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Theory of Magnetic-Field-Induced Polarization Flop in Spin-Spiral Multiferroics

The magnetic-field-induced 90-degree flop of ferroelectric polarization P in a spin-spiral multiferroic material TbMnO3 is theoretically studied based on a microscopic spin model. We find that the direction of the P-flop or the choice of +Pa or -Pa after the flop is governed by magnetic torques produced by the applied magnetic field H acting on the Mn spins and thus is selected in a deterministic way, in contradistinction to the naively anticipated probabilistic flop. This mechanism resolves a puzzle of the previously reported memory effect in the P direction depending on the history of the magnetic-field sweep, and enables controlled switching of multiferroic domains by externally applied magnetic fields. Our Monte-Carlo analysis also uncovers that the magnetic structure in the P||a phase under H||b is not a so-far anticipated simple ab-plane spin cycloid but a conical spin structure.

The magnetic-field-induced 90 • flop of ferroelectric polarization P in a spin-spiral multiferroic material TbMnO3 is theoretically studied based on a microscopic spin model. We find that the direction of the P -flop or the choice of +Pa or −Pa after the flop is governed by magnetic torques produced by the applied magnetic field H acting on the Mn spins and thus is selected in a deterministic way, in contradistinction to the naively anticipated probabilistic flop. This mechanism resolves a puzzle of the previously reported memory effect in the P direction depending on the history of the magnetic-field sweep, and enables controlled switching of multiferroic domains by externally applied magnetic fields. Our Monte-Carlo analysis also uncovers that the magnetic structure in the P a phase under H b is not a so-far anticipated simple ab-plane spin cycloid but a conical spin structure.

INTRODUCTION
Nontrivial spin textures in magnets often host intriguing physical phenomena through couplings to the charge degrees of freedom [1]. Multiferroics, i.e., magnetisminduced ferroelectricity, is a typical example [2][3][4][5][6][7]. Spiral spin orders in insulating magnets with frustrated interactions induce ferroelectric polarization P via an inverse effect of the Dzyaloshinskii-Moriya interaction, which, in the presence of finite vector spin chirality m i × m j , causes a local electric polarization where m i represents the magnetization vector on a magnetic ion, and e ij the unit vector connecting adjacent magnetic ions, [8][9][10]. The mutual coupling between magnetic and electric orders in multiferroics, the socalled magnetoelectric coupling, gives rise to rich crosscorrelation phenomena. A staggered orbital ordering in the orthorhombically distorted crystal lattice of TbMnO 3 causes significant antiferromagnetic interactions J b on the next-nearestneighbor Mn-Mn bonds along the b axis, which compete with the nearest-neighbor ferromagnetic interactions J ab [ Fig. 1(a)] [11]. This magnetic frustration stabilizes a rotating alignment of Mn spins with a propagation vector Q m b. In the absence of an external magnetic field, the Mn spins in TbMnO 3 rotate within the bc plane to form a transverse spiral (cycloidal) order [12], which is accompanied by a ferroelectric polarization P c (+P c or −P c ) where the sign depends on the rotational sense of the spin spiral [ Fig. 1(b)] [13]. If a magnetic field H Q m (+H b or −H b ) is applied, a 90 • flop of P from P c to P a (+P a or −P a ) appears in the experimental phase diagram of TbMnO 3 [see Fig. 1(c)] [14].
This P flop can be attributed to H-induced changes of the magnetic structure [15,16], but it contains several puzzles. First the emergence of P a under H b cannot be understood simply. If H is applied along Q m ( b), as illustrated in Fig. 1(d), a longitudinal conical spin order with net magnetization M ( H) is expected to be stabilized by a gain in the Zeeman energy. This conical Mn-spin order, however, cannot induce a finite P via the inverse Dzyaloshinskii-Moriya mechanism exemplified by Eq. (1) because m i × m j and e ij ( b) are parallel to each other. Instead, the ab-plane cycloidal spin state [see Fig. 1(e)] was proposed as a possible magnetic structure in the P a phase of TbMnO 3 [15,16], but its mechanism has not been fully clarified. Similar 90 • P flops with H Q m have been reported for several other spin-spiral multiferroic materials such as DyMnO 3 [17], LiCu 2 O 2 [18], MnWO 4 [19], and Ni 3 V 2 O 8 [20].
