Magnetic properties and origins of ferroelectric ... - APS link manager

PHYSICAL REVIEW B 87, 075127 (2013)

Magnetic properties and origins of ferroelectric polarization in multiferroic CaMn7 O12 J. T. Zhang,1 X. M. Lu,1,* J. Zhou,2 H. Sun,1 F. Z. Huang,1 and J. S. Zhu1 1

National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China 2 National Laboratory of Solid State Microstructures and Department of Material Science and Engineering, Nanjing University, Nanjing 210093, China (Received 10 January 2013; published 19 February 2013) The magnetic properties and the origins of ferroelectric polarization in recently discovered multiferroics material CaMn7 O12 have been investigated on the basis of model analysis and first-principles calculations. We find the noncollinear-type exchange-striction mechanism and the p-d hybridization with spin-orbit coupling coexist in the origins of polarization. The induced giant ferroelectricity can be mainly attributed to the former, related to two kinds of charge-order chains combined with the same spiral spin chirality as Mn ions. Our results confirm that the polarization induced by the two mechanisms follows the Si · Sj and Si × Sj laws, respectively. DOI: 10.1103/PhysRevB.87.075127

PACS number(s): 75.85.+t, 75.30.Et, 75.30.Gw, 77.80.−e

I. INTRODUCTION

The coupling phenomena between the electric and magnetic order in multiferroics, i.e., magnetism can be controlled via electric fields or vice versa, have attracted considerable attention recently due to the potential application in spintronics and magnetic data storage.1–4 The electric field control of magnetism in BiFeO3 provides a great opportunity to explore the low-energy-consumption microelectronic devices,5–7 while the magnetoelectric effect is usually small in the proper ferroelectrics because the polarization and magnetization originate from different sublattices. The magnetically induced improper ferroelectrics often exhibit strong magnetoelectric coupling. There have been three microscopic mechanisms proposed for magnetically induced ferroelectricity. The two most prevailing are the exchange-striction mechanism emerged in the collinear magnetic structure, such as orthorhombic RMnO3 (R = Ho-Lu, Y),8 DyFeO3 ,9 and Ca3 (Co,Mn)2 O6 ,10 and the spin-current model,11 or, equivalently, the inverse Dzyaloshinskii-Moriya (DM) mechanism12 in spiral magnets, such as the orthorhombic RMnO3 (R = Tb,Dy),13–16 CoCr2 O4 ,17 MnWO4 ,18 and some hexaferrites,19,20 etc. Recently, the p-d hybridization mechanism with spin-orbit coupling (SOC) was proposed to explain the induced ferroelectricity in triangular-lattice antiferromagnets CuFeO2 (Ref. 21) and CuCrO2 (Ref. 22) and simple antiferromagnet Ba2 CoGe2 O7 .23,24 However, the induced polarization in existing materials is too small, restricting its application in devices. Hence the search for magnetically induced ferroelectrics with large polarization is still a hot topic in current multiferroics research. Quite recently, a giant improper ferroelectric polarization has been observed in rhombohedral CaMn7 O12 ceramic and single crystals.25,26 The magnetic susceptibility and neutron power diffraction measurements indicated the existent incommensurate helical magnetic structure below 90 K. The induced electric polarization was found to be perpendicular to the spin rotation plane, which cannot be explained by the spin-current model. Subsequently, a theoretical work investigated the origin of ferroelectric polarization by extending the general spin-current model, in which they focused on calculating the coupling coefficients of different spin dimers and then attributed the polarization to the combined effect 1098-0121/2013/87(7)/075127(6)

of exchange striction and DM interaction.27 In this paper, starting from microscopic model analysis, we demonstrate how symmetric spin-exchange interactions act on the ferroelectric polarization. We show that the symmetric exchange interactions in the two kinds of Mn3+ -Mn4+ charge-order chains can cause the large ferroelectric polarization and certify that using the first-principles calculations. In addition, our results indicate that another mechanism, p-d hybridization with SOC, also exists and contributes a little ferroelectric polarization. We first calculate the magnetic properties using the firstprinciples method to understand the observed helical magnetic structure. Then we explain the calculated superexchange interaction according to the empirical rule of exchange interaction and the magnetic anisotropy using the single-ion theory. Based on the actual magnetic structure, we investigate the respective roles of noncollinear-type exchange-striction and the p-d hybridization mechanism in the polarization. Then we change the helical magnetic structure to investigate the rules of polarization in the two mechanisms and confirm the polarization follows the Si · Sj and Si × Sj laws. II. CRYSTAL AND MAGNETIC STRUCTURE

