Disorder-induced orbital ordering in doped manganites

Sanjeev Kumar1,2 and Arno P. Kampf3

1Faculty of Science and Technology, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands2Instituut-Lorentz for Theoretical Physics, Leiden University, P.O. Box 9506, 2300 RA Leiden, The Netherlands

3Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics, University of Augsburg,D-86135 Augsburg, Germany

�Received 29 January 2008; revised manuscript received 27 March 2008; published 25 April 2008�

We study the effect of quenched disorder on the ordering of orbital and magnetic degrees of freedom in atwo-dimensional, two-band double-exchange model for eg electrons coupled to Jahn–Teller distortions. Byusing a real-space Monte Carlo method, we find that disorder can induce a short-range ordering of the orbitaldegrees of freedom near 30% hole doping. The most striking consequence of this short-range ordering is astrong increase in the low-temperature resistivity. The real-space approach allows us to analyze the spatialpatterns of the charge, orbital, and magnetic degrees of freedom and the correlations among them. Themagnetism is inhomogeneous on the nanoscale in the short-range orbitally ordered state.

DOI: 10.1103/PhysRevB.77.134442 PACS number�s�: 71.10.�w, 75.47.Lx, 81.16.Rf

I. INTRODUCTION

Hole-doped perovskite manganites RE1−xAExMnO3 �RE=rare earth, AE=alkaline earth� have attracted great atten-tion from the condensed matter community over the pastdecade.1 While the initial surge of research activities on thesematerials was triggered by the discovery of the colossal mag-netoresistance effect, a rich variety of phases and phase tran-sitions was subsequently uncovered.2,3 It is now widely ac-cepted that the interplay among charge, spin, orbital, andlattice degrees of freedom is the underlying cause of thecomplexity and richness of the physical phenomena observedin manganites. Recent efforts from both experiment andtheory have highlighted the significance of quenched disor-der in these materials.4–7 Therefore, analyzing the effects ofdisorder in manganites has become an active area ofresearch.8–11

Disorder is generally viewed as an agent for suppressingthe ordering tendencies of the microscopic degrees of free-dom. Experiments on the half-doped �x=0.5� manganitesshow that quenched disorder indeed spoils the long-rangeordering of the charge, orbital, and spin variables leading, insome cases, to a short-range ordering of these microscopicdegrees of freedom.5,6 The opposite effect, however, is ob-served in manganites near 30% hole doping, where an order-ing of the orbital degrees of freedom is induced by the pres-ence of quenched disorder.7

In manganites, the average rA and the variance �2 of theA-site ionic radii are known to control the single-particlebandwidth and the magnitude of quenched disorder,respectively.12 Samples with constant rA and varying �2 wereused in the experiments of Ref. 7 with a combination of La,Pr, Nd, and Sm and Ca, Sr, and Ba at the A site, whilekeeping x=0.3. An increase in the low-temperature resistiv-ity by 4 orders of magnitude was attributed to the onset oforbital ordering, which was also evidenced from the struc-tural changes analyzed via powder x-ray diffraction. Magne-tism is strongly affected with a reduction in both the Curietemperature TC and the saturation value of the magnetization.This doping regime is also believed to be magnetically inho-

mogeneous, as independently inferred from NMR and neu-tron scattering experiments.13,14

Disorder has been previously included in models for man-ganites to study its influence on the long-range orderedphases8,15,16 especially near a first-order phase boundary or inthe vicinity of phase separation.9,11,17 The idea that quencheddisorder may lead to a partial ordering of the orbital degreesof freedom in manganite models has so far remained unex-plored.

