Magnetically induced ferroelectric polarization in rare-earth RMn2O5 manganites is believed to originate from the symmetric exchange striction associated with a specific antiferromagnetic phase in the low temperature (T) region and would be irrelevant with electropoling in the high-T paramagnetic-paraelectric phase region. In this work, we demonstrate that low-T pyroelectric polarization of GdMn2O5 single crystals along the b axis in the antiferromagnetic phase exhibits remarkable dependence on the electropoling history imposed in the high-T paramagnetic-paraelectric phase. In particular, the high-T electropoling results in a reversal of ferroelectric polarization in the low-T region, which can be flopped back by the electropoling being sustained in the low-T ferroelectric region. The existence of an electrically polarizable magnetic cluster state in the high-T paramagnetic-paraelectric region is proposed based on a combination of experimental observations and first-principles calculations. An intrinsic correlation between the low-T antiferromagnetic ordering and the high-T polarizable state is discussed. The present experiments unveil the emergent phenomena on multiferroicity of RMn2O5 and suggest an alternative scenario for electrocontrol of magnetism.

In the past decade, studies on multiferroics and underlying physics have experienced rapid progress,1–7 with a number of materials with remarkable magnetoelectric (ME) coupling.8–12 Current challenges include both the weak ME coupling in spite of high ferroelectric (FE) Curie temperature TC and large electric polarization P, as found in type-I multiferroics such as BiFeO3,13–16 and the low TC and small P in spite of strong ME coupling, as found in type-II multiferroics such as RMnO3 and RMn2O5 where R is small rare-earth species or Y.8,9,17–21 It is understood that the low TC and small P in type-II multiferroics are due to the microscopic mechanisms for polarization generation that are essentially the second-order or higher-order effects associated with specific magnetic order. The most-concerned mechanisms are: (1) the asymmetric exchange striction in noncollinear magnetic compounds where the Dzyaloshinskii-Moriya (DM) interaction is non-negligible and3,4 (2) the symmetric exchange striction in some collinear systems where spin-lattice coupling breaks the spatial inversion symmetry.22 

The orthorhombic RMn2O5 compounds are a representative family fitting to the second case.23,24 Nevertheless, it is puzzling that these compounds exhibit different FE and ME behaviors although they have highly similar lattice structures.8,11,25–30 First, R-Mn coupling can be different depending on the A-site ion R, which is strong in DyMn2O5 but negligible in GdMn2O5, making the P(T) dependence strongly material-related.11,31,32 Second, the magnetic field (H) dependence of polarization P can be distinctly different. It was found that polarization P is enhanced with increasing magnetic field H for compounds with R = Ho and Dy and suppressed for compounds with R = Tb, as well as sign reversed for compounds with R = Er and Gd.8,11,33,34 Third and unusually, the reported P(T) data for the same compound are sample-dependent.11,35 In some cases, the role of 4f electron and R-Mn coupling, that are believed to be responsible for these differences, was discussed.33,34

It comes to our attention that all the multiferroic measurements on this family of compounds were based on the scenario that the high-T lattice structure of RMn2O5 is nonpolar and ideally paramagnetic–paraelectric (PM–PE), and a nonpolar (Pbam) to polar (Pb21m) transition does not occur until the magnetic ordering from the high-T paramagnetic (PM) state. Therefore, it is believed that the system is in the ideally paramagnetic–paraelectric (PM–PE) state above the magnetic ordering point. The whole physics of multiferroicity is thus discussed in the low-T magnetically ordered region and has nothing to do with the high-T process. Nevertheless, for the P(T) data measured using the pyroelectric current method (so-called the pyroelectric polarization), one observes the distinctly different P(T) curves for R = Dy (P < 0) and for R = Ho and Tb (P > 0) while these samples were sufficiently electropoled in similar conditions.36 To avoid these confusing results, the field-cooling method was once utilized to measure the polarization and the possible uncertainties and deficiencies of this method were discussed.29 Nevertheless, these inconsistencies remain.

Furthermore, a recent work in 2015 claimed that the lattice structure of RMn2O5 at room temperature belongs to the polar Pm space group rather than nonpolar Pbam space group.37 Although several subsequent works followed this claim, the reliability of data on the structure and polarization is doubtable38–41 and the room temperature structure should fit the nonpolar Pbam space group. In spite of these disputes, possible high-T polarizable fluctuations in the high-T region and subsequent impact on the low-T multiferroicity may be an issue. In the other words, the high-T state may not be an ideal PM–PE state. This issue was initially discussed in an earlier work on TmMn2O5 with complicated Mn spin ordering and R-Mn coupling.42 In proceeding, it would be more promising if one takes GdMn2O5 as an alternative object for investigation. In this RMn2O5 family, GdMn2O5 shows a polarization as large as 0.1 μC/cm2 along the b axis (T< 30 K). Besides, a giant P-flop driven by magnetic field H, generating a change of ∼0.5 μC/cm2 was found, evidencing the large ME response.11 It is also noted that GdMn2O5 has non-negligible but relatively weak Gd–Mn coupling and the Mn spin ordering sequence from the PM–PE state is simpler than other systems.43 Our preliminary work on polycrystalline GdMn2O5 did reveal a high-T polarizable state that influences the low-T P(T) behaviors,44 based on which the present work on single crystals was stimulated.

The lattice structure of GdMn2O5 is schematically drawn in Fig. 1, where the major Mn-Mn exchanges are marked. One may refer to earlier literature on the structure.45,46 For the magnetic and dielectric properties, earlier works on GdMn2O5 single crystals reported the PM-PE state above TN1 ∼ 40 K, and an incommensurate (ICM) phase below TN1 followed by a commensurate (CM) ordering below TN2 ∼ 34 K at which a FE phase (FE1) is generated.35,43 This CM + FE1 phase region is narrow and replaced by another CM and FE phase (CM + FE2) below TC ∼ 29 K which is usually defined as the FE Curie point although the CM + FE1 phase below TN2 is also ferroelectric. Further cooling from TC does not induce more remarkable spin reordering although a peak of specific heat normalized by T, Cp/T, can be identified at TGd ∼ 10 K. This peak marks the reordering of Gd spins. The system enters the third FE phase (CM + FE3) below TGd although actually no distinct difference between FE3 and FE2 can be seen. This sequence of magnetic and FE ordering is schematically shown in Fig. 2(a) and one sees the complicated phase transition sequence.

FIG. 1.

A schematic of GdMn2O5 orthorhombic lattice structure where the three major exchanges J3, J4, and J5 are marked.

FIG. 1.

A schematic of GdMn2O5 orthorhombic lattice structure where the three major exchanges J3, J4, and J5 are marked.

Close modal
FIG. 2.

(a) A simplified drawing of the multiferroic phase diagram of GdMn2O5 along the T axis, where PM denotes paramagnetic, PE denotes paraelectric, ICM denotes incommensurate antiferromagnetic, CM denotes commensurate antiferromagnetic, FE denotes ferroelectric, FE1, FE2, FE3 are all ferroelectric phases, TN1 is the Mn spin ICM ordering point, TN2 (TC) are the Mn spin CM ordering point where the FE phase emerges defining the ferroelectric Curie point TC, TGd is the Gd spin reordering point with no essential change of magnetic structure at this point. It should be mentioned that several features appear around TN2 (TC), and thus no strict definition of them is made here. We crudely refer this point as the FE transition point and CM ordering point. (b) The emergence of two FE polarization components PMM, PGM, and total polarization Ptot as a function of T in a schematic manner. In the high-T PE–PM phase region, local magnetic clusters may appear generating local electric dipoles which may align randomly if the electropoling field EP = 0 or along the direction of EP so that a very weak macroscopic polarization p0 may be available in the PE–PM phase region. (c) The electropoling scenario employed in the present study, where Tend is the temperature at which the applied poling field EP is terminated. Subsequently, the sample is cooled down to 2 K in the short-circuited state before the pyroelectric current probing during the warming process from 2 K.