As has been known, CaMn7 O12 undergoes a first-order phase transition at ∼440 K from the high-temperature cubic ¯ to the low-temperature rhombohedral phase (space group Im3) ¯ 28 At high temperature, it can be phase (space group R3). considered the distorted perovskite with quadruple AA3 B4 O12 structure. Three Mn ions occupy the perovskite A sites, and the remaining four Mn ions occupy the B sites. When it undergoes the charge-order phase transition, the B sites develop into the 3:1 ordered occupation for Mn3+ and Mn4+ ions, accompanied by the emergence of orbital order. The Jahn-Teller distortion of the oxygen octahedral with four long and two short Mn-O bonds is uncommon in perovskite manganites, which may have a significant influence on the magnetic structure. The crystal structures of both phases are shown in Fig. 1, in which Mn1 and Mn2 denote the Mn3+ ions in A and B sites, respectively, and Mn3 denotes the Mn4+ ions. The single-crystal neutron diffraction measurements show that a helical magnetic structure with an incommensurate

075127-1

©2013 American Physical Society

ZHANG, LU, ZHOU, SUN, HUANG, AND ZHU

PHYSICAL REVIEW B 87, 075127 (2013) IV. RESULTS AND DISCUSSION A. Superexchange couplings

FIG. 1. (Color online) (a) The crystal structure of CaMn7 O12 in the cubic Im3¯ phase. The Mn2 and Mn3 ions occupy the same Wyckoff position. (b) The crystal and helical magnetic structure in the hexagonal R3¯ phase. The magnetic structure is adopted according to Ref. 26.

propagation vector (0,1,0.963) emerges in the temperature range TN2 (48 K) < T < TN1 (90 K).26 The spins of Mn1, Mn2, and Mn3 ions follow a helical modulation along the c axis. The spins in the ab plane are parallel for both Mn1 and Mn2 ions. The magnetic moments are confined to lie in the ab plane and have a 120◦ rotation angle with respect to the neighbor plane for every sublattice. For Mn1 and Mn2 ions along the c axis, their spins are parallel. The directions of the Mn3 spins are nearly perpendicular to the c axis and have an angle of 150◦ with the in-plane Mn2 spins. Most importantly, the helical magnetic structures of the three Mn sites have the same spiral chirality, breaking inversion symmetry into magnetic point group 31’ to allow the polarization arising along the c axis.

III. MODEL AND METHODS

Our first-principles calculations were performed using the projector-augmented wave method29 within the local spindensity approximation plus the on-site repulsion (LSDA + U ), as implemented in the Vienna ab initio simulation package (VASP).30 We use Ueff = 2 eV on Mn 3d states and a plane-wave cutoff of 500 eV. The calculations are performed with a 3 × 3 × 5 -centered k-point sampling. We use the hexagonal unit cell to simulate the experimental helical magnetic structure. The superexchange coefficients are calculated by comparing the energies of different magnetic structures. The magnetic anisotropy energy is calculated with the SOC included and a convergence threshold of 10−7 eV due to the small value of SOC. To exclude the influence of the exchange couplings on the single-ion anisotropy, we consider only one Mn ion in the unit cell by replacing the remnant Mn3+ and Mn4+ ions with nonmagnetic Sc3+ and Ti4+ ions, respectively. For the calculations of the ferroelectric polarization, the Berry phase method31,32 is used. To investigate the operative mechanisms and corresponding contributions to the ferroelectric polarization, the element substitution method is also used in the calculations.