In this paper, we study a two-band double-exchangemodel with quenched disorder by using a real-space MonteCarlo method. Disorder is modeled via random on-site ener-gies selected from a given distribution. We consider two dif-ferent types of distributions, which are described in detail inthe next section. Here and below, we refer to these distribu-tions as �i� binary disorder and �ii� random scatterers. Whilethe binary disorder has no significant effect on the orbitaldegrees of freedom, random scatterers lead to orbitally or-dered regions, and a sharp increase in the low-temperatureresistivity is found, as observed in the experiment.7 The mag-netic structure is inhomogeneous in a restricted doping re-gime, as observed in the NMR and the neutron scatteringexperiments. Within clusters, staggered orbital ordering isaccompanied by ferromagnetism, thus providing an exampleof Goodenough–Kanamori rules in an inhomogeneoussystem.18,19

II. MODEL AND METHOD

We consider a two-band model for itinerant eg electronson a square lattice. The electrons are coupled to Jahn–Teller�JT� lattice distortions, t2g-derived S=3 /2 core spins, andquenched disorder, as described by the Hamiltonian,

H = ��ij��

��

t��ij ci��

† cj�� + �i

�ini + Js��ij�

Si · S j − JH�i

Si · �i

− ��i

Qi · �i +K

2 �i

Qi2. �1�

Here, c and c† are annihilation and creation operators for eg

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electrons, �= ↑ ,↓ is the spin index, and �, � are summedover the two Mn eg orbitals dx2−y2 and d3z2−r2, which are la-beled �a� and �b� in what follows. t��

ij denote the hoppingamplitudes between eg orbitals on nearest-neighbor sites andhave the cubic perovskite specific forms taa

x = taay � t, tbb

x = tbby

� t /3, tabx = tba

x �−t /�3, and taby = tba

y � t /�3, where x and ymark the spatial directions.20 The eg-electron spin is locallycoupled to the t2g spin Si via the Hund’s rule coupling JH.The eg-electron spin is given by �i

�=����� ci��

† ����� ci���,

where �� are the Pauli matrices. Js is the strength of thesuperexchange coupling between neighboring t2g spins. � de-notes the strength of the JT coupling between the distortionQi= �Qix ,Qiz� and the orbital pseudospin i

�

=����ci��

† ���� ci��. K is a measure of the lattice stiffness, and

we set t=1=K as our reference energy scale.The following two forms of on-site disorder modeling are

used: �i� binary disorder, �i takes equally probable values�; �ii� random scatterers, a fraction x of the sites are takento have �i=D, while for the other sites, �i=0. Although thefirst choice of disorder is the simplest from the model pointof view, the second appears more realistic. In real materials,a fraction x of the rare-earth ions is replaced by alkaline-earth ions at random locations. Therefore, it is likely that thedisorder arising as a consequence of this substitution is con-nected to the amount of doping. This situation is modeled byplacing repulsive potentials on a fraction x of the sites, whichare randomly selected. A typical measure of the strength of adisorder distribution is its variance. For the binary distribu-tion, the variance is �, while for the finite density x of scat-terers with potential strength D, it is D�x�1−x�. These twomodels for disorder were previously employed in a study ofhalf-doped manganites.11 The JT distortions and the t2g de-rived core spins are treated as classical variables, and we set�Si�=1. Guided by earlier estimates for the JT couplingstrength in manganites, we fix �=1.5 �Ref. 21� and explorethe variation in the parameters �, D, and Js.

We further adopt the simplifying limit JH� t, which isjustified and frequently used in the context ofmanganites.9,20,22 In this limit, the electronic spin at site i istied to the orientation of the core spin Si. Transforming thefermionic operators to this local spin reference frame leads tothe following effectively “spinless” model for the eg elec-trons:

H = ��ij�

��

t̃��ij ci�

† cj� + �i

�ini + Js��ij�

Si · S j − ��i

Qi · �i

+K

2 �i

Qi2. �2�

The new hopping amplitudes t̃ have an additional depen-dence on the core-spin configurations and are given by

t̃��

t��

= cos�i

2cos

� j

2+ sin

�i

2sin

� j

2e−i� i− j�. �3�

Here, �i and i denote polar and azimuthal angles for thespin Si. From now on, the operator ci� �ci�

† � is associated

with annihilating �creating� an electron at site i in the orbital� with spin parallel to Si.

