Magnetoelectric Coupling Induced Electric Dipole Glass State in Heisenberg Spin Glass

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CHIN. PHYS. LETT. Vol. 26,No. 8 (2009) 087501

Magnetoelectric Coupling Induced Electric Dipole Glass State in Heisenberg SpinGlass *

LIU Jun-Ming(刘俊明)1,2,3**, CHAN-WONG Lai-Wa(陈王丽华)1, CHOY Chung-Loong(蔡忠龙)11Department of Applied Physics, Hong Kong Polytechnic University, Hong Kong2Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093

3International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110015

(Received 3 February 2009)Multiferroic behavior in an isotropic Heisenberg spin glass with Gaussian random fields, incorporated by magneto-electric coupling derived from the Landau symmetry argument, are investigated. Electric dipole glass transitionsat finite temperature, due to coupling, are demonstrated by Monte Carlo simulation. This electric dipole glassstate is solely ascribed to the coupling term with chiral symmetry of the magnetization, while the term associatedwith the spatial derivative of the squared magnetization has no contribution.

PACS: 75. 10. Nr, 75. 80.+q, 77. 90. +k

The discovery of a series of novel multiferroics,such as TbMnO3, DyMnO3, and Ni3V2O8 etc.,[1−3]

in which magnetic (spin) and ferroelectric (FE) orderscoexist, has recently revived new interests.[4−6] Themajor feature of these novel multiferroics is the stronginterdependence of the FE and magnetic order param-eters, so that FE-ordering and spin ordering occur si-multaneously, leading to a significant magnetoelectric(ME) effect. Theoretical approaches and experimen-tal investigations regarding the underlying mechanismproposed the so-called “ferroelectricity generated bymagnetic chirality”,[5] with which the FE-order is as-cribed to the broken spin chirality below a certain tem-perature. Subsequently, it was proposed that systemswith dislocated spin-density wave (SDW) may also of-fer multiferroicity, i.e. the FE-order can be generatedby the dislocated SDW.[7] This mechanism was ar-gued to be responsible for the ME coupling observedin RMn2O5 systems where 𝑅 represents the rare-earthelement and the spin wave is commensurate.[8−10].

This recent progress, both experimentally and the-oretically, allows us to argue that the essential ingre-dient of multiferroicity physics is the spatial inhomo-geneity of spin configuration, in either spiral (conical)form and acentric (dislocated) SDW form, or otherforms so far unknown to us, although this does notmean that such an inhomogeneity necessarily gener-ates ferroelectricity. Because such spin inhomogeneityis the typical feature of spin glass systems, an im-mediate idea to us, as extracted from this picture, isthe possible FE-ordering behavior in spin glass sys-tems with efficient ME coupling terms. In this Let-ter, based on the well-studied isotropic Heisenbergspin glass (SG) model,[11−15] we address this prob-lem and take into account the ME coupling betweenthe FE-order and magnetic order within the Landausymmetry framework.[7,16,17] It is expected that anyextension from this simplest picture would help in the

search for novel multiferroic materials.As is well established, for three-dimensional (3D)

Heisenberg SG systems, a finite temperature spin glasstransition accompanied by a chiral glass transitionwas predicted by a number of Monte Carlo simula-tions based on various efficient algorithms.[12−14] Ourmotivation is, at least, to demonstrate that an elec-tric dipole glass (DG) state is possible in an isotropicSG system. Starting from this ME coupling inducedDG state, one can immediately expect fascinating FE-orders as intrinsically generated in the Heisenberg spinsystems with various additional terms of interaction tothe isotropic Heisenberg spin model.

We study a 3D 𝑁 = 𝐿3 Heisenberg cubic latticewith periodic boundary conditions. The spin Hamil-tonian takes the form

𝐻𝑀 = −∑

⟨𝑖𝑗⟩𝐽𝑖𝑗𝑆𝑖 · 𝑆𝑗 , (1)

where 𝑆𝑖 = (𝑆𝑖𝑥, 𝑆𝑖𝑦, 𝑆𝑖𝑧) is the three-component unitvector at site 𝑟, and the sum is over all the nearest-neighbor pairs, the interactions 𝐽𝑖𝑗 are random Gaus-sian variables with zero mean and variance 𝐽0. Re-cent simulations with finite scaling analysis demon-strated that the SG transitions occur at temperature𝑇 = 𝑇𝑆𝐺 ∼ 0.16𝐽0 and the corresponding chiral-glasstransitions appear at 𝑇 ∼ 0.14𝐽0, close to 𝑇SG.[13]

