Disorder-induced orbital ordering in doped manganites

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by  Sanjeev Kumar

Disorder-induced orbital ordering in doped manganites
Sanjeev Kumar1,2 and Arno P. Kampf3
1 Faculty of Science and Technology, University of Twente,
P.O. Box 217, 7500 AE Enschede, The Netherlands
2 Instituut-Lorentz for Theoretical Physics, Leiden University,
P.O. Box 9506, 2300 RA Leiden, The Netherlands
3 Theoretical Physics III, Center for Electronic Correlations and Magnetism, Institute of Physics,
University of Augsburg, D-86135 Augsburg, Germany
We study the effect of quenched disorder on the ordering of orbital and magnetic degrees of
8 freedom in a two-dimensional, two-band double-exchange model for eg electrons coupled to Jahn-
0 Teller distortions. Using a real-space Monte Carlo method, we find that disorder can induce a
0 short-range ordering of the orbital degrees of freedom near 30% hole doping. The most striking
2 consequence of this short range ordering is a strong increase in the low temperature resistivity.
Thereal-space approach allows toanalyzethespatial patternsofthecharge, orbital, and magnetic
n
degrees of freedom, and the correlations among them. The magnetism is inhomogeneous on the
a
J nanoscale in the short-range orbitally ordered state.

9 PACS numbers: 71.10.-w, 75.47.Lx, 81.16.Rf
2

] I. INTRODUCTION also evidenced from the structural changes analyzed via
l
e powder x-raydiffraction. Magnetism is affected strongly
-
with a reduction in both, the Curie temperature T and
r Hole-doped perovskite manganites RE AE MnO C
t 1−x x 3 the saturation value of the magnetization. This dop-
s (RE=rare earth, AE=alkaline earth) have attracted
. ing regime is also believed to be magnetically inhomoge-
t great attention from the condensed matter community
a neous,asindependently inferredfromNMRandneutron
over the last decade [1]. While the initial surge of re-
m
scattering experiments [13, 14].
search activities on these materials was triggered by the
- Disorder has been included previously in models for
d discovery of the colossal magnetoresistance (CMR) ef-
manganites to study its influence on the long-range or-
n fect, a rich variety of phases and phase transitions was
deredphases[8,15,16]especiallynearafirst-orderphase
o subsequently uncovered [2, 3]. It is now widely accepted
c thattheinterplayamongcharge,spin,orbitalandlattice boundaryorinthevicinityofphaseseparation[9,11,17].
[ The idea that quenched disorder may lead to a partial
degrees of freedom is the underlying cause of the com-
ordering of the orbital degrees of freedom in manganite
1 plexity and richnessofthe physicalphenomena observed
models has so far remained unexplored.
v in manganites. Recentefforts fromboth experiment and
3 theory have highlighted the significance of quenched dis- In this paper, we study a two-band double-exchange
1 model with quenched disorder using a real-space Monte
order in these materials [4, 5, 6, 7]. Therefore,analyzing
5 Carlo method. Disorder is modelled via random on-site
theeffectsofdisorderinmanganiteshasbecomeanactive
4
energies selected from a given distribution. We consider
area of research[8, 9, 10, 11].
.
1 two different types of distributions, described in detail
Disorder is generally viewed as an agent for suppress-
0 in the next section. Here and below we refer to these
8 ing the ordering tendencies of the microscopic degreesof distributions as (i) binarydisorder and(ii) randomscat-
0 freedom. Experiments on the half-doped (x=0.5) man-
terers. Whilethebinarydisorderhasnosignificanteffect
: ganites show that quenched disorder indeed spoils the
v ontheorbitaldegreesoffreedom,randomscattererslead
long-rangeordering of the charge,orbital, and spin vari-
i to orbitally ordered (OO) regions, and a sharp increase
X ables leading, in some cases,to a short-rangeorderingof
inthelow-temperatureresistivityisfoundasobservedin
r these microscopic degrees of freedom [5, 6]. The oppo-
a the experiment [7]. The magnetic structure is inhomo-
site effect, however, is observed in manganites near 30%
geneous in a restricted doping regime as observed in the
hole doping, where an ordering of the orbital degrees of
NMR and the neutron scattering experiments. Within
freedom is induced by the presence of quenched disorder
clusters,staggeredorbitalorderingisaccompaniedbyfer-
[7].