The model given by Eq. �2� is bilinear in the electronicoperators and does not encounter the problem of an exponen-tially growing Hilbert space since all many-particle statescan be constructed from Slater determinants of the single-particle states. The difficulty, however, arises from the largephase space in the classical variables Q and S. Exact diago-nalization �ED� based Monte Carlo is a numerically exactmethod to treat such problems, and has been extensivelyused in the past.9,20,22 The classical variables are sampled bythe Metropolis algorithm, which requires the exact eigenen-ergy spectrum. Therefore, iterative ED of the Hamiltonian isneeded, which leads to N4 scaling of the required CPU time,where N is the number of lattice sites. The N4 scaling makesthis method very restrictive in terms of the achievable latticesizes, with the typical size in previous studies being 100sites. Since a study of larger lattices is essential for analyzingthe nature of inhomogeneities in manganite models, severalattempts have been made to devise accurate approximateschemes.23–25 In the present study, we employ the travelingcluster approximation,25 which indeed has been very suc-cessful in analyzing similar models in the recent past.10,11,26

III. RESULTS AND DISCUSSION

We begin with the results for bulk quantities describingthe ordering of the magnetic and the lattice degrees of free-dom. We focus on the 30% hole-doped system �x=0.3� for aclose correspondence to the experiments in Ref. 7. Figure1�a� shows the effect of binary disorder on the temperaturedependence of the magnetization m defined via m2

= ��N−1�Si�2�av. Here and below, �¯�av denotes the average

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0

0.5

1

m∆=0.0aa

∆=0.4aa

∆=0.8aa

∆=1.0aa

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aaa

aaaaaaaaaaa

a aaaaaaa

aaaaaa

aaaaaaa

aaaaaaaaa

0

0.002

0.004

DQ(π,π)

aaaaa

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aaaaaaaaa

aaaaa

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aaaaaaa

aa

aaaaaa

aaaaaaaaaaaaaaa

0 0.1 0.2T

0

0.5

1

mD=0aD=1aD=2aD=3a

aa

aa

aaaaaa

aa

aaaaaaa a

aaa

aaaaaa

aa

aaaaaaaaaa aaaaa

aaaaaaa

aaaa

aaa

aaaaaa

0 0.05 0.1T

0

0.02

0.04

DQ(π,π)

(a) (b)

Js=0.05

x=0.3

λ=1.5

(c) (d)

FIG. 1. �Color online� Temperature dependence of �a� the mag-netization m and �b� the lattice structure factor DQ�q� at q= �� ,��for various values of the disorder strength �. �c� and �d� show thesame quantities as in �a� and �b�, respectively, if the on-site disorderis modeled by random scatterers of strength D. The concentration ofscatterers is equal to the hole density, x=0.3. Note the order ofmagnitude difference in the magnitudes for DQ�q0� between panels�b� and �d�. All results are at �=1.5 and Js=0.05.

SANJEEV KUMAR AND ARNO P. KAMPF PHYSICAL REVIEW B 77, 134442 �2008�

134442-2

over thermal equilibrium configurations and additionallyover realizations of quenched disorder. Results for disor-dered systems are averaged over four to six realizations ofdisorder. Clearly, the magnetism is not affected much by thepresence of weak binary disorder. This is in agreement withprevious studies, which find that the reduction in TC is pro-portional to �2 for weak disorder.27–29

Figure 1�b� shows the temperature dependence of the q=q0��� ,�� component of the lattice structure factor,DQ�q�=N−2�ij�Qi ·Q j�ave

−iq·�ri−rj�. DQ�q0� is a measure forthe staggered distortion order in the system. The lattice or-dering leads to orbital ordering via the JT coupling. An in-crease with � in the low-temperature value of DQ�q0� sug-gests the appearance of orbital order. However, this effect istoo weak to explain the experimental resistivity data.7 More-over, the increase at low T in DQ�q0� is not monotonic,which becomes clear by comparing the results for �=0.4,0.8, and 1.0 in Fig. 1�b�.

