Anisotropic and nonlinear magnetodielectric effects in orthoferrite ErFeO3 single crystals

In rare-earth orthoferrites, strongly correlated order parameters have been thoroughly investigated, which aims to find multiple functionalities such as multiferroic or magnetoelectric properties. We have discovered highly anisotropic and nonlinear magnetodielectric effects from detailed measurements of magnetoelectric properties in single-crystalline orthoferrite, ErFeO3. Isothermal dielectric constant varies in shapes and signs depending on the relative orientations between the external electric and magnetic fields, which may be ascribed to the spin-phonon couplings. In addition, a dielectric constant with both electric and magnetic fields along the c axis exhibits two symmetric sharp anomalies, which are closely relevant to the spin-flop transition, below the ordering temperature of Er3+ spins, TEr = 3.4 K. We speculate that the magnetostriction from the exchange couplings between Er3+ and Fe3+ magnetic moments would be responsible for this relationship between electric and magnetic properties. Our results present significant characteristics of the orthoferrite compounds and offer a crucial guide for exploring suitable materials for magnetoelectric functional applications.


Results and discussion
and b depict the crystallographic structures of EFO viewed from the a-and c-axes, respectively. The corner-shared octahedral units of Fe 3+ ions are strongly distorted due to the small radius of the Er 3+ 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 χ 2 = 3.29 (see Supplementary Information S1 for details).
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 Fe 3+ magnetic moments is known to occur at T N ≈ 640 K 21,22 , below which the net magnetic moment becomes aligned along the c axis, and starts to rotate into the a axis by 90° at T SR = 113 K, evidenced with a significant increase of χ a (Fig. 2a) and a reduction of χ c (Fig. 2c) below T SR . In Fig. 2c, after completing the spin reorientation at approximately 93 K, χ c , upon further cooling increases smoothly. The sharp peak of χ c at T Er = 3.4 K indicates the long-range antiferromagnetic ordering of Er 3+ magnetic moments aligned along the c axis. According to previous studies, the Er 3+ 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 Fe 3+ spins 19,[23][24][25] . In Fig. 2a, the net moment of the Er 3+ sublattice at a low T regime follows the direction of the applied magnetic field while the smaller net moment of the Fe 3+ sublattice is aligned in the opposite direction. As T is increased, the dominant ferrimagnetic ordering between the Er 3+ and Fe 3+ sublattices is identified by the compensated magnetization at T Comp = 46 K. Further decreasing the net moment for the Er 3+ sublattice induces a sudden reversal of total magnetization at approximately 61 K. In Fig. 2b, the overall behavior of χ b reveals only weak T variation, which indicates that the Er 3+ and Fe 3+ 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 temperatures 15,25,26 . 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 properties [27][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 Fe 2+ ions.
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, M a 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  (Fig. 2d). M b exhibits a smooth increase up to 9 T with a magnetization value of 3.25 μ B /f.u., without magnetic hysteresis (Fig. 2e). In Fig. 2f, M c rises sharply at ~ 0.34 T, identified by a sharp peak in the H derivative of M c , which indicates the spin flop transition of the Er 3+ and Fe 3+ magnetic moments. The inset shows the enlarged view of M c , which indicates the absence of a hysteretic behavior at the spin flop transition. M c increases gradually above 1.5 T and reaches the largest magnetic moment of 6.98 μ B /f.u. at 9 T.
The T-dependence of ε′ is displayed in Fig. 3a, measured along the c axis (ε c ′) at f = 500 kHz for H = 0 T. ε c ′ decreases monotonously from a high T regime and exhibits an abrupt decrease below T Er = 3.4 K that corresponds to the long-range ordering of Er 3+ magnetic moments. This suggests that the intrinsic magnetic ordering of the Er 3+ moments would influence strongly on ε c ′ 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 T Er , (Fig. 3b). In contrast, no anomaly of C/T was observed at T Comp in spite of the abrupt change of χ a (Fig. 2a). It involves only a sign change of χ a due to thermal fluctuation but is not relevant to an additional entropy change. In GdFeO 3 18 and DyFeO 3 17 , multiferroicity and magnetic-field-induced ferroelectricity emerge along the c axis below the ordering temperatures of Gd 3+ and Dy 3+ ions, respectively, due to the symmetric exchange strictions. However, the clear anomaly of ε c ′ at T Er in the EFO does not involve a ferroelectric polarization in pyro-and magnetoelectric current measurements, which indicates the absence of multiferroicity (see Supplementary Information S2 for details). A similar decreasing behavior in ε' was observed in Y 2 Cu 2 O 5 below the T of antiferromagnetically ordered Cu 2+ spins, T N = 13.2 K 31 .
