Abstract
Magnetoelectric (ME) effect is recognized for its utility for lowpower electronic devices. Largest ME coefficients are often associated with phase transitions in which ferroelectricity is induced by magnetic order. Unfortunately, in these systems, large ME response is revealed only upon elaborate poling procedures. These procedures may become unnecessary in singlepolardomain crystals of polar magnets. Here we report giant ME effects in a polar magnet Fe_{2}Mo_{3}O_{8} at temperatures as high as 60 K. Polarization jumps of 0.3 μC/cm^{2} and repeated mutual control of ferroelectric and magnetic moments with differential ME coefficients on the order of 10^{4} ps/m are achieved. Importantly, no electric or magnetic poling is needed, as necessary for applications. The sign of the ME coefficients can be switched by changing the applied “bias” magnetic field. The observed effects are associated with a hidden ferrimagnetic order unveiled by application of a magnetic field.
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Introduction
A significant effort has been invested into finding new materials in which macroscopic properties, such as the magnetization and the electric polarization, are coupled and controlled by external parameters like temperature and electric or magnetic fields^{1,2,3,4,5}. Materials where these quantities are interconnected are highly desired due to their importance in developing devices with new functionalities^{6,7,8}. Examples of materials falling into this category are the pyroelectric and multiferroic materials^{9,10,11,12}. The practical aspect of pyroelectric materials is the capacity to generate a current when they are subjected to a temporal temperature gradient through heating or cooling. Due to the efficient conversion of thermal energy into electrical energy, pyroelectric materials have offered numerous device applications, for example for temperaturesensing^{13,14} and for thermoelectric applications^{15}. The practical aspect of multiferroic materials is the ability to mutual control the magnetization (polarization) by the use of external electric (magnetic) fields, the effect known as magnetoelectric (ME) effect^{16,17,18}. The ME effect can be linear or/and nonlinear with respect to the external fields and it is characterized by the appropriate ME coefficients^{19,20}. At the present time, materials with large ME coefficients are exploited for developing lowpower magnetoelectronicbased devices and new multiple state memory elements^{21,22}. Recently, materials with significant ME response associated with ferroelectricity induced by magnetic order have been identified^{23,24,25}. Unfortunately, elaborate poling procedures, such as cooling in applied electric and magnetic fields, are needed to reveal the largest ME coefficients in these systems. Finding new materials with colossal ME coefficients lacking this drawback is of primary importance for prospective applications.
Materials belonging to the polar crystallographic symmetry groups lack the inversion symmetry at all temperatures. Many of these materials contain magnetic ions and they often exhibit longrange magnetic order. We call these materials “polar magnets”. The prerequisite for nontrivial magnetoelectricity is simultaneous breaking of time reversal symmetry and space inversion symmetry. Thus, all polar magnets should exhibit nontrivial ME effects below magnetic ordering temperatures. Importantly, monodomain polar single crystals can often be grown, potentially eliminating the need for any poling procedures to reveal the largest possible ME response. While the polar magnets are numerous, the investigation of their magnetoelectricity has been extremely limited. The few examples of polar magnets whose magnetoelectricity has been studied include GaFeO_{3} (ref. 26) and Ni_{3}TeO_{6} (ref. 24). Clearly, a targeted search for enhanced ME effects among polar magnets holds significant promise.
For the ME device applications, ferro or ferrimagnetic polar magnets are some of the best candidates, as macroscopic magnetic moment is needed for their functionality. Such compounds are rare. However, in some cases macroscopic magnetic moment is “hidden” within a nominally antiferromagnetic state and can be easily revealed in a modest applied magnetic field, thereby leading to a potentially large ME response. A wellknown example of such a hidden moment is realized in La_{2}CuO_{4}, the parent compound of highT_{C} cuprate superconductors^{27}. Each CuO plane exhibits a weak ferromagnetic moment due to canting of the spins of the otherwise regular Neel order. The canting results from DzyaloshinskyMoria interaction. Weak ferromagnetism is masked in zero magnetic field because of the antiferromagnetic interplane coupling. However, spin canting is responsible for the many distinct magnetic properties of this compound, including the unusual shape of the magnetic susceptibility in the vicinity of T_{N} and in an applied field and the atomicscale giant magnetoresistence in the fieldinduced weakly ferromagnetic phase. Another layered magnet in which a small ferrimagnetic moment of each layer is hidden at zero field is multiferroic (but nonpolar at high T) LuFe_{2}O_{4} (ref. 28). The giant magnetic coercivity and the unusual ME relaxation properties of LuFe_{2}O_{4} are related to the ferrimagnetism in its FeO layers. Similar to these compounds, a hidden magnetic moment in a polar magnet could result in a strongly enhanced magnetic response, which should lead to large ME effects when the magnetic moment and crystal structure are coupled.
