Island phases and charge order in two-dimensional manganites fileIsland phases and charge order in two-dimensional manganites H. Aliaga,1 B. Normand,2 K. Hallberg,1 M. Avignon,3 and

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PHYSICAL REVIEW B, VOLUME 64, 024422

Island phases and charge order in two-dimensional manganites

H. Aliaga,1 B. Normand,2 K. Hallberg,1 M. Avignon,3 and B. Alascio11Instituto Balseiro and Centro Atomico Bariloche, Comision Nacional de Energia Atomica, 8400 San Carlos de Bariloche, Arge

2Theoretische Physik III, Elektronische Korrelationen und Magnetismus, Institut fu¨r Physik, Universita¨t Augsburg,D-86135 Augsburg, Germany

3Laboratoire d’Etudes des Proprie´tes Electroniques des Solides (LEPES), Centre National de la Re´cherche Scientifique, BP 166,F-38042 Grenoble Cedex, France

~Received 22 November 2000; published 21 June 2001!

The ferromagnetic Kondo lattice model with an antiferromagnetic interaction between localized spins is aminimal description of the competing kinetic~t! and magnetic~K! energy terms which generate the richphysics of manganite systems. Motivated by the discovery in one dimension of homogeneous ‘‘island phases,’’we consider the possibility of analogous phases in higher dimensions. We characterize the phases present atcommensurate fillings, and consider in detail the effects of phase separation in all filling and parameterregimes. We deduce that island and flux phases are stable for intermediate values ofK/t at the commensuratefillings n51/4, 1/3, 3/8, and 1/2. We discuss the connection of these results to the charge and magneticordering observed in a wide variety of manganite compounds.

DOI: 10.1103/PhysRevB.64.024422 PACS number~s!: 75.10.2b, 75.30.Vn, 75.40.Mg

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I. INTRODUCTION

Transition-metal manganite compounds have long bknown to display a broad spectrum of physical propertiesa function of temperature, filling, and counterion compotion. While the most remarkable of these is the colosmagnetoresistance1 observed in the ferromagnetic~FM!phase, the phase diagrams of both cubic perovskite andered manganite materials exhibit a rich variety of metalinsulating, magnetically ordered, and, apparently, inhomoneous or phase-separated regions.

The ferromagnetic Kondo lattice model~FKLM ! has beenused extensively as a minimal model to reproduce the phics responsible for this situation. We will study a versionthe model which includes a Heisenberg interaction betwthe localized spins. In essence, this encapsulates the cotition between the ferromagnetic polarizing effect of tdouble-exchange hopping term2 ~t! for mobile carriers in theeg orbitals of Mn31, and the antiferromagnetic~AF! interac-tion ~K! between the localized spins composed of electrin the t2g orbitals. Treatments of the model with both clascal local spins, and with fully quantum,S51/2 local spins,both return some of the features observed among the stion of manganite phase diagrams. A large number of authhas worked on many forms of the FKLM, and we wpresent in the following sections only a small selectionreferences relevant to the current approach.

Following the discovery3 in one-dimensional simulationof novel ‘‘island phases’’ near commensurate values of etron filling in the FKLM with strong Hund coupling betweelocalized and conduction electrons, we wish here to consthe possibility of higher-dimensional generalizations of thephases. By an island phase is meant a spin configuracomposed of small, regularly arranged, FM islands~clustersof 2–4 sites in Ref. 3!, with AF local spin orientations between islands~Fig. 1!. These phases are homogeneous,near the commensurate fillings maximize kinetic enewithin each island at minimal cost to the magnetic ener

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which is favored at the island boundaries. Focusing primaon the problem in two dimensions~2D!, we wish to establishthe possibility that such islands, which may be small in oor both directions, remain the most stable phase for cerfillings and parameter ratiosK/t.

A particular motivation for our study is the recent obsevation of charge-ordering phenomena, and more generahomogeneous charge and spin configurations, in a varietmanganite systems. These appear in both layered and cmaterials, and at both commensurate and incommensuvalues of the electron filling set by the counterion dopinSome of the earliest observations of charge ordering4 weremade in La12xSrxMnO3, and were followed by measurements suggesting polarons,5 phase separation,6 and pairedstripe features.7 Charge order coupled to a structural phatransition has been observed in Bi12xCaxMnO3 at incom-mensurate values of the fillingx.8,9 Among hole-doped manganites, charge ordering arose at incommensurate fillingNd12xSrxMnO3, and in a stripelike configuration at halffilling in Pr0.5Sr0.5MnO3.10,11 For the latter system, the stripfeatures could be made to ‘‘melt’’ in an applied magnefield.10 Of most interest in the current context, ordering phnomena have also appeared in 2D or layered manganitetems. In Sr22xLaxMnO3 at low doping, Baoet al.12 reportedcharge order, phase separation, and triplet bipolarons. Fo

FIG. 1. Schematic representations of the island phases (p/3,p)~a! and (p/2,p/2) ~b!.

