Highly anisotropic resistivities in the double-exchange model for strained manganites

Highly anisotropic resistivities in the double-exchange model for strained manganites

Shuai Dong,^1,2Seiji Yunoki,^3,4Xiaotian Zhang,^5,6CengizŞen,^5,6J.-M. Liu,^2,7and Elbio Dagotto^5,6

1Department of Physics, Southeast University, Nanjing 211189, China

2Nanjing National Laboratory of Microstructures, Nanjing University, Nanjing 210093, China

3Computational Condensed Matter Physics Laboratory, RIKEN, Wako, Saitama 351-0198, Japan

4CREST, Japan Science and Technology Agency (JST), Kawaguchi, Saitama 332-0012, Japan

5Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA

6Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 32831, USA

7International Center for Materials Physics, Chinese Academy of Sciences, Shenyang 110016, China 共Received 16 May 2010; revised manuscript received 28 June 2010; published 22 July 2010兲 The highly anisotropic resistivities in strained manganites are theoretically studied using the two-orbital double-exchange model. At the nanoscale, the anisotropic double-exchange and Jahn-Teller distortions are found to be responsible for the robust anisotropic resistivities observed here via Monte Carlo simulations. An unbalance in the population of orbitals caused by strain is responsible for these effects. In contrast, the anisotropic superexchange is found to be irrelevant to explain our results. Our model study suggests that highly anisotropic resistivities could be present in a wide range of strained manganites, even without共sub兲micrometer- scale phase separation. In addition, our calculations also confirm the formation of anisotropic clusters in phase-separated manganites, which magnifies the anisotropic resistivities.

DOI:10.1103/PhysRevB.82.035118 PACS number共s兲: 75.47.Lx, 71.70.Ej, 75.30.Gw

I. INTRODUCTION

Strongly correlated electronic materials, which are well known for the presence of complex phase competitions in- volving the spin, charge, and orbital degrees of freedom,¹are promising candidates to be used in new multifunctional devices.²Typically, in materials such as manganites with the colossal magnetoresistance 共CMR兲, there are several phases with free energies that are quite close to one another but their individual physical properties can be rather different.³There- fore, colossal responses to external perturbations, including the CMR 共Refs. 4 and 5兲 and colossal electroresistance 共CER兲,⁶ can and do occur in some manganites. During the past decade, theoretical studies on manganites have ad- dressed many of these colossal responses, such as CMR,^7–10 CER,^11,12 surface reconstructions,^13–16 and disorder effects.^17–25

In addition, the effects of strain on the properties of man- ganites and other complex oxides are attracting increasing attention due to the rapidly expanding research interests in complex oxides heterostructures.^26,27 In fact, phase transi- tions driven by strains have been discussed in manganite thin films for several years.^28–36The physical mechanism of these phase transitions is mostly orbital-order mediated.^37–40 For example, according to density-functional theory 共DFT兲 cal- culations, the ground states of LaMnO₃/SrMnO₃ superlat- tices can be tuned between A-type antiferromagnetic共AFM兲, ferromagnetic 共FM兲, and C-type antiferromagnetic phases when the ratioc/a is in the range 0.96–1.04, wherec共a兲is the out-of-plane 共in-plane兲 lattice constant.³⁹ Even for LaMnO₃ itself, the ground state may become FM if the 兩3x²−r²典/兩3y²−r²典-type orbital order is fully suppressed in the cubic lattice, according to both the DFT and model calculations.^37,40

Very recently, Ward et al. have observed highly aniso- tropic resistivities in strained La_5/8−xPr_xCa_3/8MnO₃

共LPCMO兲 thin films.⁴¹ LPCMO is a prototype phase- separated material.⁴² The coexistence of FM and charge- ordered-insulating 共COI兲 clusters at the 共sub兲micrometer scale can seriously affect the electric transport properties, especially the metal-insulator transition 共MIT兲. The electric conductance in the phase-separated LPCMO is dominated by the percolation mechanism.^42,43 For example, giant discrete steps in the MIT and a reemergent MIT occur in an artifi- cially created microstructure of LPCMO when the size con- finements in two directions become comparable to the phase- separated cluster sizes.^44,45 Therefore, Ward et al. proposed that the anisotropic percolation might be responsible for the highly anisotropic resistivities in strained LPCMO.⁴¹ Also, our previous simulation of CER predicted anisotropic resis- tivities due to the electric-field-driven anisotropic percolation in phase-separated manganites.¹²

Then, two interesting questions arise:共1兲how does strain drive the anisotropic percolation in the LPCMO films? And, more importantly, 共2兲 can the large anisotropies occur in more standard CMR materials with nanometer-scale phase competition or even with bicritical clean-limit phase dia- grams? Therefore, to set up a study to be used as a reference for future research, it is interesting to investigate theoreti- cally via model Hamiltonians the magnitude of the aniso- tropy in transport induced by strain in cases where phase competition is present but also where phase separation is not.