We first investigate the exchange interactions in CaMn7 O12 , which have a direct effect on the magnetic structure. We consider all the neighbor exchange paths between the Mn1, Mn2, and Mn3 ions. To obtain the exchange coupling constants, we calculate the total energies for several collinear magnetic configurations with different spin orientations on different sites. Wethen map the energies onto the Heisenberg model, E = − 12 ij Jij Si Sj . Here Si = 2 on the Mn1 and Mn2 ions and 32 on the Mn3 ions, and the sum is over all nearest-neighbor exchange paths. Our results show that the strong J11 and the weak J12 (Mn1-Mn2 along the non-c-axis direction) are antiferromagnetic exchange interactions. The rest are ferromagnetic exchange interactions, dominated by c (Mn1-Mn2 along the c direction). The the strong J13 and J12 strong antiferromagnetic interactions J11 would cause a spin c make frustration, and the strong ferromagnetic interactions J12 the spins parallel for Mn1 and Mn2 ions along the c axis. The calculated results are basically consistent with the Ref. 27, except for some quantitative differences. Additionally, it should be noted that the exchange interacc tions between Mn1 and Mn2 ions along different paths J12 and J12 are significantly different, which can be explained by the formed orbital order. The Mn1, Mn2, and Mn3 ions locate in the square plane, the axially compressed oxygen octahedral, and the regular oxygen octahedral crystal field, respectively. The crystal-field splitting of d states for three Mn ions are shown in Fig. 2(a), according to the group-theory analysis. The three Mn ions have empty dx 2 −y 2 , dz2 , and eg orbital states, respectively, which can be confirmed by the calculated deformation charge density. We try to understand the minus of the exchange interaction in CaMn7 O12 according to the semiempirical Goodenough-Kanamori-Anderson (GKA) rules.33 The hopping p electron to the empty d orbital of Mn ions makes their spins parallel due to Hund’s coupling. Hence the sign of superexchange interaction depends on the direct p-d exchange coupling between the remanent p electron and Mn ions. Under normal circumstances, the nonorthogonal mixing of the p and occupied d orbitals brings the negative exchange integral, causing the spins of two Mn ions to be parallel. Conversely, the spins are antiparallel in the case of orthogonal mixing. We first focus on the exchange coupling between Mn1 and Mn2 ions, whose bond angles are close to right angle. The characters of the bond length, bond angle, and orbital are shown in Fig. 2(b). For the couplings along the c axis, the coupled Mn2-O bond is compressed, so the bonding orbital along this bond is empty. The p-d mixing only emerges between the p orbital and dxz /dyz orbitals. This nonorthogonal mixing causes the strong ferromagnetic coupling. As for the other Mn1-O-Mn2 bonds, the coupled Mn2-O bond is not the compressed axis, so the dx 2 −y 2 electron occupies this bond, causing the orthogonal mixing with the p electron. This extra orthogonal mixing induces the antiferromagnetic coupling and competes with the nonorthogonal mixing, giving rise to the weak antiferromagnetic interaction. The above analyses explain qualitatively the causes of the strong ferromagnetic and weak antiferromagnetic interactions of the different

075127-2

MAGNETIC PROPERTIES AND ORIGINS OF . . .


(a)

(b)

FIG. 3. (Color online) Energy dependence on the direction of the Mn spin for containing single (a) Mn1, (b) Mn2, and (c) Mn3 ions severally in the unit cell. The angles are adopted according to local Cartesian coordinates and are relative to the x axis. For (d) three cases are considered: all the Mn1 ions, all the Mn2 ions, and the Mn1 and Mn2 ions simultaneously in the unit cell. The spins are parallel in the ac plane, and the angles are relative to the a axis.

c b

a

FIG. 2. (Color online) (a) Schematic illustration of the splitting of the d-orbital energies in three Mn sites and the empty orbital obtained from the deformation charge density. (b) The structural and orbital characters for the superexchange couplings.

