Multiferroic Domain Structure in Orthorhombic ... - IEEE Xplore

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IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 7, JULY 2013 3117

Multiferroic Domain Structure in Orthorhombic Multiferroics of Cycloidal Spin Order: Three-Dimensional Phase Field Simulations

P. Chu , Y. L. Wang , L. Lin , S. Dong , and J.-M. Liu

Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China Department of Physics, Southeast University, Nanjing 210008, China

Institute for Advanced Materials, South China Normal University, Guangzhou 510006, China

The three-dimensional (3-D) multiferroic domain patterns in multiferroic rare-earth manganites of the -plane cycloidal spin (CS) order and -plane CS order are investigated by means of Monte Carlo simulation based on the multiferroic Heisenberg model and phase-field simulation based on the time-dependent Ginzburg–Landau equation. It is revealed that the ferroelectric domains associated with both the -plane CS order and -plane CS order align in the 180 -domain pattern with polarization parallel with domain walls.

The mechanism dominating the domain patterns is discussed.

Index Terms—Cycloidal spin order, multiferroic domain, phasefield model.

I. INTRODUCTION

R

ECENTLY multiferroics have been receiving substantial attentions for both fundamental interests and potential applications [1]–[3]. In this type of materials, the ferroelectric (FE) and magnetic orders coexist and couple with each other, allowing a series of fascinating physical phenomena. Among these materials, the type-II multiferroics have been given em- phasis due to the intrinsic magnetoelectric (ME) coupling between the two ferroic orders, because the FE polarization arises from specific spin orders. Recent years, a series of ex- perimental and theoretical works on the underlying physics of multiferroicity have been published and one of the most ad- dressed mechanisms is associated with the inverse Dzyaloshin- skii–Moriya (DM) interaction, given a strong spin-orbit coupling in complicated transition metal oxides of cycloidal spin

(CS) order. A nonzero polarization ,

where is the unit vector connecting two neighboring spins and , is proposed and the orientation of is determined by spin chirality . This mechanism is demonstrated in several type II multiferroics such as orthorhombic rare-earth manganites ( , Dy etc) [4]–[9], and thus those magnetic materials of specific spin chirality become popular in searching for novel multiferroics.

It is noted that the complicated coupling between the two ferroic orders would result in complicated FE and magnetic domain structures. In these multiferroics of CS order, the FE domain structure is essentially controlled by the magnetic domain structure since the former is generated by the latter. For the CS order, the clockwise and counter-clockwise chiralities are degenerated, enabling the complexity of the domain structure.

Taking as an example here, if the effect of Tb spins is ignored, the -plane CS order below the lock-in CS ordering

Manuscript received October 26, 2012; revised January 10, 2013; accepted January 25, 2013. Date of current version July 15, 2013. Corresponding author:

J.-M. Liu (e-mail: [email protected]).

Color versions of one or more of thefigures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMAG.2013.2243902

point favors both the chiralities parallel and anti-parallel to the a-axis ( and ), thus the FE domains with and are developed simultaneously, generating interac- tive CS domains with opposite chiralities and FE domains [10], [11].

In a previous work, we developed a computational model to approach the structure and dynamics of two-dimensional (2-D) multiferroic domain structures in such multiferroics [12]. As- signed to the -CS in-plane thinfilms, we predicted that the FE 180 -domain pattern with head-to-head/tail-to-tail domain wall arranged along the -axis is favored, due to the specific thin film geometry. Undoubtedly, a more realistic three-dimensional (3-D) simulation which is applicable to bulk multiferroics of CS order is desirable. In the 3-D case, the spin interactions become more extensive, and also one more dimension enables one more degree of freedom to relax the total free energy. Therefore, the 3-D multiferroic domain structures can be quite different from the 2-D case. It should be mentioned that for the 3-D case, there are two CS planes, i.e., the -plane and -plane, which may excite the FE polarization along the -axis and -axis, respectively. The geometrical confinement in the 2-D thinfilm is completely relaxed.