FIG. 2.

(a) A simplified drawing of the multiferroic phase diagram of GdMn2O5 along the T axis, where PM denotes paramagnetic, PE denotes paraelectric, ICM denotes incommensurate antiferromagnetic, CM denotes commensurate antiferromagnetic, FE denotes ferroelectric, FE1, FE2, FE3 are all ferroelectric phases, TN1 is the Mn spin ICM ordering point, TN2 (TC) are the Mn spin CM ordering point where the FE phase emerges defining the ferroelectric Curie point TC, TGd is the Gd spin reordering point with no essential change of magnetic structure at this point. It should be mentioned that several features appear around TN2 (TC), and thus no strict definition of them is made here. We crudely refer this point as the FE transition point and CM ordering point. (b) The emergence of two FE polarization components PMM, PGM, and total polarization Ptot as a function of T in a schematic manner. In the high-T PE–PM phase region, local magnetic clusters may appear generating local electric dipoles which may align randomly if the electropoling field EP = 0 or along the direction of EP so that a very weak macroscopic polarization p0 may be available in the PE–PM phase region. (c) The electropoling scenario employed in the present study, where Tend is the temperature at which the applied poling field EP is terminated. Subsequently, the sample is cooled down to 2 K in the short-circuited state before the pyroelectric current probing during the warming process from 2 K.

Close modal

Earlier works on GdMn2O5 suggested the existence of two local polarization components.11,47 One is component PMM, generated by the symmetric exchange striction associated with the Mn3+–Mn4+–Mn3+ blocks and the other is component PGM, generated by the symmetric exchange striction associated with the Gd3+–Mn4+–Gd3+ blocks. The Gd–Mn coupling is insufficiently strong to strictly align the Gd spin antiparallel to the Mn spins, allowing rotation of Gd spins by magnetic field.11 These local components each deviate slightly from the b axis but the sum of these components is aligned along the b axis, making the total polarization Ptot = PMM + PGM much larger than that of other RMn2O5 members. The PMM(T) and PGM(T) dependences can be schematically drawn in Fig. 2(b) where PMM emerges at T ∼ TC and PGM appears slightly below TC. Therefore, the Ptot(T) curve would exhibit a two-steplike pattern, as observed in earlier reports and also plotted in Fig. 2(b). More discussion on Fig. 2(b) will be shown later and in the caption.

In this work, we address the effect of electropoling in the high-T PM–PE phase on the low-T multiferroic behaviors of GdMn2O5 single crystals. This is no doubt an emergent phenomenon deserved but not yet for investigation.48 We obtain the high-quality data from single crystal samples, and propose a specific roadmap for probing the pyroelectric current I as a function of T. It will be shown that the I(T) and then P(T) data below TN1 exhibit distinctly different characters depending on the high-T electropoling in the PM–PE region. The existence of an electrically polarizable magnetic cluster state in this PM–PE phase region is proposed and supported by the first-principles calculations and measured broad bump of dielectric response. It is argued that these high-T polarizable magnetic clusters lead to the unusual ferroelectric behaviors in the low-T multiferroic region.

The GdMn2O5 single crystals were grown using the flux method, as reported previously.11,49 For the present experiments, the typical size of crystals was 1.0 mm in dimension, and the orientation of these crystals was determined using the Laue diffraction method. The lattice structure at room temperature is the high-symmetry Pbam orthorhombic phase, as determined by the X-ray diffraction and will transfer into the Pb21m structure at T ∼ TN2, thus allowing for a ferroelectric polarization along the b axis.28,50

The samples were submitted to a set of characterizations and the obtained results are reproducible from sample to sample. The dc magnetization M along the three main axes (a = [100], b = [010], c = [001]) at a measuring field of 2.0 kOe under the zero field-cooled (ZFC) mode and field-cooling (FC) mode was measured using the Quantum Design Superconducting Quantum Interference Device (SQUID). The specific heat Cp was measured using the Quantum Design Physical Property Measurement System (PPMS) that also provided the cryogenic environment for dielectric and ferroelectric measurements.

The samples were cut and polished into 0.1 mm thick plates normal to one of the three main axes for electrical measurements, using the silver pads as bottom and top electrodes. Each platelike sample was preannealed at 500 °C for 120 min in the air ambient before the silver pads were deposited on the top and bottom surfaces. This annealing was done to avoid possible external effect on the virgin sample and/or possible oxygen deficiency on the surface layers.

The dielectric constants along the three main axes were measured, respectively, using the HP4294A Precision Impedance Analyzer with the stimulating ac signal of 50 mV, during the sample cooling from room temperature. It was found that the sharp dielectric peak around TN2 was only observed along the b axis, consistent with earlier reports on the ferroelectric polarization direction. Subsequently, the high-T electropoling treatments and pyroelectric current measurements were done mainly along the b axis, unless stated elsewhere.

In our experiments, the electropoling scheme is plotted in Fig. 2(c). Each platelike sample was cooled down slowly at a rate of 1.0 K/min from room temperature to an assigned ending temperature Tend under a DC electric field EP  =  10 kV/cm unless stated elsewhere. After an isothermal equilibration at this Tend, the poling field was removed and the sample was electrically short-circuited for 30 min and then further cooled down to the lowest temperature (2 K) in the short-circuited mode. It implies no more electric poling of the samples below Tend, which is a scheme of specific advantage. The pyroelectric current I(T) was measured during the sample heating from 2 K up to 50 K at a fixed warming rate vt between 1.0 K/min and 5.0 K/min, using a high-resolution Keithley 6514 electrometer. It is noted that in our experiment, the current was probed without any electric bias, and thus no any contribution from leaky current was possible. The polarization P was evaluated from the I(T) data.

For a consideration of integrity, Tend in our measurements was varied step by step from 200 K down to 2 K in sequence, and our focus was on the region of Tend > TN1, i.e., on the poling only in the high-T PM–PE phase region. In these cases (Tend > TN1), the sample before entering the low-T CM  + FE phase region was no longer under any electric bias. Our common understanding is that a pyroelectric current is nonmeasurable unless the sample is prepoled electrically from the PE phase region into the FE phase region. Nevertheless, non-negligible low-T pyroelectric current was found even if Tend > TN1.

The reliability of the measured pyroelectric current data was checked carefully from various aspects by excluding those possible other sources. The most-concerned sources are from those possible charged defects. To our best knowledge, there are possibly two types of charged surface defects in our single crystals. The first type includes those charged defects generated during the sample polishing. If these defects would be generated, the charge-induced electric field would polarize the sample during the sample cooling and then low-T current signals during the warming sequence should be observed. Our checking excluded this possibility. The second type includes those injected or trapped defects during the electropoling. For this side effect, extensive discussion in the literature was given and we checked every particular to exclude its influence. For example, we set various warming rates vt and compared the measured I(T) curves which should be overlapped with each other if no other sources contribute. The validity of this technique and related data reliability are established and no more discussion on the details will be given here.

We first present the M(T) curves measured along the three main axes, as shown in Fig. 3(a) where the H/M vs T data are plotted too. For each case, no separation between the data under the ZFC and FC modes can be identified at the cooling and measuring fields of 2.0 kOe. It is seen that the M(T) along the b axis and c axis increases monotonously with decreasing T with no identifiable anomaly. However, the a axis M(T) curve does show a broad bump around ∼10 K, reflecting the 4f-spin reordering along the a axis, as evidenced previously.43 Since the Gd3+ has much larger moment than Mn3+/Mn4+ ions, the measured M is mainly from the contribution of Gd3+ spins and that from Mn ions is more or less covered. The Mn antiferromagnetic ordering cannot be clearly identified only from the M(T) data.

FIG. 3.