Mn1-O-Mn2 bonds. Now we turn to the antiferromagnetic interaction between Mn1 ions. The two neighbor Mn1O4 planes are nearly perpendicular and have no shared oxygen ion. The occupied dz2 orbital of one Mn1 ion points directly to the other Mn1 ion and forms the orthogonal mixing with the p orbitals along the two Mn1-O bonds, inducing the strong antiferromagnetic interaction. B. Magnetic anisotropy

The spin orientations of the magnetic structure depend on the magnetic anisotropy of three ions. It has been shown that in the triangular-lattice antiferromagnets, the spin spiral plane is confined, depending on the sign of the anisotropy constant.22 We investigate the single-ion anisotropies of the three Mn ions, including the SOC in the calculation. The results for the three Mn ions are shown in Figs. 3(a)–3(c), respectively. It is clearly shown that the Mn1 ion has the easy-axis anisotropy with the easy axis perpendicular to the Mn1O4 plane, which is consistent with the previous results.27 However, for Mn2 ion, we find that the easy-plane anisotropy evolves into the easy axis (along the longest Mn-O bond) due to the discrepancy of the bond length in the plane. The Mn3 ion has a small magnetic anisotropy, with its easy axis along the c axis. Now we consider the causes of the different magnetic anisotropies of Mn1 and Mn2 ions. The single-ion anisotropy

derives from the combination of the crystal-field effect and SOC. The ground multiplet of Mn3+ (3d 4 ) ions is 5 D. Under the respective D4h crystal fields, the ground states of Mn1 and Mn2 ions are |5 B1g = |dx 2 −y 2 ,|5 A1g = |dz2 , respectively. The first-order perturbation of SOC vanishes in these singlet states. Accurate to the second-order perturbation approximation, the spin Hamiltonian can be written as34 HS(2) = −λ2 αβ Sα Sβ , αβ

where αβ =

0|Lα |nn|Lβ |0 n

En − E0

(α,β = x,y,z),

λ is the SOC constant, and |n represents the crystal-field excited state in the ground multiplet. Considering the additional in-plane distortion on the single-axis D4h symmetry, the spin Hamiltonian develops into HS(2) = DSz2 + E Sx2 − Sy2 , where D = λ2 [(xx + yy )/2 − zz ] is the zero-field splitting constant for the D4h crystal field and E = λ2 (yy − xx )/2 is the in-plane anisotropy. A negative D determines the easy-axis anisotropy. Through detailed calculation according to the splitting energy level of the crystal field,35 we find the values of D for the Mn1 and Mn2 sites are negative and positive, respectively, which means the easy-axis anisotropy of Mn1 and easy-magnetization-plane anisotropy of Mn2 ions under the D4h crystal field. Meanwhile, the negative E values indicate the easy axis of the Mn2 ions is along the longest Mn2-O bond. Next, we focus on the final anisotropy induced by all the Mn1 and Mn2 ions. The total magnetic anisotropy in the ab plane should be canceled due to the threefold rotation symmetry, which is confirmed by the first-principles calculations.

075127-3



As for the origins of the ferroelectric polarization in CaMn7 O12 , it has been found that the direction of polarization is perpendicular to the spin rotation plane of the helical magnetic structure, ruling out the spin-current mechanism.26 By examining carefully the possible models of induced ferroelectricity, we find the exchange-striction mechanism working in two charge-order chains is responsible for the large ferroelectric polarization, which is illustrated in Fig. 4(a). The Mn2-Mn3 and the Mn1-Mn3 ions form the Mn3+ -Mn4+ chains along the three equivalent [100] and diagonal (excluding the [111]) directions of the cubic unit cell, respectively. For the Mn2-Mn3 chain, the Mn4+ spins make an angle of 30◦ with the left Mn3+ spins and 90◦ with the right Mn3+ spins. Due to the ferromagnetic exchange-coupling interaction, the neighbor Mn3+ -Mn4+ pairs with 30◦ spin angle would attract each other, causing the ions to shift away from centrosymmetric positions and generating the ferroelectric polarization along the opposite direction of the chain. Considering the existent charge-order chains along the three pseudocubic directions, the total polarization would orientate along the [111] direction, i.e., the threefold rotation axis. An identical mechanism exists in the Mn1-Mn3 chain, while the polarization is the reverse relative to the Mn2-Mn3 chain. (b) 2

P Mn2-Mn3

C. Origins of ferroelectric polarization

Polarization (µC/cm )

(a)

large DM interaction in CaMn7 O12 is interesting and may provide new insight into the search for materials with colossal magnetoelectricity.