II. MODEL

In the present work, we extend our 2D phase field model to the 3-D cubic-like lattice and perform extensive simulation on the magnetic domains and FE domains, in order to inves- tigate the 3-D multiferroic domain pattern in multiferroics of the -CS orders. For a self-contained description, we assume that the FE polarization in type-II multiferroics is a second-order order parameter generated by the primary order parameter, i.e. magnetic moment , via the ME coupling. In a good approximation, the Landau potential can be written as

(1) where is the energy for magnetic order parameter , stands for the ME coupling [13], and the last term is the self-energy of with the dielectric permeability. Re- garding term in a CS ordered lattice, the invariance upon the time reversal and spatial inversion sequence must be

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3118 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 7, JULY 2013

maintained. When undergoes a transition from sinusoidal spin phase to the CS ordered phase, a nonzero appears as the consequence of the ME coupling and thus the FE order related energy is weak with respect to the magnetic energy. Following the Ginzburg–Landau (GL) theory on the CS ordered lattice, one has [14]:

(2) where and stand for the ferromagnetic and antiferromagnetic interactions, and denote the nearest-neighbor (NN) and next-nearest-neighbor (NNN) pairs, respectively, is the ME coupling coefficient. The third term in defines a phenomenological magnetocrystalline anisotropy, which deter- mines which of the -CS order or -CS order is favored below the CS ordering temperature. We choose as the -plane ferromagnetic interaction and antiferromagnetic interaction along the -axis, and as the antiferromagnetic interaction along the -axis, so that a 1/6 CS order along the -axis can be generated.

For details of this choice, readers may refer to [15].

Since the Landau potential only allows roughly isotropic polarization to be generated, a series of free energy terms associated with the polarization should be taken into account, re- ferring to the standard GL theory on tetragonal FE lattice [16].

It is noted that the multiferroic manganites considered here are orthorhombic instead of tetragonal, however, the difference between the two structures in terms of the GL phenomenological theory is not critical and we still follow the theory on tetragonal FE lattice. The total free energy for this multiferroic lattice can be written as

(3) where , , , , and are the gradient energy, dipole- dipole interaction, elastic energy, and electrostrictive energy, respectively. For , the lowest-order expression is

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where and coefficients , , , and

are the gradient energy coefficients [17]. Terms can be written as

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which favors the antiparallel alignment of and is of long- range. A realistic calculation is done either by Fourier transform orfinite truncation treatment [17]. Considering the balance of calculating speed and precise, we choose the truncating distance

in our simulation.

As described above, the inverse DM interaction induced polarization is microscopically represented by the spatial shift between the negative and positive ions. We assume that the FE domains are coherent on the domain walls, and elastic strain energy will become important on the walls during the domain switching [18], [21]. Term elastic energy term is

(6) where is the displacement vector. The electrostrictive energy is

(7)

where ,

, and are the electrostrictive coefficients.

Our simulations are performed for a 3-D cubic lattice in the coordinate system. Therefore, for manganites, the -axis, -axis, and -axis point respectively to the [1 0], [110], and [001] directions. To clearly display the domain structure, we employ lattice with dimension of for the -CS structure and that of for the -CS structure. The detail simulation processes and choice of the parameters can be found in our previous work [12].

III. RESULTS

Fig. 1 plots the simulated -components as functions of when all the anisotropic free energy terms , , , and

are excluded. In this case, a single domain will develop. In Fig. 1(a), coefficient in (2) is positive and thus only polarization along the -axis appears below the phase transition point, indicating that the -CS order is completely suppressed.

In Fig. 1(b), we take and thus only the -CS order is available since only polarization along the -axis appears below the transition point. These dependencesfit well with mea- sured for typical multiferroics of CS order [19], [22], and are associated with the second order transitions from the sinusoidal spin state to the CS state.

When all the anisotropic FE energy terms in (3) are included, a multidomain pattern appears. One set of typical results are shown in Fig. 2(a) and (b), and correspondingly the magnetic domains show the -CS order and -CS order, respectively.

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CHUet al.: MULTIFERROIC DOMAIN STRUCTURE IN ORTHORHOMBIC MULTIFERROICS OF CYCLOIDAL SPIN ORDER 3119

Fig. 1. (Color online) Simulated temperature profiles of polarization in the mono-domain lattice: (a) for the -CS phase and (b) for the -CS phase.

Fig. 2. (Color online) Simulated FE domains: (a) the -CS phase and (b) the -CS phase. The arrows indicate the polarization orientations inside the domains.