Measured physical quantities of GdMn2O5 as a function of T: (a) magnetization M along the three axes (a, b, c) respectively, with the measuring field of 2.0 kOe. No difference of the data between the ZFC and FC modes is shown. The H/M data measured under H//b axis mode are plotted too with a linear fitting of the high-T data by the Curie-Weiss law. The deviation of data from the linear dependence appears roughly at Tm ∼ 100 K. (b) (dM/dT) data along the a axis with two clear anomalies, defining TN2h and TN2l for the Mn spin ICM–CM transition and induced Gd spin ordering. Hereafter, we no longer distinguish the delicate difference among TN2l and TN2h, and denote them by TN2 unless stated specifically. (c) Specific heat data normalized by T, CP/T, where Tm, TN1, TN2, TGd are marked, respectively. (d) Dielectric constant data along the b axis, ɛb, at several frequencies.

FIG. 3.

Measured physical quantities of GdMn2O5 as a function of T: (a) magnetization M along the three axes (a, b, c) respectively, with the measuring field of 2.0 kOe. No difference of the data between the ZFC and FC modes is shown. The H/M data measured under H//b axis mode are plotted too with a linear fitting of the high-T data by the Curie-Weiss law. The deviation of data from the linear dependence appears roughly at Tm ∼ 100 K. (b) (dM/dT) data along the a axis with two clear anomalies, defining TN2h and TN2l for the Mn spin ICM–CM transition and induced Gd spin ordering. Hereafter, we no longer distinguish the delicate difference among TN2l and TN2h, and denote them by TN2 unless stated specifically. (c) Specific heat data normalized by T, CP/T, where Tm, TN1, TN2, TGd are marked, respectively. (d) Dielectric constant data along the b axis, ɛb, at several frequencies.

Close modal

Nevertheless, one can still find some features related to the magnetic ordering. First, a replotting of the H/M vs T data, following the Curie-Weiss law, reveals a clear deviation from the linear dependence at T = Tm ∼ 100 K, a roughly estimated value. The linear extrapolation down to H/M = 0 marks a negative T-intercept, suggesting the antiferromagnetic background.51 Furthermore, a plotting of the derivative of M(T) curve along the a axis over T, as shown in Fig. 3(b), does disclose two anomalies at T = TN2h ∼ 33 K (weak anomaly) and T = TN2l ∼ 26 K (strong anomaly). They are most likely generated from the ICM–CM transition for Mn spins and the induced Gd spin ordering. This feature is quite weak and cannot be clearly identified due to the much larger Gd3+ moment than Mn3+/Mn4+ moments. The two transitions indicate that Mn and Gd spins order into the CM structure roughly along the a axis, respectively, whereas no such anomalies in the Cp/T vs T curves along the b axis and c axis are observed. The two transitions allow the generation of PMM and PGM, respectively. Hereafter, for simplified consideration, we no longer distinguish the delicate difference between TN2l and TN2h, and denote them by TN2 unless stated specifically, since our attention is paid to the high-T region.43,51

The specific heat data, plotted by the Cp/T vs T curve in Fig. 3(c), also exhibit several anomalous features. First, a broad bump appearing around Tm echoes the feature in the H/M vs T curve. While the value of Tm may not have definite physical meaning and the bump is broad and weak, it suggests a broad T-range around Tm, say from 200 K down to TN1, within which the magnetic state may no longer be an ideal PM state, while it is believed in literature that the system above TN1 is in the PM state.43,46 An intermediate state consisting of small magnetic clusters embedded in the PM matrix likely appears inside this T-range and thus the magnetic state is distorted from ideal PM state. This state is dynamic likely with strong spin fluctuations. Second, a sharp peak and a weak kink appear respectively at TN1 and TN2. It is believed that the ICM ordering occurs at TN1 and the weak kink marks the ICM–CM ordering at TN2. Third, an additional peak at ∼5 K is the consequence of the Gd spin reordering and the position of the valley is defined as TGd ∼ 10 K.

The measured dielectric constants along the three main axes show different features. The dielectric constant data along the b axis, ɛb(T), at different frequencies (f) and plotted in Fig. 3(d), do show a clear and large anomaly from TN1 and peaked at TN2. This anomaly does mark the FE transition at which an electric polarization emerges.52 Besides this main peak, a small bump around TGd and a broad/weak bump around Tm can be seen but these features are minor with no convincing speculation at this stage. In addition, the dielectric constants along the a axis and c axis, ɛa(T) and ɛc(T), increase monotonously with increasing T over the whole T-range, although there are indeed extremely weak bumps around TN2.35 It should be mentioned that the broad/weak bump around Tm is a signature of the magnetic cluster fluctuations and we shall come back to this issue in Sec. IV.

Based on the data about the magnetic and ferroelectric transitions, one can discuss the measured I(T) and evaluated P(T) data measured along the b axis if not stated elsewhere. The data show high quality with negligible background noise. We present in Fig. 4(a) the data with Tend = 2 K, i.e., the sample was electrically poled over the whole T-range. Here, four I(T) curves are plotted, and three were obtained by setting vt = 1.0, 3.0, 5.0 K/min at EP = 10 kV/cm and one by setting vt = 3.0 kV/cm at EP = −10 kV/cm, respectively. A quantitative comparison of these curves clearly excludes any dominant contribution of charge release other than the pyroelectric effect. For details, it is shown that nonzero current signals are detectable from 2 K until ∼33 K above which the current signals are smaller than 0.05 pA. All the I(T) curves have two consecutive sharp peaks around ∼31 K (∼TC). For details, the starting point is TN2h ∼ 33 K, the first peak is at TN2m ∼ 31 K, and the second peak is at TN2l ∼ 26 K. These peaks remain nonshifted for different vt and the area under curve is proportional to vt, indicating that the current is indeed from the pyroelectric release without other contributions.

FIG. 4.

Measured pyroelectric current I and evaluated polarization P along the b axis as a function of T, respectively. (a) The I(T) curves measured at various EP and vt, (b) the P(T) curves evaluated at EP = ± 10 kV/cm, respectively.

FIG. 4.

Measured pyroelectric current I and evaluated polarization P along the b axis as a function of T, respectively. (a) The I(T) curves measured at various EP and vt, (b) the P(T) curves evaluated at EP = ± 10 kV/cm, respectively.

Close modal

Given the dominant contribution from the pyroelectric effect, one can check the polarization reversal upon the reversed poling field. The evaluated P(T) curves at EP = ± 10 kV/cm are plotted in Fig. 4(b). They are symmetric with respect to the axis P = 0. The two-steplike FE transition around TN2 (from TN2h to TN2l) is identified and no other remarkable anomaly is observable. Below TN2l, the polarization continues to increase with decreasing T and becomes much less T-dependent or saturated.

Now, we look at the evolutions of I(T) and P(T) curves with varying Tend. The experiments were carried out following the sequence shown in Fig. 2(c) step by step, starting from T = 200 K. As T > 200 K, the sample was leaky and a poling field of EP = 10 kV/cm can be applied when T was lower than 200 K. Therefore, for each cycle of measurements at a fixed Tend, the sample was warmed to room temperature before cooling-down to 200 K and then the electropoling was applied. For Tend = 2 K to Tend = 140 K, some of the measured curves are presented in Figs. 5(a)5(k), given vt = 3 K/min.

FIG. 5.

(a)–(k) The I(T) and P(T) curves measured along the b axis at different Tend as marked, where EP = 10 kV/cm, vt = 3 K/min. (l) Evaluated polarization P(T = 2 K) as a function of Tend, marking the polarization reversal at Tend ∼ 50 K.

FIG. 5.

(a)–(k) The I(T) and P(T) curves measured along the b axis at different Tend as marked, where EP = 10 kV/cm, vt = 3 K/min. (l) Evaluated polarization P(T = 2 K) as a function of Tend, marking the polarization reversal at Tend ∼ 50 K.

