Multiferroic domain structure in orthorhombic multiferroics of cycloidal spin order: Phase field simulations

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Multiferroic domain structure in orthorhombic multiferroics of cycloidal spin order: Phase field simulations

P. Chu, D. P. Chen, and J.-M. Liu

Citation: Appl. Phys. Lett. 101, 042908 (2012); doi: 10.1063/1.4739426 View online: http://dx.doi.org/10.1063/1.4739426

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Multiferroic domain structure in orthorhombic multiferroics of cycloidal spin order: Phase field simulations

P. Chu,¹D. P. Chen,¹and J.-M. Liu^1,2,a)

1Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China

2Institute for Advanced Materials, School of Physics, South China Normal University, Guangzhou 510006, China and International Center for Materials Physics, Chinese Academy of Sciences, Shenyang, China (Received 15 May 2012; accepted 11 July 2012; published online 27 July 2012)

The multiferroic domains and their responses to external electric and magnetic fields in orthorhombic multiferroics of cycloidal spin order, taking nonmagnetic rare-earth manganites as examples, are investigated using a phase-field model based on the time-dependent Ginzburg- Landau equation and Monte Carlo simulation. The ferroelectric 180-domain pattern with head-to- head/tail-to-tail domain wall along theb-axis, assigned to the in-planeab-CS thin film, is revealed.

The domain switching in response to both electric and magnetic fields is simulated, consistent with experiments. The magnetic origin for the 180 head-to-head/tail-to-tail domain walls is discussed.

V^C 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4739426]

In multiferroics, the ferroelectric (FE) and magnetic orders coexist and couple with each other, allowing fascinating physical phenomena and potential applications.^1–3 In particular, those type-II multiferroics, in which the FE polarization (P) is believed to arise from specific spin orders, have been given emphasis due to the intrinsic magnetoelectric (ME) coupling between the two ferroic orders. One of the most addressed mechanisms for multiferroicity is the inverse Dzyaloshinskii- Moriya (DM) interaction associated with the spin-orbit coupling in complicated transition metal oxides of cycloidal spin (CS) order, which leads to a nonzeroP¼Arij(SiSj), where rijis the unit vector connecting two neighboring spinsSiandSj, Ais a constant depending mainly on the spin-orbit coupling,⁴ and the orientation of P is determined by spin chirality C¼SiSj. This mechanism is demonstrated in several type II multiferroics such as orthorhombic rare-earth manganites RMnO₃(R¼Tb, Dy,…).^4–9

One possible consequence of the coupling between the two ferroic orders is the complicated FE and magnetic domain structure at low temperature (T). For the CS order, the clockwise and counter-clockwise chiralities are degenerated, enabling the com- plexity of the domain structure. Taking TbMnO₃as an example here, thebc-plane CS order below the lock-in CS ordering point favors both the chiralities parallel and anti-parallel to thea-axis (C//a and C//þa); thus, the FE domains with P//c and P//þcare developed simultaneously, generating interactive CS domains with opposite chiralities and FE domains. Upon a magnetic fieldHapplied along theb-axis, these CS domains may flop in accompanying with the variation of chiralityC; thus, the FE domain switching occurs. Similarly, an electric fieldEcan also switch the FE domains and thus the CS domains.^10,11

In this work, we propose a phase-field model based on the Ginzburg-Landau (GL) phenomenological theories.

While various multiferroic phase field models have been developed recently,¹² we address particularly the FE domains/CS domains and their coupling in response toHand E in multiferroics of CS order. Although the microscopic

mechanism was linked to the DM interaction,⁹a prominent GL model was proposed even earlier.¹³

We start from a two-dimensionalLLsquare FE lattice in thex-ycoordinates with the periodic boundary conditions, as a model to an epitaxial multiferroic thin film with the ab-CS plane and the c-axis normal to the film surface. The b-axis aligns along the [11] direction, and thea-axis points to the [11] direction. Here,P¼(Px,Py) is a second-order order parameter generated by the primary order parameter, i.e., magnetic momentM, via the ME coupling. In a good approximation, the Landau energy can be written as

Fld ¼FðMÞ þFðM;PÞ þP²=2pe; (1) where F(M) is the energy for magnetic order parameterM, F(M, P) stands for the ME coupling,¹⁴ and the last term is the self-energy ofPwithethe dielectric permeability.