Now, we explore the results for the disorder arising fromrandom scatterers of strength D. Since the disorder originatesfrom the replacement of RE3+ by AE2+ ions, the density ofrandom scatterers is kept equal to the doping concentration x.Since the two models for binary disorder and random scat-terers, respectively, are identical at x=0.5, the two modelsare compared for x�0.5. m�T�, shown in Fig. 1�c�, is af-fected strongly upon increasing D, with a decrease in thesaturation value of the magnetization pointing toward a mag-netically inhomogeneous ground state. More importantly, amonotonic increase with D is observed in the low-temperature values of DQ�q0� see Fig. 1�d��. The rise inDQ�q0� clearly indicates the emergence of orbital ordering inthe system, with the area and/or strength of the ordered re-gions increasing with increasing D.

It is expected that these orbital ordering correlations arereflected in the transport properties. We therefore computethe dc resistivity � approximated by the inverse of low-frequency optical conductivity, which is calculated by usingexact eigenstates and eigenenergies in the Kubo–Greenwoodformula.30 Figures 2�a� and 2�b� show � as a function oftemperature for the two disorder models described above.The low-temperature resistivity increases upon increasingthe binary disorder strength � see Fig. 1�a��. For small val-ues of �, the resistivity curves appear parallel to each otherbelow T0.1. The resistivity therefore follows Mathiessen’srule, i.e., ��T� for the disordered system is obtained from��T� for the clean system by simply adding a constant con-tribution arising from the scattering off the disorder poten-tial. d� /dT remains positive at low temperature, indicating ametallic behavior. This oversimplified description, however,does not take into account the disorder-induced changes inthe orbital ordering correlations and the related changes inthe density of states �discussed below�.

Random scatterers lead to a drastically different behavior.The low-temperature rise in ��T� covers several orders ofmagnitude see Fig. 2�b��. The negative sign of d� /dT forD�1 signals an insulating behavior. Upon increasing thedisorder strength D, we therefore observe a metal to insulatortransition. For x=0.3, both disorder models have the samevariance if �0.46D holds. Comparing, therefore, the re-sults for D=2 and �=1, we have to conclude that the drastic

rise in the resistivity for random scatterers cannot be attrib-uted to the strength of the disorder potential. In fact, a largeincrease in the low-T resistivity was one of the experimentalindications for the onset of disorder-induced orbitalordering.7

Figures 2�c� and 2�d� highlight the difference between thedensities of states �DOSs� for the two choices of disordermodeling. The DOS is defined as N���= �N−1�i���−Ei��av,where Ei denotes the eigenvalues of the Hamiltonian. Here,we approximate the � function by a Lorentzian with width�=0.04,

��� − Ei� ��/�

�2 + �� − Ei�2�. �4�

The DOS for the clean system has a pseudogap structure nearthe chemical potential. For binary disorder, the pseudogapslowly fills up with increasing �. In contrast, it deepens uponadding random scattering centers and even leads to a clean

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-2 -1 0 1 2

CQ

0

1

2

P(CQ)

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

-2 -1 0 1 2

CQ

0

1

2

Js=0.05

λ=1.5

x=0.3

(a) (b)

D=0

D=1

D=2

∆=0

∆=0.6

∆=1.0

T=0.01

FIG. 3. �Color online� Low-temperature distribution functionsgenerated from the Monte Carlo data for the nearest-neighbor cor-relations CQ of the lattice distortions for �a� binary disorder and �b�for random scatterers. The curves for different � are off-set alongthe y axis for clarity. CQ is positive �negative� for ferro-�antiferro�distortive patterns of the lattice variables.