For ME coupling in multiferroics, the Landau sym-metry consideration requires the invariant ME cou-pling terms against both the time reversal (𝑡 → −𝑡)which changes the sign of the magnetization 𝑀(𝑟),and the spatial inversion (𝑟 → −𝑟) which changes thesign of the polarization 𝑃 (𝑟), where 𝑟 is the spatialcoordinates of site 𝑖. These considerations lead to thelowest-order (third-order) ME coupling terms[7,16]

𝐻ME(𝑟) = 𝐻ME1 + 𝐻ME2

= 𝛾1 · 𝑃 · ∇(𝑀2) + 𝛾2 · 𝑃 ·[𝑀(∇ ·𝑀)− (𝑀 · ∇)𝑀

],

𝑀 = (𝑆𝑥, 𝑆𝑦, 𝑆𝑧), 𝑃 = (𝑃𝑥, 𝑃𝑦, 𝑃𝑧), (2)

*Supported by the Hong Kong Polytechnic University (1-BB84), and the National Natural Science Foundation of China un-der Grant Nos 50832002 and 10674061, the National Basic Research Program of China under Grant No 2009CB925101 and2006CB921802, and the 111 Project of the MOE of China.

**To whom correspondence should be addressed. Email: liujm@nju.edu.cnc○ 2009 Chinese Physical Society and IOP Publishing Ltd

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CHIN. PHYS. LETT. Vol. 26,No. 8 (2009) 087501

where the first term 𝐻ME1 on the right side includesthe contribution of the spatial inhomogeneity of thesquare of 𝑀 , while the second term 𝐻ME2 dealswith the spatial inhomogeneity of magnetization 𝑀in terms of both magnitude and orientation. Theircontributions to the ME coupling are scaled respec-tively by coefficients 𝛾1 and 𝛾2.

Here, we only deal with the FE-order solely gen-erated by the spatial inhomogeneity of the spin align-ment, so the free energy associated with the FE-orderis simply written as

𝐻𝑃 (𝑟) =1

2𝜒𝐸(𝑟)𝑃 2(𝑟), (3)

where 𝜒𝐸(𝑟) is the dielectric susceptibility. The sys-tem Hamiltonian is

𝐻 = 𝐻𝑀 +∑⟨𝑟⟩

(𝐻𝑃 + 𝐻ME). (4)

For the Heisenberg SG model, the SG order parame-ter generalized to wave vector 𝑘, 𝑞𝑀

𝑢𝑣(𝑘), and the wavevector dependent SG susceptibility, 𝜒𝑆𝐺(𝑘), can bewritten as[13]

𝑞𝑀𝑢𝑣(𝑘) =

1𝑁

∑𝑖

𝑆(1)𝑖𝑢 · 𝑆(2)

𝑖𝑣 exp(𝑖𝑘 · 𝑟𝑖),

𝜒𝑆𝐺(𝑘) = 𝑁∑𝑢,𝑣

[⟨𝑞𝑀𝑢𝑣(𝑘)

2⟩]𝑎𝑣

, (5)

where (𝑢, 𝑣) = (𝑥, 𝑦, 𝑧) are the spin components, su-perscripts (1) and (2) represent the two replicas ofthe lattice with the same interaction distribution, 𝑟𝑖

is the site coordinates, ⟨· · ·⟩ denotes a thermal averageand [· · ·]𝑎𝑣 the average over the disorder distribution.Consequently, the SG correlation length is written as

𝜉SG(𝐿) =1

2 sin(𝑘min/2)

(𝜒SG(0)

𝜒SG(𝑘min)− 1

)1/2

,

𝑘min = (2𝜋/𝐿)(1, 0, 0). (6)

For the polarization 𝑃 (𝑟), correspondingly, one candefine the electric DG order parameter, susceptibilityand correlation length, respectively,

𝑞𝑃𝑢𝑣(𝑘) =

1𝑁

∑𝑖

𝑃(1)𝑖𝑢 · 𝑃 (2)