romagnetism,thusprovidinganexampleofGoodenough-
In manganites, the average r and the variance σ2 of
A Kanamorirules in an inhomogeneous system [18, 19].
the A-site ionic radii are known to control the single-
particle bandwidth and the magnitude of quenched dis-
order, respectively [12]. Samples with constant r and
A
varying σ2 were used in the experiments of Ref. [7] with II. MODEL AND METHOD
a combination of La, Pr, Nd, Sm, and Ca, Sr, Ba at the
A-site, while keeping x = 0.3. An increase in the low- We consider a two-band model for itinerant e elec-
g
temperature resistivity by four orders of magnitude was trons on a square lattice. The electrons are coupled to
attributed to the onset of orbital ordering, which was Jahn-Teller (JT) lattice distortions, t -derived S = 3/2
2g
2
core spins and quenched disorder as described by the e electrons:
g
Hamiltonian:
αβ
αβ H = Xt˜αiβjc†iαcjβ +Xǫini+JsXSi·Sj
H = X tiαjβ c†iασcjβσ +Xǫi ni+JsXSi·Sj hiji i K hiji
hijiσ i Khiji −λXQi.τi+ 2 XQ2i. (2)
−JHXSi.σi−λXQi.τi+ 2 XQ2i. (1) i i
i i i The new hopping amplitudes t˜ have an additional de-
pendence on the core-spin configurations and are given
Here, c and c† are annihilation and creation operators by:
for e electrons, σ = , is the spin index and α, β are
g
summedoverthetwo↑M↓n-eg orbitalsdx2−y2 andd3z2−r2, t˜αβ =cosθi cosθj +sinθi sinθj e−i(φi−φj). (3)
whicharelabelled(a)and(b)inwhatfollows. tiαjβ denote tαβ 2 2 2 2
the hopping amplitudes between e orbitals on nearest-
g
neighbor sites and have the cubic perovskite specific Here,θi andφi denotepolarandazimuthalanglesforthe
form: txaa = tyaa ≡ t, txbb = tybb ≡ t/3, txab = txba ≡ −t/√3, spinSi. Fromnowontheoperatorciα (c†iα)isassociated
ty = ty t/√3, where x and y mark the spatial di- with annihilating (creating) an electron at site i in the
raebctionsba[2≡0]. The eg-electron spin is locally coupled to orbital α with spin parallel to Si.
the t spin S via the Hund’s rule coupling J . The e - The model given by Eq. (2) is bilinear in the elec-
2g i H g
electronspinisgivenbyσiµ =Pασσ′c†iασΓµσσ′ciασ′,where tarnoneixcpoopneernattiaolrlsyagnrdowdionegsHnioltbeerntcospuanctee,rstihnecepraollbmlemanyo-f
Γµ are the Pauli matrices. J is the strength of the
s
particle states can be constructed from Slater determi-
superexchange coupling between neighboring t spins.
2g nants of the single-particle states. The difficulty, how-
λ denotes the strength of the JT coupling between the
ever, arises from the large phase space in the classical
distortion Q = (Q ,Q ) and the orbital pseudospin
i ix iz variables Q and S. Exact diagonalization based Monte
τµ = αβc† Γµ c . K is a measure of the lattice
i Pσ iασ αβ iβσ Carlo (ED-MC) is a numerically exact method to treat
stiffness, and we set t = 1 = K as our reference energy
suchproblems,andhasbeenusedextensivelyinthepast
scale.