Close modal

Here, instead of describing in detail each set of I(T) and P(T) curves, we highlight the similarity and difference among these curves. First, each I(T) curve shows a group of sharp peaks around TN2, and some peaks are positive and the others are negative. These peaks are sharp and mark the appearance of PMM and PGM, and in most cases they are both positive or both negative, so are PMM and PGM. They are aligned along the same direction for each case, marking the two-step FE transition around TN2. Second and surprisingly, nonzero macroscopic polarization is observed in the low-T region, although the sample is electrically poled only in the PE phase region. This phenomenon has never been possible in conventional ferroelectrics because a macroscopic polarization can be detected only if the sample under poling enters the FE phase region. For conventional ferroelectrics, this is understandable because the domains in the FE phase region are randomly oriented if the sample is not prepoled electrically. Here, a nonzero polarization is still detectable even if Tend100KTN2(TC) while EP is only ∼10 kV/cm.

Third and even more surprisingly, one sees a positive–negative reversal of the P(T) curve in response to increasing Tend. In detail, we take the value of P at T = 2 K, P(T = 2 K), as a function of Tend and plot the data in Fig. 5(l). The P(T = 2 K) is positive as Tend < ∼ 50 K and zero as Tend > 140 K, while it is negative at 50 K < Tend < 140 K. A reversal occurs at Tend ∼ 50 K. This polarization reversal strongly suggests some unknown physics in the PM–PE region, which controls the ferroelectric behaviors in the low-T FE phase. It is seen that the positive maximal P(T = 2 K) value is ∼0.27 μC/cm2 at Tend = 2 K, much larger in magnitude than the negative maximal P(T = 2 K) value of ∼−0.05 μC/cm2 at Tend = 85 K. This is reasonable since the sample in the FE state should not be in a fully poled state if Tend is high, and thus the macroscopic polarization is small. Instead, the polarization is much larger if Tend is inside the FE phase region.

The emergent phenomenon, as described above, is unusual to our conventional understanding of ferroelectricity. To understand this behavior, the first issue is the mechanism for polarization components PMM and PGM, the second is why an electropoling in the PE region induces a polarization in the FE region, and the third is why the polarization is negative when the poling field in the PE region is positive. It can be a challenge to resolve all these issues in this work, considering the complexity of underlying physics. We intend to present a preliminary scenario as a reference for future investigation.

The first issue was once discussed in earlier work and a multiferroic model for the two polarization components was proposed.11 Since the net polarization is aligned along the b axis, one may map the low-T lattice and magnetic structures of GdMn2O5 onto the ab-plane, as shown in Fig. 6(a) so that the underlying mechanism can be illustrated more clearly.

FIG. 6.

A schematic of the measured spin structure below TN2, referring to neutron scattering data available in the literature. (a) The spin structure projected on the ab-plane with the square pyramidal Mn3+–O2− unit and octahedral Mn4+–O2− unit shown in (b). The structural block A, composed of one Mn4+–O2− octahedra connected by two Mn3+–O2− pyramids roughly along the b axis, is shown in (c). The structural block B, composed of one Mn4+–O2− octahedra, connected by two Gd3+ roughly along the b axis, is shown in (d). The proposed polarizations PMM and PGM generated by the two types of blocks due to the symmetric exchange strictions, are labeled in (c) and (d), respectively.

FIG. 6.

A schematic of the measured spin structure below TN2, referring to neutron scattering data available in the literature. (a) The spin structure projected on the ab-plane with the square pyramidal Mn3+–O2− unit and octahedral Mn4+–O2− unit shown in (b). The structural block A, composed of one Mn4+–O2− octahedra connected by two Mn3+–O2− pyramids roughly along the b axis, is shown in (c). The structural block B, composed of one Mn4+–O2− octahedra, connected by two Gd3+ roughly along the b axis, is shown in (d). The proposed polarizations PMM and PGM generated by the two types of blocks due to the symmetric exchange strictions, are labeled in (c) and (d), respectively.

Close modal

It is accepted that PMM is generated by the Mn3+–Mn4+–Mn3+ symmetric exchange striction associated with chainlike blocks each consisting of one Mn3+O5 pyramid unit, one Mn4+O6 octahedral unit, and one Mn3+O5 pyramid connected in sequence, as schematically drawn in Fig. 6.31 In the PM–ICM transition at TN1, the Mn spins are aligned from the PM state into an antiferromagnetic order along the a axis with small out-of a axis canting, and subsequently the ICM–CM transition around TN2 occurs, characterized by the smaller peak of (dM/dT)aaxis at TN2h, as shown in Fig. 3(b). The Mn spin structure is drawn in Fig. 6(a). Simultaneously, component PMM emerges via the mechanism in Fig. 6(c) where a Mn3+–Mn4+–Mn3+ block (block A) with the ↑↑↓ or ↓↓↑ configuration is drawn. In this block, the Mn3+ and Mn4+ ions with roughly parallel spin alignment tends to be away from each other, and the Mn3+ and Mn4+ ions with roughly antiparallel spin alignment tends to be close with each other. If the Mn4+ ion is set as the fixed point (zero point), a local polarization pointing downward, i.e., PMM, is generated.

In spite of the Gd–Mn coupling in GdMn2O5 is relatively weak, the Mn spin ordering can induce the Gd spin ordering which occurs at a lower T, marked by the larger peak of (dM/dT)aaxis at TN2l, as shown in Fig. 3(b). As proposed in earlier work,31 the Gd3+-Mn4+-Gd3+ chainlike block, as shown in Fig. 6(d), may generate a local upward or downward polarization. The macroscopic component over the whole lattice would generate a polarization PGM which is parallel to PMM, making the total polarization Ptot = PMM + PGM. Here, it should be mentioned that for RMn2O5 systems with strong R-Mn coupling, PGM may not be parallel to PMM and a ferrielectric behavior was observed for DyMn2O5 where PMM is antiparallel to PGM.31 

The second and third issues are challenging and in fact the underlying mechanism is complicated. The high-T PM–PE phase must be an electrically polarizable state that is the source or seed for the domain alignment in the low-T FE phase region, resulting in the macroscopic polarization. Along this line, an earlier theoretical investigation based on the first-principles calculations comes to our attention.53 This work predicted that for an RMn2O5 structure where the 4f-spin's role is not considered, the polarization (here it is PMM) contains the ionic contribution Pion and electronic contribution Pele. Unfortunately, the two contributions are antiparallel, leading to a largely canceled polarization that is consistent with measured value. A basic character here is that Pion and Pele are on the same order of magnitude of ∼μC/cm2 but Pion is larger than Pele in magnitude.

Nevertheless, the ionic polarization Pion cannot exist in the PE region which is a high-symmetry Pbam phase, but the electronic polarization Pele can still exist in this high-symmetry phase with a proper magnetic structure. In the other words, if the PE phase region contains local magnetic clusters that have the ↑↑↓ or ↓↓↑-like configuration, these clusters may exhibit local polarization Pele. These are the basic ingredients of physics for understanding the emergent phenomenon in the present work. In fact, the M(T) and Cp/T data shown in Fig. 3 do show a broad anomaly around Tm ∼ 100 K, hinting the existence of magnetic clusters in the PM–PE phase region, as discussed previously. These clusters may be sufficiently small in this wide region which is macroscopically occupied by the PM and PE phase matrix with embedded magnetic clusters.