©2001 The American Physical Society22-1

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ALIAGA, NORMAND, HALLBERG, AVIGNON, AND ALASCIO PHYSICAL REVIEW B 64 024422

same system atx50.5, Moritomoet al.13 related charge or-dering to lattice effects by substitution for La, and Murakaet al.14 made direct measurements of charge and orbital ofor the commensurate La member. Finally, we mention athe observation15 of charge order in the layered 327 compound LaSr2Mn2O7.

The paper is organized as follows. In Sec. II we presthe model in the form we wish to consider, and outline tmethods by which it is analyzed. In Sec. III we discussavailable means to characterize the phases which appearillustrate these with examples. Section IV contains a detadiscussion of the issue of phase separation, and a glphase diagram for the augmented FKLM which delimitsregimes of interest for island phases. We return in Sec. Vthe robust flux and island phases, discuss their propertiestheir charge order, and consider their relevance to the abexperiments. Section VI gives a summary and conclusio

II. MODEL AND METHODS

We consider the FKLM in the form

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tor creating an electron of spins in the soleeg orbital; si

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tron, and its mobility depends on the orientation of the locized t2g spins according to the double-exchange mechanis2

The second term is the Hund coupling,JH.0, which favorsa FM orientation of spins on the same site. Following Re16,17, we will be concerned with the limit of largeJH ; whilein real systemsJH is of the same order as the bandwidth, thsimplifying approximation has been found to give reasonaresults. The limit corresponds to a situation where the cduction electron is bound to follow the spin texture of tlocalized system, while antialigned electrons occupy a bwith energy higher byJH . The projecting effect of the largeHund coupling allows one to neglect direct Coulomb intactions of theeg electrons. The final term, withK.0, ex-presses the AF interactions between the localt2g spins,whose competition with the FM spin alignment required

FIG. 2. Schematic representation of the Hamiltonian~1! for twosites.

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In the following we will consider the properties of thmodel ~1! over the full parameter range. Bare values of tratio K/t deduced to date for real systems are rather smalrecent study of the double-exchange phase diagram~see Ref.18, and references therein! outlines this situation for the(La,Ca)MnO3 system, and summarizes the reasons (eg elec-tron contributions, direct exchange enhancement! why themeasured ratioK/t.0.005 may be raised to effective valueon the order of 0.1. We note further that the manganperovskite structure offers a wide variety of counterions, asystems such as (Bi,Ca)MnO3 ~Ref. 8! have a significantlysmaller lattice constant than (La,Ca)MnO3. This may be ex-pected to give rise to a marked increase in the ratioK/t, andindeed superexchange values~12.6 meV! larger by a factorof 20 have been found in the former compound.8 Thus K/tvalues in excess of 0.2 would appear to be physically rsonable.

We will analyze the model primarily by a classical MonCarlo ~MC! procedure for the localized spins, in conjunctiowith exact diagonalization of the conduction electrsystem.19,20The localized spins are thus taken to be classican approximation to the true situation ofS53/2 which isfound not to invalidate the connection to real systems. Tconduction electrons are taken to occupy a singleeg orbital,or band, and from the condition onJH only one spin projec-tion need be considered. This part of the process is the stion of the single-electron problem with hopping set constently by the localized spin configuration. In the limits olargeS andJH , this is16

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The MC simulation proceeds from the FKLM partitiofunction with classical spins,

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whereN is the system dimension. Positivity of the integraassures that the sign problem is absent. Updates of theconfiguration$u i ,f i% are accepted or rejected accordingthe Glauber algorithm. In simulations with these spheriangles we were unable to find in the 2D system any cawhere noncoplanar spin configurations appear. Becausthe large degeneracy of coplanar phases, the simulatcould be accelerated by fixingu i5p/2, and varying only theangles$f i%. The number of MC steps per site forN58 istaken as 2000 to equilibrium and 3000 for measuremewhile for N512 the corresponding numbers are 500 a1000. The equilibrium criterion was taken from the numbof steps required to ensure a relative standard deviationthe energies per site smaller than 531024. Systems of sizeup to 12312 are accessible by this method, and thus

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ISLAND PHASES AND CHARGE ORDER IN TWO- . . . PHYSICAL REVIEW B 64 024422

supplement the MC results by a variety of classical analytconsiderations, which afford considerable insight and allodetailed assessment of finite-size effects. The simulatmay be pursued down to temperatures ofT50.005t, whichunless otherwise stated will be the relevant value for theresults displayed. This temperature is sufficiently low thcomparison with the zero temperature, analytical calculatiis meaningful, and in most cases quantitatively so. Tmethod is the same as that used by Dagotto, Yunoki,co-workers in a series of papers.19–24 We will reproducesome of the same results, and comment on the similarand differences in the context of our island phase analysewhat follows.

Because the classical MC method has been used befothe literature, we comment only briefly on further technicissues in order to focus on the physics of the model. Incases the boundary conditions used were periodic. Finite-effects are known to be very strong for small cluste(434, 636), and we will show only results for the largesystems (838, 12312) which we believe from commensurability and comparison with the infinite system to be repsentative for the phases illustrated. We performed simtions using a variety of initial spin configurations; while thmost unbiassed starting point is a paramagnetic~PM! spinconfiguration, convergence in this case may be very loThe majority of our simulations at the lowest temperatuillustrated here were performed with a starting state obtaifrom MC at a higher temperature. This ensured convergein a reasonable number of steps, and agreed in all casetested with the results from the PM start. Finally, we haobtained data over a range of temperatures with a viewanalyzing the thermodynamic properties of the model. Whfinite temperatures may stabilize interesting excited sstates, further expanding the space of configurations todiscussed below,25 we will restrict our considerations here tthe ground-state properties of the model~1!.