𝑖𝑣 exp(𝑖𝑘 · 𝑟𝑖),

𝜒DG(𝑘) = 𝑁∑𝑢,𝑣

[⟨𝑞𝑃𝑢𝑣(𝑘)

2⟩]𝑎𝑣

,

𝜉DG(𝐿) =1

2 sin(𝑘min/2)

(𝜒DG(0)

𝜒DG(𝑘min)− 1

)1/2

,

𝑘min = (2𝜋/𝐿)(1, 0, 0). (7)

With these definitions, one can investigate the SGand DG transitions, giving the non-zero ME couplingterms defined in Eq. (2). For simplification of the cal-culation, we take 𝐽0 = 1.0. The maximal value of𝑃 (𝑟) is 0.5 and 2𝜒𝐸(𝑟) = 10.0.

The Monte Carlo simulation employed for thepresent work was widely described in Refs. [13,14] andno details will be given here. We utilize the paral-lel tempering algorithm for accelerating the processtoward the equilibrium state.[18]

Fig. 1. Simulated correlation length 𝜉𝐿, divided by 𝐿for the isotropic Heisenberg spin glass (a) and for electricdipole glass (b), as a function of temperature 𝑇 , at differ-ent values of 𝐿, without any ME coupling (𝛾1 = 𝛾2 = 0).𝑇SG is the spin glass transition point, i.e. the crossingpoint of the curves. The inset shows the finite size scalingof the data (see text).

We simulate the two correlation lengths, 𝜉SG(𝐿)and 𝜉DG(𝐿), as a function of lattice dimension 𝐿from 4 to 10, for different ME coefficients 𝛾1 and𝛾2. Then we check the possible SG and DG transi-tions by performing a finite size scaling analysis onthe correlation length. First, we demonstrate the SGtransitions of the lattice in the case of 𝛾1 = 0 and𝛾2 = 0, i.e. without ME coupling. The results areshown in Fig. 1, where 𝜉SG(𝐿)/𝐿 and 𝜉DG(𝐿)/𝐿 asa function of 𝑇/𝐽0 at different values of 𝐿 (𝐿 = 4,6, 10) are plotted. It is clearly seen that all the𝜉SG(𝐿)/𝐿 ∼ 𝑇 curves intersect at the same point(𝜉SG(𝐿)/𝐿 ∼ 0.45 and 𝑇/𝐽0 ∼ 0.187), indicating thepreferred SG state at 𝑇 < 𝑇SG. We also performedfinite-size scaling by fitting the data using the formula𝜉SG(𝐿)/𝐿 ∼ 𝐿1/𝑣(𝑇 −𝑇SG), and the rescaled data areplotted in the inset of Fig. 1(a) with the obtained ex-ponent 𝑣 ∼ 1.15. Both the simulated values of 𝑇SG

and 𝑣 are roughly consistent with earlier reports fromdifferent authors,[13,14] demonstrating the SG transi-tions occurring at 𝑇 = 𝑇SG ∼ 0.187𝐽0. The 𝑇SG ob-tained here, which is slightly higher than the previ-ous result (𝑇SG/𝐽0 ∼ 0.16) might probably be due tothe statistical errors or the unreachable equilibriumstate using the Metropolis algorithm. In Fig. 1(b), thesimulated 𝜉DG(𝐿)/𝐿 data are shown as a function of𝑇 for different values of 𝐿. As expected, no signifi-cant dependence of 𝜉DG(𝐿)/𝐿 on 𝑇 is observed for allthe values of 𝐿, indicating no DG transition occurring

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CHIN. PHYS. LETT. Vol. 26,No. 8 (2009) 087501

over the whole 𝑇 -range. This result is understand-able since energetically the nonzero polarization 𝑃 (𝑟)is unfavored (referring to Eq. (3)) if no ME couplingis available.

Fig. 2. Simulated 𝜉𝐿/𝐿 for the Heisenberg spin glass andthe dipole glass as a function of 𝑇 , at different values of𝛾1. 𝐿 = 6 and 𝛾2 = 0.

Fig. 3. Simulated 𝜉𝐿/𝐿 for the Heisenberg spin glass (a)and the dipole glass (b) as a function of 𝑇 , at differentvalues of 𝐿, for 𝛾1 = 0 and 𝛾2 = 0.4. 𝑇SG is the spinglass transition point and 𝑇DG is the dipole glass tran-sition point, i.e. the crossing points of the curves. Theinsets show the finite size scaling of the data (see text).