Following the GL theory on the CS ordered lattice, one has¹³

FðMÞ ¼ J1

X

hi;ji

SiSjJ2

X

½i;j

SiSj; FðM;PÞ ¼ cP ½Mðr MÞ ðM rÞM:

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where J1 and J2 stand for the ferromagnetic and antiferro- magnetic interactions, h…i and […] denote the nearest- neighbor (NN) and next-nearest-neighbor (NNN) pairs, respectively, c is the ME coupling coefficient. We choose J1>0 and 0.8<jJ2/J1j<1.2 to generate a 1/6 CS order along theb-axis, referring to TbMnO3as an example.¹⁵

Since the Landau potentialFldonly allows roughly iso- tropic polarization to be generated, a series of free energy terms associated with the polarization should be taken into account, referring to the standard GL theory.¹⁶The total free energy for this multiferroic lattice can be written as

F¼FldþFgþFddþFelþFesþFse; (3) where Fg, Fdd, Fel, Fes, and Fse are the gradient energy, dipole-dipole interaction, elastic energy, electrostrictive

a)Electronic mail: liujm@nju.edu.cn.

0003-6951/2012/101(4)/042908/4/$30.00 101, 042908-1 VC2012 American Institute of Physics

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energy, and electrostatic energy, respectively. For Fg, the lowest-order expression is

Fg ¼X

hii

1

2½G11ðP²_x;xþP²_y;yÞ þG12Px;xPy;y

þG44ðPx;yþPy;xÞ²þG⁰44ðPx;yPy;xÞ²; (4)

wherePi,j¼@Pi/@xjand coefficientsG11,G12,G44, andG⁰44

are the gradient energy coefficients.¹²TermsFddandFsecan be written, respectively, as

Fdd¼ 1 4pe

X

hii

X

hji

PðriÞ PðrjÞ jrirjj³

3PðriÞ ðrirjÞ½PðrjÞ ðrirjÞ

jrirjj⁵ ; (5) Fse¼ EX

hii

Pi; E¼ ðEx;EyÞ; (6)

whereFddfavors the anti-parallel alignment of Pi and is of long-range. A realistic calculation is done either by Fourier transform or by finite truncation treatment¹⁷while for 2D lattice this treatment is precise as long as the truncating dis- tanceRis big (R¼8 in our simulation).

The lattice strain is associated with nonzero P. The stress-free strain caused byP¼(Px,Py) is

e⁰₁₁¼Q11P²_xþQ12P²_y; e⁰₂₂¼Q11P²_yþQ12P²_x; e⁰₁₂¼Q44PxPy:

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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 The strain can be expressed as

eij¼eije⁰_ij;ði;j¼1;2Þ;

Fe¼1 2

X

hijkli

cijkleijekl¼1 2

X

hiji

cijklðeije⁰_ijÞðekle⁰_klÞ; ðk;l¼1;2Þ; (8) whereeijis the total strain,cijklis the elastic stiffness tensor which only have three independent elastic constants C11, C12, and C44 in a cubic lattice. A combination of Eqs. (7) and(8) allowsFe¼FelþFes. TermFelis the elastic energy term

Fel¼X

hii

1

2C11ðu²_x;xþu²_y;yÞ þC12ux;xuy;yþ1 2C44u²_x;y; ui;i¼@ui=@ri; ui;j¼@ui=@rjþ@uj=@ri:

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whereu(r)¼(ux,uy) is the displacement vector. The electrostrictive energyFesis

Fes¼ X

hii

ux;xðq11P²_xþq12P²_yÞ

þuy;yðq11P²_yþq12P²_xÞ þux;yq44PxPy; (10)

whereq11¼C11Q11þ2C12Q12,q12¼C11Q12þ2C12(Q11þQ12), andq44¼2C44Q44are the electrostrictive coefficients.

In our simulation, each lattice site is assigned with an electric diploe P(r)¼(Px, Py), a displacement vector u(r)¼(ux,uy), and a classic spin vectorS(r)¼(Sx, Sy) with jS(r)j ¼1. For parameteru, one may assume that the lattice reaches its mechanical equilibrium instantaneously for a given polarization distribution. In that view, we can use a static condition of mechanical equilibrium to degrade the displacement field to a function of the polarization field.¹⁹ The mechanical equilibrium equation is expressed as r_ij,j¼0, and stress vectorr_ijis related to straineijaccording to the Hook’s law via rij¼cijklekl¼cijkl(ekle⁰_kl). Therefore, the equilibrium equation can be rewritten as

cijkluk;lj¼cijkle⁰_ki;j; (11) which has a solution in the Fourier space given a periodic boundary conditions

^

ui ¼ ffiffiffiffiffiffiffi p1

gijcjmkle^⁰_klf_m; ðm¼1;2Þ; (12) where gij1¼cijklf_kf_l andfis the coordinate in the Fourier space, u^i and e^⁰_kl are the Fourier transform of ui and e⁰_ij, respectively. We can use the inverse Fourier transform to obtainuiin real space.

We use Monte Carlo simulation with the annealing algo- rithm to track the evolution of the CS domain structure. We start from a paramagnetic lattice at a high T0and gradually anneal it down to a lowTdbelow theab-CS ordering point.