[9, 20, 22]. The classical variables are sampled by the
The following two forms of on-site disorder modelling Metropolis algorithm, which requires the exact eigenen-
are used: (i) binary disorder: ǫi takes equally probable ergy spectrum. Therefore iterative ED of the Hamilto-
values ∆, (ii) random scatterers: a fraction x of the nianis needed, which leads to N4 scalingofthe required
±
sites are taken to have ǫi = D, while for the other sites cpu time, N is the number of lattice sites. The N4 scal-
ǫi = 0. Although the first choice of disorder is the sim- ing makes this method very restrictive in terms of the
plest from the model point of view, the second appears achievable lattice sizes, with the typical size in previous
more realistic. In real materials a fraction x of the rare- studies being 100sites. Since a study oflargerlattices
∼
earth ions is replaced by alkaline-earth ions at random isessentialforanalyzingthenatureofinhomogeneitiesin
locations. Therefore, it is likely that the disorder aris- manganite models, several attempts have been made to
ing as a consequence of this substitution is connected devise accurate approximateschemes [23, 24, 25]. In the
to the amount of doping. This situation is modelled by present study we employ the travelling cluster approxi-
placing repulsive potentials on a fraction x of the sites, mation(TCA)[25],whichindeedhasbeenverysuccessful
which are randomly selected. A typical measure of the inanalyzingsimilarmodelsintherecentpast[10,11,26].
strengthofadisorderdistributionisitsvariance. Forthe
binary distribution the varianceis ∆, while for the finite
density x of scatterers with potential strength D, it is III. RESULTS AND DISCUSSION
Dpx(1 x). These two models for disorder were pre-
−
viously employed in a study of half-doped manganites
Webeginwithresultsforbulkquantitiesdescribingthe
[11]. The JT distortions and the t derived core spins
2g ordering of the magnetic and the lattice degrees of free-
are treated as classical variables, and we set S = 1.
i dom. We focus on the 30% hole-doped system (x=0.3)
| |
Guided by earlierestimates for the JT coupling strength
foraclosecorrespondencetotheexperimentsinRef. [7].
in manganites, we fix λ= 1.5 [21], and explore the vari-
Fig. 1(a) shows the effect of binary disorder on the tem-
ation in the parameters ∆, D, and J .
s peraturedependenceofthemagnetizationm,definedvia
WefurtheradoptthesimplifyinglimitJ >>t,which m2 = (N−1 S )2 . Here and below ... denotes
H P i av av
h i h i
isjustifiedandfrequentlyusedinthecontextofmangan- the average over thermal equilibrium configutations and
ites[9,20,22]. Inthislimittheelectronicspinatsiteiis additionally over realizations of quenched disorder. Re-
tied to the orientationof the core spin S . Transforming sults for disordered systems are averaged over 4 6 re-
i
−
the fermionicoperatorsto this localspin referenceframe alizations of disorder. Clearly, the magnetism is not af-
leads to the following effectively ’spinless’ model for the fected much by the presence of weak binary disorder.
3
m1 (a) ∆∆==00..04 (b) Jλ==01..055 80 (a) Js=x=0.00.53 (b) DDD===012 104
s 0.004 D=4
∆=0.8 x=0.3 ρ 3
0.5 ∆=1.0 D (π,π) 10 ρ
Q
0.002 40 ∆=0.0 102
∆=0.4
∆=0.6 101
∆=1.0
0 0
0
1 0 0.1 0.2 0.3 0 0.1 0.2 0.3
T T
(c) D=0 (d)
m D=1
D=2 0.04 0.3 ∆=0.0 (c) D=0 (d) 0.3
∆=0.6 D=2
D=3 D (π,π) ∆=1.0 D=4
0.5 Q 0.2 0.2
T=0.01
T=0.01
0.02 N N
0.1 0.1
0 0 0 0
0 0.1 0.2 0 0.05 0.1 -4 0 4 -4 0 4
T T
E - E E - E
f f
FIG. 1: (Color online) Temperature dependence of (a) the
magnetization m and (b) the lattice structure factor DQ(q) FIG. 2: (Color online) Temperature dependent resistivity
at q = (π,π) for various values of the disorder strength ∆. ρ(T) (in units of ~/πe2) for varying strength of (a) binary
(c)and(d)showthesamequantitiesasin(a)and(b),respec- disorder ∆, and (b) random scatterers D. Note the logarith-
tively,iftheon-sitedisorderismodelledbyrandomscatterers mic scale in (b). (c)-(d) Low-temperature density of states
ofstrengthD. Theconcentrationofscatterersisequaltothe for the two typesof disorder.
