Probing magnetoelectric effect in the spin-modulated magnet Fe2GeO4

The distinct spin amplitude wave was reported in a highly frustrated magnetic compound Fe2GeO4, which is very different from observations on other members of the M2GeO4 (M = Fe, Co, and Ni) family, raising interest in this compound for some additional emergent phenomena. In particular, this non-uniform spin order allows the intrinsic connection between ferroelectric polarization and magnetically gradient structure to probe the potential linear magnetoelectric (ME) effect. In this work, we address this issue and investigate the magnetism of Fe2GeO4 single crystal that hosts two successive anomalies at antiferromagnetic (AFM) Néel temperatures T N1 ∼ 7.5 K and T N2 ∼ 6.7 K, respectively. Our results reveal a remarkable metamagnetic transition in the magnetization as a function of the magnetic field, occurring at a critical magnetic field H c ∼ 4.1 T when applied along the [110] and [1–10] directions, while such transition along the [001] direction is pointedly absent. Further exploration uncovers two predominant off-diagonal ME coefficients αyz and αzy in the incommensurate AFM phase between T N1 and T N2. Additionally, all components of the linear ME tensor remain non-vanishing in the canting AFM phase below T N2. This indicates the ME mechanisms for the two phases that may be driven by different magnetic structures. All these presented results are sufficient for us to draw a non-trivial ME phase diagram, which is beneficial to understanding the ME behavior of Fe2GeO4. Therefore, our study implies that Fe2GeO4, an unusual frustrated magnet, provides a platform for manipulating the fascinating ME effect in the spinel structure.


Introduction
The magnetoelectric (ME) effect, which allows for the ability to control magnetization (M) of a material using an electric field (E) or its polarization (P) applying a magnetic field (H), plays an important role in condensed matter physics because of the promising avenue for next generation of data storage [1][2][3][4].Especially several signature conclusions on this crossing-coupling effect have been realized by a large number of experimental and theoretical studies in the last decades.First, materials with linear or nonlinear ME effect can be consulted from the viewpoint of ME tensor analysis in principle, assuming the magnetic point group is available [5].Subsequently, focusing on versatile multiferroics with magnetism-driven ferroelectricity, three microscopic mechanisms of such ferroelectricity can be classified into: (i) spin-current mechanism [6], (ii) exchange-striction mechanism [7], and (iii) p-d hybridization mechanism [8].These iconic summarizations are authentical contributions to the interpretation and analysis of ME behaviors in the majority of ME compounds [1,9].
Recently, some highly frustrated magnets with complicated spin order, including magnetic multipole, skyrmions, and magnetic chirality, have been widely concerned due to the emerging unconventional phenomena [10][11][12][13][14].Among them, some prominent situations, which are associated with multiferroicity [15,16], ME monopoles [17], dichroism of elementary excitations [14,18], and so on, should be considered from the viewpoint of physics origin.For instance, symmetry operational similarity is a significant attempt that provides a roadmap to explain the ME effect from the viewpoint of symmetry operation, and indeed, some particular spin orders can be the major sources [19,20].In addition, it is also reasonable to consider that the geometrical contributions in magnetic moments are important for explaining the ME effect based on the issue from the theories of magnetic point group and magnetism-induced ferroelectricity.Based on the models mentioned above, the only known so far is that the ME effect in magnetically ordered systems necessitates the simultaneous breaking of both time-reversal symmetry and spatial inversion symmetry.Undoubtedly, more theoretical and experimental studies on these ME compounds deserve to be presented.