[100]

20 1.0 15

0.5

10

0.0

5

-0.5

0

-1.0 0

Mn1-Mn3

120

180

240

300

360

Angle (degree) (d) 2

Polarization (µC/m )

100

a

b

2.5

80

2.0

c

60

1.5

40

1.0

20

0.5

0 0

30

60

90

120

150

Energy (meV/unit cell)

(c)

60

[-111]

Energy (meV/unit cell)

Meanwhile, we find the total anisotropy of all Mn1 or Mn2 ions always develops into the easy-axis (c-axis) anisotropy or the easy-plane (ab plane) anisotropy, depending on the angles between the local z axis of the crystal field and the c axis.35 The real angles for Mn1 and Mn2 ions both meet the conditions of the easy-plane anisotropy, ( 25 sin2 θ − 1)D < 0. Hence, the observed ab-plane spin orientations are the combined effect of exchange couplings and the collaborative single-ion anisotropy of Mn1 and Mn2 ions. These are further confirmed by the first-principles results shown in Fig. 3(d), in which the energy reaches minimum when spins are orientated in the ab plane for the three cases and the total anisotropy induced by Mn2 ions is predominant. The Mn3 ion has a small easy-c-axis anisotropy due to the cubic crystal-field symmetry. Therefore, the ab-plane orientations of Mn3 spins mainly depend on the exchange interactions with Mn1 and Mn2 ions. It has been shown in Ref. 27 that the DM interaction between Mn2 and Mn3 ions is anomalously large and responsible for the in-plane Mn3 spin direction. The DM interaction emerged widely in the theoretical study of the magnetic phase diagram of perovskite manganites36,37 and plays an important role in the form of the spiral spin order of BiFeO3 .38 Most importantly, DM interaction provides the microscopic mechanism for magnetoelectric coupling, such as causing polarization in transverse spiral spin order and ferroelectrically inducing weak ferromagnetism,39 which is the only known mechanism for electric field control of magnetization reversal in a single-phase material. However, the induced polarization or magnetization is usually small due to the dependence on SOC. The observed

0.0 180

Angle (degree) FIG. 4. (Color online) (a) Diagrammatic sketch of the noncollinear-type exchange-striction mechanism in the two chains. The double arrows mean the neighbor ions shift towards each other. (b) The polarization and energy dependence on the spin angles in the noncollinear-type exchange-striction mechanism. (c) Perspective view of an isosurface calculated for the electron density difference between the SOC being included and excluded. The arrows denote the spin direction of Mn1 ions in one ab plane with the helical spin structure. (d) The polarization and energy dependence on the spin angles with the spin model in the inset. 075127-4

MAGNETIC PROPERTIES AND ORIGINS OF . . .