Subsequently, we set different 𝛾1 but keep 𝛾2 = 0(𝐻ME2 = 0), in order to check the contribution to theDG transition from the spatial inhomogeneity of thesquared 𝑀 (𝐻ME1). It should be noted that 𝑀2 = 1.0is always satisfied over the whole lattice and thus nocontribution of 𝐻ME1 to the DG transition is expected.The results are shown in Fig. 2, where 𝜉SG(𝐿)/𝐿 and𝜉DG(𝐿)/𝐿 as a function of 𝑇 , respectively, at differentvalues of 𝛾1 for 𝐿 = 6 are plotted. Indeed, neithersignificant dependence of 𝜉SG(𝐿)/𝐿 on 𝛾1, nor signif-icant dependence of 𝜉DG(𝐿)/𝐿 on 𝑇 is observed forall values of 𝛾1, indicating no DG transition occurringover the whole 𝑇 range. Therefore, we are allowed toargue that the ME coupling term

∑𝑃 · ∇(𝑀2) has

no effect on the ordering of the electric dipoles. Infact, a spatial inhomogeneity of 𝑀2 may be possiblyinduced by either doping of additional spin moment

or generating interface or surface.We then set different 𝛾2 but keep 𝛾1 = 0, in order

to check the contribution of the chiral inhomogene-ity of magnetization 𝑀 . As an example, we showin Fig. 3 the simulated 𝜉SG(𝐿)/𝐿 ∼ 𝑇 relations and𝜉DG(𝐿)/𝐿 ∼ 𝑇 relations at different values of 𝐿, for𝛾2 = 0.4. When a similar analysis as depicted in Fig.1 is performed, one observes from Fig. 3(a) that theSG transition point shifts towards a high 𝑇 = 𝑇SG =0.34𝐽0, slighter higher than 𝑇SG ∼ 0.187𝐽0 for thecase of no ME coupling (𝐻ME = 0). Fitting with fi-nite size scaling produces an exponent of 𝑣 = 1.2, alsoslightly higher than 𝑣 = 1.15 for 𝐻ME = 0. Moreimportant is the DG transition at a nonzero temper-ature, as shown in Fig. 3(b), where the crossing point𝑇 = 𝑇DG ∼ 0.11𝐽0 is clearly identified. Similarly, thefinite size scaling analysis gives an exponent of 𝑣 = 1.3.

Fig. 4. Simulated 𝜉𝐿/𝐿 for the Heisenberg spin glass (a)and the dipole glass (b) as a function of 𝑇 , at differentvalues of 𝛾2. 𝐿 = 6 and 𝛾1 = 0.

The results shown in Fig. 3 demonstrate that thespatial inhomogeneity of 𝑀 , a generic feature of spinglass systems, does induce the electric DG transi-tion at 𝑇/𝐽0 ∼ 0.11. This seems to be the firstsimulation evidence of ME coupling induced elec-tric DG transitions. Subsequently, we simulate the𝜉SG(𝐿)/𝐿 ∼ 𝑇 and 𝜉DG(𝐿)/𝐿 ∼ 𝑇 relations at dif-ferent values of 𝛾2 but keeping 𝛾1 = 0, and the re-sults are plotted in Figs. 4(a) and 4(b), respectively.While the correlation length for the spins, 𝜉SG, al-ways increases monotonously with decreasing 𝑇 , thisincreasing tendency is gradually suppressed with in-creasing 𝛾2. However, this suppression is very weak for𝛾2 < 0.8. On the other hand, the correlation lengthfor the dipoles, 𝜉DG, as a function of 𝛾2 first increasesand then decreases. Referring to the low 𝑇 data, themaximal 𝜉DG is reached at 𝛾2 ∼ 0.4. In terms ofME coupling induced DG transitions, the most sig-nificant effect is achieved at an optimized coefficient𝛾2 ∼ 0.4, beyond which the induced DG state may besuppressed, as shown in Fig. 4(b).