For a better interpretation of the existence of Pele, we consider a high-symmetry Pbam lattice where the Mn spins take the CM order, same as the low-T CM phase. Such an assumption is made only for the convenience of subsequent computation, and in fact we have no evidence for the CM magnetic structure of those magnetic clusters. In the other words, we don't believe that the spin configuration of those magnetic clusters in the high-T PM phase region follows strictly the CM configuration. Certainly, no ionic polarization Pion is allowed in such a lattice. We perform the first-principles calculations on this lattice and investigate whether a nonzero Pele exists or not. Our calculations are based on the generalized gradient approximation (GGA) with the Perdew-Burke-Ernzerhof (PBE) functional54 using the projector augmented wave (PAW) method55 as implemented in Vienna ab initio simulation package (vasp).56 The 5p65d16s2, 3p63d54s2 and 2s22p4 valence electron configurations for Gd, Mn and O atoms are used respectively, and the Gd-4f electrons are treated as core electrons. The cutoff energy for the plane wave basis is set at 550 eV. A Γ-centered Monkhorst-Pack k-point sampling method with a 2 × 4 × 6 k-mesh is used. The convergence thresholds are set at 1.0 × 10−6 eV in energy and 0.005 eV/Å in force. The strong on-site Coulomb interaction on the Mn 3d states is treated by the DFT + U scheme57 and the effective U, expressed as Ueff, is set to 5.2 eV, which have been successfully used in the manganese oxide calculations.58 We has a pretesting on the cases for Ueff = 2.0 eV to 5.2 eV, and only a slight difference in polarization is found. The polarization is calculated using the Berry phase method.59 Here, only the collinear magnetism without the spin-orbit coupling is considered.

The structure used in the calculations is constructed by extending the unit cell of GdMn2O5 in the Pbam space group to a 2 × 1 × 1 supercell. The lattice constants are taken from experimentally determined values. We first optimize the supercell with ferromagnetic (FM) configuration for Mn atoms, which has the centrally inversion symmetry. Then the atomic positions are fixed to maintain the central symmetry. Subsequently, the antiferromagnetic (AFM) configuration is used for calculating the polarization.11 The calculated results show that both Pion and Pele along the a- and c-axes are zero. While the ionic polarization along the b axis, as expected, must be zero, a remarkable electronic polarization Pele ∼ 0.20 μC/cm2 along the b axis is obtained. This confirms that nonzero electronic polarization can be available in the “PM–PE phase” of Pbam symmetry for RMn2O5.53 

For a clear illustration, we present in Figs. 7(a) and 7(b) the charge distributions in the Mn3+-O2− plane near Mn4+ ions, given the FM order and collinear AFM order as reported experimentally.23 A first glance at the charge pattern shows no difference in the charge profile between Figs. 7(a) and 7(b), or the difference between them is too weak to be identifiable. This is reasonable since the absolute charge density is so large in comparison with the charge difference associated with the possible electronic polarization (Pele ∼ 0.20 μC/cm2). In order to illustrate clearly the charge profile difference associated with Pele, we extract the charge difference between the two magnetic configurations and the results are plotted in Fig. 7(c). Since the FM configuration corresponds to the centrosymmetric case and the charge profile distribution is of high symmetry and can be set as a reference distribution, so that any deviation of the charge distribution representing a reduction in charge distribution symmetry can be obtained by extracting the charge profile difference between the FM and AFM lattices. This difference should be the electronic polarization. In fact, Fig. 7(c) does show an asymmetric profile in the charge density difference along the b axis, where the yellow contours indicate the excess electron density while the blue contours mark the deficient electron density. The red arrow indicates the local electric polarization p that is purely electronic. Here, it should be mentioned that the charge profile associated with such an electronic polarization can also be obtained by extracting the difference between the charge profile and the spatially averaged charge density for this AFM lattice unit, and the as-obtained charge profile is similar to that shown in Fig. 7(c).

FIG. 7.

The calculated charge (ρ) distributions in the Mn3+–O2− plane near Mn4+ ions for the chosen structural unit, given the ferromagnetic (FM) order (a) and collinear antiferromagnetic (AFM) order (b), respectively. The lattice symmetry is fixed by the Pbam group. The magnetic structure in (b) is taken from the experimentally determined spin configuration, ignoring the noncollinear component. The difference in charge profile (ρAFM–ρFM) between the FM and AFM lattices is plotted in (c), where the excess negative charge in the upper region (marked with ϴ) and excess positive charge in the bottom region (marked with ⊕) are shown. Clearly, a local polarization p as indicated by the red arrow is generated from the electronic polarization.

FIG. 7.

The calculated charge (ρ) distributions in the Mn3+–O2− plane near Mn4+ ions for the chosen structural unit, given the ferromagnetic (FM) order (a) and collinear antiferromagnetic (AFM) order (b), respectively. The lattice symmetry is fixed by the Pbam group. The magnetic structure in (b) is taken from the experimentally determined spin configuration, ignoring the noncollinear component. The difference in charge profile (ρAFM–ρFM) between the FM and AFM lattices is plotted in (c), where the excess negative charge in the upper region (marked with ϴ) and excess positive charge in the bottom region (marked with ⊕) are shown. Clearly, a local polarization p as indicated by the red arrow is generated from the electronic polarization.

Close modal

It is now understandable that the Pbam GdMn2O5 lattice with the magnetic clusters can accommodate a nonzero electronic polarization (Pele) even if the ionic polarization (Pion) is zero. The existence of such magnetic clusters in the high-T PM-PE phase is hinted by the data shown in Fig. 3, considering the antiferromagnetic background. This allows us to propose a physical scenario drawn in Fig. 8. In Fig. 8(a) are drawn two small clusters as examples and each cluster consists of the Mn3+–Mn4+–Mn3+ blocks that have the local ↑↑↓ or ↓↓↑-like configuration. Consequently, an electric dipole is created with such a block. These clusters are randomly aligned and no macroscopic polarization is available if no poling field is applied in this PE phase. Therefore, one has Pele ∼ 0 and Pion = 0 if EP = 0. However, if a nonzero EP is applied, the local electric dipole of each cluster will be aligned with the direction of EP. The poling process is illustrated in Fig. 8(b), and thus one has Pele > 0 and Pion = 0. This state can be called the electrically polarizable state, and of course, this state is dynamic upon external field or varying temperature. For instance, with decreasing T, these clusters may grow and interact with each other, and then long-range magnetic and ferroelectric orders gradual develop.

FIG. 8.

The proposed evolution of magnetic and polarization structures upon high-T electropoling. (a) Magnetic clusters randomly embedded in the PM matrix at EP = 0 and these clusters take the spin alignment similar to that in the low-T CM phase and thus have local polarizations from the electronic contribution. However, in statistics, Pele = 0 and Pion = 0. (b) Under EP > 0, pointing downward, these clusters are reorganized so that their local polarizations are aligned along the direction of EP. In statistics, Pele > 0 but Pion = 0. (c) At T ∼ TN2 (TN1) < Tend, the lattice symmetry changes from the nonpolar Pbam structure to the polar Pb21m group, and the ionic polarization Pion is generated in parallel to Pele. Since Pion > Pele in magnitude, the total polarizations of these clusters point upward and the macroscopic polarization PMM are negative. (d) At TTN2(TN1)<Tend, the FE domains are randomly aligned except those clusters frozen from the high-T poling, the macroscopic polarization PMM is only contributed from those clusters and is thus negative. (e) At T ∼ TN2 (TN1) > Tend, the lattice symmetry changes from the nonpolar Pbam structure to the polar Pb21m group, and the ionic polarization Pion is generated in parallel to Pele. Since Pion > Pele in magnitude, the total polarization of these clusters is reversed by EP and point downward and the macroscopic polarization PMM is positive. (f) At TTN2(TN1)>Tend, the FE domains including those clusters frozen from the high-T poling are all aligned along the direction of EP, and the macroscopic polarization PMM is thus positive.

FIG. 8.