Herein, we report giant ME effects in a monodomain polar magnet Fe_{2}Mo_{3}O_{8}, that possesses both multiferroic and pyroelectric characteristics. Below T_{N} ≈ 60 K, it exhibits a layered collinear magnetic structure with a small ferrimagnetic moment in each layer^{29}. As in La_{2}CuO_{4}, this moment is “hidden”, but can be revealed in a modest applied magnetic field^{28}. As a result of field and temperatureinduced magnetic transitions in Fe_{2}Mo_{3}O_{8}, the electric polarization exhibits changes as large as 0.3 μC/cm^{2}. As the hidden ferrimagnetism is converted to a bulk moment by an applied magnetic field, giant differential ME coefficients approaching 10^{4} ps/m are achieved. The observed effects are significantly larger than those previously reported in polar magnets, such as Ni_{3}TeO_{6} (ref. 24). The ME control is mutual, as both the magnetization and electric polarization can be tuned by the electric and magnetic field, respectively. Importantly, no electric or magnetic poling is needed and the sign of the differential ME coefficients can be switched by simply changing the applied “bias” magnetic field. Using first principles calculations, we show that exchange striction is the leading mechanism responsible for the observed ME effects. Our results demonstrate the promise of polar magnets as ME systems and indicate that their functional properties could be further enhanced by presence of a local (“hidden”) magnetic moment that can be easily converted to macroscopic magnetization by an applied field.
Results
Fe_{2}Mo_{3}O_{8}, known as the mineral kamiokite^{30,31}, consists of honeycomblike FeO layers separated by sheets of Mo^{4+} ions, See Fig. 1(a). The layers are stacked along the c axis. The FeO layer is formed in the ab plane by cornersharing FeO_{4} tetrahedra and FeO_{6} octahedra, as shown in Fig. 1(b). In this layer, the tetrahedral (Fe_{t}) and octahedral (Fe_{O}) triangular sublattices are shifted along the c axis by 0.614 Å with respect to each other^{31}, leading to short and long interlayer FeFe distances, see Fig. 1(a). The vertices of the FeO_{4} tetrahedra point along the positive c axis, reflecting the polar structure of Fe_{2}Mo_{3}O_{8} (ref. 31). The Mo kagomelike layer is trimerized. The Mo trimers are in the singlet state and do not contribute to magnetism^{32}. Below T_{N} ≈ 60 K, the Fe^{2+} moments exhibit the antiferromagnetic (AFM) order in the honeycomb layers, see Fig. 1(c). As discussed below, Fe_{O} has larger spin than Fe_{t} and therefore each of the FeO layers is ferrimagnetic^{33}. Along the c axis, the nearest Fe spins are aligned in the same direction, implying ferromagnetic interlayer coupling. The resulting stacking of the ferrimagnetic FeO layers along the c axis leads to vanishing macroscopic magnetic moment and we call this state AFM.
The temperature variation of DC magnetic susceptibility χ in zero fieldcooled (ZFC) and fieldcooled (FC) processes is shown in Fig. 1(e) for the magnetic field both parallel and normal to the c axis. The shapes of the curves are consistent with the transition to the AFM order shown in Fig. 1(c) at T_{N} = 61 K, with Fe^{2+} spins pointing along the c axis. The large difference between the caxis and inpane susceptibilities in the paramagnetic state demonstrates appreciable anisotropy of the Fe^{2+} spins. No thermal hysteresis is observed, see Supplementary Fig. 1. A large specific heat (C_{P}) anomaly is present at the magnetic transition, see Fig. 1(f). To account for the phonon part, the specific heat was fit to a double Debye model for T > T_{N} (90 to 200 K). The best fit, shown in Fig. 1(f), was obtained for the Debye temperatures θ_{D1} = 174 K and θ_{D2} = 834 K. It fails for T < T_{N} as it implies a negative magnetic contribution for T < 50 K. This indicates an additional lattice contribution for these temperatures, suggesting a structural transition associated with the magnetic order. This suggestion is corroborated by the temperature dependence of the dielectric constant ε(T) and the variation of the electric polarization ΔP(T) ∫ P(T)P(T = 120 K), both along the c axis, shown in Fig. 2(a,b). In particular, the jump of ΔP at T_{N} clearly indicates simultaneous magnetic and structural transitions. The magnitude of this jump, ~0.3 μC/cm^{2}, is larger than the values typically observed in multiferroics and is the largest measured value in polar magnets, to our knowledge. Importantly, no poling is needed in an already polar material to observe the changes shown in Fig. 2(a,b). In particular, ΔP was measured by integrating the pyroelectric current on warming after cooling down to T = 5 K in zero electric field (see Supplementary Fig. 2 for details). In our measurements, the direction of the ΔP vector (along or opposite to the positive direction of the c axis defined above) is undetermined. First principles calculations described below indicate that ΔP points in the positive c direction, hence we adopt this convention here.