To certify the above analysis of the origin of ferroelectric polarization, we calculate the ferroelectric polarization using the Berry phase method. For the Mn2-Mn3 chain and the Mn1Mn3 chain, we replace the Mn1 and Mn2 ions, respectively, with the nonmagnetic Sc3+ ions. We relax the positions of magnetic ions with the helical spin structure and find the polarizations induced by ion displacement are −4508 and 5196 μC/m2 , respectively, along the c axis, which confirms the above analysis. However, with the electronic shielding effect considered and its contribution included, the total polarizations are 3492 and 743 μC/m2 , respectively. Then we focus on the purely electric mechanism by recalculating the polarization under the unrelaxed structure. We find P = 4755 and −435 μC/m2 , respectively, for the two chains, whose values have the same order of magnitude as the relaxed structure, but the direction of polarization for the Mn1-Mn3 chain is switched. The discrepancy in polarization between the relaxed structure and the purely electronic effect has been observed in the calculations for other magnetically induced ferroelectrics.15,16 The polarization values of the Mn2-Mn3 chain in both cases are close to the observed polarization of 2870 μC/m2 , which indicates that the exchange-striction interaction in the Mn2-Mn3 chain is responsible for the large ferroelectric polarization of CaMn7 O12 . Since such a noncollinear-type exchange-striction mechanism is rare and has not been studied, we investigate this mechanism by changing the spin angle between Mn3+ and Mn4+ ions in Mn2-Mn3 chain. We first reverse the spin spiral chirality of Mn3+ ions and find the polarization disappears. This indicates the emergence of the polarization depends on the same spiral chirality of the two kinds of ions. Then we keep the spiral chirality and rotate the Mn3+ spins in the ab plane to change the relative spin angles. We calculate the polarization and energy dependence on the spin angles, and the results are shown in Fig. 4(b). The polarization and energy exhibit sine-wave modulation relative to the spin angles. The polarization disappears at 60◦ , accompanied by the energy reaching its minimum. This dependence of polarization on the spin angles can be interpreted by the modifications of the superexchange interaction caused by displacement of magnetic ions. Through detailed calculation based on the two-magnetic-ions model,35 we can obtain the polarization, P ∝ cos(θl + 30◦ ), where θl is the left spin angle, which is consistent with the first-principles results. The above results indicate the polarization follows the Si · Sj law in the noncollinear-type exchange-striction mechanism. In addition to the above calculation, we also investigate the effect of a single helical spin order on polarization since it has been reported recently that the helical spin order can induce polarization, as found in triangular-lattice antiferromagnets.22,40,41 When we turn on the SOC, the helical spin orders of three sublattices induce polarizations of −29, −248, and 10 μC/m2 , respectively. Apparently, neither the exchange-striction mechanism nor the spin-current mechanism can cause the polarization in this case. Recently, spin-dependent p-d hybridization42,43 with SOC has been proposed to explain the ferroelectricity driven by helical

spin-spiral order. Furthermore, it becomes a single-ion case in Ba2 CoGe2 O7 ,23 rather than the spin-correlation one. Subsequently, a general theory is proposed to generalize the polarization induced by the helical spin spiral order, in which the polarization is divided into a single-spin case and the spin-exchange case.44 Because of the local inversion symmetry of the Mn sites in the ligand field, the single-spin case should disappear. Since the spin-dependent p-d hybridization requires the SOC, we calculate the electron density difference between including and excluding the SOC. We find the SOC causes different electron gains and losses for the three Mn1O4 planes in same ab plane [shown in Fig. 4(c)] due to the different angles between the spin direction and the Mn1-O bonds. The SOC makes the charge gather along the Mn-O bonds which have the spin component to increase the covalent. The charge transfer on O sites is asymmetric and has a uniform component along the c axis. The above results confirm the spin-dependent p-d hybridization mechanism exists, which operates as the spin-correlation effect. In order to make certain how this mechanism affects the polarization, we construct a spin model, as shown in Fig. 4(d). We set the Mn1 spins in the up and down ab planes antiparallel and change the direction of the spins in the middle plane. The dependence of polarization and energy on the spin angle are depicted in Fig. 4(d). The near-sinusoidal curve about the polarization indicates the polarization obeys P ∝ Si × Sj , and the energy curve indicates that the coupling vector and spins Si ,Sj form the right-hand chirality. It should be noted that the emergence of polarization does not depend on the inversion symmetry at the center of two spin sites since inversion symmetry only exists at the center of two Mn3 ions. This is different from the DM interactions. The polarization term P ∝ Si × Sj has been referred to as the generalized spin-current model in Ref. 44. The spin-current and bond polarization models both consider the p-d hybridization and SOC in the M-O-M cluster model, but the spin-current model treats the p-d hybridization as perturbation, and the bond polarization model treats the SOC as perturbation. In addition, the actual microscopic process should be more complicated than the M-O-M cluster model in the bond polarization and spin-current models. The extended p-d hybridization mechanism with SOC has been proposed to explain the polarization in CuFeO2 ,40 in which the Fe4 O2 cluster model is considered with the SOC on O sites included. As for the same mechanism in CaMn7 O12 , why it affects the antisymmetric exchange coupling is still unclear and needs further investigation. The above results and analysis of the polarization indicate both the noncollinear-type exchange-striction mechanism and p-d hybridization with SOC contribute to the polarization. The exchange-striction mechanism induces the large polarization close to that experimentally observed. This mechanism operates in two kinds of charge-order chains, in which the Mn2-Mn3 chain plays a big role in the large polarization. The p-d hybridization mechanism can induce a small polarization. The induced polarizations in the two mechanisms follow the Si · Sj and Si × Sj laws, respectively, which confirms the expression proposed in the general theory for ferroelectric polarization.44