Fe 2 GeO 4 belongs to the M 2 GeO 4 (M = Fe, Co, and Ni) family and features a cubic spinel lattice structure at room temperature.A lattice symmetry-based mechanism for ferroelectricity is unlikely, only considering the lattice structure in these compounds.Consequently, the inherent spontaneous electric polarization is absent.Instead, the ME effect could be triggered if one considers the unconventional frustration wave order, which deserves attention [21][22][23].Here, several features of Fe 2 GeO 4 should be mentioned.First, similar to Co 2 GeO 4 and Ni 2 GeO 4 , Fe 2 GeO 4 also manifests a face-centered cubic spinel structure with an Fd-3 m space group, as depicted in figure 1(a).One can see that the Ge 4+ cations occupy the tetrahedral sites, and the Fe 2+ (3d 6 , S = 2) cations are on the octahedral sites located on the corners of vertex-sharing tetrahedra [21].Second, powder neutron diffraction data reveals that the moments of Fe1 and Fe2 adopt frustration wave orders, aligning parallel to the (110) and (1-10) planes below T N1 , respectively, as shown in figure 1(b).This magnetic structure differs from the situations observed in Ni 2 GeO 4 and Co 2 GeO 4 , where a three-dimensional propagation vector is present [24].In detail, Co 2 GeO 4 exhibits a transition at T N ∼ 21 K, corresponding to the triangular plane moments perpendicular to the [111] direction.While Ni 2 GeO 4 displays two continuous anomalies at T N1 ∼12 K and T N2 ∼ 11 K, and its moments are aligned parallel to the [111] direction below T N1 [25,26].It is noted that the multiferroic compound Mn 2 GeO 4 with the olivine structure (space group Pnma) presenting successive magnetic transitions at T N1 ∼ 47 K, T N2 ∼ 17 K, and T N3 ∼ 5.5 K, respectively, is prominently different from the family of M 2 GeO 4 with a spinel structure [27].As for the case of Fe 2 GeO 4 , it also features two successive antiferromagnetic (AFM) transitions at T N1 ∼ 9 K and T N2 ∼ 7 K, respectively.A narrow temperature window of ∼ 2 K spans the whole regions of the incommensurate AFM (IC-AFM) phase, while a canting AFM (C-AFM) phase emerges below T N2 .In general, these phases host a similar propagation vector, as indexed by k i = (2/3 + δ 1 , 2/3 + δ 1 , 0) below T Ni (i = 1 or 2), showcasing a noncollinear spin ordering with the moments of Fe1 and Fe2 varying between fully saturated 4 µ B and 0 [21].
Here, our consideration is whether any ME effect can be activated in Fe 2 GeO 4 , which demonstrates a center-symmetry structure but exhibits a significant amplitude modulation of the moment in the magnetically ordered phases.This evidences a previously non-recognized class of ground states for orbitally degenerate spins on frustrated lattices.This issue can be considered from two perspectives.On one hand, those magnets hosting complex spin textures, such as helimagnets, spin vortices, or skyrmions, possess consistent moment amplitudes while their orientations change spatially.On the other hand, it was reported that Ca 3 Co 2 O 6 is, until now, the only close analog to Fe 2 GeO 4 .To the best of our knowledge, the chains of collinear moments are modulated between 0 and 5 µ B with a sizable orbital contribution, indicating a varied moment amplitude, as illustrated in the right side of figure 1(b) [28].Notably, recent investigations on spinel Fe 2 GeO 4 also unveiled a noncollinear long-range spin amplitude wave, which is ascribed to the orbital-frustrated order and is related to the magnetic exchange interactions [21].
The emergence of spontaneous electric polarization is usually based on the spatial inversion symmetry breaking of the lattice.For the ferroelectricity generation, one understands that it is challenging to probe remarkable ME behaviors in Fe 2 GeO 4 with a highly symmetric spinel lattice at room temperature.Meanwhile, it is an opportunity, from the viewpoint of excitation, that a ferroelectricity-active state with both broken spatial-inversion symmetry and broken time-reversal symmetry in a strongly frustrated magnet should be possibly driven by extrinsic perturbation, e.g.electric and magnetic stimuli, similar to the issues in Cu 2 OSeO 3 and LaMn 3 Cr 4 O 12 , and so on [12,[29][30][31][32].