Magnetic field (H) induces a metamagnetic transition signaled by sharp magnetization (M) jumps, see Fig. 2(c). It is accompanied by a structural transition indicated by the corresponding jumps in the electric polarization, as shown in Fig. 2(d). A small hysteresis is observed in the latter transition. The ΔP ∫ P(H)P(H = 0 T) vector is in the negative c axis direction and its value at T = 50 K is roughly twice as small as the ΔP induced at T_{N} for H = 0 T. No poling of any kind is needed. Replacement of Fe with Mn, as well as Zn doping on the Fe site are known to convert the AFM state observed in Fe_{2}Mo_{3}O_{8} into a ferrimagnetic (FRM) state^{32,33}, in which the AFM order in the FeO layers is preserved, but the spins in every second layer are flipped, see Fig. 1(c). In the FRM state, the ferrimagnetic moments of the FeO layers are coaligned, giving rise to a macroscopic magnetization. The extrapolation of the highfield M(H) data of Fig. 2(c) to zero field gives a positive intercept of ~0.5 μ_{B}/f.u. at T = 50 K, indicating the ferrimagnetic character of the highfield state, which we assume to have the same FRM structure as shown in Fig. 1(c). This assumption is corroborated by the FeO net ferrimagnetic moment of 0.6 μ_{B}/f.u. for a single layer, expected from the Moessbauer measurements of the Fe_{O} and Fe_{t} moments^{33} (4.83 μ_{B} and 4.21 μ_{B}, respectively), as well as by the results of the first principles calculations described below.
Discussion
To understand the microscopic origin of the observed ME effects, we have carried out abinitio calculations in the framework of density functional theory adding an onsite Coulomb selfinteracting potential U (DFT + U). For the DFT part, the generalized gradient approximation PerdewBurkeErnzerhof (GGAPBE) functional was used. For U = 0, the ground state is metallic with the FRM structure, but moderate correlation strength (U = 4 eV) leads to an AFM insulating ground state. While U of the order of 4 eV is required to obtain the correct ground state, its exact value was found to be unimportant for the magnetic exchange energies relevant to this work. The details of the DFT calculations can be found in the Methods section. The ionic positions were optimized for two imposed magnetic structures, the AFM and FRM. The FRM structure was found marginally higher in total energy (less than 10 meV/f.u.), indicating that this phase is expected to be induced in modest magnetic fields, consistent with our experimental data.
The calculated ionic shifts for the transitions from the paramagnetic (PARA) to the AFM state and from AFM to FRM, are shown in Figs 1(c) and 3(c,d). The ionic shifts for every ion in the unit cell are given in Supplementary Table I. The experimental paramagnetic structure and the calculated AFM and FRM structures were used. The ionic shifts can be utilized for an estimate of the magneticallyinduced electric polarization change ΔP. While the total polarization is a multivalued quantity, the difference ΔP between two structures is a welldefined quantity^{34}. For a qualitative comparison with experiment, it is sufficient to use the ioniclike formula for ΔP given by , where and are the caxis ionic coordinates for the initial and the final structures, respectively, are the formal ionic charges, V is the unit cell volume and the sum is taken over the unit cell. For the PARA to AFM and AFM to FRM transitions, we obtain ΔP values of 0.60(11) μC/cm^{2} and −0.55(11) μC/cm^{2}, respectively. The calculated magnitudes and the relative signs of ΔP are in good qualitative agreement with our experiments. The positive sign of ΔP for the PARA to AFM transition indicates that the ΔP vector points along the positive c axis, justifying the convention used in our work.