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0 0.1 0.2 0.3T

0

40

80

ρ

∆=0.0aa∆=0.4aa∆=0.6aa∆=1.0aa

aaaaa

aaaa

aaaaaaaa

aa

aaa

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aaaaa

aaa

aaaaaa

aaaaaaa

aaa

a

aa

0 0.1 0.2 0.3T

101

102

103

104

ρ

D=0aD=1aD=2

aD=4

aa

-4 0 4

E - Ef

0

0.1

0.2

0.3

N

∆=0.0

∆=0.6

∆=1.0

-4 0 4

E - Ef

0

0.1

0.2

0.3

N

D=0

D=2

D=4

(a) (b)x=0.3Js=0.05

(c) (d)

T=0.01 T=0.01

FIG. 2. �Color online� Temperature dependent resistivity ��T��in units of � /�e2� for varying strength of �a� binary disorder � and�b� random scatterers D. Note the logarithmic scale in �b�. �c� and�d�� Low-temperature density of states for the two types of disorder.

DISORDER-INDUCED ORBITAL ORDERING IN DOPED… PHYSICAL REVIEW B 77, 134442 �2008�

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gap for D�3. This opposite behavior is partly responsiblefor the drastically different low-temperature resistivity dis-cussed above. The three-peak structure for large values of Din Fig. 2�d� can be understood as follows: a fraction 2x ofelectronic states split off and form a narrow impurity bandcentered at an energy D above the Fermi level of the un-doped system. The lower band now contains a fraction 2�1−x� of the states with the Fermi level located in the middle ofthe band. This leads to a situation similar to the undopedsystem, and an energy gap originating from staggered orbitalordering opens at the Fermi level.

To gain further insight into the nature of the states in thepresence of the two types of disorder, we plot the distributionfunctions for the lattice variables in Fig. 3. Panel �a� showsthe distribution of the nearest-neighbor lattice correlations,CQ�i�= �1 /4���Qi ·Qi+� for binary disorder; here, � denotes

the four nearest-neighbor sites of site i. A negative value ofCQ�i� indicates an antiferro pattern of JT distortions and,hence, a pattern of staggered orbital ordering. The distribu-tion function for CQ is defined as P�CQ�= �N−1�i�CQ−CQ�i���av; the � function is again approximated by aLorentzian with width 0.04. A peak in P�CQ� centered nearCQ=0.8 for �=0 indicates that the clean system has weakferrodistortive and hence ferro-orbital correlations. Tails go-ing down to CQ−1.4 arise in the distribution function uponincluding binary disorder.

The distribution function P�CQ� for random scattererslooks qualitatively different. We recall that the strengths ofthe two types of disorder are related via �0.46D. The low-temperature distributions P�CQ� are plotted in Fig. 3�b� forrandom scatterers. A qualitative change in the shape of thedistribution function occurs for D=2, where a second peak

0

0.5

1

1.5

2

ε(i)

0.2

0.6

1

n(i)

-1.2

-0.8

-0.4

0

0.4

CQ(i)

-1

-0.5

0

0.5

1

ε(i)

0.2

0.6

1

n(i)

-1.2

-0.8

-0.4

0

0.4

CQ(i)

FIG. 4. �Color online� Real-space patterns of the disorder po-tential ��i�, charge density n�i�,and lattice correlations CQ�i�. Toprow: binary disorder with �=1;bottom row: random scattererswith D=2. The patterns in bothcases are shown on a 24�24 lat-tice for a single disorder realiza-tion at T=0.01 and x=0.3.

x=0.05

CQ

x=0.10

-1.2

-0.8

-0.4

0

0.4

x=0.20

x=0.05

ni

x=0.10

0.4

0.6

0.8

1

x=0.20

FIG. 5. �Color online� Dopingevolution of the local charge den-sity ni and the local lattice corre-lations CQ�i� for a single realiza-tion of random scatterers withstrength D=1 on a 24�24 lattice.