To understand the underlying physics, one may re-fer to

∑𝐻ME2(=

∑𝛾2 ·𝑃 · [𝑀(∇·𝑀)−(𝑀 ·∇)𝑀 ]) in

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CHIN. PHYS. LETT. Vol. 26,No. 8 (2009) 087501

Eq. (2). It was identified that an incommensurate andspiral (helical or conical) spin configuration favors spa-tially homogeneous polarization 𝑃 , so that a non-zeronet polarization (ferroelectricity) is generated.[5,7,16]

This picture has a microscopic origin based on thequantum gauge approach to the M-O-M (M is transi-tion metal element) perovskite structure in transitionmetal oxides like manganites.[19] The local polariza-tion 𝑃 associated with the M-O-M chain can be writ-ten as

𝑃 ∼ 𝑒𝑖𝑗 × (𝑆𝑖 × 𝑆𝑗), (8)

where 𝑒𝑖𝑗 represents the connecting vector pointing tospin 𝑆𝑗 from 𝑆𝑖. It is noted that Eq. (8) is actuallyone solution to the Hamiltonian 𝐻0 = 𝐻ME2 + 𝐻𝑃 :

𝐻0 = 𝛾2 ·𝑃 ·[𝑀(∇ ·𝑀)− (𝑀 · ∇)𝑀

]+

12𝜒𝐸

𝑃 2, (9)

if one proceeds with 𝜕𝐻0/𝜕𝑃 = 0. For an isotropicHeisenberg spin glass, it is clear that no long-rangespin ordering (transitional or chiral type) exists.Therefore, no long-range FE (dipole) ordering is pos-sible within the present framework. Instead, SG glassordering would obviously lead to dipole glass orderingdue to Eq. (8).

To illustrate the fact that 𝑇SG at 𝛾1 = 0 and 𝛾2 > 0is higher than 𝑇SG at 𝛾1 = 𝛾2 = 0, one refers to Eq. (9)once more. In a qualitative sense, the term

∑𝐻ME2

is necessarily negative because∑

𝐻𝑃 is always posi-tive. The negative term is equivalent to an additionaleffective field applied to the Heisenberg spin lattice(Eq. (1)), thus leading to an increase of 𝐽0. Conse-quently, the SG ordering point at 𝛾2 > 0 must behigher than that at 𝛾2 = 0. There is no doubt that𝑇DG < 𝑇SG is also obvious because the DG orderingis induced by the SG ordering.

Nevertheless, it is still unclear why both the SG or-dering and the DG ordering would be suppressed if 𝛾2

becomes very large (𝛾2 > 0.4). One possible origin isagain from Eq. (8). On the one hand, when 𝛾2 is verylarge, the ME coupling term becomes the dominantone in the system Hamiltonian. In order to minimizethe system free energy, the spin configuration prefersto take spiral (helical or conical) symmetry as definedby Eq. (8), although the local polarization 𝑃 cannot belarge due to 𝐻𝑃 ∼ 𝑃 2. A preference of this symmetrywill break the SG order and thus destabilize the SGstate. On the other hand, if the term

∑𝐻ME2 dom-

inates, one can infer from Eq. (8) that the lattice netpolarization should no longer be zero, i.e. the systemhas a tendency to generate homogeneous polarization,although this tendency cannot be significant. This ar-gument is confirmed by the increasing net polarization

of the whole lattice with increasing 𝛾2, while the netpolarization for an ideal DG state should be zero. Thesimulated data are shown in Fig. 5 and in the inset.The 𝑥-axis component of the net polarization, 𝑃𝑥, asa function of 𝛾2 at 𝐿 = 6 and 𝑇 = 0.05, does behaveas argued above, noting that 𝑃𝑥 seems saturated as𝛾2 > 1.0.

Fig. 5. Simulated 𝑥-axis component of the net polariza-tion, 𝑃𝑥, as a function of 𝑇 at different values of 𝛾2. 𝐿 = 6and 𝛾1 = 0. The arrow highlights the data for increasing𝛾2 and the inset shows the dependence of 𝑃𝑥 on 𝛾2.

In conclusion, we have investigated electric dipoleglass behavior in an isotropic Heisenberg spin glasssystem with Gaussian random fields and incorpo-rated two Landau magnetoelectric coupling interac-tion terms. The ME coupling induced electric dipoleglass transitions have been demonstrated in such aspin glass system.

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