Simultaneously, polarization P is computed in parallel. We investigate the evolution of the time-dependent GL theory forP

@Pðr;tÞ

@t ¼ D dF

dPðr;tÞ; (13)

whereDis the kinetic coefficient. We carefully balance the value of D for the P-evolution and total Monte Carlo step number for the CS domain evolution, so that the former is much slower than the latter.

Unfortunately, so far neither phenomenological GL theory on multiferroics of the CS order (e.g., TbMnO₃) is available nor physical parameters for these materials have been measured. As a qualitative approximation, we choose the set of parameters for BaTiO₃ for the calculation in the present work, and this issue is therefore only technically rel- evant. All the parameters for the present calculation are listed in TableI, where parameterP0is the polarizationPat the lowest Td, generated by Eq. (3) without considering terms Fg, Fdd, Fel,Fes, andFse. An evaluation of these parameters can be found in Ref.18.

Fig.1shows four snapshoted images of the FE domains on thexy(ab)-plane in four temperatures. A near-equilibrium domain pattern is reached at30 000 time steps, accompa- nied with domain growth from the irregular pattern at higher Tto the stripe-like pattern in the lowT. It is seen that the FE domains are 180-stripe-like with head-to-head and tail-to- tail domain walls (DWs) align along the b-axis, which is referred as the A-type 180-DWs (Fig. 1(e)). Alternatively,

042908-2 Chu, Chen, and Liu Appl. Phys. Lett.101, 042908 (2012)

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the 180-domain wall may also align along thea-axis, parallel toP itself, as shown in Fig.1(e), referred as the B-type 180-DW.

To understand the mechanism underlying the A-type 180-DWs, we first examine the effect of FE energy terms Fg, Fdd,Fel,Fes, andFse. If all these terms are excluded, a mono-domain with P//a-axis is generated, shown in Fig.

2(a). An exclusion of only Fg favors the FE stripe-like domains with the A-type 180-DWs, shown in Fig. 2(b).

This is attributed to the depolarization field induced byFdd. For normal FE systems such as BaTiO₃, electrostrictive term Fesis crucial in leading to the 90-domains but becomes less important in multiferroic manganites.²⁰ As shown in Fig.

2(c), by includingFgin addition to all the other terms, one sees that the FE DWs become straighter and broader than those DWs shown in Fig. 2(b), and the dipoles inside each domain deviate slightly from the6a-axis. It seems that two neighboring domains tend to slightly re-align their dipoles.

In order to characterize this tendency, we calculate the corresponding spin structure function in the wave-vector plane (kx,ky), and the results are shown in Fig.2(d), with the left and right patterns for the cases excluding and includingFg, respectively. The inclusion ofFgmakes the single peak at (kx, ky)¼(6, 6) splits into two sub-peaks along the a-axis.

Our extensive calculations over the whole parameter space, given that the spin order related energy is dominant over the

FE order related energy and the CS order is maintained, indicate no B-type 180-DWs available. This suggests that the A-type 180-DW is one of the intrinsic characteristics for the present multiferroic lattice.

The stability of the A-type 180-DWs is assured by the CS domain structure. Fig.3(a)shows the simulated spin lattice corresponding to Fig. 2(c). In the present lattice, a CS domain may have its chiralityCpositive or negative (C>0 or C<0), and the two types of CS domains are degenerate and, thus, they may coexist. The as-generated CS domain wall may align along two different directions. In one case where each CS chain along the b-axis may be broken into two parts, the spiral wave-vector in one part is along the b- axis, opposite to the other along the -b-axis. The as- generated CS domain wall would be very irregular, as an example schematically illustrated in Fig. 3(b), although the overall orientation of the wall may be parallel to thea-axis.

This corresponds to the B-type 180FE domain wall. In the other case, two neighboring CS chains have opposite spiral wave-vectors along the 6b-axis, generating a CS domain wall along the b-axis, as showed in Fig.3(c), corresponding to the A-type 180FE domain wall.

It is seen that the CS domain wall along thea-axis can- not be straight but twisted.²¹On the wall, the NN interaction (J1) is similar in magnitude to the NNN interaction (J2).

Therefore, the domain wall roughly along the a-axis would have much higher energy than the domain wall along theb- axis. This explains why the CS domain walls align along the b-axis rather than thea-axis, i.e., favoring the A-type 180- FE domain walls.

TABLE I. Physical parameters chosen for the simulation (a¼1/2pe,kB¼1 is assumed) (Ref.18).