The proposed evolution of magnetic and polarization structures upon high-T electropoling. (a) Magnetic clusters randomly embedded in the PM matrix at EP = 0 and these clusters take the spin alignment similar to that in the low-T CM phase and thus have local polarizations from the electronic contribution. However, in statistics, Pele = 0 and Pion = 0. (b) Under EP > 0, pointing downward, these clusters are reorganized so that their local polarizations are aligned along the direction of EP. In statistics, Pele > 0 but Pion = 0. (c) At T ∼ TN2 (TN1) < Tend, the lattice symmetry changes from the nonpolar Pbam structure to the polar Pb21m group, and the ionic polarization Pion is generated in parallel to Pele. Since Pion > Pele in magnitude, the total polarizations of these clusters point upward and the macroscopic polarization PMM are negative. (d) At TTN2(TN1)<Tend, the FE domains are randomly aligned except those clusters frozen from the high-T poling, the macroscopic polarization PMM is only contributed from those clusters and is thus negative. (e) At T ∼ TN2 (TN1) > Tend, the lattice symmetry changes from the nonpolar Pbam structure to the polar Pb21m group, and the ionic polarization Pion is generated in parallel to Pele. Since Pion > Pele in magnitude, the total polarization of these clusters is reversed by EP and point downward and the macroscopic polarization PMM is positive. (f) At TTN2(TN1)>Tend, the FE domains including those clusters frozen from the high-T poling are all aligned along the direction of EP, and the macroscopic polarization PMM is thus positive.

Close modal

When the electropoling is ended at a Tend > TN1, i.e., the sample poling is terminated in the PE region, some of those aligned clusters will be eventually frozen into the FE region with decreasing T, generating a macroscopic electronic polarization Pele. Once the system enters the FE region that has the Pb21m symmetry, ionic polarization Pion that is larger than but opposite to Pele emerges. Consequently, the measured polarization is actually polarization PMM = Pion + Pele < 0. As discussed above, since polarization PGM is induced by PMM and parallel to PMM, the total polarization would be negative, as schematically drawn in Fig. 8(c).

It should be mentioned that, as T < TN2, those well-aligned and frozen clusters only occupy a fraction of the whole sample and the other volume is occupied with randomly oriented FE domains. Therefore, the measured polarization is small in magnitude in spite of negative in sign. For simplicity consideration, we present in Fig. 8(d) the domain structure where only two types of domains, upward and downward domains, are sketched. In each domain are there antiparallel Pion and Pele, resulting in polarization PMM parallel to Pion in this domain.

When the electropoling applied to the sample sustains into the FE region, the situation is very different. We first look at the case of T ∼ TN2, at which the magnetic clusters grow in size and in degree of ordering, and then eventually a long-range of magnetic order is developed. The ionic polarization Pion is generated. Since the sample is under the electropoled state and Pion is larger than Pele in magnitude and opposite to Pele in sign, the poling field will switch these FE domains, leading to a reversal of Pion and thus Pele. Consequently, the effective polarization PMM = Pion + Pele becomes positive. This is the reason why we measured a positive polarization at Tend ∼ TN2 and below, as shown in Fig. 8(e). When Tend falls far below TN2, the whole sample under the electropoling field can be of single-domain, as shown in Fig. 8(f). In this case, the measured polarization is much larger than that case shown in Fig. 8(d).

The discussion in above two subsections gives a qualitatively reasonable scenario with which the P(T = 2 K) behavior as a function of Tend, as shown in Fig. 5(l), can be understood. It should be mentioned that the positive-to-negative reversal of P(T = 2 K) occurs at Tend ∼ 50 K. This value should not be much concerned in quantitative sense, considering the measurement uncertainties and the fact that this is only a preliminary and simplified scenario. A quantitative comparison seems to be overcritical at this stage.

To this stage, this scenario seems to explain the observed phenomena in a crude and qualitative way. Nevertheless, it can still be questioned from various aspects. First, one may question the existence of electrically polarized magnetic clusters because the magnetic state around Tm is believed to be paramagnetic. Nevertheless, RMn2O5 manganites are typically strongly correlated systems and the short-range ordered clusters in the paramagnetic phase is highly possible. The second and also the most critical question is what is the evidence with the existence of such electronic polarization Pele in the high-T PE region of the Pbam symmetry. A direct observation of these clusters experimentally is beyond the scope of this work, but any indirect signature of them would be appreciated.

As we assume, these magnetic clusters, if existing, are electrically polarizable. They may be reflected from the dielectric response owing to the local polarization relaxation. We come back to the dielectric response data along the main axes and check their behaviors in the high-T PE region. For an illustration, the measured dielectric constant along the b axis, ɛb(T), and the constant along the c axis, ɛc(T), at frequency f = 100 kHz, are plotted in Fig. 9(a). Since the ferroelectric polarization is aligned along the b axis, it is understandable that ɛb(T) exhibits a sharp peak around TN2 whole ɛc(T) does not. Furthermore, one sees clearly the difference between ɛb(T) and ɛc(T) curves in the high-T-range, and a broad bump of ɛb(T) covers the high-T-range but no such a broad bump for ɛc(T). This difference suggests that the broad bump of ɛb(T) may be somehow related to the electrically polarizable magnetic clusters in the high-T-range.

FIG. 9.

(a) Measured dielectric constants along the b axis and c axis, ɛb(T) and ɛc(T) at f = 100 kHz. (b) Measured ɛb(T) curve at f = 100 kHz replotted and the dashed curve is the assumed dielectric response contributed by the thermally activated dipole relaxations. The difference between the two curves, Δɛb(T), exemplifies the peak around TN2 and the broad bump in the high-T region. (c) The as-evaluated Δɛb(T) curves at different frequencies, where the curves are vertically shifted for illustration.

FIG. 9.

(a) Measured dielectric constants along the b axis and c axis, ɛb(T) and ɛc(T) at f = 100 kHz. (b) Measured ɛb(T) curve at f = 100 kHz replotted and the dashed curve is the assumed dielectric response contributed by the thermally activated dipole relaxations. The difference between the two curves, Δɛb(T), exemplifies the peak around TN2 and the broad bump in the high-T region. (c) The as-evaluated Δɛb(T) curves at different frequencies, where the curves are vertically shifted for illustration.

Close modal

In proceeding, it is known that the dielectric response of a homogeneous dielectric media can be characterized by thermally activated relaxation time. Just for a guide of eyes, we replot the ɛb(T) data in Fig. 9(b) where the dashed curve can be a reference if GdMn2O5 would be a pure and ideal homogeneous dielectric. This dashed curve should overlap with the measured dielectric data in the high-T region where GdMn2O5 is a paraelectric. In fact, GdMn2O5 is not an ideal paraelectric in the low-T region but a ferroelectric where a clear dielectric peak is identified below TN2.

When the difference curve, Δɛb(T), between the ɛb(T) curve and this reference curve is extracted, one can see two anomalies. This extraction is certainly crude and shows no quantitative significance. The difference shows not only the sharp peak at TN2 but also a broad bump well above Tm, as shown in Fig. 9(c) at f = 100 kHz. This broad bump is not from release of any thermally trapped charges, since such a release would generate much stronger dielectric response. Similar data processing is applied to other ɛb(T) curves at different f and the results are summarized in Fig. 9(c) where the Δɛb(T) curves at different f are shifted upward for easy illustration.

Although the validity of the obtained Δɛb(T) curves may be questioned due to the somehow arbitrary data processing for Δɛb(T), it is shown that each curve does show a broad bump which shifts gradually toward the high-T side with increasing f. This feature is a consequence of magnetic cluster formation in the high-T region. These clusters are polarizable electrically, contributing to the dielectric bump. On the other hand, the bump shifting with varying frequency is the reasonable for polarizable cluster state.

Nevertheless, there remain some issues need to be given more careful consideration. One issue is the convincing and direct evidence with the existence of such clusters. Second issue is the discrepancy between this model prediction and measured results regarding Fig. 5(l) where the positive-negative crossing point appears at roughly Tend ∼ 50 K, while the model predicted crossing point should be at TN2. This discrepancy may be contributed by a series of measuring uncertainties and unknown sources. If one assumes that this proposed model is valid, the observed phenomena are the features of an interesting example for electrocontrol of multiferroicity. An electropoling within a properly chosen T-range can “define” the low-T magnetic structure and thus the electric polarization. The last but far not the least issues is whether this scenario can be extended to other member compounds of this family. These issues and questions are open for future investigation.