In other words, it is important to study regimes where in the clean limit 共no quenched disorder兲, a first-order transition separates the two competing states, typically a metal and an insulator, inducing a CMR effect in a narrow range of pa- rameters but where phase separation it not present. This cal- culation will allow us to disentangle the effects of mere strain on a clean-limit model in the regime of phase compe- tition from the effects of strain on a truly phase-separated state. More basically, these investigations are important to move beyond the micrometer scale to find the microscopic

1098-0121/2010/82共3兲/035118共6兲 035118-1 ©2010 The American Physical Society

origin of anisotropic resistivities in generic strained manga- nites.

II. MODELS AND TECHNIQUES

In this paper, the two-orbital double-exchange 共DE兲 model will be employed to study the anisotropic resistivities in strained manganites. In the past decade, the DE model has been extensively studied and it proved to be a quite reason- able model to describe perovskite manganites.³ In Ward et al.’s experiments, the anisotropic strain field splits the in- plane lattice constants along the关100兴and关010兴axes in the pseudocubic convention共or the关101兴and关101¯兴axes in the orthorhombic Pnma convention兲. Thus, a modified model has to be developed to reflect the features of this strained lattice since most previous model studies were done on cubic or square lattices.

As a well-accepted approximation for manganite models, an infinite Hund coupling is here adopted. With this useful simplification, the DE model Hamiltonian reads

H= −

兺

具ij典

␣␤

t_␣␤^rជ 共⍀ijc_i␣^†c_j␤+ H.c.兲+

兺

具ij典

J_AF^rជ Sជ_i·Sជ_j +␭

兺

_i ^{共−Q1ini+Q2i␶xi+Q3i␶zi兲}

+ 1

2

兺

_i ^{共2Q1i2 +Q2i2 +Q3i2兲. 共1兲}

In the above model Hamiltonian, the first term is the stan- dard DE interaction. ␣^and␤denote the two Mne_g orbitals a共兩x²−y²典兲 and b共兩3z²−r²典兲. cia共ci†␣兲 annihilates 共creates兲 an e_g electron at orbital␣ ^{of site}i, with its spin parallel to the localizedt_2gspinSជ_i. The nearest-neighbor共NN兲hopping di- rection is denoted byrជ. The Berry phase⍀ijgenerated by the infinite Hund coupling equals cos共␪i/2兲cos共␪j/2兲 + sin共␪i/2兲sin共␪j/2兲exp关−i共␾i−␾j兲兴, where ␪ ^and ␾ ^{are the} polar and azimuthal angles of the t_2gspins, respectively. In strained manganites, an elongated lattice constant gives rise to more straight Mn-O-Mn bonds, thus enhancing the FM DE interaction. To mimic this effect, the in-plane DE hop- ping amplitudest_␣␤^rជ have to be set as

t^x=

冉

^ttaaxba ^tabx

x t_bb^x

冊

^=t40x

冉

⁻³

^冑

^{3 −1}

^冑

³

冊

^,

t^y=

冉

^ttaayba ^taby

y t_bb^y

冊

^=t40y

冉 冑

³3

^冑

1³

冊

^{. 共2兲}

In the rest of the paper,t₀^x is taken as the energy unitt₀ and At=t₀^y/t₀^x− 1 is defined to characterize the degree of aniso- tropy of the DE interaction.

The second term of the model Hamiltonian is the AFM superexchange 共SE兲 interaction between NN t_2g spins. The SE coefficient J_AF could also become anisotropic in the strained lattices, which is here characterized by AJ

=J_AF^x /J_AF^y − 1.