075127-5



V. CONCLUSIONS

ACKNOWLEDGMENTS

In summary, we have investigated the magnetic properties and the origins of the ferroelectric polarization in CaMn7 O12 . We explain the calculated superexchange interaction and the magnetic anisotropy by the empirical rule of exchange interaction and the single-ion theory, respectively. As for the origins of polarization, we find the noncollinear-type exchange-striction and the p-d hybridization mechanism coexist in this material. The former operates in two kinds of Mn3+ -Mn4+ chains, and one of them causes the giant ferroelectric polarization. We find the polarization induced in the two mechanisms follows the Si · Sj and Si × Sj laws, respectively.

Computer resources provided by the High Performance Computing Center of Nanjing University are gratefully acknowledged. This work was supported by the State Key Program for Basic Researches of China under Grant No. 2009CB929501, the National Science Foundation of China (Grants No. 61271078, No. 51002075, No. 51102133, and No. 51225201), the Priority Academic Program Development of Jiangsu Higher Education Institutions, and the Fundamental Research Funds for the Central Universities.

*

23

[email protected] N. A. Spaldin and M. Fiebig, Science 309, 391 (2005). 2 M. Fiebig, J. Phys. D 38, R123 (2005). 3 S.-W. Cheong and M. Mostovoy, Nat. Mater. 6, 13 (2007). 4 K. F. Wang, J.-M. Liu, and Z. F. Ren, Adv. Phys. 58, 321 (2009). 5 Y. H. Chu, L. W. Martin, M. B. Holcomb, M. Gajek, S. J. Han, Q. He, N. Balke, C. H. Yang, D. Lee, W. Hu et al., Nat. Mater. 7, 478 (2008). 6 D. Lebeugle, A. Mougin, M. Viret, D. Colson, and L. Ranno, Phys. Rev. Lett. 103, 257601 (2009). 7 G. Catalan and J. F. Scott, Adv. Mater. 21, 2463 (2009). 8 I. A. Sergienko, C. Sen, and E. Dagotto, Phys. Rev. Lett. 97, 227204 (2006). 9 Y. Tokunaga, S. Iguchi, T. Arima, and Y. Tokura, Phys. Rev. Lett. 101, 097205 (2008). 10 Y. J. Choi, H. T. Yi, S. Lee, Q. Huang, V. Kiryukhin, and S.-W. Cheong, Phys. Rev. Lett. 100, 047601 (2008). 11 H. Katsura, N. Nagaosa, and A. V. Balatsky, Phys. Rev. Lett. 95, 057205 (2005). 12 I. A. Sergienko and E. Dagotto, Phys. Rev. B 73, 094434 (2006). 13 T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature (London) 426, 55 (2003). 14 T. Goto, T. Kimura, G. Lawes, A. P. Ramirez, and Y. Tokura, Phys. Rev. Lett. 92, 257201 (2004). 15 H. J. Xiang, S. H. Wei, M.-H. Whangbo, and J. L. F. Da Silva, Phys. Rev. Lett. 101, 037209 (2008). 16 A. Malashevich and D. Vanderbilt, Phys. Rev. Lett. 101, 037210 (2008). 17 Y. Yamasaki, S. Miyasaka, Y. Kaneko, J.-P. He, T. Arima, and Y. Tokura, Phys. Rev. Lett. 96, 207204 (2006). 18 K. Taniguchi, N. Abe, T. Takenobu, Y. Iwasa, and T. Arima, Phys. Rev. Lett. 97, 097203 (2006). 19 T. Kimura, G. Lawes, and A. P. Ramirez, Phys. Rev. Lett. 94, 137201 (2005). 20 Y. Kitagawa, Y. Hiraoka, T. Honda, T. Ishikura, H. Nakamura, and T. Kimura, Nat. Mater. 9, 797 (2010). 21 T. Kimura, J. C. Lashley, and A. P. Ramirez, Phys. Rev. B 73, 220401(R) (2006). 22 S. Seki, Y. Onose, and Y. Tokura, Phys. Rev. Lett. 101, 067204 (2008). 1