hole density x=0.3. Note the order of magnitude difference
in the magnitudes for DQ(q0) between panels (b) and (d).
All results are at λ=1.5 and Js =0.05. the ordered regions increasing with increasing D.
It is expected that these orbital ordering correlations
are reflected in the transport properties. We therefore
This is in agreement with previous studies, which find compute the dc resistivity ρ approximated by the in-
that the reduction in TC is proportional to ∆2 for weak verse of low-frequency optical conductivity, which is cal-
disorder [27, 28, 29]. culated by using exact eigenstates and -energies in the
Fig. 1(b) shows the temperature dependence of the Kubo-Greenwoodformula[30]. Fig. 2(a)-(b)showρasa
q = q (π,π) component of the lattice structure fac- function of temperature for the two disorder models de-
0
tor DQ(q≡)=N−2PijhQi·Qjiav e−iq·(ri−rj). DQ(q0) is scribed above. The low-temperature resistivity increases
a measure for the staggered distortion order in the sys- upon increasing the binary disorderstrength ∆ (see Fig.
tem. Thelatticeorderingleadstoorbitalorderingviathe 1(a)). For small values of ∆ the resistivity curves ap-
JTcoupling. Anincreasewith∆inthe low-temperature pear parallel to each other below T 0.1. The resistiv-
value of DQ(q0) suggests the appearance of orbital or- ity therefore follows Mathiessen’s ru∼le, i.e. ρ(T) for the
der. However, this effect is too weak to explain the ex- disordered system is obtained from ρ(T) for the clean
perimental resistivity data [7]. Moreover,the increase at system by simply adding a constant contribution arising
low T in DQ(q0) is not monotonic, which becomes clear from the scattering off the disorder potential. dρ/dT re-
by comparingthe results for ∆=0.4, 0.8 and1.0 in Fig. mains positive at low temperature indicating a metallic
1(b). behavior. This oversimplified description, however, does
Nowweexploretheresultsforthedisorderarisingfrom nottakeintoaccountthedisorderinducedchangesinthe
randomscatterersofstrengthD. Sincethedisorderorig- orbital ordering correlations and the related changes in
inates from the replacement of RE3+ by AE2+ ions, the the density of states (discussed below).
density of random scatterers is kept equal to the doping Random scatterers lead to a drastically different be-
concentration x. m(T), shown in Fig. 1(c), is affected havior. The low-temperature rise in ρ(T) covers several
strongly upon increasing D, with a decrease in the sat- orders of magnitude (see Fig. 2(b)). The negative sign
uration value of the magnetization pointing towards a ofdρ/dT forD >1signalsaninsulatingbehavior. Upon
magnetically inhomogeneous groundstate. More impor- increasing the disorder strength D we therefore observe
tantly, a monotonic increase with D is observed in the a metal to insulator transition. For x = 0.3 both disor-
low-temperature values of D (q ) (see Fig. 1(d)). The der models have the same variance, if ∆ 0.46D holds.