III. PHASE CHARACTERIZATION

In this section we will present some results for typicphases which emerge from MC simulations performed atcommensurate fillingsn51/2, 1/3, and 1/4, and for the fulrange of values ofK/t. The results of the simulations for thlocalized spin system may be characterized by three sepbut related quantities: the spin structure factor

S~k!5(i , j

Si•Sjeik(r i2r j ), ~4!

a histogram of the distribution of angles between all nearneighbor spin pairs, which we choose to present as a funcof cosQij , and a simple ‘‘snapshot’’ of the spin configurations at a representative step late in the MC process. Notethe histogram thatQ i j is the full angle between spins giveby cosQij5(Si•Sj )/S

2 for the classical case, and is not to bconfused with the on-site azimuthal angleu i in Eq. ~2!. Fi-nally, one may compute in addition the charge distributfunction

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As a guide to understand the variety of possibilities whis contained in these quantities, we first calculate the clacal, ground-state energies of a multiplicity of possible spconfigurations. This may be carried out for an infinite 2system by straightforward extension from the arguments psented for the 1D case in Ref. 3. For each spin configuratthe magnetic energy per spin is a simple function ofaverage of the angles across each bond, which varies f2K for the FM case to22K for the AF. The kinetic energyat this level is a readily calculable function of the spin cofiguration which varies from 0 in the AF case, wherekinetic processes are excluded, to the average energy o2D nearest-neighbor bandek522t(coskx1cosky), for therelevant band filling, in the FM case where it is maximanegative. The results of this exercise are illustrated in Figfor n51/2, n51/3, andn51/4.

All of the phases denoted by (kp/m,lp/m) have neigh-boring spins only either parallel or antiparallel, in both diretions. The rational fractionsk/m,l /m may be understood aindicating that the spin direction turns overk or l times in 2mlattice constants. Figure 1 shows two small-m possibilities,the (p/3,p) ~a! and (p/2,p/2) ~b! phases. As a more complex example, the phase (3p/4,p), which appears over awide range ofK/t at filling n51/4 @Figs. 3~c!,6#, would becomposed of chains with repeat unit↑↑↓↑↓↓↑↓ in the xdirection, and AF alignment in they direction. In addition tothese phases, which include the FM (0,0) and AF (p,p) endpoints, we include also the ‘‘flux phase,’’26,24 which will bediscussed in more detail below, and a ‘‘double spiral’’~DS!phase, by which is meant a single phase where the neaneighbor spins rotate by the same angle 0<Q<p in both xand y directions. In this last case, the optimal angleQ isobtained by minimizing a function ofK/t, and the doublespiral may be expected to be more favorable than any varof single-spiral phases combined with other forms of modlation in the transverse direction. Although we have consered many possible phases of the above types, in Fig. 3include for clarity only those which are the ground statesome range ofK/t.

The calculation of all of these phase energies is straigforward. In brief, calculation of the only 2D band at (0,0proceeds as above, with the filling determining the chempotential up to which the filled band is integrated. For thestructures (0,lp/m), one may consider the bandek522t cosk in the continuous direction, split appropriateinto 2, 3, or 4 ~the maximum included here! by an equalinterchain hoppingt. Integration over the filled parts of thesbands up to the chemical potential yields the average kinenergy. For the ‘‘0D’’ structures (kp/m,lp/m), the kineticenergy is a simplem2/kl-site diagonalization problem to obtain the discrete levels. These phases are particularly faable when the filling exactly matches a large gap in the felevel spectrum, e.g., (p/3,p) for n51/3 @Fig. 1~a!# or

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(p/2,p/2) for n51/4 @Fig. 1~b!#. The calculation of the ki-netic part for the double-spiral phase follows the 2D caabove, with reduction of the bandwidth by a factorcosQ/2, while the magnetic part varies as cosQ. We do notfind that canted states are favored in these consideratFinally, two special configurations which require separconsideration are the (p/2,p)1(p,p) phase, to which wereturn in Fig. 10, and flux phases.

Flux phases26 are an important feature of the model in adimension higher than 1. From Eq.~2! it is clear that thehopping term also contains a phase factor, and that fortain spin textures this phase may differ depending on

FIG. 3. Energies of selected spin configurations forn51/2 ~a!,n51/3 ~b!, andn51/4 ~c! at all values ofK/t. Note in~a! the clearsuccession of the ground state with increasingK/t from flux phaseto (p/2,p) to (p/2,p)1(p,p). Note in~b! the competition of sev-eral phases aroundK/t50.1, and in~c! the dominance of the phas(p/2,p/2) at intermediateK/t.