As shown in Fig. 2(a), the -CS order results in the FE domains with 180 -domain walls on the -plane and the points to the -axis, respectively, as shown by the arrows. The FE domains with 180 -domain walls on the -plane are shown in Fig. 2(b), where the points to the -axis. In our simulations, no matter how to adjust the parameters in (3), no FE domain patterns other than the 180 -domain walls can be obtained unless some physically unreasonable values for those parameters are taken. However, it must be mentioned that in these multiferroic manganites of the CS order, the polarization is small , and thus those anisotropic FE free energy terms are all weak with respect to the spin interaction terms. This is the core physics for the multiferroic domain struc-

tures in .

To explain the mechanism underlying the 180 FE domain walls, the CS domain wall structures are schematically illus- trated in Fig. 3, respectively. As mentioned above, the CS wave spreads along the -axis with two degenerate chiralities. There- fore there are two geometry schemes accommodating the two coexisting chiralities. In one case, where each CS chain along the -axis may be broken into two parts, i.e., the spiral wave- vector in one part is along the -axis, and the other along the -axis. It was suggested that this kind of spin arrangement leads to irregular CS domain walls which cost high domain wall energy. In the other case, two neighboring CS chains have opposite spiral wave-vectors along the -axis, respectively, thus leading to a straight domain wall parallel with the -axis. In our 2-D simulation, dual to confinement of the geometry degree of freedom in the -axis oriented thinfilms, all domains must share the same CS-plane, the 180 head-to-head and tail-to-tail domain walls become inevitable in order to be accommodated with the -CS order.

However, for the bulk systems, one more spatial degree of freedom allows those domain arrangements that further relax

Fig. 3. (Color online) Schematic drawing of the -CS domain viewing on (a) the -plane, (b) the -plane, and (c) the -plane. To obtain the -CS domain, one can just exchange the -axis and -axis in thefigure.

Fig. 4. (Color online) Simulated evolution of a preset bc-plane 180 FE domain structure at four different times: (a) 0, (b) 1000 mcs, (c) 2000 mcs, and (d) 50000 mcs. The preferred domain structure is with the -plane 180 FE domain walls. The arrows indicate the -plane domain wall and -plane domain walls.

the total free energy. For the -CS order, as an example, where the points to the -axis, to further reduce the total free energy by minimizing the depolarization energy, the FE polarization must be parallel with domain walls. As shown in Fig. 3(a) and Fig. 3(c), the two layers, which are vertical with -axis, have opposite polarizations along -axis, developing a domain wall parallel with polarization. We then consult to the CS domain of the two layers as shown in Fig. 3(b). The spin chains share the same wave-vector in each layer, but have opposite wave-vectors between layers. We can also explain the domain structure of our simulated -CS order in the same way by just exchanging the -axis and -axis in Fig. 3.

To further check the stability of the simulated domain patterns, we can start from preset initial lattices and observe the domain evolutions. For one case, we confirm that the domain pattern given in Fig. 2 is stable against thermalfluctuations and no identifiable evolution of the pattern is observed upon suffi- cient long simulation cycling. For another case where the -CS domain structure is the ground state, we start from a preset lattice with assigned -CS domain walls, as shown in Fig. 4(a). It is seen that the initial -CS domain walls becomes destabilized against thermalfluctuation cycling and theyfirst evolve into the broken and irregular walls [see Fig. 4(b) and (c) at 1000 mcs and 2000 mcs respectively]. Eventually, the -CS domain walls ap- pear at 50000 mcs, as shown in Fig. 4(d), indicating the gradual switching of the -CS domains into the -CS domains.

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3120 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 7, JULY 2013

IV. CONCLUSION

In summary, we have employed the phenomenological model to simulate the FE domains in bulk multiferroics of CS order.

The 180 FE domain wall with polarization parallel with domain wall is obtained in the lattices of either the -CS order or the -CS order. We conclude that for bulk multiferroics of CS order, the 180 FE domain wall with FE polarization parallel with domain wall will be developed and the FE domain walls are parallel with the CS-plane. Unfortunately so far there has not been much direct evidence with the FE domain structures in , mainly due to the substantial challenges to probe these domains, while the ferroelectricity is weak and appears only at low temperature. A recent work [20] on bulk samples

finds the 180 FE domain structure on the -plane, which is

similar with our simulation results.

ACKNOWLEDGMENT

This work was supported by the National 973 Projects of China (Grants 2011CB922101 and 2009CB623303), the Natural Science Foundation of China (Grants 11234005 and 11074113), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.

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