**2. Experiment**

The DyFeO3 polycrystalline sample was synthesized by the solid-phase synthesis method from Dy2O3 (99.99%) and Fe2O3 (99.99%), and the DyFeO3 single crystal was grown by using a Four Mirror Optical Floating Zone Furnace (Crystal Systems Corp., Salem, MA, USA). The crystal structure and purity of both the DyFeO3 powder (crushed single crystal) and the single-crystal samples were measured with an X-ray diffractometer (XRD, X'Pert MPD Powder-DY3734, PANalytical B.V., Almelo, NL) using Cu Ka radiation (λ = 1.5406 Å). The scan ranged from 10 to 90◦, the step size was 0.013◦, and the scan rate was 0.042◦/s. The directions of three principal axes were determined according to the Laue X-ray diffractometer measurement results. The low magnetic-field magnetization was measured by using a superconducting quantum interference device (SQUID VSM, Quantum Design, San Diego, CA, USA). Dynamic behaviors of both the *M* and the *P* were measured under a pulsed high magnetic field at the Wuhan National High Magnetic Field Center. The pulsed-high-magnetic-field *M* was detected by the standard inductive method employing two concentric pick-up coils connected in series with opposite polarity [7]. The electrical polarization *Pc* measurement schematic is shown in Figure S1 of the supplementary material. The silver electrodes are evenly distributed on the upper and lower surfaces of the sample. When the magnetic field is applied to the sample, the current signal (corresponding to the change in charge density induced by magnetic fields) in the sample is converted into an electrical signal (V) at the reference resistor (R), and then the V is obtained after being processed by a preamplifier. Finally, the change in the electric polarization induced by the pulsed magnetic fields is obtained.

### **3. Results and Discussion**

The XRD pattern of the DyFeO3 powder and its fitting by the general structure analysis system (GSAS) are shown in Figure 2a. All the diffraction peaks are well indexed by a distorted orthorhombic structure with *Pbnm*. No impurity peaks are observed within the diffraction resolution, indicating the single-phase nature of the sample. The lattice parameters *a* = 5.3031 Å, *b* = 5.5983 Å, and *c* = 7.6228 Å and the detailed crystal parameters are listed in the Table S1 of the supplementary material, which are close to the values in the Inorganic Crystal Structure Database (ICSD 27280). In the diffraction pattern of the single-crystal sample, only the (002), (004), and (006) diffraction peaks are observed, which confirms the high quality and accurate *c*-axis orientation of the DyFeO3 singlecrystal sample.

**Figure 2.** (**a**) The XRD patterns of DyFeO3 powder and single crystal (right inset) and the DyFeO3 single-crystal sample morphology (left inset). (**b**–**d**) The temperature dependence of the magnetization of the DyFeO3 single crystal with magnetic fields along the *a*-, *b*-, and *c*-axes, respectively. The partial magnification near the transition temperatures of ZFC and FC curves measured at 0.05 T are shown in the corresponding insets.

The temperature dependence of magnetization measured in various magnetic fields (0.01 T, 0.05 T, 1 T, and 5 T, respectively) is shown in Figure 2b–d. In the lower-temperature region (below ~60 K), the zero-field-cooled (ZFC) and field-cooled (FC) magnetization curves have slight deviation, and the difference is presented in the insets of Figure 2b–d. In the higher-temperature region (above ~60 K), the difference between ZFC and FC becomes indiscernible. When the magnetic field is applied along the *c*-axis (see Figure 2d), there is an obvious transition with the magnetization jumps of ~0.13 μB/f.u. at *T*SR(Fe) ~57 K, which is related to the spin-flop transition of the Fe3+-sublattice. As the temperature decreases to *TN*(Dy), an obvious drop can be observed (see the inset of Figure 2d), which indicates that the Dy3+-sublattice undergoes a transition from a paramagnetic state to an AFM state G*x*A*y*. Since the magnetic moment of the Ising Dy3+ ions is localized in the *ab* plane, it is difficult to disturb the magnetic field (0.05 T, along the *c*-axis) or change the direction of the magnetic anisotropy (or the anisotropy energy) of the Dy3+ spin. With the magnetic field increasing, *T*SR(Fe) moves to the low-temperature region. However, with the magnetic field applied along the *c*-axis, no obvious movement of *TN*(Dy) is observed, which confirms the strong magnetic anisotropy and localization in the *ab* plane of the Dy3+ spins. The temperature and magnetic field dependence of the transitions are shown in the magnetic phase diagram of Figure S2 in the supplementary material. In DyFeO3, there is the magnetic anisotropy of Dy3+ ions, field-induced spin flop of Fe3+ ions, temperature-driven spin reorientation of Fe3+ ions, AFM interaction between Dy3+ and Fe3+ ions, and thermal fluctuation [28], and these lead to the complex dependence of both *T*SR(Fe) and *TN*(Dy) on temperatures and magnetic fields.