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path through the lattice taken between two points. The splest flux phase is that appearing at half-filling over a brorange of intermediateK/t, as discussed in Ref. 24, anshown in the snapshot in Fig. 9~c! below. The term ‘‘fluxphase’’ is used here to refer to any spin configuration wthis nontrivial topological property, which can be quantifieby a nonzero spin current.24 In principle, a variety of fluxphases may exist, but we have not yet been able to findothers which are ground states at any filling. At the analytilevel, the semimetallic density of states26 of the dispersion

ek56Acoskx21cosky

2 ~6!

of the simplest flux phase, which is zero precisely at hfilling, accounts for its particularly low energy atn51/2. Wewill characterize this phase in detail in Sec. V.

While these classical, zero-temperature pictures turnto be rather valuable, and also not quantitatively unreasable, for understanding the 2D pictures to follow, they alimited by the imagination of the authors as further possibties may not be excluded. We have obtained many ofphases proposed in Fig. 3 in MC simulations, and the folloing Figs. 4–6 illustrate some representative results.

In Fig. 4 is shownS(k), histogram and snapshot information for a phase at fillingn51/2 and for the ratioK/t50.22. We see a single peak inS(k) @Fig. 4~a!# only at(p/2,p), indicating an island phase of FM pairs~the ‘‘is-lands’’! arranged in an AF pattern. The histogram@Fig. 4~b!#shows essentially only angles of 0 andp, ruling out a pos-sible interpretation as ap/2 spiral in one direction; the ratioof angles 0 to anglesp is approximately 1:3 as expectedFinally, the instantaneous spin configuration in Fig. 4~c! il-lustrates that the simulation has in fact converged quite wto the expected phase. Comparison with Fig. 3~a! indicatesthat for the 2D case, the value ofK/t for a robust (p/2,p)phase is that expected from the infinite system atT50.

Figure 5 illustrates the same quantities for fillingn51/3andK/t50.25. For this relatively large parameter ratio, tdominant (2p/3,p) phase inS(k) @Fig. 5~a!# consists of AFchains with spin configuration↑↑↓↑↑↓↑↑↓•••.3 This is oneof the primary types of island phase which we will mentioagain in Secs. IV and V. Both parts of Fig. 5 show in adtion that this phase is not pure in the small-system MC simlation, with spin misalignments across the cluster manifesresidual components inS(k). As in Fig. 4~b!, the histogram~omitted! shows an absence of intermediate angles fromkind of spiral phase.

Figure 6 characterizes the phase arising forn51/4 atK/t50.20. From Fig. 3~c! we expect the phase (3p/4,p) asground state, and indeed this is the dominant componenS(k) @Fig. 6~a!#. The rather stronger admixture of other components arises because the chosen value ofK/t is close to aphase crossover, and so other possible 838 phases are noentirely absent. These are not reflected in the histogram~notshown! because all the pure phases present have angleonly 0 or p, but the snapshot@Fig. 6~b!# does show a smalamount of misalignment between the predominantly Aoriented spins. We note that the expected pure configura~see below Fig. 3! remains rather hard to observe in Fi

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6~b!, and ascribe this to the mixing problem, and to thefects of fluctuations on the small-cluster MC calculatioThis example illustrates both the need for careful considation of finite-size effects, and the fact that for all commesurate fillings there exist regions ofK/t ~close to the linecrossings in Fig. 3! where the MC results show strong mixtures of different phases. We note in passing that forfillings we find pure FM phases at small but finiteK/t ratios,in accord with zero-temperature, infinite-system expectatibased on Fig. 3. These straightforward cases are not shhere. At large values ofK/t, small-cluster calculations arunable to access the double spiral phase, and show insthe AF. We defer a more detailed characterization ofmost interesting phases in these figures, namely thephase atn51/2, the (p/3,p) phase atn51/3, and the(p/2,p/2) phase atn51/4, until Sec. V, after addressing thquestion of phase separation.

The results of Figs. 4 –6 were obtained for small systewhere finite-size effects are of paramount importance.

FIG. 4. MC phase forn51/2 at K/t50.22, calculated for an838 system.~a! Structure factor.~b! Angle histogram.~c! Configu-ration snapshot.

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FIG. 5. MC phase forn51/3 at K/t50.25, calculated for a12312 system.~a! Structure factor.~b! Configuration snapshot.

FIG. 6. MC phase forn51/4 at K/t50.20, calculated for an838 system.~a! Structure factor.~b! Configuration snapshot.

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fillings n51/2 and n51/4 we may compare 434 with838 MC results, and forn51/3 636 with 12312. Thesecomparisons give already a good indication of where,example, certain of the many possible phases are anolously favored by the location of the chemical potential retive to a gap between sets of degenerate states. Evenvaluable information is provided by comparison with tinfinite-system results: these may be augmented by perfoing the same calculation, placing spins in a fixed configution and deducing the magnetic and kinetic energies, forsystem sizes 434 to 12312 of the simulations~and furtherfor 16316). An effective calibration of the MC results ithen possible, by which is meant a renormalizationaccount for effects arising only from system sizwhich is particularly important in discussing phase trantions ~Sec. V!.