In this paper, as elaborated above, our major motivation is to probe any potential ME effect in Fe 2 GeO 4 single crystals, which is yet to be checked or confirmed for the whole M 2 GeO 4 family.Concretely, we present a systematic investigation of Fe 2 GeO 4 single crystals, including the magnetism, specific heat, ferroelectricity, and magnetoelectricity.Then a possible magnetic field-excited ME effect will be probed and discussed, focusing on the whole ME tensor matrix, which has not been well addressed so far.All these presented results are sufficient for us to describe the non-trivial ME phase diagram.Our investigations reveal that Fe 2 GeO 4 , as a specific member in the M 2 GeO 4 family, exhibits intriguing linear ME responses.This work establishes a comprehensive platform for understanding and extending our knowledge of the linear ME effect.

Experimental details
The Fe 2 GeO 4 single crystals were grown by the chemical vapor transport technique.The polycrystalline precursor with nominal Fe 2 GeO 4 composition was prepared via the solid-state reaction processing as described earlier [21,22].In detail, the stochiometric amount of GeO 2 , Fe 2 O 3 , and Fe powder was thoroughly ground in an agator mortar and sealed in evacuated silica tubes at 900 • C for 60 h.Then the obtained powder and transporting agent I 2 were sealed in one evacuated silica tube filled with a small amount of argon at 780 • C in the change zone (T 1 ) and 690 • C in the growth zone (T 2 ) for 14 d.
The shiny black crystal with a typical size of 2 × 3 × 2 mm 3 was shown in the inset of figure 1(c).The crystal structure was checked using the x-ray diffraction (XRD, D8 Advanced, Bruker) in the θ-2θ mode with Cu Kα source (λ = 1.5406Å) at room temperature.The back-reflection Laue detector (MWL120, Multiwire Laboratories, Ltd) was used to check the quality of single crystals and determine the crystallographic orientation.The electron dispersion spectroscopy (EDS) attached to the scanning electron microscopy (Quanta 200, FEI) was used to probe the chemical composition.
On the other hand, a set of well-prepared and aligned single crystal samples was used for characterizations on magnetism, specific heat, dielectric constant, electric polarization, etc.In particular, these samples were well aligned so that the ME tensor could be as accurate as possible determined.In detail, the temperature (T) dependence of dc magnetic susceptibility (χ) under the zero-field cooling (ZFC) and field cooling (FC) modes with cooling magnetic field H = 0.1 T were measured from T = 2 K to 300 K using the Quantum Design superconducting quantum interference device magnetometer.Simultaneously, the magnetization (M) as a function of H at selected T was also measured.The specific heat (C P ) under different H was measured from T = 2-100 K using the Quantum Design physical property measurement system (PPMS).
For the electrical and ME measurements, three single crystals aligned respectively with surface normal to the [110, 1-10], and [001] directions were polished into disk-like plates of ∼0.2 mm thickness and coated with Au electrodes.The dielectric constant (ε) as a function of T was measured using the LCR meter IM3536 analyzer integrated with PPMS.The electric polarization (P) was evaluated from the standard pyroelectric current method, which is widely used to probe such small remnant polarization.Prior to the polarization (P) and ME measurements, each sample was prepared under the ME pooling fields of E = 2 kV cm −1 and H = 9 T with E⊥H (or //H) arrangements during the cooling sequence down to 4 K. Subsequently, the applied electric and magnetic fields were removed at this temperature.Each sample was short-circuited for about one hour and submitted to a slow warming process at a fixed heating rate of 4 K min −1 from 4 to 20 K under the selected magnetic field H.During the measurements, the current released from the sample was collected using the Keithley 6514 electrometer connected to the PPMS, and this current should be the polarization current in the low-T region.The P can be obtained by performing the integration of current with time.

Crystal structure
The crystal structure of Fe 2 GeO 4 was checked by performing the XRD measurements at room temperature.The inset of figure 1(c) shows the photo-image of a typical single crystal.The naturally developed surface was first scanned using the slow-scan XRD mode, and it produces sharp θ-2θ reflections, which are calibrated by the standard XRD diffraction pattern.The natural surfaces are well indexed by the (111) reflection, and the lattice structure fits the cubic Fd-3 m space group well with lattice constants a = b = c = 8.4136 Å.One unit cell consists of 16 Fe 2+ ions with a standard oxygen octahedral configuration surrounded by eight Ge 4+ ions located in the normative oxygen tetrahedra.Among them, each Fe 2+ cation overlaps with another in the 90 • Fe-O-Fe pathway, which provides an important environment for orbital moments and leads to amplitude modulation of the moment [21].