Second, a memory effect in the direction of H-induced P flop has been reported [21,22]. Naively, it is believed that the direction of P after the flop, i.e., +P a or −P a , should be selected in a probabilistic way because these two states are degenerate in energy. Several experiments, however, have reported that if one increases and/or decreases a magnetic field, the P -flop direction is seemingly selected in a deterministic way depending on the history of the H sweep. The deterministic nature of the flop with a unique correspondence +P c →+P a and −P c →−P a has been confirmed by real-space imaging of ferroelectric domains using second harmonic generation (SHG) microscopy [23]. This memory effect was previously attributed to f -d coupling between Mn spins and Tb f -electron moments [21,24]. However, a similar memory effect was reported also for the H-induced P flop in CuFeO 2 [25] and MnWO 4 [26,27] without f -electron moments, where the effect was attributed to extrinsic origins such as repopulation of multiple Q domains [25,27] or nuclei growth of ferroelectric embryos [26].
As observations of the H-induced P flop and the memory effect have been reported for several spin-spiral multiferroic materials, we expect that there is a general physical origin intrinsically underpinning them. In this paper, we theoretically study both the statics and dynamics of the H-induced P flop in TbMnO 3 under H b by utilizing a microscopic spin model. A Monte-Carlo calculation of this spin model reproduces the observed P flop from P c to P a, and uncovers that the magnetic structure in the P a phase is not a naively believed, simple ab-plane cycloid but a nearly transverse conical state. Moreover, numerical simulations of the phase-transition dynamics reveal that the flop direction of spin-spiral plane under an external H field is governed by magnetic torques from H acting on local Mn spins, which results in a deterministic P flop. Our finding may enable controllable switching of P direction via magnetic fields towards potential applications of multiferroic materials to storage devices.

SPIN MODEL
To describe the Mn-spin system in TbMnO 3 , we employ a classical Heisenberg model on the orthorhombic lattice containing frustrated exchange interactions, Dzyaloshinskii-Moriya interactions, single-ion magnetic anisotropies, and Zeeman coupling [28][29][30][31]. The Mn spins are treated as classical magnetization vectors m i (=−S i /h) with norm of m=2. The full Hamiltonian consists of five terms: where Here i x , i y , and i z are the integer coordinates of the ith Mn ion with respect to the pseudocubic x, y, and z axes.
The first term H ex describes the exchange interactions [see Fig. 1(f)]. The frustration between the nearestneighbor ferromagnetic exchange J ab and the nextnearest-neighbor antiferromagnetic exchange J b results in transverse spiral (cycloidal) orders of Mn spins [11,32], whereas the inter-plane antiferromagnetic coupling J c causes their staggered stacking. Their expressions are given by The terms H D sia and H E sia stand for single-ion magnetic anisotropies, defined with tilted local axes ξ i , η i and ζ i attached to the ith MnO 6 octahedron. H D sia renders the magnetization along the c axis hard. On the other hand, H E sia causes an alternate arrangement of the local hard and easy magnetization axes in the ab plane because of the staggered orbital ordering. Using the experimentally measured crystal parameters in Ref. [37], the directional vectors ξ i , η i and ζ i are calculated with respect to the a, b, and c axes in the P bnm setting [29,31]. Explicit formula of the directional vectors are given by, Here axis with a wave number Q m =0.3π at H=0, which coincides with the observed bc-plane spin cycloid in TbMnO 3 (Q m =0.28π) [12]. The optical absorption spectra associated with electromagnon excitations for TbMnO 3 have been perfectly reproduced with respect to lightpolarization selection rule, spectral structure with dou- ble peaks, and resonance frequencies [38]. The Monte-Carlo analysis has also reproduced the phase diagrams of TbMnO 3 in plane of temperature and magnetic field as well as the H-induced P flop from P c to P a under H b [39]. These quantitative reproductions of experimental results guarantee the validities of the spin model and the parameter set used in the calculations and, hence, the validity of the obtained results.