On these finite systems we are unable to observe ptransitions, which are replaced by crossovers occurringfinite range ofK/t. As we will show in Sec. V, however, acertain amount of care is required in interpreting two-pefeatures inS(k), because some robust, single phases arisat particular values of filling andK/t do indeed have morethan one characteristic wave vector in small systems.other feature requiring particular attention is the possibiof large-unit-cell phases, which cannot be accessed inMC simulations. An example already mentioned is tdouble spiral, which is expected from Fig. 3 to be the mfavorable phase on approaching the AF limit, into whithis phase in fact passes continuously. However, at inmediate to large values ofK/t we must also considercompeting, large-unit-cell~large-m) phase of the type„(m2k)p/m,p…, k!m, with only 0 andp angles betweenthe spins, in which the kinetic energy gain comes from spshared between rare FM pairs in an otherwise AF structThese phases are compared in the next section.

To conclude this section, we find that island-like phasare quite ubiquitous at all intermediate values ofK/t ~Fig. 3!.The FM islands may be restricted in one direction, givirise to stripelike features, or in both to give true islandepending on the filling. These states are also accompaby flux phases, of nontrivial spin texture, in certain paraeter regimes. These novel, homogeneous phases arise oa result of the competition between the first and last termEq. ~1!, without recourse to additional physics~a discussionof which is deferred to a later section!. However, we haveworked in a canonical ensemble and considered only theergy of the emerging phases at zero or the lowest temptures. We now turn to the question of phase separation withe model.

IV. PHASE SEPARATION

In the previous section we have considered a canonensemble, meaning fixed particle number, and deducedground states on the basis of minimal internal energy~or freeenergy at very low temperature!. To ensure the global stability of these phases we must consider the possibility of thseparation into regions of distinct and different filling. Thpropensity has been shown in the same model applie

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1D,20 by working in a grand canonical ensemble and obseing discontinuities in filling on varying the chemical potetial. Here we choose to characterize phase separation fthe energy in the canonical ensemble, by observing thevature of this quantity as a function of filling. In Fig.is shown the energy for fillingsn between 0 and 1/2, alow, intermediate, and higher values ofK/t. We note thatthe energy is a symmetrical function for 1/2<n<1 byelectron-hole transformation, and do not comment furtherthis region. In these figures are included data from 12312and 16316 systems, and infinite-system values for the fland double spiral phases.

FIG. 7. Energy as a function of filling at fixedK/t50.04 ~a!,K/t50.12 ~b!, andK/t50.24 ~c! for a variety of phases. In~a!, thetangent to the curve indicates the regime of phase separation bMaxwell construction. Solid lines in~b! are Maxwell constructions.Solid line in ~c! is a guide to the eye.

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In Fig. 7~a! we see a convex~up! region at low filling, theimplication of which is a preference for phase separation itwo regions, one of zero hole content and the other whfilling n is given by a Maxwell construction using the tangeto the concave part of the curve. The empty region wohave AF spin configuration, while for this low value ofK/tthe partially filled region would be FM. This result confirmthat phase separation is an important property of the moand agrees qualitatively with Ref. 20. In the absence of Clomb interaction terms, on which we comment further in SV, a complete separation into just two domains is expecin the presence of Coulomb interactions, the separashould proceed to a characteristic length scale determinetheir strength.3,22,27

For intermediateK/t @Fig. 7~b!# the situation is morecomplex. The convex regime extends over a much broarange of filling, but the ‘‘curve’’ is much less smooth, asresult of the particularly favorable island phases which cbe established at the commensurate fillings. In fact, Maxwconstructions applied to Fig. 7~b! yield for this value ofK/ta separation only into phasesn50 and 1/4, or inton51/4and n close to 1/2. Inspection of Fig. 3 shows that forK/t

FIG. 8. Phase diagram of augmented FKLM for the full rangefilling n and ratioK/t. PS denotes phase separation, the thick, vtical lines the island phases, and the shaded region the regimlarge-unit-cell phases.

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50.12 the phases at these two fillings are particularly robwhereas atn51/3 a crossing between two phases occurs;contrast, ann51/3 phase would be expected as an end poof such separation forK/t50.15, and indeed emerges~Fig.8, below!. This result highlights the dominant role of thcommensurately filled phases, and suggests both ‘‘hicontrast’’ and ‘‘low-contrast’’ phase separation. By thismeant in the former case the abrupt split into zero- and pfilled regions, and in the latter a finer phase separationcertainK/t where incommensurate fillings 1/4,n,1/2 mayundergo separation into regions with closely neighborimore commensurate fillings. These statements are madetematic in the summary phase diagram presented as Fig

At large K/t @Fig. 7~c!# the picture changes again. Hethe finite-system points for commensurate phases showintriguing feature of lying on a straight line connecting zerand 1/2-filling. These are the„(m2k)p/m,p… phases intro-duced above, for those values ofm small enough for the unitcell to fit within the system studied. Simple considerationfixed spin configurations suggests that, in principle, phaof arbitrarily large unit-cell size are possible, and their engies will fall on the same line. From above, the naturethese phases is an AF configuration of spin chains witkup-spin andk down-spin pairs contained in an otherwise Asystem with unit-cell size 2m. In a fully classical systemthere would be no phase separation with filling in the thmodynamic limit at largeK/t, but instead a continuous evolution of the unit-cell dimension to accommodate the addcharges. In fact the values ofk andm are fixed rather simplyby the filling n, because the phases of this type appearingthe ground state are„(12n)p,p…, and their energy is givenfrom the number of FM pairs and AF bonds as

E522K1n~K2t ! ~7!

per site. For the commensurate fillingsn51/m51/2, 1/3,and 1/4, we recover the island phases of Fig. 3. These phappear to have been overlooked in Ref. 20, althoughauthors were little concerned with the high-K regime.