SANJEEV KUMAR AND ARNO P. KAMPF PHYSICAL REVIEW B 77, 134442 �2008�

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centered around CQ−1.2 emerges. This is a direct indica-tion that a significant fraction of the system becomes orbit-ally ordered. This perfectly correlates with the strong rise inDQ�q0� at low temperatures see Fig. 1�b�� and the anoma-lous increase in the resistivity see Fig. 2�b��.

A real-space picture for the emergence of orbital orderingis presented in Fig. 4, which displays the disorder potential�i, the electronic density ni, and the lattice correlations CQ�i�.The top row for binary disorder shows that the charge den-sity closely follows the disorder potential. The local latticecorrelations are centered around CQ=0, which is also evidentfrom the peak in the distribution P�CQ� shown in Fig. 3�a�.The bottom row in Fig. 4 shows the corresponding results forthe disorder potential arising from random scatterers. Sincethe doping concentration in this case coincides with the con-centration of the scatterers, the holes are trapped at the im-purity sites. This leaves the surrounding effectively undopedand thereby induces orbital ordering. This is apparent fromthe spread of the dark-blue regions and their cross correlationwith the charge density distribution in the bottom row of Fig.4. Such a picture with orbitally ordered regions coexistingwith orbitally disordered patches perfectly describes the

double peak structure of the distribution function in Fig.3�b�.

Although we are primarily interested in the experimen-tally relevant case x=0.3, it is useful to see how the real-space patterns evolve as one moves from low to high holedensities. The undoped system is an orbitally ordered insula-tor, which turns into an orbitally disordered metal upondoping.31 We show real-space patterns at three different dop-ing concentrations in Fig. 5. The density of random scatterersis kept equal to the doping fraction x. The charge densitydistribution is largely controlled by the disorder distribution.At low doping, disconnected orbitally disordered regions areessentially tied to the trapped holes. With increasing x, theorbitally disordered regions begin to connect in one-dimensional snakelike patterns. By further increasing thedoping and the concentration of scattering centers, the orbit-ally disordered regions grow. Since the low-doping regime ofthe present model is phase separated,1,10 the phenomenon ofdisorder-induced orbital ordering occurs only for x�0.25.The upper critical value of doping concentration, beyondwhich this phenomenon does not occur, depends on thestrength of the disorder potential used. The inhomogeneousstructures shown in Fig. 5 arise from the combined effects ofdisorder and phase separation tendencies.

It is worthwhile to point to a similarity between the ef-fects of disorder in the present study and in a model analysisfor d-wave superconductors with nonmagnetic impurities. InRef. 32, it was found that the impurities nucleate antiferro-magnetism in their near vicinity. Upon increasing the impu-rity concentration, static antiferromagnetism is observed.There seems to be a perfect analogy between the two situa-tions if one interchanges antiferromagnetism by orbital or-dering; both are ordering phenomena with the staggered or-dering wave vector �� ,��. The �� ,�� ordering phenomenaare partially triggered by the charge inhomogeneities in bothcases. An additional complication in the present case arisesfrom the spin degrees of freedom in addition to the orbitalvariables and from the anisotropy of the hopping parameters.

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0 0.1 0.2T

0

0.5

1

m

Js=0.00

aa

Js=0.05aJs=0.10

aa

aa

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a

aaaaa

aa

aa

aaa

aa

aaa

a

0 0.05 0.1T

0

0.1

0.2

0.3

DQ(π,π)

(a) (b)

D=4x=0.3

FIG. 6. �Color online� Temperature dependence of the magneti-zation m and the staggered lattice structure factor DQ�� ,�� forvarying superexchange coupling strength Js. The results are for ran-dom scatterers with strength D=4.

CS -1

-0.5

0

0.5

1

Js = 0.02

CQ

Js = 0.06

-1.5

-1

-0.5

0

Js = 0.10

FIG. 7. �Color online� Real-space patterns for the lattice corre-lations CQ�i� and the analogouslydefined spin correlations CS�i� forvarying superexchange couplingJs. The patterns are shown for asingle disorder realization at T=0.01. Orbitally ordered regionstend to maintain ferromagnetism,while the orbitally disordered re-gions are more susceptible towardantiferromagnetism with increas-ing Js.