Q 0
∼
rise in D (q ) clearly indicates the emergence of orbital Comparing, therefore, the results for D = 2 and ∆ = 1,
Q 0
ordering in the system, with the area and/or strength of we have to conclude that the drastic rise in the resis-
4
(a) (b)
J=0.05
s
λ=1.5
2 2
x=0.3
P(C ) T=0.01
Q
∆=0 D=0
1 1
∆=0.6 D=1
∆=1.0 D=2
0 0
-2 -1 0 1 2 -2 -1 0 1 2
C C
Q Q
FIG. 3: (Color online) Low-temperature distribution func-
tions generated from the Monte Carlo data for the nearest-
neighbor correlations CQ of the lattice distortions for (a) bi-
nary disorder and (b) for random scatterers. The curves for FIG. 4: (Color online) Real-space patterns of the disorder
different∆areoff-setalongthey-axisforclarity. CQ isposi- potential ǫ(i), charge density n(i), and lattice correlations
tive(negative)for ferro- (antiferro)-distortivepatternsof the CQ(i). Top row: binary disorder with ∆ = 1, bottom row:
lattice variables. random scatterers with D = 2. The patterns in both cases
areshownona24×24latticeforasingledisorderrealization
at T =0.01 and x=0.3.
tivity for randomscattererscan not be attributed to the
strengthofthedisorderpotential. Infact,alargeincrease
inthe low-T resistivitywasoneoftheexperimentalindi- sites of site i. A negative value of CQ(i) indicates an an-
cationsfor the onsetofdisorder-inducedorbitalordering tiferro pattern of JT distortions, and hence a pattern of
[7]. staggeredorbital ordering. The distribution function for
Figs. 2(c)-(d) highlight the difference between the CQisdefinedasP(CQ)=hN−1Piδ(CQ−CQ(i))iav;the
densities of states (DOS) for the two choices of dis- δ-function is again approximated by a Lorentzian with
order modelling. The DOS is defined as N(ω) = width 0.04. A peak in P(CQ) centered near CQ =0.8
∼
N−1 δ(ω E ) , where E denotes the eigenvalues for∆=0indicatesthatthecleansystemhasweakferro-
hof thePHaimilt−oniain.iaHvere we apiproximatethe δ-function distortive/ferro-orbitalcorrelations. Tails going down to
by a Lorentzian with width γ =0.04: CQ 1.4 arise in the distribution function upon in-
∼ −
cluding binary disorder.
γ/π The distribution function P(C ) for random scatter-
Q
δ(ω E ) . (4)
− i ≃ [γ2+(ω E )2] ers looks qualitatively different. We recall that the
i
− strengths of the two types of disorder are related via
The DOS for the clean system has a pseudogap struc- ∆ 0.46D. The low-temperature distributions P(C )
Q
∼
ture near the chemical potential. For binary disorder, are plotted in Fig. 3(b) for random scatterers. A qual-
the pseudogap slowly fills up with increasing ∆. In con- itative change in the shape of the distribution function
trast, it deepens upon adding random scattering centers occurs for D = 2, where a second peak centered around
and even leads to a clean gap for D 3. This opposite C 1.2emerges. Thisisadirectindicationthatasig-
Q
≥ ∼−
behaviorispartlyresponsibleforthedrasticallydifferent nificantfractionofthesystembecomesorbitallyordered.
low-temperature resistivity discussed above. The three- This correlates perfectly with the strong rise in D (q )
Q 0
peak structure for large values of D in Fig. 2(d) can be at low temperatures (see Fig. 1(b)), and the anomalous
understood as follows: A fraction 2x of electronic states increase in the resistivity (see Fig. 2(b)).