Our conclusions are summarized in the global phasegram of Fig. 8. The properties of the minimal form of thFKLM @Eq. ~1!# fall broadly into four regions, determinelargely by the ratioK/t of the super- and double-exchangenergy scales. For the lowest values ofK/t, the system separates into AF and FM phases. For small to intermediatetios, 0.08,K/t,0.2, there is large-scale phase separatinto only the island phases appearing at the commensufillings n51/4, 1/3, 3/8, and 1/2. An exception here is tflux phase, which occupies a finite doping region close ton51/2. We note in passing that within our classical formution, only the FM and flux phases offer the possibilityhopping of conduction electrons throughout the system; othese phases would have metallic properties, and all otwill be insulating.

For intermediate ratios 0.2,K/t,0.28 we find the large-unit-cell phases discussed above. The hierarchy of possstates exists across the full doping range only when no cpeting phase falls below the straight-line energy funct@Fig. 7~c!, Eq. ~7!# for any filling, and it is this conditionwhich sets the limits inK/t of the shaded region in Fig. 8We have marked~vertical dashed lines! the small-m phaseswhich are compatible with the finite clusters considered,

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stress again that from the present calculations we expefind all phases of the form„(12n)p,p… for the infinite sys-tem. All states in the shaded region are a form of two-sFM island phase, which would show charge-ordering pein N(k) ~Sec. V!, while the small-m members at the commensurate fillings provide examples which may be studon small clusters~Figs. 4–6!. At intermediate to large valueof K/t, the large-unit-cell phases are replaced by a wregion of ‘‘high-contrast’’ phase separation due to the etraordinary stability of the (p/2,p)1(p,p) phase atn51/2. We have found only this phase, which is considein more detail in the following section, and the AF phawith zero filling, to be stable in this regime ofK/t, but stressthat we cannot fully exclude the possibility of simila„(m2k)p/m,p…1(p,p) phases at other commensurate fiings. A search for these is limited by the available clussize, and remains a topic for future investigation. Finally,large values ofK/t we recover the conventional, spiraordered DS phase, which passes smoothly to an AF pha

V. ISLAND PHASES

With the results of the previous section concerning phstability and separation, we may now turn in more detailthe regime of interest for island phases. This is largely liited to the commensurate fillingsn51/2, 1/3, and 1/4, and tothe parameter range 0.1,K/t,0.3, which ~Fig. 8! encom-passes both the isolated phases which are PS end pointsthe large-unit-cell phases. Forn51/2, this region is domi-nated first by the flux phase, shown in Fig. 9. In Fig. 9~a!, wesee the double-peak structure ofS(k) with equal weight in(0,p) and (p,0) components which is the hallmark24 of thisspin configuration. We stress that the real-space spin stture @Fig. 9~c!# of this uniform phase contains both compnents simultaneously and equally, and there is no senswhich these arise as a superposition of two degenerate sor domains. In the MC simulation the peaks inS(k) can beseen to grow together towards the value of 0.5 in the pstate. Figure 9~b! provides a rare example of a phase whethe angles between neighboring spins are distributedaround the FM and AF configurations, but aroundp/2; ourdistribution is narrower than that in Ref. 24 because oflarger lattice size employed. This spin configuration givrise to a uniform charge distribution with no inhomogeneoordering.

By contrast, for the same filling at largerK/t, it is pos-sible to find inhomogeneous charge structures. The (p/2,p)phase of Fig. 4 exists as an end point both of phase seption and of the large-unit-cell series~Fig. 8!. In this structure,electrons are delocalized across every second bondequivalently every FM bond in thep/2 direction, and aremuch more weakly present on the alternate AF bonds. Tsimple picture implies a stripelike charge order with wavector (p,0), and the phase would give peaks in x-ray dfraction or electron microscopy experiments, which measthe charge distributionn(r ). However, because the chargdensityni is the same on all sites, there is no structure inquantityn(k), which is defined in Eq.~5! and readily calcu-lated on a finite cluster. This situation arises only for perio

02442

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icities p/2 in the spin structure factor; for all higherm val-ues, n(k) and N(k) computed from the site charges aindeed suitable indicators of charge order. We note briehere that by translational invariance one may in fact expto find a linear superposition of equivalent island phaswith a uniform mean value ofni , and a charge order discernible only inN(k). In the classical MC simulations wehave shown results only for one such phase, which is serated by thermal barriers from its degenerate counterpar

In Fig. 10 we show a further stable configuration, whiwe call the (p/2,p)1(p,p) phase. As with the flux phas~Fig. 9! the two peaks in the structure factor shown in F10~a! do not indicate a mixture of phases. While the hisgram information@Fig. 10~b!# can be used only to rule ouintermediate angles, it is the instantaneous MC spin confiration @Fig. 10~c!# which reveals the true nature of this homogeneous phase. Once again one expects a 1D charge

FIG. 9. MC results forn51/2 atK/t50.12, characterizing theflux phase on an 838 lattice.~a! Structure factor.~b! Angle histo-gram.~c! Configuration snapshot.