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As inferred above from the results for the temperaturedependent magnetization m�T�, the magnetic ground stateappears to be homogeneous for binary disorder but may beinhomogeneous in the case of doped scatterers see Figs. 1�a�and 1�c��. Since the magnetism is partially controlled by theantiferromagnetic superexchange coupling Js, we study theeffect of increasing Js for a fixed large disorder strength ofrandom scatterers. Figure 6�a� shows the result for m�T� andFig. 6�b� shows the result for the temperature dependence ofDQ�q0�. The saturation value of m�T� as well as the onsetscale for ferromagnetism decrease with increasing Js. Moreimportantly, DQ�q0� at low temperatures increases with in-creasing Js, indicating an enhancement in the orbital order-ing. For a homogeneous system, this would mean that orbitalordering and antiferromagnetism are both enhanced with in-creasing Js. This is a contradiction to the Goodenough–Kanamori rules, which state that an orbitally antiferro systemshould be magnetically ferro. The contradiction is resolvedby analyzing the microscopic details of this complicatedstate providing an example where the real-space structuresare essential for a comprehensive understanding.

We show in Fig. 7 the effect of the superexchange cou-pling on the real-space patterns of lattice and spin variables.The lattice correlations are shown in the top row, andthe analogously defined spin correlations, CS�i�= �1 /4���Si ·Si+�, in the bottom row. For Js=0.02, the systemcontains orbitally ordered nanoscale regions, but magneti-cally, it appears homogeneous. For Js=0.06, the area of theorbitally ordered regions is enlarged and magnetic inhomo-

geneities appear. The orbitally ordered regions remain ferro-magnetic, while the orbitally disordered regions become an-tiferromagnetic upon increasing Js. In the orbitally orderedclusters of this inhomogeneous system in the selected param-eter regime, the Goodenough–Kanamori rules are thereforefulfilled. However, upon increasing Js further to 0.1, the an-tiferromagnetic regions start to extend also into the orbitallyordered clusters. The charge density patterns �not shownhere� are insensitive to the increase in Js.

IV. CONCLUSIONS

Our analysis for a two-band double-exchange model formanganites leads us to conclude that the disorder-inducedorbital ordering in manganites near x=0.3 is properly de-scribed if the density of scattering centers tracks the holeconcentration. Within this specific model of quenched disor-der, the induced staggered orbital ordering is responsible forthe orders of magnitude increase in the low-temperature re-sistivity, as observed in the experiments in Ref. 7.

ACKNOWLEDGMENTS

S.K. acknowledges support by “NanoNed,” a nanotech-nology programme of the Dutch Ministry of Economic Af-fairs. A.P.K. gratefully acknowledges support by the Deut-sche Forschungsgemeinschaft through SFB 484. Simulationswere performed on the Beowulf Cluster at HRI, Allahabad�India�.

1 For overviews, see Nanoscale Phase Separation and ColossalMagnetoresistance, edited by E. Dagotto �Springer-Verlag, Ber-lin, 2002� and Refs. 2 and 3.

2 Colossal Magnetoresistive Oxides, edited by T. Chatterji �Klu-wer, Dordrecht, 2004�.

3 Colossal Magnetoresistive Oxides, edited by Y. Tokura �Gordonand Breach, Amsterdam, 2000�.

4 L. M. Rodriguez-Martinez and J. P. Attfield, Phys. Rev. B 58,2426 �1998�; 54, R15622 �1996�.

5 D. Akahoshi, M. Uchida, Y. Tomioka, T. Arima, Y. Matsui, andY. Tokura, Phys. Rev. Lett. 90, 177203 �2003�.

6 Y. Tomioka and Y. Tokura, Phys. Rev. B 70, 014432 �2004�.7 L. M. Rodriguez-Martinez and J. P. Attfield, Phys. Rev. B 63,