H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and Y. Tokura, Phys. Rev. Lett. 105, 137202 (2010). 24 H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and Y. Tokura, Phys. Rev. B 85, 174106 (2012). 25 G. Q. Zhang, S. Dong, Z. B. Yan, Y. Y. Guo, Q. F. Zhang, S. Yunoki, E. Dagotto, and J.-M. Liu, Phys. Rev. B 84, 174413 (2011). 26 R. D. Johnson, L. C. Chapon, D. D. Khalyavin, P. Manuel, P. G. Radaelli, and C. Martin, Phys. Rev. Lett. 108, 067201 (2012). 27 X. Z. Lu, M.-H. Whangbo, S. Dong, X. G. Gong, and H. J. Xiang, Phys. Rev. Lett. 108, 187204 (2012). 28 B. Bochu, J. L.Buevoz, J. Chenavas, A. Collomb, J. C. Joubert, and M. Marezio, Solid State Commun. 36, 133 (1980). 29 P. E. Bl¨ochl, Phys. Rev. B 50, 17953 (1994). 30 G. Kresse and J. Furthm¨uller, Phys. Rev. B 54, 11169 (1996). 31 R. D. King-Smith and D. Vanderbilt, Phys. Rev. B 47, 1651 (1993). 32 D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48, 4442 (1993). 33 J. B. Goodenough, Magnetism and the Chemical Bond (Interscience, New York, 1963). 34 B. Bleaney and K. W. H. Stevens, Rep. Prog. Phys. 16, 108 (1953). 35 See Supplemental Material at http://link.aps.org/supplemental/ 10.1103/PhysRevB.87.075127 for (i) the calculation details for single-ion anisotropy constants, (ii) the criterion for the easyplane anisotropy caused by the collaborative effect of single-ion anisotropy, and (iii) the rule of polarization in the noncollinear-type exchange-striction mechanism. 36 M. Mochizuki and N. Furukawa, Phys. Rev. B 80, 134416 (2009). 37 S. Dong and J.-M. Liu, Mod. Phys. Lett. B 26, 1230004 (2012). 38 J. T. Zhang, X. M. Lu, J. Zhou, H. Sun, J. Su, C. C. Ju, F. Z. Huang, and J. S. Zhu, Appl. Phys. Lett. 100, 242413 (2012). 39 N. A. Benedek and C. J. Fennie, Phys. Rev. Lett. 106, 107204 (2011). 40 T. Arima, J. Phys. Soc. Jpn. 76, 073702 (2007). 41 T. Kurumaji, S. Seki, S. Ishiwata, H. Murakawa, Y. Tokunaga, Y. Kaneko, and Y. Tokura, Phys. Rev. Lett. 106, 167206 (2011). 42 C. Jia, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B 74, 224444 (2006). 43 C. Jia, S. Onoda, N. Nagaosa, and J. H. Han, Phys. Rev. B 76, 144424 (2007). 44 H. J. Xiang, E. J. Kan, Y. Zhang, M.-H. Whangbo, and X. G. Gong, Phys. Rev. Lett. 107, 157202 (2011).

075127-6