractometer (D/Max 2500, Rigaku Corp.). The oxygen vacancy of the EFO crystals was measured utilizing a WDS (Wavelength Dispersive X-ray Spectrometer) in an EPMA (Electronic Probe Micro-Analyzer, JEOL JXA-8530F). The incident electron beam was applied with an acceleration voltage of 15 kV and a current of 20 nA. The composition ratio was determined by analyzing the characteristic x-rays of each element measured in the four WDS channels with different wavelength ranges. The T and H dependences of the DC magnetization were measured using a vibrating sample magnetometer at T = 2–150 K and H = –9–9 T with a Physical Properties Measurement System (PPMS, Quantum Design, Inc.). The T dependence of specific heat was measured with the standard relaxation method in the PPMS. The T and H dependences of the dielectric constant were observed at f = 500 kHz using an LCR meter (E4980, Agilent).