RESULTS AND DISCUSSIONS
In the Monte-Carlo calculation, we find that a magnetic structure in the P a phase is not a simple ab-plane cycloid as commonly believed but a conical structure [ Fig. 2(a)]. The cones in this state are slightly slanted from the c axis towards the H direction ( b). The orthorhombic TbMnO 3 crystal has two different MnO layers (layers 1 and 2) stacking alternately along the c axis. The averaged magnetizations M 1 and M 2 for layers 1 and 2 are calculated from M γ = (1/N x N y ) i m iγ , where the summation i is taken over the Mn sites within the γth layer. The orientations of M 1 and M 2 are depicted by thick arrows in Fig. 2(a). Their c-axis components are staggered, whereas their b-axis components uniformly point in the H direction. This spin structure results from a compromise of several competing interactions. The rotating spin alignment is caused by the frustrated exchange interactions H ex , whereas the spin rotation residing nearly within the ab plane is favored by the easy-plane magnetic anisotropy H Peculiarity of this magnetic structure will show up in optical spectra. Electromagnon excitations in multiferroic Mn perovskites with spiral spin orders such as TbMnO 3 , DyMnO 3 , and Eu 1−x Y x MnO 3 are ascribed to the symmetric exchange striction mechanism due to coupling between the ac electric field E ω of light and local electric polarizations p ij = Π ij m i · m j on the distorted Mn-O-Mn bonds [40,41]. (Note that these exchange-striction driven local polarizations are different from those given by Eq. (1), which cancel out in total and never contribute to the ferroelectricity.) They can be activated when E ω is parallel to the a axis irrespective of the spiral-plane orientation as long as cycloidal spin orders in RMnO 3 are considered. In Ref. [42], electromagnon spectra for the P a phase of TbMnO 3 under H b was investigated theoretically. In this study, although the detailed magnetic structure was not explicitly argued, we expect the ab-plane conical phase because the spin model and the parameter set used in Ref. [42] are totally the same as those used in the present study. It was confirmed that the light-polarization selection rule holds even for the P a phase with the ab-plane conical order, that is, the electromagnons are activated only with E ω a via the symmetric exchange striction. Moreover, it was predicted that the higher-energy spectral peak splits into two peaks. This prediction was indeed confirmed by the Raman spectroscopy [42]. The coincidence between the theoretical prediction and the experimental observation strongly supports the validity of the predicted ab-plane conical spin structure because splitting of the higherlying electromagnon spectral peak can be expected only for the ab-plane conical state but not for the ab-plane cycloidal state [41].
Next, we investigate the dynamics of m i and p ij for the H-induced P flop in TbMnO 3 using the Landau-Lifshitz-Gilbert equation given by Here α G is the dimensionless Gilbert-damping coefficient, which is fixed at 0.05 in the calculations [41]. The effective local magnetic field H eff i acting on the ith Mn magnetization m i is derived from the Hamiltonian H as We solve this equation numerically using the fourth-order Runge-Kutta method with a system of 40×40×36 sites on which periodic boundary conditions are imposed. Spatiotemporal distributions of local polarizations p ij are calculated from simulated configurations of m i using Eq. (1) by considering the inverse Dzyaloshinskii-Moriya mechanism as their origin [43].
Repeating the simulation with different initial configurations of m i , we find that the flop occurs in a deter-ministic way. In Fig. 3(a) and (b), we show simulated time profiles of the a-, b-, and c-axis components of ferroelectric polarization P (t) = (1/N ) i p i (t) for the Hinduced P flop when +H b of 17 T is applied. It turns out that +P c (−P c ) always flops to +P a (−P a ) under +H b , in contradistinction to the naively anticipated probabilistic flop. This deterministic P flop can be understood as follows: if H is applied, each Mn magnetization m i experiences a magnetic torque τ i ∝−m i × H. For the bc-plane spin cycloids with +P c and −P c , the directions of the torques under +H b are marked by solid arrows pointing in the +a and −a directions in Fig. 3(c) and (d), respectively. For both cases, the torques cooperatively contribute to rotate the cycloidal plane in a counterclockwise fashion around the b axis, resulting in the deterministic switching from +P c to +P a or from −P c to −P a . Figure 3(e) shows simulated spatiotemporal dynamics of local p ij vectors for the switching from +P c to +P a , whereas Fig. 3(f) shows that from −P c to −P a under +H b of 17 T.