Here, the [110, 1-10], and [001] orientations of the as-grown Fe 2 GeO 4 crystals can be indexed by an analysis based on the consistency between the Laue diffraction spots and XRD patterns.For example, unambiguous and symmetrical Laue diffraction spots can be obtained by rotating the samples and simulated as [110] (or [1][2][3][4][5][6][7][8][9][10]) direction.Then, this labeled sample was checked using the XRD and calibrated for the validity of this [110] direction assignment in comparison with the standard XRD diffraction pattern, as shown in figure 1(c).In addition, the single crystals were used for chemical composition mapping using the EDS technique, producing the atomic Fe: Ge ratio of 2.007, quite close to the nominal ratio 2, as plotted in the inset of figure 1(c).Therefore, the above results demonstrate the good quality of our crystals.
Based on the above results, several issues regarding on the magnetic properties, as shown in figures 2(a)-(c), should be highlighted.First, by fitting the 1/χ(T) data between T = 100 K and 300 K to the Curie-Weiss law, as shown in the insets, we can obtain the Curie constant C, Curie-Weiss temperature θ CW , and effective moment µ eff for each set of data, respectively.The detailed values fitted from the Curie-Weiss law are listed in table 1.Here, one can note that the frustrated fact f (|θ CW |/T N ) along the three directions are 5.89, 6.07, and 6.56, respectively, indicating a relatively high level of magnetic frustration.In addition, we observed that the effective paramagnetic moments (µ eff ) along the three directions are 4.49 µ B /f.u., 4.74 µ B /f.u., and 4.41 µ B /f.u., respectively.These values surpass the anticipated spin-only moments of 4 µ B /f.u. for high spin Fe 2+ (S = 2) in Fe 2 GeO 4 , indicating the incomplete quenching of the orbital moment.Furthermore, the observed two successive magnetic transitions for the present single crystals are consistent with previous observations on polycrystalline samples under a measuring field of 0.5 T, corresponding to an incommensurate AFM (IC-AFM) phase below T N1 and a canting AFM (C-AFM) phase below T N2 , as shown in the top of figures 2(a) and (d) [21,22].Here, it is noted that the observed T Ni (i = 1, 2) in single crystals are slightly lower than that in polycrystalline samples, and this difference is attributed to the difference in measuring H.To evidence this claim, the measured χ(T) and specific heat C P (T) data under several selected H // [1-10] up to 7 T are plotted in figures 2(d) and (g), respectively.For example, the peak of χ [1-10] (T) in It is worth noting that the T N usually shifts toward low-T regions with increasing H in the AFM system since the fields destabilize the AFM order.Nevertheless, the opposite behavior observed here may be ascribed to the strong ferromagnetic (FM) interaction reflecting the important role of Fe 2+ in competing with the AFM interaction.This is evident from the distinct deviation observed in all the FC and ZFC curves, as plotted in figures 2(a)-(c), indicating the potential existence of FM interaction in the pyrochlore Fe 2+ spins.Similar phenomena have been observed in several strongly frustrated magnets, such as FeCl 2 [33][34][35].Here, it should be noted that other members in this family, for instance, Co 2 GeO 4 and Ni 2 GeO 4 , exhibit diverse behaviors and uncover a much weaker tendency towards FM interaction [25].