The calculated ionic shifts also allow to get an insight into the mechanism of the ME effect. The atoms shift to maximize the magnetic energy gains in the AFM and FRM states. Oxygen ions exhibit the largest shifts and therefore the ME energy gains should be associated with the modifications of the superexchange paths between the interacting Fe^{2+} spins. Lattice structure, as well as preservation of the inplane magnetic order in applied magnetic field imply that the largest magnetic coupling (J) is between the nearest Fe^{2+} ions, see Fig. 1(b). The calculations show that upon the transition from the paramagnetic to the AFM state, the FeOFe angle (θ) between the nearest Fe^{2+} increases from 109° to ~111°, mostly due to the oxygen shifts, see Fig. 1(d). The inplane antiferromagnetic J increases with increasing θ due to the more favorable FeOFe orbital overlap, resulting in the magnetic energy gain. Thus, we ascribe the ionic shifts, as well as the accompanying ΔP, to the exchange striction in the AFM state.
The FRM state can be induced both by a positive and a negative magnetic field along the c axis. The two states differ only by 180° rotation of every spin in the system. While the fieldinduced magnetizations should be opposite for the opposite fields, ΔP induced by exchange striction should be identical. This prediction is clearly confirmed by the data of Fig. 3(a,b). The calculated ionic shifts for the AFM to FRM transition, shown in Fig. 3(d), are opposite (but smaller) to those occurring at the PARA to AFM transition, see Fig. 3(c). In other words, the lattice partially relaxes towards the paramagnetic structure in the FRM state. This is consistent with the magnetic energy loss due to the interlayer interactions in the FRM phase and corresponding relaxation of the lattice distortion realized in the AFM state. As a result, ΔP is negative in the AFM to FRM transition. Thus, the data of Fig. 3(b), in combination with our first principles calculations, show that exchange striction underlies the ME effect in the transition to the FRM state, as in the AFM transition discussed above.
The sharpness of the fieldinduced transitions shown in Figs 2(d) and 3(a,b) gives rise to giant values of the differential ME coefficient dP/dH in the vicinity of the transition field, reaching almost −10^{4} ps/m for T = 55 K. (Consult Supplementary Fig. 2(d) for the fielddependent dP/dH for different temperatures). Combined with absence of poling requirements and the small hysteresis (0.02 T at 55 K, 0.007 T at 58 K), it leads to giant, reproducible and almost linear variation of P with H, as shown in Fig. 4(a) for T = 55 K. In the range shown, ΔP oscillates, varying by 0.08 μC/cm^{2} as H goes from 3.25 to 3.5 T and back. The inverse effect, in which an applied electric field (E) changes the magnetization is also giant, reproducible and linear, as shown in Fig. 4(b). At T = 55 K and H = 3.345 T, the magnetization varies by 0.35 μ_{B}/f.u. in the field oscillating between ±16.6 kV/cm, resulting in the dM/dE of −5700 ps/m. Similarly large differential ME coefficients dP/dH and dM/dE are observed at other points on the AFMFRM transition boundary shown in Fig. 4(c). These coefficients are more than an order of magnitude larger than those reported for the polar magnet Ni_{3}TeO_{6} (ref. 24), see Fig. 4(d).
When both external fields H and E are collinear and their direction coincides with the positive c axis of the crystal, both ΔP and ΔM are negative in applied positive H and E, respectively. Thus, both dP/dH and dM/dE are negative. The data of Fig. 3(a,b) show that, consistent with exchange striction mechanism, ΔM changes sign in negative H, while ΔP does not. As a result, both dP/dH and dM/dE change their sign and become positive in negative H and E, while retaining the same magnitudes. This sign reversal is illustrated in Supplementary Fig. 4.
In conclusion, polar magnets clearly possess a great potential as ME materials. The absence of poling requirements makes possible utilization of giant ME coefficients associated with sharp metamagnetic transitions practical, because reproducible, hysteresisfree linear responses can be achieved, as necessary for applications. In Fe_{2}Mo_{3}O_{8}, hidden ferrimagnetism of the FeO layers strongly enhances the magnetic response at the transition field, providing explanation for the observed giant differential ME coefficients. Exchange striction mechanism of the ME effect in Fe_{2}Mo_{3}O_{8} provides an additional functional capability of controlling the sign of these coefficients by the direction of the applied “bias” magnetic field. Therefore, studies of other polar magnets, especially with exchange striction ME mechanism and local ferrimagnetism, are, in our opinion, of significant promise.