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for the same reasons as above. It is this phase, whose enfalls below the function given in Eq.~7! for n51/2 andK/t.0.28, which breaks the large-unit-cell sequence, anresponsible for the wide region of high-contrast PS inphase diagram of Fig. 8.

We dwell only briefly on the case of 3/8 filling. The results from the previous section show a (3p/8,p) phase to bea stable end point in the PS regime, while the large-unit-region contains a (5p/8,p) member. The properties of thesconfigurations are readily deduced by comparison withother examples presented, and both have charge-ordewave vectors of (p/4,0). Certain anomalies have been oserved in experiment for fillingn53/8, but these appear tbe restricted to 3D systems.

Turning to n51/3, the most robust island phase in tintermediate parameter range is (p/3,p), illustrated sche-matically in Fig. 1~a!, and forK/t50.15 in Fig. 11. At thisvalue ofK/t, Fig. 11~a! shows a rather strong (p/3,p) peak,while the histogram~omitted! suggests a 1:2 ratio betwee

FIG. 10. MC results forn51/2 atK/t50.32, characterizing the(p/2,p)1(p,p) phase on an 838 lattice.~a! Structure factor.~b!Angle histogram.~c! Configuration snapshot.

02442

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FM and AF bond angles despite the weak presence o(p/3,2p/3) component. Figure 11~b! shows the actual spinstructure, which gives rise to a charge order at the wvector (2p/3,0), due to the higher population of every thisite in thep/3 direction. This ordering is present in the sicharge distribution functionni , which is shown in Fig. 12.From Fig. 12~a! it is clear that the charge contrast betwethe center and edge sites of each island approaches thesical ratio of 2:1.3 Very similar results are obtained for th‘‘ „(12n)p,p…’’ phase (2p/3,p) as K/t is raised beyond0.2, as already shown in Fig. 5. In this state the charordering wave vector remains (2p/3,0). We have not beenable to find a novel flux phase for 1/3 filling which might ba ground state anywhere in the intermediateK/t regime.

Finally, for n51/4 the energy diagram@Fig. 3~c!# in theregion of small to intermediateK/t is dominated by a singleand very robust, island phase. The extraordinary stabilitythe (p/2,p/2) phase@Fig. 1~b!# at this filling is clear to seeby diagonalizing the 4-site square cluster with hoppingt.This exercise yields energy levels of22t,0,0,2t, the loca-tion of the gaps demonstrating immediately why the phasefavors 1/4 filling, but is so unfavorable atn51/2. Figure 13requires little commentary, and we note only that the AFFM angle ratio here is 1:1. As in Fig. 10, the chargequivalence of all sites results in a homogeneousn(k) @Eq.~5!#, but the delocalization of charge within the 232 squareswould give a peak at (p,p) in experiments measuringn(r ).

Returning to the question of phase transitions, these m

FIG. 11. MC phase forn51/3 atK/t50.15, characterizing the(p/3,p) phase on a 12312 system.~a! Structure factor.~b! Con-figuration snapshot.

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be considered as a function ofK/t or as a function of filling.In the former case the results are essentially those of FigThe only robust phases preceeding those in Figs. 9, 11,13 are FM phases, and at higherK/t a short cascade ofurther states leads to the AF configuration. As describedSec. III, the phases arising from MC simulations requirerenormalization of their final energies to account for systsize, and when this is performed the crossovers are fconsistent with the infinite-system results. In the experimtally more relevant case of fixedK/t and variable filling, theresults of Sec. IV imply that, for all but the smallest valuof K/t, ‘‘transitions’’ take the form of a differential occupation of undoped and commensurately filled states, withexception of the regime 0.2,K/t,0.28 where they are replaced by a continuous evolution in the period of a largunit-cell phase.

Returning to the experiments presented in the introdtion, our results justify certain, rather broad conclusioManganite systems which are structurally layered, or hav2D electronic structure as a result of orbital ordering incubic system, may indeed be susceptible to the island-pphenomena, with resultant charge and spin order, discuhere. The effect of interlayer double-exchange and supechange terms is rather involved: while weak interactionsinvoked to discuss the stability of 2D phases in the true,

FIG. 12. Charge distribution functionni for n51/3 at K/t50.15, illustrating charge order of (p/3,p) phase on a 12312 sys-tem. ~a! Site charge densities: site numbers 1–12 label thecolumn from bottom to top@see ~b! and Fig. 11~b!#, 13–24 thesecond column from bottom to top, and so on.~b! Charge contourplot: high densities in white, low in gray.