Subsequently, the specific heat C P as a function of T under H = 0 also demonstrates two consecutive anomalies in consistency with the results of χ(T), as depicted in figure 2(e).The λ-type C P anomalies in Fe 2 GeO 4 indicate that the second-order transition generally occurs in the magnetic phase transition.As shown in figure 2(e), two remarkable enhancements of magnetic specific heat at T Ni are 18.5 J mol −1 K −1 and 14.2 J mol −1 K −1 , which can be approximated by subtracting the value of the phonon contribution calculated with a combined Debye model: where R is the gas constant, x D = θ D /T, and θ D is the Debye temperatures, respectively.This term is much smaller than the value deduced from the mean-field theory, where the magnetic contribution C M (T) can be obtained from where R is the gas constant and S = 2.The difference between the measured and evaluated values implies the presence of short-range magnetic interaction above T Ni in Fe 2 GeO 4 [36].Beyond, the measured C M data are well fitted by the power law C M ∼ T 3.2 in the low-T region, consistent with the consequence of a three-dimensional AFM ordering [36].The T dependence of magnetic entropy ∆S M (T) is also plotted in figure 2(f).Obviously, the entropy loss occurs at T N1 and T N2 with a saturation value ∆S M (T ∼ 40 K) ∼ 12.84 J mol −1 K −1 , just half of the saturated value (2Rln5 ∼ 26.75 J mol K −1 ).This value is much closer to recent reports on polycrystalline compound Fe 2 GeO 4 [21].It is worthwhile to point out that the peak of C M (T)/T at T N1 obviously shifts toward the high-T region with increasing H up to 7 T, as shown in figure 2(g).The fitted factor κ is 1.6 at H = 7 T obtained by using the power law C M /T ∼ T κ .Here, one can see that the κ is 2 for the three-dimensional AFM system while κ = 1 for an FM structure [36].Hence, it is reasonable to consider that the applied magnetic field as high as 7 T would remarkably enhance the competition of the FM interaction against the AFM ones in this frustrated magnet Fe 2 GeO 4 , coincident with the measured magnetic behaviors.Motivated by the χ(T) data, we proceed to examine the measured M(H) data, with a focus on M [110] , M [1][2][3][4][5][6][7][8][9][10] , M [001] , and the M responses below T N1 along the [110, 1-10], and [001] directions, respectively.For clarity, the dM/dH data are included in the plots for reference, as depicted in figures 3(a)-(c).Linear M [110] and M [1][2][3][4][5][6][7][8][9][10] dependences of H are seen in the low-H region, a typical feature for AFM behavior, while a clear anomaly in the (dM/dH) data appears at H ∼ critical field H c .For instance, the anomaly occurs at H c ∼ 4.1 T at T = 2 K in the M [110] curve.In the case of M [1][2][3][4][5][6][7][8][9][10] , a similar anomaly can also be found below T N2 , as shown in figure 3(b).These metamagnetic transitions may be related to the spin reorientations (SR) induced by H, noting that the spins prefer to align in the ab plane, as plotted at the top of figures 3(a) and (b).In addition, similar behaviors with a metamagnetic transition also emerge in Co 2 GeO 4 and Ni 2 GeO 4 [23,37].One also should believe that the SR transition observed in figures 3(a) and (b) indeed implies anisotropy within the ab plane in Fe 2 GeO 4 .In addition, the observed anomalies indicate the changing in the direction of the spins within the ab plane due to the magnetic anisotropy rather than varying in the population of spins.
Finally, it deserves to be mentioned that the measured M [001] as a function of T does show a linear dependence over the entire range of magnetic fields covered here, as shown in figure 3(c).This linear behavior signifies field-driven magnetic anisotropy within Fe 2 GeO 4 , an unexpected outcome given the highly symmetric cubic pyrochlore structure of Fe 2+ spins.One plausible explanation for this distinctive spin structure may be attributed to orbital-frustrated order, which is intricately connected to magnetic exchange interactions [21].

Ferroelectricity and ME response
Given the lattice symmetry argument, it is known that spontaneous ferroelectricity is forbidden in Fe 2 GeO 4 because it hosts a highly symmetric Fd-3 m space group.However, this rule does not exclude the linear ME response if the applied magnetic field allows the emergence of an ME-active state in some frustrated magnets with specific spin structures.In the present case, the single crystal samples were submitted to the proper initial state, and then dielectric and pyroelectric current measurements in the standard mode can be performed to probe possible magnetic field-induced electric polarization.