In conclusion, we have carefully investigated the multiferroic properties of GdMn2O5 single crystals by measuring the pyroelectric current and polarization along the b axis, upon the high-T electropoling from room temperature down to a given Tend. It has been revealed that an electropoling in the high-T PM-PE region can induce a negative electric polarization in the low-T multiferroic CM-FE region, while a positive electric polarization is obtained if the electropoling is sustained in the low-T CM-FE region. It is suggested that this electric polarization reversal, in response to reduced poling-ending temperature, is related to the antiparallel electronic polarization and ionic polarization. While the ionic polarization arises from the Mn–Mn symmetric exchange striction, the electronic polarization can be sustained to the high-T PE phase due to the existence of high-T magnetic clusters. This high-T electronic polarization in coexistence with the low-T ionic polarization is the reason for the electric polarization reversal induced by the high-T electropoling. A model to account for the observed emergent phenomenon is proposed based on the first-principles calculations. The present work represents a substantial forward step into the emergent phenomena of multiferroicity in the RMn2O5 family.

This work was supported by the National Key Research Projects of China (Grant No. 2016YFA0300101) and by the National Natural Science Foundation of China (NNSFC) (Grant Nos. 51431006, 11834002, and 51721001). The work at Rutgers University was supported by the Department of Energy (DOE) under Grant No. DE-FG02-07ER46382.