The third term of the model stands for the electron-lattice coupling.⁴⁶ ␭ is a dimensionless coefficient and ni is thee_g

electronic density at site i. Q’s are phonons, including the Jahn-Teller共JT兲modes共Q2andQ₃兲and the breathing mode 共Q1兲:Q₁=共␦x+␦y+␦z兲/

冑

3, Q₂=共␦x−␦y兲/

冑

2, and Q₃=共−␦x

−␦y+ 2␦z兲/

冑

6, where ␦ stands for the length change in the oxygen coordinates in the Mn-O-Mn bonds along the axes directions.⁴⁶ ␶is the orbital pseudospin operator, namely, ␶x

=c_a^†cb+c_b^†ca and ␶z=c_a^†ca−c_b^†cb. The last term is the lattice elastic energy. Note that the model used here induces coop- erative distortions of the oxygen positions.

The above model Hamiltonian is numerically solved via the Monte Carlo共MC兲simulation on a two-dimensional共2D兲 8⫻8 lattice. The reason for this restriction to a two- dimensional geometry is simply practical: simulations in three-dimensional lattices are very demanding computation- ally. Thus, here␦zis set to zero and our effort will only focus on the in-plane anisotropy. Using standard periodic boundary conditions 共PBCs兲 具Q1典,具Q2典, and具Q3典 共if具 典 stands for av- erages over the whole lattice兲 equals to zero. However, to simulate the strain effect in the JT distortion, anisotropic PBCs 共aPBCs兲 should be introduced to the lattice. In the aPBCs for 2D lattices,具Q2典is set as a constant which can be nonzero while 具Q1典 and 具Q3典 remain zero. To characterize this anisotropic JT distortion, the quantity AQis defined as

−具Q₂典/共2

冑

3兲.

In Ward et al.’s experiments, the difference between the in-plane lattice constants is small 共⬃0.2– 0.3 %兲.⁴¹ Corre- spondingly, the anisotropies of interactions should be weak, implying that A_t,A_J, andA_Qmust be small quantities in our study.

In our MC simulations, the averagee_gdensity具n典is cho- sen as 0.75. As discussed in previous literature, to obtain the MIT and CMR effects, the parameters 共JAF,␭兲 should be chosen to be near the phase boundaries between FM and AFM COI phases.^9,10This fine tuning of couplings could be avoided by introducing quenched disorder but our study will be conducted in the clean limit to set up a benchmark to decide on the origin of strain-induced transport anisotropies that are investigated experimentally. According to the phase diagram of the two-orbital DE model for 具n典= 0.75,⁴⁷ the parameters J_AF= 0.09 and ␭= 1.2 are suitable and they are here adopted as the default ones in our simulation, unless other parameters are explicitly used. In fact, other sets of parameters near the default ones have also been partially tested and no qualitative differences have been found. Thus, this choice of parameters does not alter the general validation of our results and conclusions, at least qualitatively. In the MC simulation, the first 10⁴ MC steps are used to reach thermal equilibrium and another 2⫻10⁴ MC steps are used for measurements.

The dc conductances, which are calculated using the Kubo formula, are in units ofe²/h, whereeis the elementary charge and h is the Planck’s constant.⁴⁸The resistivities are the reciprocals of MC averaged conductances. The normal- ized magnetization 共M兲 is obtained from the spin structure factorS共kជ_{兲, at}kជ=共0 , 0兲.¹⁶

III. RESULTS AND DISCUSSION

To start the discussion of results, the original state without any anisotropic contribution is simulated as a reference. The

resistivities along both thexandydirections共␳xand␳y兲are calculated as a function of temperature共T兲, as shown in Fig.

1. As expected,␳xand␳yare almost identical in the wholeT range. The small differences between ␳x and ␳y are from statistical fluctuations during the MC simulation, and these differences should converge to zero with increasing MC simulation times. With this set of parameters, both␳xand␳y

show a MIT with increasing temperature at T_MI⬃0.045t₀, which is the same approximate location as our estimation for the Curie temperature共TC兲, according to theM-Tcurve. For a typical manganite with a MIT under zero magnetic field,t₀ is roughly estimated to be in the range 0.4–0.5 eV.^16,40Thus, T_MI⬃200– 260 K in agreement with bulk measurements.

Therefore, the set of parameters 共JAF= 0.09, ␭= 1.2兲 used here is suitable to describe typical manganites, such as La_1−xCa_xMnO₃.