First, without losing the generality, we measure the dielectric constant (ε), e.g.along the [1-10] direction, as a function of T under the selected H, which is aligned along the [001] direction.The activating electric signal for the dielectric measurement is fixed at the amplitude of 100 mV and frequency f = 10 kHz, and the measured ε(T) curves are plotted in figure 4(a).These curves present the H-driven anomaly at T ∼ T N1 , implying the H-driven ME response along with the AFM transition, and the anomaly point shifts toward the high-T side, corresponding to the right-side shift of T N1 as extracted from the χ(T) curves.These observations evidence unambiguously the connection of the dielectric anomaly with the magnetic ordering, a clear indication of an unusual ME response in which induced electric polarization becomes possible.
To further probe the ME effect over the whole tensor components, the orientation-dependent P as a function of T along [110, 1-10], and [001] axes was measured under the H // P (or H ⊥ P) in different geometries, as plotted in figures 5(a)-(i).Given the knowledge of the magnetic structure, we define the Cartesian coordinates x//[110], y//[1-10], and z//[001], respectively, as shown in figure 1(b).All the subscripts of these ME tensors α ij are named in this Cartesian coordinate, and several features are highlighted here.
First, the intriguing ME effect with all of the nine non-vanished components is identified at the lowest T ∼ 4 K in these experiments, while a remarkable difference exists in the magnitude of the measured P under H up to 9 T.Among them, the largest ME coefficients are α yz = α zy = 4.7 ps m −1 , as calculated from the dP/dH data at T = 4 K, but the diagonal ME coefficients are much smaller, with α xx ∼ α yy = 1.6 ps m −1 , and α zz ∼ 0.7 ps m −1 , respectively.Here, it should be mentioned that the four remaining components α xy , α xz , α yx , and α zx , are all indeed nonzero despite their small values.The uncertainties and leakage effects are safely excluded in the process of the measurements, as discussed in the paragraphs below [39].It is reasonable to consider that an obvious distortion will rise in the cycloid spin structure when an external magnetic field is applied along the [1-10] direction below T N1 .One considers a cycloid with h along [1][2][3][4][5][6][7][8][9][10].Then, for Q along [110], the spin current mechanism gives a polarization P = h × Q along [0, 0, 1], as observed in figure 5(h).
Second, upon increasing T, there exist anomalies in the measured P(T) data at T = T N2 for all the nine components α ij , suggesting possible ferroelectric transition under the applied magnetic fields at T N2 .Further towards the higher T N1 , only the two largest off-diagonal ME components α yz and α zy remain nonzero beyond T N2 , while the others are zero.The persistence of non-zero off-diagonal ME components suggests the presence of a different magnetic structure within this range, a topic to be explored in subsequent discussions.It is noted in these figures that the transition points T N1 and T N2 become ∼ 11 K and ∼ 8 K, respectively, under H up to 9 T, higher than the values mentioned above.The results are self-consistent, considering the measured χ(T) and C P (T) in figures 2(d) and (g), and T Ni is identified to shift toward the high-T region with increasing H.
With the neutron diffraction information in [21], one knows that the ferroelectric phase transition at T N1 (H = 9 T) ∼ 10 K is related to the incommensurate spin order, which hosts a frustration wave order corresponding to propagation vector k 1 = (2/3 + δ 1 , 2/3 + δ 1 , 0) with refining δ 1 ∼ −0.025 (1).The ME effect with all nonzero ME coefficients can be ascribed to the transition of magnetic structure to a more commensurate one below T N2 (H = 9 T) ∼ 8 K in Fe 2 GeO 4 .The symmetry of magnetic structure with decreasing T may go from the IC-AFM phase down to a canting AFM phase, which presents a nearly commensurate spin order with propagation vector k 2 = (2/3 + δ 2 , 2/3 + δ 2 , 0).The observed ME tensor, if all components are nonzero, is fully consistent with the −1 ′ magnetic point group below T N2 .However, if we assume that the four small off-diagonal components are ascribed to the possible sample misalignment, i.e. α xy = α xz = α yx = α zx = 0, we cannot find any ME coupling tensor corresponding to nonzero α xx , α yy , α yz , α zy and α zz in all of the 90 magnetic point groups.Surely, the precise magnetic structure below T N2 remains unclear so far, as reported in the supplementary materials of [21]., which exhibits two obscure situations using the powder neutron scattering.It requires more experiments to determine the magnetic structure, e.g.single crystal neutron scattering, which is beyond the scope of this work.