1.
S.-W.
Cheong
and
M.
Mostovoy
,
Nat. Mater.
6
,
13
(
2007
).
2.
S.
Dong
,
J.-M.
Liu
,
S.-W.
Cheong
, and
Z.
Ren
,
Adv. Phys.
64
,
519
(
2015
).
3.
H.
Katsura
,
N.
Nagaosa
, and
A. V.
Balatsky
,
Phys. Rev. Lett.
95
,
057205
(
2005
).
4.
M.
Mostovoy
,
Phys. Rev. Lett.
96
,
067601
(
2006
).
5.
W.
Ratcliff
,
J. W.
Lynn
,
V.
Kiryukhin
,
P.
Jain
, and
M. R.
Fitzsimmons
,
NPJ Quantum Mater.
1
,
16003
(
2016
).
6.
J.
Hlinka
,
J.
Privratska
,
P.
Ondrejkovic
, and
V.
Janovec
,
Phys. Rev. Lett.
116
,
177602
(
2016
).
7.
L. C.
Peng
,
Y.
Zhang
,
M.
He
,
B.
Ding
,
W. H.
Wang
,
H. F.
Tian
,
J. Q.
Li
,
S. G.
Wang
,
J. W.
Cai
,
G. H.
Wu
,
J. P.
Liu
,
M. J.
Kramer
, and
B. G.
Shen
,
NPJ Quantum Mater.
2
,
30
(
2017
).
8.
N.
Hur
,
S.
Park
,
P. A.
Sharma
,
J. S.
Ahn
,
S.
Guha
, and
S.-W.
Cheong
,
Nature
429
,
392
(
2004
).
9.
T.
Kimura
,
T.
Goto
,
H.
Shintani
,
K.
Ishizaka
,
T.
Arima
, and
Y.
Tokura
,
Nature
426
,
55
(
2003
).
10.
Y.
Geng
,
H.
Das
,
A. L.
Wysocki
,
X.
Wang
,
S.-W.
Cheong
,
M.
Mostovoy
,
C. J.
Fennie
, and
W.
Wu
,
Nat. Mater.
13
,
163
(
2014
).
11.
N.
Lee
,
C.
Vecchini
,
Y. J.
Choi
,
L. C.
Chapon
,
A.
Bombardi
,
P. G.
Radaelli
, and
S.-W.
Cheong
,
Phys. Rev. Lett.
110
,
137203
(
2013
).
12.
N. A.
Benedek
and
C. J.
Fennie
,
Phys. Rev. Lett.
106
,
107204
(
2011
).
13.
T.
Choi
,
S.
Lee
,
Y. J.
Choi
,
V.
Kiryukhin
, and
S.-W.
Cheong
,
Science
324
,
63
(
2009
).
14.
Y.
Li
,
Y.
Jin
,
X.
Lu
,
J.-C.
Yang
,
Y.-H.
Chu
,
F.
Huang
,
J.
Zhu
, and
S.-W.
Cheong
,
NPJ Quantum Mater.
2
,
43
(
2017
).
15.
T.
Choi
,
Y.
Horibe
,
H. T.
Yi
,
Y. J.
Choi
,
W.
Wu
, and
S.-W.
Cheong
,
Nat. Mater.
9
,
253
(
2010
).
16.
J. R.
Arce-Gamboa
and
G. G.
Guzmán-Verri
,
NPJ Quantum Mater.
2
,
28
(
2017
).
17.
S.
Petit
,
V.
Balédent
,
C.
Doubrovsky
,
M. B.
Lepetit
,
M.
Greenblatt
,
B.
Wanklyn
, and
P.
Foury-Leylekian
,
Phys. Rev. B
87
,
140301
(
2013
).
18.
Y.
Tokunaga
,
S.
Iguchi
,
T.
Arima
, and
Y.
Tokura
,
Phys. Rev. Lett.
101
,
097205
(
2008
).
19.
T.
Kubacka
,
J. A.
Johnson
,
M. C.
Hoffmann
,
C.
Vicario
,
S.
de Jong
,
P.
Beaud
,
S.
Grubel
,
S.-W.
Huang
,
L.
Huber
,
L.
Patthey
,
Y.-D.
Chuang
,
J. J.
Turner
,
G. L.
Dakovski
,
W.-S.
Lee
,
M. P.
Minitti
,
W.
Schlotter
,
R. G.
Moore
,
C. P.
Hauri
,
S. M.
Koohpayeh
,
V.
Scagnoli
,
G.
Ingold
,
S. L.
Johnson
, and
U.
Staub
,
Science
343
,
1333
(
2014
).
20.
S.
Ghara
,
E.
Suard
,
F.
Fauth
,
T. T.
Tran
,
P. S.
Halasyamani
,
A.
Iyo
,
J.
Rodríguez-Carvajal
, and
A.
Sundaresan
,
Phys. Rev. B
95
,
224416
(
2017
).
21.
J.
Varignon
,
M. N.
Grisolia
,
J.
Íñiguez
,
A.
Barthélémy
, and
M.
Bibes
,
NPJ Quantum Mater.
2
,
21
(
2017
).
22.
Y. J.
Choi
,
H. T.
Yi
,
S.
Lee
,
Q.
Huang
,
V.
Kiryukhin
, and
S.-W.
Cheong
,
Phys. Rev. Lett.
100
,
047601
(
2008
).
23.
G. R.
Blake
,
L. C.
Chapon
,
P. G.
Radaelli
,
S.
Park
,
N.
Hur
,
S.-W.
Cheong
, and
J.
Rodríguez-Carvajal
,
Phys. Rev. B
71
,
214402
(
2005
).
24.
C.
Wang
,
G.-C.
Guo
, and
L.
He
,
Phys. Rev. Lett.
99
,
177202
(
2007
).
25.
C.
Vecchini
,
L. C.
Chapon
,
P. J.
Brown
,
T.
Chatterji
,
S.
Park
,
S.-W.
Cheong
, and
P. G.
Radaelli
,
Phys. Rev. B
77
,
134434
(
2008
).
26.
M.
Tachibana
,
K.
Akiyama
,
H.
Kawaji
, and
T.
Atake
,
Phys. Rev. B
72
,
224425
(
2005
).
27.
W.
Ratcliff
,
V.
Kiryukhin
,
M.
Kenzelmann
,
S.-H.
Lee
,
R.
Erwin
,
J.
Schefer
,
N.
Hur
,
S.
Park
, and
S.-W.
Cheong
,
Phys. Rev. B
72
,
060407
(
2005
).
28.
Y.
Ishii
,
S.
Horio
,
M.
Mitarashi
,
T.
Sakakura
,
M.
Fukunaga
,
Y.
Noda
,
T.
Honda
,
H.
Nakao
,
Y.
Murakami
, and
H.
Kimura
,
Phys. Rev. B
93
,
064415
(
2016
).
29.
M.
Fukunaga
,
Y.
Sakamoto
,
H.
Kimura
,
Y.
Noda
,
N.
Abe
,
K.
Taniguchi
,
T.
Arima
,
S.
Wakimoto
,
M.
Takeda
,
K.
Kakurai
, and
K.
Kohn
,
Phys. Rev. Lett.
103
,
077204
(
2009
).
30.
S.
Wakimoto
,
H.
Kimura
,
Y.
Sakamoto
,
M.
Fukunaga
,
Y.
Noda
,
M.
Takeda
, and
K.
Kakurai
,
Phys. Rev. B
88
,
140403
(
2013
).
31.
J.-M.
Liu
and
S.
Dong
,
J. Adv. Dielectr.
05
,
1530003
(
2015
).
32.
S.
Chattopadhyay
,
V.
Balédent
,
F.
Damay
,
A.
Gukasov
,
E.
Moshopoulou
,
P.
Auban-Senzier
,
C.
Pasquier
,
G.
André
,
F.
Porcher
,
E.
Elkaim
,
C.
Doubrovsky
,
M.
Greenblatt
, and
P.
Foury-Leylekian
,
Phys. Rev. B
93
,
104406
(
2016
).
33.
D.
Higashiyama
,
S.
Miyasaka
, and
Y.
Tokura
,
Phys. Rev. B
72
,
064421
(
2005
).
34.
D.
Higashiyama
,
S.
Miyasaka
,
N.
Kida
,
T.
Arima
, and
Y.
Tokura
,
Phys. Rev. B
70
,
174405
(
2004
).
35.
S. H.
Bukhari
,
Th.
Kain
,
M.
Schiebl
,
A.
Shuvaev
,
A.
Pimenov
,
A. M.
Kuzmenko
,
X.
Wang
,
S.-W.
Cheong
,
J.
Ahmad
, and
A.
Pimenov
,
Phys. Rev. B
94
,
174446
(
2016
).
36.
N.
Hur
,
S.
Park
,
P. A.
Sharma
,
S.
Guha
, and
S.-W.
Cheong
,
Phys. Rev. Lett.
93
,
107207
(
2004
).
37.
V.
Balédent
,
S.
Chattopadhyay
,
P.
Fertey
,
M. B.
Lepetit
,
M.
Greenblatt
,
B.
Wanklyn
,
F. O.
Saouma
,
J. I.
Jang
, and
P.
Foury-Leylekian
,
Phys. Rev. Lett.
114
,
117601
(
2015
).
38.
S.
Mansouri
,
S.
Jandl
,
M.
Balli
,
J.
Laverdière
,
P.
Fournier
, and
D. Z.
Dimitrov
,
Phys. Rev. B
94
,
115109
(
2016
).
39.
S. I.
Vorob’ev
,
D. S.
Andrievskii
,
S. G.
Barsov
,
A. L.
Getalov
,
E. I.
Golovenchits
,
E. N.
Komarov
,
S. A.
Kotov
,
A. Yu.
Mishchenko
,
V. A.
Sanina
, and
G. V.
Shcherbakov
,
J. Exp. Theor. Phys.
123
,
1017
(
2016
).
40.
J.
Ahmad
,
S. H.
Bukhari
,
M. T.
Jamil
,
M. K.
Rehmani
,
H.
Ahmad
, and
T.
Sultan
,
Adv. Condens. Matter Phys.
2017
(
8
),
5389573
(
2017
).
41.
G.
Yahia
,
F.
Damay
,
S.
Chattopadhyay
,
V.
Balédent
,
W.
Peng
,
E.
Elkaim
,
M.
Whitaker
,
M.
Greenblatt
,
M.-B.
Lepetit
, and
P.
Foury-Leylekian
,
Phys. Rev. B
95
,
184112
(
2017
).
42.
L.
Yang
,
X.
Li
,
M. F.
Liu
,
P. L.
Li
,
Z. B.
Yan
,
M.
Zeng
,
M. H.
Qin
,
X. S.
Gao
, and
J.-M.
Liu
,
Sci. Rep.
6
,
34767
(
2016
).
43.
C.
Vecchini
,
A.
Bombardi
,
L. C.
Chapon
,
N.
Lee
,
P. G.
Radaelli
, and
S.-W.
Cheong
,
J. Phys. Conf. Ser.
519
,
012004
(
2014
).
44.
X.
Li
,
S.
Zheng
,
L.
Tian
,
R.
Shi
,
M.
Liu
,
Y.
Xie
,
L.
Yang
,
N.
Zhao
,
L.
Lin
,
Z.
Yan
,
X.
Wang
, and
J.
Liu
,
Chin. Phys. B
28
,
027502
(
2019
).
45.
J. A.
Alonso
,
M. T.
Casais
,
M. J.
MartinezLope
,
J. L.
Martinez
, and
M. T.
FernandezDiaz
,
J. Phys. Condens. Matter
9
,
8515
(
1997
).
46.
P. G.
Radaelli
and
L. C.
Chapon
,
J. Phys. Condens. Matter
20
,
434213
(
2008
).
47.
Z. Y.
Zhao
,
M. F.
Liu
,
X.
Li
,
L.
Lin
,
Z. B.
Yan
,
S.
Dong
, and
J.-M.
Liu
,
Sci. Rep.
4
,
3984
(
2015
).
48.
S.
Kivelson
and
S. A.
Kivelson
,
NPJ Quantum Mater.
1
,
16024
(
2016
).
49.
R. A.
de Souza
,
U.
Staub
,
V.
Scagnoli
,
M.
Garganourakis
,
Y.
Bodenthin
,
S.-W.
Huang
,
M.
García-Fernández
,
S.
Ji
,
S.-H.
Lee
,
S.
Park
, and
S.-W.
Cheong
,
Phys. Rev. B
84
,
104416
(
2011
).
50.
I.
Kagomiya
,
S.
Matsumoto
,
K.
Kohn
,
Y.
Fukuda
,
T.
Shoubu
,
H.
Kimura
,
Y.
Noda
, and
N.
Ikeda
,
Ferroelectrics
286
,
167
(
2003
).
51.
C. L.
Lu
,
J.
Fan
,
H. M.
Liu
,
K.
Xia
,
K. F.
Wang
,
P. W.
Wang
,
Q. Y.
He
,
D. P.
Yu
, and
J.-M.
Liu
,
Appl. Phys. A
96
,
991
(
2009
).
52.
N.
Poudel
,
M.
Gooch
,
B.
Lorenz
,
C. W.
Chu
,
J. W.
Kim
, and
S. W.
Cheong
,
Phys. Rev. B
92
,
144430
(
2015
).
53.
G.
Giovannetti
and
J.
van den Brink
,
Phys. Rev. Lett.
100
,
227603
(
2008
).
54.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
55.
P. E.
Blöchl
,
Phys. Rev. B
50
,
17953
(
1994
).
56.
G.
Kresse
and
J.
Furthmuller
,
Comput. Mater. Sci.
6
,
15
(
1996
).
57.
I. A.
Vladimir
,
F.
Aryasetiawan
, and
A. I.
Lichtenstein
,
J. Phys. Condens. Matter
9
,
767
(
1997
).
58.
T.-R.
Chang
,
H.-T.
Jeng
,
C.-Y.
Ren
, and
C.-S.
Hsue
,
Phys. Rev. B
84
,
024421
(
2011
).
59.
R. D.
King-Smith
and
D.
Vanderbilt
,
Phys. Rev. B
47
,
1651
(
1993
).