The magnetization curves as functions of different magnetic fields along three principal axes are shown in Figure 3. Below *T*N(Dy) (taken 2 K as an example), when the magnetic field is applied along the *a*-axis (see Figure 3a)**,** a transition is observed at *B*C(Dy) ~0.8 T (labeled with a red arrow), which is attributed to the spin-flop transition of the Dy3+ sublattice (as shown in the inserted cartoon). With the magnetic field increasing, the magnetic field drives the spins of the Dy3+-sublattice toward the *a*-axis as much as possible, resulting in a sharp increase in *M.* With the magnetic field further increasing, the AFM coupling of Dy3+-sublattices may be partially broken. For the magnetic field parallel to the *b*-axis, similarly, a slightly smaller critical magnetic field of 0.5 T can be observed due to the smaller deviation (the angle ~33◦) of the Ising vector of the Dy3+ spins from the *b*-axis. Above 0.5 T, a saturated magnetic moment of ~8.3 μB/f.u. is obtained, which indicates that the Dy3+ moment was almost magnetized to saturation by the magnetic field (as shown in the inserted cartoon in Figure 3b). The saturated magnetization shows that the Ising Dy3+ moment is mainly localized in the *ab* plane with the *b*-axis as the easy axis [18]. In the case with the magnetic field along the *c*-axis, a magnetization jump (Δ*M*) of ~0.13 μB/f.u. is observed around 3 T. The value of the Δ*M* is the same as the value of the magnetization jump in the *M*-*T* curves shown in Figure 2d, which confirms that the transition is mainly associated with the spin-flop transition of the Fe3+-sublattice; the result is also consistent with previous studies [4]. In Figure 3c, two transitions are observed around 3 T and 3.2 T (labeled as black and red triangles, respectively). The lower one may be related to the spin-flop transition of the Fe3+ moment, while the higher one may be caused by the destruction of the AFM coupling between the FM components of the Dy3+ moment (induced by the magnetic field) and the Fe3+ moment (canted AFM) [29]. The magnetization behaviors measured in a pulsed higher magnetic field are shown in Figure 3d–f. Besides the low-field transitions, no additional transitions are induced up to 45 T, and the slopes of the *M*-*B* curves are almost constant, indicating that there is a strong AFM interaction in the Fe3+-sublattice [4].

**Figure 3.** (**a**–**c**) Magnetization as a function of the static magnetic field along *a*-, *b*-, and *c*-axes, respectively. (**d**–**f**) The magnetization curves measured along the *a*-, *b*-, and *c*-axes under pulsed high magnetic field, respectively. The curves in the (**b**,**e**,**f**) are offset for clarity. *J*AFM and *J*<sup>D</sup> are the AFM interaction strength and anisotropy energy of the Dy3+-sublattice, respectively.

In order to further investigate magnetic-field-induced transitions and the dynamic magnetization behavior of DyFeO3, the magnetization results were investigated by pulsed magnetic fields. In this work, the waveform of the pulsed magnetic field is shown in Figure 4a, which is a full-wave pulsed magnetic field (including four quadrants, QA, QB, QC, and QD) with a maximum field sweep rate of about 104 T/s. When the magnetic field is along the *a*-axis and the temperature is below 4.2 K (see Figure 4b), in the field-increasing branch (quadrant QA), the transitions resulting from the Dy3+-sublattice (marked with red arrows) and the field-induced spin flop in the Fe3+-sublattice (marked with black arrows) are observed. When the temperature ranges from 4.2 K to 50 K, only the transition, corresponding to the spin flop in the Fe3+-sublattice, could be observed, and its critical magnetic field decreases drastically with increasing temperature. Above 50 K, no obvious transition is observed (not shown). In the field-decreasing branch (quadrant QB), the transition field moves to the low-magnetic-field region (lower than 1 T). In the fieldincreasing and field-decreasing branches of the negative magnetic field (quadrants QC and QD), the magnetization behaviors are similar to those in QA and QB. As shown in Figure 4c, the magnetization behaviors of the magnetic field along the *b*-axis are similar to those along the *a*-axis.