ME phase diagram
To summarize the effect of H on the magnetic and electric polarization properties, we present the ME phase diagrams of Fe 2 GeO 4 for H applying along the three directions in figures 6(a)-(c).The phase boundaries are determined by the observed magnetic, specific heat, and electric polarization anomalies as a function of H at various temperatures.The phase diagrams demonstrate the strong interplay of the magnetism and electric polarization, suggesting an intrinsic ME coupling in Fe 2 GeO 4 .
Though T N2 shows little T dependence and vanishes under H > H c along [110] and [1-10] directions, T N1 shifts obviously toward higher T with increasing H.The most striking feature in these phase diagrams is the phase boundary's strong dependence on H and T, simultaneously forming four different phase regions, as shown in figures 6(a) and (b).Nevertheless, slight variations in transition temperatures are inevitable, but the overall trends remain consistent.Considering these complicated phase regions, the possible origin is coupled to the competition between the FM and AFM interaction of Fe 2+ located in a strong frustrated site and hosting several degenerate states with similar energy.The ME states can cross the barrier to other ME states along with the magnetic field and temperature changes.In the low-H (H < H c ) region, Fe 2 GeO 4 hosts two successive AFM transitions at T N1 ∼ 9 K and T N2 ∼ 7 K, respectively.Besides, there is a narrow T-window of ∼2 K for the incommensurate AFM phase (IC-AFM), while the canting AFM (C-AFM) phase appears below T N2 [21].It is noted that an obvious SR induced by H rises in the low-T region, while it goes indefinable with increasing T up to T N1 .In the next section, we will discuss the mechanism of the field-driven ME effect for systems with successive phase transitions in Fe 2 GeO 4 .

Discussion
To this stage, one believes that the ferroelectricity in Fe 2 GeO 4 is generated by the magnetic field-driven spin ordering of Fe 2+ .We then focus our attention on the magnetic structure variation to better understand the ME mechanism.Based on the phase diagrams, as shown in figures 6(a)-(c), we would like to discuss the possible microscopic mechanism for the ME effect in Fe 2 GeO 4 .First, at zero magnetic fields, the lattice structure hosts a centrosymmetric space group Fd-3 m with decreasing T down to T = 5 K, indicating that a microscopic net electric polarization cannot be induced upon a lattice deformation [22].Here, one should believe that it is very different from the case of multiferroic FeTe 2 O 5 Br with amplitude modulation waves [40,41], which has been verified that a ferroelectric order due to exchange striction involving polarizable Te 4+ lone-pair electrons develop perpendicular to the wave vector q = (0.5, 0.463, 0) and Fe 3+ magnetic moments.
Secondly, in the IC-AFM state, a notable observation from the polarization results, as depicted in figure 5, is the anisotropy of the Fe 2 GeO 4 moments in the xy-plane.This anisotropy suggests the presence of canted moments in the xy-plane, which in turn implies the potential generation of a net magnetization ∆M(//H) upon the application of a magnetic field.Drawing upon the mechanisms of magnetic field-driven electric polarization [6][7][8], the breaking of symmetry along the z-axis allows for polarization generation in the z direction through fluctuations.Consequently, it can be inferred that anti-symmetric exchange interactions may contribute to the generation of electric polarization, with polarization occurring in both the z and y directions.
However, it should be noted that the C-AFM state characterized by a spin canting order encompasses nine non-zero components within the ME tensor matrix.An alternative magnetic structure could exist, wherein the Fe 2+ moments also exhibit spin canting in the z direction (not illustrated here).Considering the magnetically frustrated nature of the structure at temperatures below T N2 , it is plausible to attribute it to a triclinic phase with a magnetic point group of −1 ′ , corresponding to the results of polarization in figures 5(a)-(i).Based on the polarization in both the xy-plane and the z direction, it becomes evident that multiple mechanisms may contribute to the polarization P x (H//x, or H//y), P y (H//x, or H//y), and P z (H//x, or H//y), including the spin current mechanism, exchange-striction mechanism.Importantly, the presence of polarization in both the xy-plane and the z direction enables the involvement of these two mechanisms.Additionally, it is worth noting that the weak disturbance to ferroelectricity caused by the SR state under the applied magnetic field suggests that the mechanism driving the field-induced ME effect is akin to that observed in the case of the C-AFM state.