024424 �2000�.8 G. Alvarez, H. Aliaga, C. Sen, and E. Dagotto, Phys. Rev. B 73,

224426 �2006�.9 A. Moreo, M. Mayr, A. Feiguin, S. Yunoki, and E. Dagotto,

Phys. Rev. Lett. 84, 5568 �2000�.10 S. Kumar, A. P. Kampf, and P. Majumdar, Phys. Rev. Lett. 97,

176403 �2006�.11 K. Pradhan, A. Mukherjee, and P. Majumdar, Phys. Rev. Lett.

99, 147206 �2007�.12 For precise definitions of rA and �2, see, e.g., J. P. Attfield, Int. J.

Inorg. Mater. 3, 1147 �2001�.13 G. Papavassiliou, M. Pissas, M. Belesi, M. Fardis, J. Dolinsek,

C. Dimitropoulos, and J. P. Ansermet, Phys. Rev. Lett. 91,

147205 �2003�.14 M. Hennion, F. Moussa, G. Biotteau, J. Rodriguez-Carvajal, L.

Pinsard, and A. Revcolevschi, Phys. Rev. Lett. 81, 1957 �1998�.15 Y. Motome, N. Furukawa, and N. Nagaosa, Phys. Rev. Lett. 91,

167204 �2003�.16 C. Sen, G. Alvarez, and E. Dagotto, Phys. Rev. B 70, 064428

�2004�.17 S. Kumar and P. Majumdar, Phys. Rev. Lett. 92, 126602 �2004�.18 J. B. Goodenough, Magnetism and the Chemical Bond �Inter-

science, New York, 1963�; J. Kanamori, J. Phys. Chem. Solids10, 87 �1959�.

19 M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys. 70,1039 �1998�.

20 T. Hotta, S. Yunoki, M. Mayr, and E. Dagotto, Phys. Rev. B 60,R15009 �1999�.

21 E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1 �2001�;A. J. Millis, B. I. Shraiman, and R. Mueller, Phys. Rev. Lett. 77,175 �1996�; P. B. Allen and V. Perebeinos, ibid. 83, 4828�1999�.

22 J. A. Vergés, V. Martín-Mayor, and L. Brey, Phys. Rev. Lett. 88,136401 �2002�.

23 J. L. Alonso, L. A. Fernandez, F. Guinea, V. Laliena, and V.Martin-Mayor, Nucl. Phys. B 596, 587 �2001�.

24 G. Alvarez, C. Sen, N. Furukawa, Y. Motome, and E. Dagotto,Comput. Phys. Commun. 168, 32 �2005�.

25 S. Kumar and P. Majumdar, Eur. Phys. J. B 50, 571 �2006�.

SANJEEV KUMAR AND ARNO P. KAMPF PHYSICAL REVIEW B 77, 134442 �2008�

134442-6

26 S. Kumar and P. Majumdar, Phys. Rev. Lett. 96, 016602 �2006�.27 J. Salafranca and L. Brey, Phys. Rev. B 73, 214404 �2006�.28 Y. Motome and N. Furukawa, Phys. Rev. B 68, 144432 �2003�.29 G. Bouzerar and O. Cépas, Phys. Rev. B 76, 020401�R� �2007�.30 S. Kumar and P. Majumdar, Eur. Phys. J. B 46, 237 �2005�.

31 B. B. Van Aken, O. D. Jurchescu, A. Meetsma, Y. Tomioka, Y.Tokura, and T. T. M. Palstra, Phys. Rev. Lett. 90, 066403�2003�.

32 B. M. Andersen, P. J. Hirschfeld, A. P. Kampf, and M. Schmid,Phys. Rev. Lett. 99, 147002 �2007�.

DISORDER-INDUCED ORBITAL ORDERING IN DOPED… PHYSICAL REVIEW B 77, 134442 �2008�

134442-7