In the following, we will apply the aforementioned three anisotropic interactions one by one into the model simulation to clarify their respective roles. First, let us consider the an- isotropic DE interaction. For this purpose, At is set to 0.1 while other parameters are kept the same as the original ones. In other words, the DE hopping amplitude along they direction is made 10% larger than that along thexdirection because of the presence of more straight Mn-O-Mn bonds along the y axis. The resistivities and magnetization of this strained lattice are shown in Fig.2共a兲 as a function ofT.␳x

shows a MIT similar to the original one while ␳y is now considerably suppressed in magnitude. Thus, a high degree of anisotropic resistivities can be obtained using an aniso- tropy A_t in the hoppings which is only 0.1. Interestingly, although the differences between␳xand␳yare substantial, a difference in the T_MI’s shown in␳xand␳yis not obvious in thisAt= 0.1 case. The anisotropy inT_MIappears to be weak and it must be hidden by the sparse steps in temperature used in our simulations 共=0.05t₀⬃20– 30 K兲. Comparing with the original one,T_CandT_MIactually simultaneously raise to

⬃0.055t0 due to the increase int₀^y.

Next, the anisotropy in SE is taken into account. A_t is restored to 0, whileAJis set to 0.1. In this case,J_AF^y has to be weakened slightly to preserve the presence of a MIT, other- wise the system becomes insulating in the wholeT range if J_AF^y remains at 0.09. Thus, for this case the new valuesJ_AF^y

= 0.085 andJ_AF^x = 0.0935 are adopted. The MC simulated re- sistivities and magnetization for this strained lattice are shown in Fig. 2共b兲, as a function of T. The T_MI remains

isotropic and coincides with T_C⬃0.05t₀. In contrast to the DE case, the differences between␳xand␳yare much smaller, especially below T_C 共or T_MI兲:␳y is only slightly lower than

␳xabove 0.04t₀, and they are almost identical below 0.04t₀. Then we conclude that the effect of an anisotropic SE is much weaker than the case of an anisotropic DE, when their anisotropic ratios are the same.

Finally, it is necessary to address the effect of anisotropies in the JT sector, for completeness. In a distorted oxygen oc- tahedron, the twoe_gorbitals are not degenerate anymore. For instance, when the lattice constants along the xand y axes are different, as in Ward et al.’s strained manganites thin films, 具Q2典 is no longer zero. This nonzero 具Q2典 mode in- duces an orbital-state preference over the whole lattice. With At= 0, AJ= 0, and AQ= 0.01, the MC simulated resistivities and magnetization are shown in Fig.2共c兲, as a function ofT.

Similarly to the case of an anisotropic DE, there is now a substantial difference between ␳x and ␳y. In addition, the T_MI’s of the␳xand␳ycurves become anisotropic: the lower resistivity curve has a higher T_MI, in agreement with the experiments.⁴¹ However, it should be noted that the aniso- tropy ofT_MIis not large.

To further clarify the anisotropic resistivities observed here, the relative percentage difference共␦兲between␳xand␳y

关defined as␦^=共␳x−␳y兲/␳y⫻100%兴is calculated for each of FIG. 1. 共Color online兲MC simulated resistivities共triangles兲and

magnetization 共dots兲for a square 8⫻8 isotropic lattice 共A_t= 0,A_J

= 0, andA_Q= 0兲, as a function ofT.

FIG. 2. 共Color online兲MC simulated resistivities共triangles兲and magnetization 共dots兲 for strained lattices, as a function of T. 共a兲 Only the anisotropic DE interaction is here considered 共A_t= 0.1, A_J= 0, andA_Q= 0兲.共b兲 Only the anisotropic SE interaction is con- sidered共A_t= 0,A_J= 0.1, and A_Q= 0兲. To maintain the presence of a metal-insulator transitionJ_AF^y must be slightly reduced to 0.086.共c兲 Only the strained JT distortion is considered 共A_t= 0, A_J= 0, and A_Q= 0.01兲.

the three cases discussed above, as shown in Fig. 3共a兲. For the original isotropic and for theA_J= 0.1 cases, the values of

␦ are very small共⬍⫾20%兲in the whole temperature range, as expected from Figs. 1 and 2共b兲. In contrast, for the At

= 0.1 andAQ= 0.01 cases, the situation is different. With in- creasingTfrom low temperatures, the␦’s first increase. After each case reaches a robust peak of 200– 300 %, then they decrease with further increases in T. Interestingly, for both these two cases, the correspondingT’s of the peaks found in

␦are slightly lower than the correspondingT_C’s andT_MI’s, in agreement with the experimental results.⁴¹

To understand the physical mechanism leading to the an- isotropic resistivities, the orbital properties of the strained states, characterized by the average values of the pseudospin- orbital operator 具␶x典, are also calculated, as shown in Fig.