**Figure 4.** (**a**) The waveform of the pulsed magnetic field. (**b**–**d**) Magnetization derivative (d*M*/d*B*) as a function of the magnetic field measured along the three principal axes. The inset in (**c**) is the enhancement of the d*M*/d*B* around the zero-magnetic-field regions. The curves of (**b**–**d**) are offset for clarity. QA, QB, QC, and QD are the four quadrants of the pulsed magnetic field; all the transitions are labeled with arrows.

In Figure 4b,c, the temperature dependence of the transition field of the spin flop in the Fe3+-sublattice is different above and below 4.2 K. Below 4.2 K, the critical field (corresponding to the spin flop of the Fe3+ moment) is essentially unchanged with the temperature increasing. Above 4.2 K, the transition field (labeled with arrows) has a strong dependency on temperature. These results suggest the pinning of the Dy3+-sublattice on the spin flop of Fe3+ ions. With the temperature increasing, the pinning gradually becomes weaker due to the destruction of the long-range order of the Dy3+-sublattice. The transition field (labeled with arrows) of the Dy3+-sublattice has a strong temperature dependence, which indicates that the direction of the magnetic anisotropy (or anisotropy energy) of Dy3+ spins may also be disturbed by the magnetic field in the *ab* plane. The critical field moves toward the lower-magnetic-field region with the temperature increasing. To present the moving trend more clearly, the phase diagrams with the magnetic field parallel to the *a*and *b*-axes are shown in Figure S2a,b of the supplementary material, respectively. When the magnetic field is applied along the *c*-axis, only the transitions related to the spin flop of Fe3+ ions are observed (see Figure 4d). Since the Dy3+ ion has strong Ising behavior and is localized in the *ab* plane, no obvious transition related to the Dy3+-sublattice was observed for lower magnetic fields. The critical field (corresponding to the spin-flop of the Dy3+ moment) moves toward the lower-magnetic-field region with the temperature increasing, and the temperature and magnetic field dependence of the critical behaviors are shown in Figure S2c of the supplementary material.

In the measurement of the electric polarization, Δ*Pc* is a relative value, and it shows the change in *Pc* induced by magnetic fields, and the applied magnetic field includes four quadrants: QA, QB, QC, and QD. As shown in Figure 5a, both the magnetic field and the electric field are parallel to the *c*-axis; with the magnetic field increasing (quadrant QA), Δ*Pc* jumps (labeled with red pentacles) are observed at *BP*(Fe) and a temperature below *TN*(Dy). At 2 K, the transition is observed in *BP*(Fe) ~3 T (labeled with a red pentacle), and the critical magnetic field is coincident with the result of magnetization measurement in the field-increasing branch (quadrant QA). As the temperature increases, the critical field *BP*(Fe) moves to the low-field region, and a similar temperature dependence of the transition field is also observed in the magnetization curves (see Figure 4d). In the field-decreasing branch (quadrant QB), Δ*Pc* becomes zero at *B*'*P*(Fe) (marked with black triangles). In quadrant QC (the field-increasing branch of the negative magnetic field), the transitions are also observed at −*BP*(Fe) (marked with black diamonds). In the field-decreasing branches of the negative-magnetic-field region (quadrant QD), a transition is observed at −*B*'*P*(Fe) (marked with crosses). Particularly, a metastable state (indicated by solid circles) and Δ*Pc* reversal (marked with a cross) are observed around 3.1 K in the negative-magnetic-field region. The transitions are affected by magnetic fields and temperatures, which may originate from the complicated interactions between the anisotropy energy of the Dy3+-sublattice, the coupling energy between the Dy3+ and Fe3+-sublattices, and Zeeman energy.