splitoffandformanarrowimpuritybandcenteredatan A real-space picture for the emergence of orbital or-
energy D above the Fermi level of the undoped system. deringis presentedinFig. 4,whichdisplaysthe disorder
The lower band now contains a fraction 2(1 x) of the potentialǫ ,theelectronicdensityn ,andthelatticecor-
i i
−
states with the Fermi level located in the middle of the relations C (i). The top row for binary disorder shows
Q
band. This leads to a situation similar to the undoped that the charge density closely follows the disorder po-
system, and an energy gap originating from staggered tential. Thelocallatticecorrelationsarecenteredaround
orbital ordering opens at the Fermi level. C = 0, which is also evident from the peak in the dis-
Q
To gain further insight into the nature of the states tribution P(C ) shown in Fig. 3(a). The bottom row
Q
in the presence of the two types of disorder, we plot the in Fig. 4 shows the corresponding results for the disor-
distribution functions for the lattice variables in Fig. 3. der potential arising from random scatterers. Since the
Panel (a) shows the distribution of the nearest neighbor doping concentrationin this case coincideswith the con-
lattice correlations, C (i) = (1/4) Q Q for bi- centration of the scatterers,the holes are trapped at the
Q Pδ i · i+δ
nary disorder; here δ denotes the four nearest neighbor impurity sites. This leaves the surrounding effectively
5
1
(a) J=0.00 (b)
s 0.3
J=0.05
s
m Js=0.10
0.2
0.5 D=4
x=0.3 D (π,π)
Q
0.1
0 0
0 0.1 0.2 0 0.05 0.1
T T
FIG.6: (Coloronline)Temperaturedependenceofthemagne-
tizationmandthestaggeredlatticestructurefactorDQ(π,π)
for varying superexchange coupling strength Js. The results
are for random scatterers with strength D=4.
FIG. 5: (Color online) Doping evolution of the local charge
densityni andthelocallatticecorrelationsCQ(i)forasingle
realization of random scatterers with strength D = 1, on a
24×24 lattice. (π,π). The (π,π) ordering phenomena is partially trig-
gered by the charge inhomogeneities in both cases. An
additional complication in the present case arises from
the spin degrees of freedom in addition to the orbital
undoped and thereby induces orbital ordering. This is
variablesandfromthe anisotropyofthe hopping param-
apparent from the spread of the dark-blue regions and
eters.
their cross correlation with the charge density distribu-
Asinferredabovefromthe resultsforthe temperature
tion in the bottom row of Fig. 4. Such a picture with
dependent magnetization m(T), the magnetic ground-
orbitally ordered regions coexisting with orbitally disor-
stateappearstobehomogeneousforbinarydisorder,but
dered patches describes perfectly the double peak struc-
may be inhomogeneous in the case of doped scatterers
ture of the distribution function in Fig. 3(b).
(see Figs. 1(a),(c)). Since the magnetism is partially
Althoughweareprimarilyinterestedintheexperimen- controlled by the antiferromagnetic superexchange cou-
tally relevant case x = 0.3, it is useful to see how the pling J , we study the effect of increasing J for a fixed
s s
real-spacepatternsevolveasonemovesfromlowto high large disorder strength of random scatterers. Fig. 6(a)
hole densities. The undoped system is an orbitally or- shows the result for m(T) and Fig. 6(b) the result for
dered insulator, which turns into an orbitally disordered the temperature dependence of D (q ). The saturation
Q 0
metal upon doping [31]. We show real-space patterns value of m(T) as well as the onset scale for ferromag-
at three different doping concentrations in Fig. 5. The netism decrease with increasing J . More importantly,
s
density of random scatterers is kept equal to the dop- D (q )atlowtemperaturesincreaseswithincreasingJ
Q 0 s
ing fraction x. The charge density distribution is largely indicating an enhancement in the orbital ordering. For
controlled by the disorder distribution. At low doping, a homogeneous system this would mean that orbital or-
disconnected orbitally disordered regions are essentially dering and antiferromagnetism are both enhanced with
tiedtothetrappedholes. Withincreasingxtheorbitally increasingJ . ThisisacontradictiontotheGoodenough-
s
disordered regions begin to connect in one-dimensional Kanamori rules, which state that an orbitally antiferro
snake-likepatterns. Byfurtherincreasingthedopingand system should be magnetically ferro. The contradiction
theconcentrationofscatteringcenterstheorbitallydisor- is resolved by analyzing the microscopic details of this
deredregionsgrow. Thephenomenonofdisorder-induced complicated state providing an example where the real-
orbital ordering is likely to be present only in a narrow spacestructuresareessentialforacomprehensiveunder-
doping range near and above 0.25, because for x<0.25, standing.