So far, the intrinsic proof for the ME coupling mechanisms in Fe 2 GeO 4 is still lacking, specifically in terms of low-temperature neutron scattering and high-resolution XRD performed on single crystals [42].Such difficulty remains true for a number of linear ME compounds reported recently in literature [43][44][45].Moreover, for the present case, keeping in mind that the orbital angular momentum is not quenched completely because of its strong lattice frustration, it is believed that the orbital moments can also contribute to the linear ME effect [45].Further investigation in the future is motivated to elucidate the microscopic origin of the ME phenomena in Fe 2 GeO 4 .

Conclusion
In conclusion, we have investigated extensively the magnetism and ME coupling properties of single crystal Fe 2 GeO 4 with a spinel structure, focusing on the measurements and discussion on all components of the ME tensor coefficients.It is noted that this is the first experimental report on the magnetic field-activated ME effect in the whole M 2 GeO 4 (M = Fe, Co, Ni) family, and it is surprising to demonstrate the remarkable ME responses as stimulated by the magnetic field.Two consecutive AFM transitions take place at T N1 ∼7.5 K and T N2 ∼ 6.7 K with the appearance of electric polarization anomalies, which display the intrinsic correlation of magnetism and H-induced electric polarization.Especially the existence of off-diagonal ME coefficients in Fe 2 GeO 4 indicates the formation of ferrotoroidic order, which may give rise to intriguing phenomena related to the coupling between magnetic order and ferroelectricity in pyrochlore structure.

Figure 1 .
Figure 1.(a) A schematic crystal structure of Fe2GeO4.(b) The schematic magnetic structure of Fe2GeO4 below TN1 viewed down [001].Red arrows and blue arrows represent Fe1 and Fe2 spins propagating sinusoidally along [110] and [−110] axises, respectively.The set of red arrows on the right side is a schematic diagram of the moment-modulated collinear spin chain in Ca3Co2O6.(c) The XRD patterns onto the cleavage planes of [110] and [111] directions.The insets are the photo-image of a typical single crystal, and Laue diffraction spots obtained along each direction.(d) The chemical composition analyzed with EDS.

Figure 2 .
Figure 2. (a-c) The dc magnetic susceptibilities χ of Fe2GeO4 as a function of temperature in the sample's geometry aligned along [110], [1-10], and [001] axes with a measuring field of 0.1 T. The light-colored curves and solid curves represent data obtained under zero field cooling (ZFC) and field cooling (FC) conditions.The insets are Curie-Weiss fitting of the inverse magnetic susceptibilities 1/χ.(d) The dc magnetic susceptibilities χ along [1-10] axis with various measuring magnetic fields.(e) The specific heat variations of Fe2GeO4 at zero field with magnetic contribution fitted by power-law CM ∼ T 3.2 .The inset is the Debye model fitting from 2 K to 100 K. (f) The T dependence of magnetic entropy ∆SM (T).(g) The T variations of magnetic contribution CM/T measured under different magnetic fields H up to 7 T along [1-10].The canting antiferromagnetic (C-AFM) phase, incommensurate antiferromagnetic (IC-AFM) phase, and paramagnetic (PM) phase are labeled on the top of figures.

Figure 3 .
Figure 3.The H dependence of magnetization M at selected temperatures with fields applied along (a) H // [110], (b) H // [1-10], and (c) H // [001], respectively.The insets are dM/dH curves for a plot at T = 2 K.The canting antiferromagnetic (C-AFM) and spin reorientations (SR) phases are marked on the top of figures.

Figure 6 .
Figure 6.H-T phase diagrams of Fe2GeO4 for magnetic fields applied along [110], [1-10], and [001] directions.Closed symbols represent the data obtained by the measurements of magnetism, specific heat, and electric polarization, respectively.