In Fig. 4, the MD effect, described by the variation of ε' by applying H and defined as MD × 100 was measured at f = 500 kHz and T = 2 K along the a, b, and c axes (MD a (Fig. 4a-c),   (Fig. 4d-f), and MD c (Fig. 4g-i) at H a , H b and H c , 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. MD a varies slightly at H a with the value of approximately 0.1% at 9 T (Fig. 4a), and it shows only a negligible H b dependence (Fig. 4b). In Fig. 4c, a small peak in MD a was observed at low H c , after which MD a starts decreasing to a negative value with the change in slope at H c ≈ 2.0 T and reaches approximately − 0.25% at 9 T. Applying both H a and H b (Fig. 4d,e), the initial curve of MD b 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. MD b at 9 T was found to be 0.32 and 0.74%, respectively, for H a and H b . In contrast, the MD b (Fig. 4f) tends to behave similarly to MD a at H c (Fig. 4c), with the maximum variation of − 0.65% at 9 T. Starting from the linear decrease upon increasing H a , MD c changes in slope and shows a broad minimum at 3.7 T with − 0.4% variation (Fig. 4g). In Fig. 4h, MD c begins to increase linearly at H b ≈ 1.9 T and maintains the plateau mostly above 4 T. In Fig. 4i, the initial curve of MD c increases with a slight curvature at low H c regime and shows a kink at approximately 0.7 T, above which it reduces gradually, becomes almost linear above H c = 2.2 T, and crosses zero at H c ≈ 3.2 T. The maximum variation of MD c is found to be approximately − 0.47% at 9 T. Among a variety of MD responses, as shown in Fig. 4, MD c at H c appears to be strongly correlated to the isothermal M c . The T evolution of MD c examined this intriguing aspect at H c compared with the T dependence of dM c /dH c . Figure 5 displays the H c -dependence of MD c (Fig. 5a-f) and dM c /dH c (Fig. 5g-l), measured up to ± 9 T at T = 2, 2.5, 3, 3.5, 5, and 10 K. Additionally, dM c /dH c at 2 K is plotted in Fig. 5g for a precise comparison of the H c dependence of MD c at 2 K in Fig. 4a. dM c /dH c demonstrates two sharp peaks, which coincide with the spin-flop transitions and sharp features in MD c . At 2.5 K, the characteristics of both MD c and dM c /dH c at 2 K are almost maintained (Fig. 5b,h). At 3 K, the anomalies in MD c , shown in Fig. 5c, are considerably diminished as small kinks along with the reduction of dM c /dH c (Fig. 5i). In Fig. 5d, the kinks disappear at 3.5 K and a cusp occurs at H c = 0 T. The value of MD c at 9 T decreases continuously from − 0.47% at 2 K to − 0.74% at 3.5 K. At 5 and 10 K above T Er , the linear regime of MD c is progressively curved with further suppression of dM c /dH c . The highly nonlinear H c -dependence of MD c and its close correlation to dM c /dH c below T Er 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 Er 3+ and Fe 3+ magnetic moments along the c axis below T Er . This may lead to a change in phonon energies, which are relevant to the displacement modes of Er 3+ ions and consequently modifies ε c ' based on the Lydanne-Sachs-Teller (LST) relation 32,33 .
Although a positive and negative MD has been previously presented 34 , it is unconventional that both the MD effects arise in a single-phase material such as EFO, depending on relative orientations between the electric www.nature.com/scientificreports/ 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 ω ≈ S i · S j 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 materials [37][38][39][40] . Further, the spin-spin correlation function can be related to the magnetic contribution of heat capacity (C m ) obtained after subtracting the phonon contribution proportional to T 3 : where 8N A and J 1 denote the number of bonds per mole and exchange constant for the Er-Fe pairs, respectively, and 3 N A and J 2 do the number of bonds per mole by considering the double counting and exchange constant for the Er-Er pairs, respectively 37 . Recent experiment of low-T Raman spectroscopy for the GdFeO 3 reveals that Raman shift for the mode relevant to the motion of Gd 3+ ions occurs below the T Gd and is found to be ∆ω ≈ 1 cm −138 . In a theoretical work on the GdFeO 3 , J 1 = 0.03 meV and J 2 = 0.05 meV were also calculated 41 . 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 Sr 4 Ru 3 O 10 40 . 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 ErFeO 3 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   www.nature.com/scientificreports/ 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).