Parameter (unit) Value Parameter (unit) Value Parameter (unit) Value

T0(aP²₀) 4.0 Td(aP²₀) 0.1 J1(J₁/aP²₀) 1.0

J2a(J_2a/aP²₀) 0.01 J2b(J_2b/aP²₀) 1.0 c(aP0) 1.0

G44(G₁₁/a³) 1.28 G12(G₁₂/a³) 0.0 G44(G₄₄/a³) 0.64

G⁰44(G⁰₄₄/a³) 0.64 C11(C₁₁/aP²₀) 2.75 C12(C₁₂/aP²₀) 1.79

C44(C₄₄/aP²₀) 0.54 q11(q₁₁/a) 0.142 q12(q₁₂/a) 0.0074

Q44(q₄₄/a) 0.0157 L 36–64

FIG. 1. Simulated snapshots of the FE domains at various times (temperatures) of the simulation: (a)T¼0.48, (b) T¼0.43, (c) T¼0.39, and (d) T¼0.1. The arrows indicate the dipole orientations inside the domains. (e) A schematic of the two types of 180-domain walls along theb-axis.

FIG. 2. Simulated dipole configurations by (a) excluding all FE energy terms exceptFld, (b) excluding only termFg, and (c) including all the energy terms. (d) The corresponding spin structure factors mapping on the (kx,ky) coordinates obtained by excluding termFg(left) and including all the energy terms (right).

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Nevertheless, a recent work²²on MnWO₄single crystal using second harmonic generation technique images the FE domain and CS domain simultaneously on a macroscopic scale, where the B-type 180-FE domain wall was revealed, inconsistent with our results on the 2D lattice. As we men- tioned earlier, our simulation actually refers to an epitaxial thin film with the out-of-plane along thec-axis. Because the magnetic energy is dominant over the FE energy, the A-type 180-FE domain walls would be favored, while the B-type domain wall along theab-plane would generate much higher wall energy due to the in-plane geometry. Thus, the observed FE domain wall should have the 180 head-to-head/tail-to- tail configuration. The main features reported in Ref. 22 would be explainable qualitatively if one extends the simulation to 3D lattice by including more interactions.

Finally, we address the multiferroic response of the lattice to T, H, and E, so as to compare with experimental observations. In Fig.4(a)is plotted the component ofPalong thea-axis, Pa, as a function ofT. This dependence fits well with measured P(T) for typical multiferroics of CS order²³ and is associated with the second order transitions from the sinusoidal spin state to the CS state. In response to a nonzero H along the b-axis, Hb, Pa for the mono-domain and multi-domain structures shows a clear transition from nonzero value to zero, due to the gradual suppression of the CS

state by Hb, consistent with experiments and earlier simulation.^24,25

When a nonzeroEalong the negative a-axis,Ea, would switch P, the CS chirality C also switches in consequence via the ME coupling, as exampled in Fig.4(c). The value of Cgradually increases with the magnitude ofEaand a switching occurs at Ea 0.6 above which the lattice becomes mono-domained. In correspondence, the component of the lattice moment along thea-axis,Sa, is featured with a clear valley at the switching field. Fig.4(d)shows the response of PatoEa, confirming theP-switching. We also plot two FE energy termsFgandFelas a function ofEa, respectively, as shown in Fig.4(e). The gradient energy arising from the ex- istence of FE domain walls remains roughly constant before the domain switching and then tends to zero due to the for- mation of mono-domain. The elastic energy, however, shows a peak in association with the switching, owing to the coupling betweenPand strain in the dynamic process of FE domain walls.

This work was supported by the National Key Projects for Basic Research of China (Nos. 2009CB623303 and 2011CB922101), the Natural Science Foundation of China (Nos. 11074113 and 50832002), and the Priority Academic Program Development of Jiangsu Higher Education Institu- tions, China.

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FIG. 3. (a) Simulated spin configuration with CS order corresponding to Fig.2(c). (b) A schematic drawing of a CS domain wall where each spin chain along theb-axis is split into two domains with opposite chiralities.

The domain wall is twisted along the a-axis, corresponding to the B-type 180FE domain wall. (c) A schematic drawing of a CS domain wall corresponding to the A-type 180FE domain wall. The two spin chains aside the CS domain wall have opposite chiralities and the wall is straight along the b-axis.

FIG. 4. SimulatedPa(T) (a),Pa(Hb) in the mono-domain lattice and multi- domain lattice atT¼0.1 (b), chiralityC(Ea) and magnetic moment along thea-axis,Sa(Ea) atT¼0.1 (c),Pa(Ea) atT¼0.1 (d),Fg(Ea) andFel(Ea) at T¼0.1 (e).

042908-4 Chu, Chen, and Liu Appl. Phys. Lett.101, 042908 (2012)

Multiferroic domain structure in orthorhombic multiferroics of cycloidal spin order: Phase field simulations

Multiferroic domain structure in orthorhombic multiferroics of cycloidal spin order: Phase field simulations

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Multiferroic domain structure in orthorhombic multiferroics of cycloidal spin order: Phase field simulations