the system is orbitally ordered even in the clean limit. WeshowinFig. 7,theeffectofthesuperexchangecou-
It is worthwhile to point to a similarity between the pling on the real-space patterns of lattice and spin vari-
effects of disorder in the present study and in a model ables. The lattice correlations are shown in the top row
analysis for d-wave superconductors with non-magnetic and the analogously defined spin correlations C (i) =
S
impurities. In Ref. [32] it was found that the impuri- (1/4) S S in the bottom row. For J = 0.02
Pδ i · i+δ s
ties nucleate antiferromagnetism in their near vicinity. the system contains orbitally ordered nanoscale regions,
Upon increasing the impurity concentration static anti- butmagneticallyitappearshomogeneous. ForJ =0.06
s
ferromagnetismis observed. There seems to be a perfect the area of the orbitally ordered regions is enlarged and
analogy between the two situations, if one interchanges magnetic inhomogeneities appear. The orbitally ordered
antiferromagnetism by orbital-ordering; both are order- regions remain ferromagnetic, while the orbitally disor-
ing phenomena with the staggered ordering wavevector dered regions become antiferromagnetic upon increasing
6
ever,uponincreasingJ furtherto0.1,theantiferromag-
s
netic regions start to extend also into the orbitally or-
dered clusters. The charge density patterns (not shown
here) are insensitive to the increase in J .
s
IV. CONCLUSIONS
Our analysis for a two-band double-exchange model
for manganites leads us to conclude that the disorder
induced orbital ordering in manganites near x = 0.3 is
properly described, if the density of scattering centers
trackstheholeconcentration. Withinthisspecificmodel
ofquencheddisordertheinducedstaggeredorbitalorder-
ing is responsible for the ordersof magnitude increase in
the low-temperatureresistivity as observedin the exper-
FIG. 7: (Color online) Real-space patterns for the lattice
iments in Ref. [7].
correlations CQ(i), and the analogously defined spin correla-
tions CS(i) for varying superexchange coupling Js. The pat-
terns are shown for a single disorder realization at T =0.01.
ACKNOWLEDGMENTS
Orbitally ordered regions tend to maintain ferromagnetism,
while the orbitally disordered regions are more susceptible
towards antiferromagnetism with increasing Js. SK acknowledges support by ”NanoNed”, a nanotech-
nology programme of the Dutch Ministry of Economic
Affairs. APK gratefully acknowledges support by
J . In the orbitally ordered clusters of this inhomo- the Deutsche Forschungsgemeinschaftthrough SFB 484.
s
geneous system in the selected parameters regime the Simulations were performed on the Beowulf Cluster at
Goodenough-Kanamorirulesarethereforefulfilled. How- HRI, Allahabad (India).
[1] ForoverviewsseeNanoscalePhaseSeparationandColos- Carvajal, L. Pinsard, and A. Revcolevschi, Phys. Rev.
sal Magnetoresistance by E. Dagotto (Springer-Verlag, Lett. 81, 1957 (1998).
Berlin, 2002) and [2] and [3]. [15] Y. Motome, N. Furukawa, and N. Nagaosa, Phys. Rev.
[2] Colossal Magnetoresistive Oxides,editedbyT.Chatterji Lett. 91, 167204 (2003).
(Kluwer, Dordrecht,2004). [16] C. Sen, G. Alvarez, and E. Dagotto, Phys. Rev. B 70,
[3] Colossal Magnetoresistive Oxides, edited by Y. Tokura 064428 (2004).