**Figure 5.** (**a**–**c**) Electric polarization as a function of pulsed magnetic fields, measured under the pulsed magnetic field along *c*-axis (**a**), *a*-axis (**b**), and *b*-axis (**c**), where the electric fields (*E* = 1.5 kV/cm) are along *c*-axis. (**d**) The magnetic field dependence of d*P*c/d*B* measured at various temperatures with the applied pulsed magnetic field along *b*-axis and *E* = 0. The curves are offset for clarity. The various symbols of red pentacle, black triangle, black diamond, cross, solid circle, blue pentacle, red triangle, and purple triangle represent the transitions in the curves. The sweep directions of the magnetic field are labeled by black arrows.

According to the exchange striction model, *Pc* is related to the spin flop of both the Fe3+-sublattice and the Dy3+-sublattice [18]; to reverse the *Pc*, it is necessary to change the phase (0 or π) of the magnetic vector of either the Fe3+ or the Dy3+ ions. The magnetic vector of the Fe3+ ions is directly connected to the direction of the wFM of the Fe3+-sublattice. Thus, the field-induced Δ*Pc* is observed when a large magnetic field is antiparallel to the *c*-axis. With the magnetic field decreasing and the temperature increasing, the interaction between the Dy3+ and Fe3+ ions, as well as the Zeeman energy, becomes weaker, and the magnetic anisotropy energies of the Dy3+-sublattices gradually dominate, which leads to the Δ*Pc* reversal and metastable polarization states in the negative field (quadrants QC and QD of Figure 5a). On the other hand, the strong magnetic anisotropy of the Dy3+ ions becomes dominant, which drives the Dy3+ spins to its easy axis and leads to the change in the exchange striction and Δ*Pc* reversal to a lower value. For the observed metastable state (marked with solid circles in Figure 5a), we assume that this is due to the spin-pinning effect of Dy3+ ions on the change in the wFM of the Fe3+-sublattice. The fact that metastable polarization behavior is more obvious when the temperature approaches *T*N(Dy) indicates that magnetic anisotropic Dy3+ is more easily magnetized by the magnetic field when the temperature approaches *T*N(Dy) than at lower temperatures.

For the pulsed magnetic field within the *ab* plane (as shown in Figure 5b–d), the change in the Dy3+ spins induced by the magnetic field also causes the change in exchange striction. The Δ*Pc*-*B* curves with magnetic fields along the *a*- and *b*-axes were measured (where the electrical polarization is along the *c*-axis). As shown in Figure 5b (the magnetic field along the *a*-axis) and Figure 5c (the magnetic field along the *b*-axis), the sign of Δ*Pc* is unchanged. This is totally different from the case with a magnetic field along the *c*-axis. The unchanged Δ*Pc* suggests the simultaneous flop of both the Fe3+ and Dy3+ spins when the magnetic field (in the *ab* plane) reverses. At 2 K, with the magnetic field along the *a*-axis and *b*-axis and the field increasing (quadrant QA), the field-induced Δ*Pc* are observed at *BP*(Dy) ~0.8 T (*a*-axis, marked with a blue pentacle) and 0.5 T (*b*-axis, marked with red triangles), respectively. Both the critical fields are lower than that along the *c*-axis, and with the increasing temperature, the transition fields move further to the lower magnetic fields. The effects of the temperature and magnetic field on the critical behaviors are shown in Figure S2 of the supplementary material.

In DyFeO3, the AFM interaction in the Dy3+-sublattice is weak and mainly localized within the *ab* plane. The lower magnetic field in the *ab* plane will disturb the direction of the magnetic anisotropy (or the anisotropy energy) of Dy3+ ions; that is, the Dy3+ moments are easily magnetized by the magnetic field, resulting in a higher magnetic field sensitivity. At zero fields, the magnetic vector (the Ising axis) of the Dy3+ ion deviates by about 33◦ from the *b*-axis (as shown in the inset of Figure 3a), which leads to the different critical field of the field-induced Δ*Pc* between the magnetic field along the *a*- and *b*-axes (see Figure 5b,c). In order to confirm the intrinsic effect of the magnetic field on polarization, the Δ*Pc* was also investigated with zero electric fields (see Figure 5d), and a weaker change of the electrical polarization was observed in the d*Pc*/d*B*-*B* curves, where the critical fields (marked with purple triangles) are coincident with those observed in the Δ*Pc*–*B* curves (see Figure 5c). With the temperature increasing, the transition peaks of the Δ*Pc* shift to the lower-field region. These experimental results indicate that the Δ*Pc* is an intrinsic behavior and can be induced by the magnetic field alone.