3共b兲. The occupation difference between the 兩3y²−r²典 and 兩3x²−r²典 components is in proportion to具␶x典. The values of 具␶x典 for the original isotropic case fluctuate around zero in the whole temperature range analyzed, implying that the weights of the 兩3y²−r²典 and兩3x²−r²典 orbitals are equal, as expected by symmetry. For the AJ= 0.1 cases, 具␶x典 remains very small, implying that the anisotropicJ_AFused is not rel- evant to affect substantially the orbital composition of the state. In fact, both these two cases give rise to共almost兲iso- tropic resistivities. In clear contrast, for the At= 0.1 and AQ

= 0.01 cases, finite values for 具␶x典 are observed at high T, which are gradually suppressed by the FM transitions with decreasingT. For theAQ= 0.01 case, the finite具␶x典is mainly caused by the JT distortion, which remains finite at lowTas long as the lattice is anisotropically distorted. However, the finite具␶x典for theA_t= 0.1 case is caused by the enhanced DE process along the y direction. Namely, it is a DE mediated polarization of the orbital occupancy. Thus, for the fully FM state at low T, this DE mediated orbital rearrangement is largely suppressed to near zero, which is different from the results obtained for the JT distortion case. In summary, in our simulation the large anisotropy of the resistivity emerges in those cases where there is an unbalance in the orbital-state population, as sketched in Fig.3共c兲, although the value of␦ is not linearly dependent on 具␶x典 in the whole T range. In simple terms, the orbitals that increase their overlaps due to strain are now more populated than the other ones.

Note that all the above simulations were carried out on relatively small 8⫻8 clusters using clean-limit models and still the anisotropy observed is comparable to that found ex- perimentally. This implies that clean-limit strained mangan- ites can be as anisotropic as phase-separated compounds.

Therefore, the classical percolation at the 共sub兲micrometer scale does not appear to be essential to obtain highly aniso- tropic resistivities in manganites but of course in the clean limit, the strain induced by substrates must be sufficiently large, e.g., large enough to generate a A_t= 0.1 as used here.

Thus, the highly anisotropic resistivities should be a general property of manganites and even other complex oxides, as long as the bond lengths/angles are tuned to be sufficiently anisotropic by strain. A recent experiment on strained Sm_0.5Ca_0.5MnO₃ films 关without the 共sub兲micrometer phase separation兴, showed highly 共in-plane兲 anisotropic conduc- tances under magnetic fields.⁴⁹More experimental studies on strained oxide films are needed to further verify our results.

However, it is important to clarify what occurs in the particular case of large-length-scale phase-separated manga- nites. It is well known that the A-site disorder can drive a manganite into a phase-separated state, if it is close to certain bicritical boundaries.^3–5,42In this case, the classic percolation mechanism can certainly also contribute to the anisotropic resistivities if the shapes of the FM metallic clusters become anisotropic, as suggested in Ref. 41. In fact, our model can also qualitatively explain the formation of anisotropic FM clusters. To study an individual phase-separated FM cluster embedded in the AFM COI matrix, the ground-state energies of FM lattices with openboundary conditions can be calcu- lated directly. For simplicity, all Q_2i are set to be uniform 共and equal to 具Q2典兲 and all spins are aligned to be perfectly FM. Then, the shape of the FM clusters can be determined FIG. 3. 共Color online兲 共a兲 Relative percentage differences be-

tween␳xand␳y, as a function ofT, for the cases indicated.Ostands for original isotropic.共b兲The average values of具␶x典, as a function of T. 共c兲 Sketch of the effect of strain on the orbitals. To better distinguish the orbital leaves along the x and y directions, here 具␶x典/具n典is magnified to 0.1. In the sketch, the overlap of electronic clouds becomes stronger along theydirection and weaker along the xdirection. Thus, the conductances, which are in proportion to the overlaps, become anisotropic. Note that the real overlaps are indi- rect and mediated by oxygen共not shown here兲. Inset: sketch of an oxygen octahedron’s in-plane distortion.

by varying the lattice’s shape but keeping a constant lattice area. For instance, the energies of lattices with the same area size共L_x⫻L_y= 900,L_xandL_yare side lengths along thexand yaxes, respectively兲are shown in Figs.4共a兲–4共c兲, as a func- tion of Lx. The energy of the 25⫻36 lattice, which is elon- gated along the y direction, can be obviously more stable than that of a 30⫻30 one when At⬎0.3, or AQⱖ0.04, or A_tⱖ0.2 andA_Qⱖ0.02 simultaneously. This process is quali- tatively sketched in Fig. 4共d兲. FM clusters with other sizes 共e.g.,Lx⫻Ly= 576 andLx⫻Ly= 1764兲have also been tested, reaching the same conclusion. Thus, it is reasonable to ex- pect similar effects when FM clusters expand to the共sub兲mi- crometer scale, although our microscopic model can not be directly used on such large lattices with the currently avail- able computational capabilities.