(Gordon and Breach, Amsterdam, 2000). [17] S.KumarandP.Majumdar,Phys.Rev.Lett.92,126602
[4] L.M.Rodriguez-Martinez andJ.P.Attfield,Phys.Rev. (2004).
B 58, 2426 (1998); Phys. Rev.B 54, R15622 (1996). [18] J. B. Goodenough, Magnetism and the Chemical Bond,
[5] D.Akahoshi,M.Uchida,Y.Tomioka,T.Arima,Y.Mat- Interscience, New York, 1963; J. Kanamori, J. Phys.
sui, and Y. Tokura, Phys.Rev.Lett. 90, 177203 (2003). Chem. Solids 10, 87 (1959).
[6] Y. Tomioka and Y. Tokura, Phys. Rev. B 70, 014432 [19] M.Imada,A.Fujimori,andY.Tokura,Rev.Mod.Phys.
(2004). 70, 1039 (1998).
[7] L.M.Rodriguez-Martinez andJ.P.Attfield,Phys.Rev. [20] T. Hotta, S. Yunoki, M. Mayr, and E. Dagotto, Phys.
B 63, 024424 (2000). Rev. B 60, R15009 (1999).
[8] G. Alvarez, H. Aliaga, C. Sen, and E. Dagotto, Phys. [21] E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1
Rev.B 73, 224426 (2006). (2001);A.J.Millis,B.I.Shraiman,andR.Mueller,Phys.
[9] A. Moreo, M. Mayr, A. Feiguin, S. Yunoki, and E. Rev.Lett.77,175(1996);P.B.AllenandV.Perebeinos,
Dagotto, Phys.Rev.Lett. 84, 5568 (2000). ibid. 83, 4828 (1999).
[10] S. Kumar, A. P. Kampf, and P. Majumdar, Phys. Rev. [22] J. A. Verg´es, V. Mart´ın-Mayor, and L. Brey, Phys. Rev.
Lett.97, 176403 (2006). Lett. 88, 136401 (2002).
[11] K.Pradhan,A.Mukherjee,andP.Majumdar,Phys.Rev. [23] J. L. Alonso, L. A. Fernandez, F. Guinea, V. Laliena,
Lett.99, 147206 (2007). and V.Martin-Mayor, Nucl. Phys.B 596, 587 (2001).
[12] ForprecisedefinitionsofrAandσ2 seee.g.J.P.Attfield, [24] G. Alvarez, C. Sen, N. Furukawa, Y. Motome, and E.
Int.J. Inorg. Mater. 3, 1147 (2001). Dagotto, Comput. Phys. Commun. 168, 32 (2005).
[13] G. Papavassiliou, M. Pissas, M. Belesi, M. Fardis, J. [25] S. Kumar and P. Majumdar, Eur. Phys. J. B 50, 571
Dolinsek, C. Dimitropoulos, and J. P. Ansermet, Phys. (2006).
Rev.Lett. 91, 147205 (2003). [26] S.KumarandP.Majumdar,Phys.Rev.Lett.96,016602
[14] M. Hennion, F. Moussa, G. Biotteau, J. Rodriguez- (2006).
7
[27] J. Salafranca and L. Brey, Phys. Rev. B 73, 214404 (2005).
(2006). [31] B. B. Van Aken, O. D. Jurchescu, A. Meetsma, Y.
[28] Y. Motome and N. Furukawa, Phys. Rev. B 68, 144432 Tomioka, Y. Tokura, and T. T. M. Palstra, Phys. Rev.
(2003). Lett. 90, 066403 (2003).
[29] G. Bouzerar and O. C´epas, Phys.Rev.B 76, 020401(R) [32] B. M. Andersen, P. J. Hirschfeld, A. P. Kampf, and M.
(2007). Schmid, Phys.Rev.Lett. 99, 147002 (2007).
[30] S. Kumar and P. Majumdar, Eur. Phys. J. B 46, 237