Methods
Single crystal preparation and structure analysis
Fe_{2}Mo_{3}O_{8} single crystals were grown using a chemical vapor transport method at 1000 °C for 10 days, followed by furnace cooling. They are black hexagonal plates with typical size ~1 × 1 × 0.5 mm^{3}. Powder Xray diffraction measurement was performed on crushed powders of Fe_{2}Mo_{3}O_{8} single crystals. Refinement shows that the roomtemperature space group is P6_{3}mc. a and c lattice constants are 5.773(3) and 10.054(3) Å, respectively.
Measurements
All measurements of magnetic properties M(H), χ(T) and M(E) were performed in a Quantum Design MPMSXL7. The dielectric constant ε(T), specific heat C_{P}(T), electric polarization P(T) and P(H) properties were performed using Quantum Design PPMS9. ε(T) was measured with 1 V a.c. electric field applied along the c axis using a Quadtech 7600 LCR meter at 44 kHZ. Specific heat measurements were conducted using the standard relaxation method. P(T) and P(H) were obtained by integrating the pyroelectric current J(T) and magnetoelectric current J(H), which were measured using Keithely 617 programmable electrometer at 5 K/min warming rate and ramping magnetic field with 200 Oe/s.
First principles calculations
Abinitio calculations were performed using the fullpotential linearized augmented plane wave (FPLAPW) method as implemented in the WIEN2k code^{35} within the framework of density functional theory^{36,37}. The electronic, magnetic and structural properties of Fe_{2}Mo_{3}O_{8} were calculated using the generalized gradient approximation (GGA) for the exchangecorrelation potential, in the form of Perdew, Burke and Ernzerhof^{38,39} (PBE) plus an onsite Coulomb selfinteraction correction potential (U) treated by DFT + U and the doublecounting in the fully localized limit^{40}. Since the symmetry of low temperature crystal structure is not known, the point group symmetry of the hexagonal paramagnetic space group P6_{3}mc (ref. 31) was artificially reduced for the purpose of optimizations of internal parameters (OIP). All the calculations were done in the triclinic space group P1, with the lattice parameters kept fixed to a = b = 5.773 Å, c = 10.054 Å, α = 90°, β = 90°, γ = 120°. OIP were performed with imposed AFM and FRM magnetic configurations, using as the initial guess the experimentally determined internal parameters of the paramagnetic phase^{31}. The search for equilibrium ionic positions was carried out by means of the PORT method^{41} with a force tolerance ≤0.5 mRy/Bohr. The calculations were performed with more than 200 kpoints in the irreducible wedge of the Brillouin zone (10 × 10 × 4 mesh). The total energy, charge and force convergence criteria were ~10^{−4} Ry, ~10^{−4} electrons and 0.25 mRy/Bohr, respectively. The muffintin radii R_{MT} were chosen as 1.90, 1.93 and 1.66 bohr for Mo, Fe and O, respectively. To ensure that no charge leaks outside the atomic spheres, we have chosen the energy which separates the core and the valence states to be −10 Ry, thus treating the Mo(4s, 4p, 4d, 5s), Fe(3s, 3p, 3d, 4s) and O(2s, 2p) electrons as valence states. All other input parameters were used with their default values.
Additional Information
How to cite this article: Wang, Y. et al. Unveiling hidden ferrimagnetism and giant magnetoelectricity in polar magnet Fe_{2}Mo_{3}O_{8}. Sci. Rep. 5, 12268; doi: 10.1038/srep12268 (2015).
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PORT is a method implemented in the WIEN2K code for finding the equilibrium positions by minimization of the total energy; for further details see the WIEN2K User Guide.
Acknowledgements
This work was supported by DOE under Grant No. DEFG0207ER46382. K.H. (theory) was supported by NSFDMR 1405303. Heat capacity measurements at NJIT (T.A.T.) were supported by DOE under Grant DEFG0207ER46402. The PPMS (used for heat capacity measurements) was acquired under NSF MRI Grant DMR0923032 (ARRA award). Discussions with Karin M. Rabe are gratefully acknowledged.
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Y.Z.W. carried out magnetoelectric, magnetic and dielectric measurements. T.A.T. performed the heat capacity measurement. B.G. synthesized single crystals and performed X.R.D. G.L.P. developed the theoretical model under the supervision of K.H., Y.Z.W., G.L.P., V.K. and S.W.C. wrote the manuscript. S.W.C. initiated and supervised the research.
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Wang, Y., Pascut, G., Gao, B. et al. Unveiling hidden ferrimagnetism and giant magnetoelectricity in polar magnet Fe_{2}Mo_{3}O_{8}. Sci Rep 5, 12268 (2015). https://doi.org/10.1038/srep12268
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DOI: https://doi.org/10.1038/srep12268
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