Finally, it is important to estimate how large should be the lattice mismatch required for the highly anisotropic resistiv- ities observed here to appear in strained manganites with nanoscale phase separation or in the case of a bicritical phase diagram. According to the well-known Harrison’s formula,⁵⁰ the DE hoppingt and SE exchangeJ_AFcan be estimated to be in proportional tor⁻⁷ andr⁻¹⁴共ris the Mn-O-Mn’s bond length兲, respectively. Thus, t/J_AF is in proportion to r⁷. To obtain the valuesAt= 0.1 andAJ= 0.1 used in our simulations, the required lattice mismatch is about 1.4%. Similarly, by comparing experimental data共rl−rs⬃0.6 Å, whererlandrs

are long and short bonds, respectively兲⁵¹and theoretical pa- rameters 共␭兩Q2兩= 1.5兲 共Ref. 52兲 for the JT distortions in RMnO₃, the parameter AQ= 0.01 used here is estimated as

⬃0.45%. It should be noted that the required strain 共lattice

mismatch in real films or 共A_t,A_Q兲 in our simulations兲 de- pends on the particular materials under study 共or, equiva- lently, the actual values of the parameters 共JAF,␭兲 in our simulations兲. The anisotropies are more sensitive to strain when the system moves closer to the phase boundary be- tween the FM and COI phases. With this idea in mind, it is natural that the anisotropies of LPCMO films can be notori- ous even if the lattice mismatch is small in average 共0.2– 0.3 % in Wardet al.’s experiments兲because LPCMO is precisely at the FM-COI phase boundary. According to our simulations, the highly anisotropic resistivities are also ex- pected in other strained CMR manganites films共even with- out phase separation兲, although the required strain might be somewhat larger than for the LPCMO case.

IV. CONCLUSIONS

In conclusion, the highly anisotropic resistivities of strained manganite films were studied using microscopic models. For this purpose, the two-orbital double-exchange model was modified to include the strain contributions. In this revised model, the anisotropic Jahn-Teller distortion was emphasized, in addition to the anisotropic exchanges. The results of our MC simulation show that the highly aniso- tropic resistivities are associated with an unbalance in orbital populations which is driven by the anisotropic double- exchange and anisotropic Jahn-Teller distortions. In contrast, the anisotropic superexchange was not found to be a domi- nant driving force for the anisotropic resistivities. The ob- served highly anisotropic resistivities in our simulation did not rely on phase separation at the 共sub兲microscopic scale.

Therefore, it is expected that this anisotropic state could be realized in a variety of manganites and other complex oxides as well, if a sufficiently large lattice mismatch can be achieved in the growth of the manganite films. In addition, for the particular case of phase-separated manganites, our model investigations suggest that the anisotropic double- exchange and strained Jahn-Teller distortions could indeed reshape the ferromagnetic clusters, thus inducing an aniso- tropic percolation and concomitant anisotropic resistivity that further enhances these effects.

ACKNOWLEDGMENTS

We thank T. Z. Ward and J. Shen for fruitful discussions.

Work was supported by the 973 Projects of China 共Grants No. 2006CB921802, No. 2009CB623303兲 and the National Science Foundation of China 共Grant No. 50832002兲. S.Y.

was supported by CREST-JST. X.T.Z, C.S., and E.D. were supported by the USA National Science Foundation under Grant No. DMR-0706020 and by the Division of Materials Science and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy.

FIG. 4.共Color online兲Energy differences of the ground states of L_x⫻L_y= 900 lattices, as a function of L_x. The energy of the L_x

= 30 lattice is set as the reference point. 共a兲Results obtained with the anisotropic DE interactions. 共b兲 Results obtained with the strained JT distortions. 共c兲 Results obtained with both the aniso- tropic DE interactions and strained JT distortions simultaneously active. The values of共A_t,A_Q兲in共c兲are simultaneously stepped the same as in共a兲and共b兲, respectively.共d兲Sketch of the formation of an anisotropic FM cluster, according to共a兲–共c兲.

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