02442

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lattice, strong interactions give rise to a new phase diagfor the 3D system,25 containing a rich variety of states whicincludes generalized island, flux and skyrmion configutions. The fundamental ingredient for this is only the copetition betweenK andt intrinsic to all materials in the classHowever, we have emphasized throughout the crude naof the model we consider, and close with a brief discussof the possible extensions which may be required to repduce more closely the physics of real materials.

One of the fundamental features of manganite systemthe doubly degenerate nature of theeg orbital. This has beenincluded by a number of authors, and has been argued28 to beessential in accounting for the CE-type~planar in 3D! chargeorder observed in La12xSrxMnO3.7 A further important in-gredient in manganite systems is Jahn-Teller distortion oflocal structural environment of each Mn ion,29 which mayact to lift the eg orbital degeneracy, and also to promocharge order. Both terms have been included in a classMC study of the type performed here,21 albeit on very smallsystems. Island phases, in the orbital or spin degrees of fdom, were not among the already very rich variety of phaconsidered. When twoeg orbitals are considered, on-sitCoulomb interactions were found30 to lead to the formationof an upper Hubbard band, and to cause significant speweight shifts and broadening. As mentioned in Sec. IV, aother term in many models of strongly correlated electron

st FIG. 13. MC phase forn51/4 atK/t50.12, characterizing the(p/2,p/2) phase on an 838 lattice. ~a! Structure factor.~b! Con-figuration snapshot.

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a possible Coulomb repulsion between nearest-neighsites, conventionally denoted asV. This contribution acts tosuppress phase separation, and to promote a charge ordwhenV competes with the hopping energy scalet, as notedin the 1D system.3 In higher dimensions, sufficiently stronV may lead to anisotropic charge order if the hoppinganisotropic, and more generally for weakV one expects amoving of phase boundaries to favor homogeneous stsuch as the stripes and islands considered here. Preciselphysics was found in Ref. 22, where the terminology ‘‘islaphase’’ is applied to mean a shrinking of the size of phaseparated regimes. We stress that the island phasescharge order in our study are intrinsic to the physics ofcompeting double exchange and superexchange, and thadditionalV term is not required for their appearance.

Finally, one of the major restrictions of the current aproach is the limitation to small system sizes, which becosmaller still on addition of the further terms discussed inprevious paragraph, and then still to largely classical conerations. The method of classical MC with diagonalizationthe one-electron problem is in fact not particularly sophiscated, and we highlight here only two rather recent contritions which have the potential to reveal many more featuon systems large enough to be considered thermodyncally representative. These are the variational mean fie31

and hybrid Monte Carlo32 techniques, both introduced for thdouble-exchange problem by the same group of authwhich allow extensions in the former case to 963 systemswith appropriate approximations, and in the latter to 163 siteswith rather fewer. A last important point is the questioncorrections to the above results due to the effects of quanfluctuations. In 1D, it was found3 that the boundaries between phases were moved to significantly larger valuesK/t than predicted classically. While the methods presenherein do little to allow an assessment of fluctuation effethese should be significantly smaller in 2D, both direcbecause of the higher dimensionality, and because theresults were obtained with a localized (t2g) spin S51/2,whereas the classical limit may be no less representativthe physical situation (S53/2). Thus our phase diagrams ca

.

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be expected to be qualitatively quite accurate. One mayask if quantum fluctuations would act to destroy the cohence of the large-unit-cell phases: because these phasenot spiral-ordered, and already possess the AF or FM lospin alignment favored by fluctuations, they may be assumto be robust in this respect.

VI. SUMMARY

In conclusion, we have considered the possibility of ‘‘iland’’ phases and associated charge order in 2D systeusing as a model the augmented FKLM with strong Hucoupling. Indeed we find that stripelike and island phasesstable at intermediate values ofK/t for each of the commensurate fillingsn51/2, 1/3, and 1/4. This result includes stbility against global phase separation, even in the absencadditional Coulomb terms. Spiral magnetic order appenear the antiferromagnetic regimes at low filling or at larK/t. A variety of ‘‘flux’’ phases is possible, because thelectron phase factor is nontrivial in all dimensionsd.1, butwe find only one to be a stable ground state and this an51/2. While the flux phase has a homogeneous chargetribution, the majority of the island phases show a chamodulation. Thus even the simple form~1! of the FKLMreproduces some of the most important experimental featof manganite charge and spin order. The critical valuesK/t for transitions between ordered phases, and betweendered and separated phases, may be identified rather arately from classical considerations augmenting smsystem studies.

ACKNOWLEDGMENTS

We are grateful to A. Aligia, C. Balseiro, C. Batista, DGarcia, K. Held, and D. Poilblanc for helpful discussionThis work was supported by the Consejo Nacional de Invtigaciones Cientificas y Tecnicas~CONICET! of Argentina,and by the Deutsche Forschungsgemeinschaft through GNo. SFB 484~B.N.!. We acknowledge also the supportprogram ECOS-SETCIP A97EO5 for bilateral cooperatibetween France and Argentina.

.

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tate

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Island phases and charge order in two-dimensional manganites fileIsland phases and charge order in two-dimensional manganites H. Aliaga,1 B. Normand,2 K. Hallberg,1 M. Avignon,3 and

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