We also find that if H is reversed, the relationship is reversed, as indicated in Fig. 4(a); +P c flops to −P a , whereas −P c flops to +P a under −H b . This occurs because the directions of the magnetic torques are determined by the sign of H and become opposite if the sign of H is reversed. Subsequently, the spin-spiral plane rotates in a clockwise fashion under −H b , again, irrespective of the spin-spiral sense.
The experimentally observed memory effect in the Hinduced P flop in TbMnO 3 can be explained by the above mechanism. Figure 4 shows the measured evolution of the a-axis component of P (P a ) during the sweep of the magnetic field H b, previously reported in Ref. [21]. Starting from +H b of +9 T, Abe et al gradually decreased H b to −9 T and then increased H b from −9 T to +9 T. The P a phase appears in the range of H b > 5 T and H b < 5 T. They found that the P a always has positive values if they start from the +P a state. This behavior is a consequence of the clockwise rotation of the spiral plane under +H b at stages (2) and (8), and the counterclockwise rotation under −H b at stages (4) and (6), which can be experimentally confirmed by simultaneous measurement of the c-axis component P c .
Here it is noteworthy that the memory effect on the P direction can be observed not only as a function of H field but also as a function of temperature as reported for thermally induced ferroelectric transitions in MnWO 4 [44] and CuO [45] at H=0. Since the magnetic torque from external H field is absent in these cases, the observed effect can be attributed to other mechanisms such as nuclei growth of tiny domains survived due to pinning. This extrinsic mechanism can work also in the H-induced P flop even while the magnetic torque is a major source of the memory effect. We expect that comprehensive understanding of memory effects in multiferroics can be achieved by taking both intrinsic and extrinsic mecha- nisms into account. The deterministic P flop in TbMnO 3 results in the emergence of nominally charged domain walls as observed by the SHG microscopy [23]. At H=0, multiferroic domain walls between +P c and −P c domains in the ac plane tend to run parallel to the c axis, because of the strong inter-plane antiferromagnetic coupling J c along the c axis. Under this condition, 180 • domain walls between +P c and −P c at H=0 necessarily turn into headto-head or tail-to-tail domain walls between +P a and −P a upon the P -flop driven by H b. Across the domain wall, we expect that the spin-spiral plane continuously changes from +ab (clockwise) to −ab (anticlockwise) via bc [46,47]. Namely the bc-plane spin cycloid appears between the two different ab-plane cycloids. Consequently, the electric polarization gradually varies from +P a to −P a via P c to form a Neel wall of p ij . Thus, the domain wall between +P a and −P a states can be regarded as head-to-head or tail-to-tail only in the macroscopic scale, and there is no classical divergence of the electrostatic potential at the center of the wall. However, we can still expect small but finite −∇ · P between sites in the domain-wall area owing to the discreteness of the crystal lattice, which causes nominally charged multiferroic domain walls.

SUMMARY
In summary, we theoretically studied the magneticfield-induced 90 • flop of ferroelectric polarization P in the spin-spiral-based multiferroic material TbMnO 3 . Our Monte-Carlo analysis of a microscopic spin model revealed that the magnetic structure in the field-induced P a phase is not the simple ab-plane spin cycloid as commonly believed, but a conical spin structure, which is stabilized by the Zeeman coupling and the Dzyaloshinskii-Moriya interactions associated with vectors on the interplane Mn-O-Mn bonds. Our simulations of the flop dynamics indicated that the flop direction of P or the rotation direction of the spiral plane is governed by the sign of the applied magnetic field via the magnetic torques acting on local Mn magnetizations. Consequently, the choice of +P a or −P a after the flop is selected in a deterministic way depending on the magnetic-field direction. This contrasts the naively believed probabilistic flop and accounts for the reported memory effect; that is, the P direction depends on the history of the magnetic-field sweep. This finding enables controllable switching of the P direction by application of magnetic fields and paves the way to technical application of spin-spiral multiferroics.