Complex State Found in the CMR Regime of Models for Colossal Magnetoresistive Manganites

0.42 MB·English

by  Cengiz Sen

Complex state found in the colossal magnetoresistance regime of
models for manganites
Cengiz S¸en,1,2 Shuhua Liang,1,2 and Elbio Dagotto1,2
1Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996
2Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 32831
(Dated: January 4, 2012)
Thecolossalmagnetoresistance(CMR)effectofmanganitesiswidelybelievedtobecausedbythe
competition between a ferromagnetic (FM) metallic state induced by the double-exchange mecha-
2 nismandaninsulatorwithcomplexspin,charge,andorbitalorder. Recentcomputationalstudiesin
1 smallclustershaveindeedreportedaCMRpreciselynearthefrontierbetweenthosetwostatesata
0 realisticholedensityx=1/4. However,thedetailedcharacteristicsofthecompetinginsulatorwere
2 not fully understood in those previous investigations. This insulator is expected to display special
properties that lead to the CMR, otherwise any competition between ferromagnetic and antiferro-
n
magnetic states would induce such an effect, which is not the case experimentally. In this report,
a
J thecompetinginsulatoratelectronicdensityx=1/4andintheCMRregimeisstudiedindetailus-
ing the double-exchange two-orbital model with Jahn-Teller lattice distortions on two-dimensional
3
clusters, employing a careful large-scale cooling down process in the Monte Carlo simulations to

] avoid being trapped in metastable states. Our investigations show that this competing insulator
i hasanunexpectedcomplexstructure,involvingdiagonalstripeswithalternatingregionsdisplaying
c
FMandCE-likeorder. Thelevelofcomplexityofthisnewstateevensurpassesthatoftherecently
s
- unveiledspin-orthogonal-stripestatesandtheirassociated highdegeneracy. Thisnewstatecomple-
l
r mentsthelong-standingscenarioofphaseseparation,sincethealternatingFM-CEpatternappears
t even in theclean limit. The present and recent investigations are also in agreement with themany
m
“glassy” characteristics of the CMR state found experimentally, due to the high degeneracy of the
. insulating states involved in theprocess. Resultsfor thespin-structurefactor of thenew states are
t
a also hereprovided to facilitate theanalysis of neutron scattering experiments for these materials.
m
- PACSnumbers: 75.47.Lx,75.30.Mb,75.30.Kz
d
n
o I. INTRODUCTION temveryclosetothe transitionfromthemetaltothein-
c sulatoratlowtemperatures. Quencheddisorderisknown
[
to enhance considerably the CMR effect and avoid the
The colossal magnetoresistance (CMR) effect of the
2 fine tuning of couplings, rendering the presence of the
manganites provides a fascinating example of the col-
v effect more universal. However, it is still an important
7 lective behavior and unexpected nonlinearities that can conceptual issue to study the CMR in the clean limit
9 emerge in complex materials, such as transition metal
using finite clusters, even if a fine-tuning of couplings is
7 oxides, when several degrees of freedom are simultane-
needed. Such studies provide insight on what makes the
1 ously active.1,2 Reaching a complete theoretical under-
manganite oxides special, since few other complex ma-
.
9 standing of the CMR effect is certainly important in the
terials present such a large magnetoresistance effect in
0 contextofMn oxides,but alsoto providea paradigmfor
so many members of the same oxide family and in wide
1 rationalizing related complex phenomena, or to predict
ranges of electronic composition and bandwidth. More-
1 similar effects in other materials.3 The current widely
: over,adeepunderstandingoftheCMReffectwouldcon-
v accepted scenario to understand the CMR relies on the
tribute to new areas of research where manganites play
Xi existence of insulating states competing with the ferro- a fundamentalrole,suchas oxideheterostructures20 and
magnetic(FM)metallicstateinducedbythe well-known manganite multiferroics.21
ar double-exchange mechanism.2 In addition, particularly There have been two interesting recent conceptual de-
inthepresenceofsmallamountsofquencheddisorderor
velopments in the study of models for manganites in the
other effects such as strain, intriguing nanometer-scale
cleanlimitthatareofrelevanceforthepresenteffort. As
electronicstructureshavebeenfoundinbothexperimen-
already briefly mentioned, in Ref. 19 a study of the con-
tal efforts and theoretical studies.1–17
ductancevs. temperature,varyingthet superexchange
2g
Considerableprogressinthetheoreticalanalysisofthe coupling to interpolate between the FM metal and the
CMR effect was recently reached when the CMR effect AFM insulator at x = 1/4, unveiled a clear CMR effect
was numerically found in a standard two-orbitaldouble- andafirst-ordermagnetictransitionthatisingoodqual-
exchange model for manganites at x = 1/4 doping, in- itative agreementwith experiments. The AFM insulator
cluding Jahn-Teller distortions, in the clean limit.18,19 was characterized as a C E state in Ref. 19 due to
1/4 3/4
Theabsenceofimpuritiesinthisstudybringstheneedto the existence in the static spin structurefactor ofapeak
finetunecouplings,suchastheantiferromagnetic(AFM) atmomenta(π/2,π/2)whichisoftentakenasevidenceof
superexchangeamongthet electrons,tolocatethesys- CE-like states in experimental investigations. Since the
2g
2
dominant state at x = 1/2 is precisely a well-known CE surprise, this insulating state reveals a remarkable com-
state,2 thensuchanassignmentwasnatural,butthisas- plexitythatrivalsthatofthe SOSstatesandtheircorre-
sumptionwillbe revisitedbelow. Thesecond,evenmore sponding degenerate states. The CE peak at (π/2,π/2)
recent, development was presented in Refs. 22 and 23 that was reported in Ref. 19 certainly is confirmed, and
where surprisingly highly-degenerate diagonal “stripe” in fact the overall insulator does have regions with the
phaseswerereportedatspecialholedensitiesintherange characteristicCEzigzagchains. However,andthisisthe
between0and1/2. Theseinvestigationsshowedthatthe mainresultofoureffort,thecompletestatealsocontains
competing insulators to the FM metal states at interme- FM regions of equal length size, such that the overall
diate densities are far more complex than previously an- stateisamosaicofCEandFMregionsregularlyspaced.
ticipated, particularly in the range of hole doping which Both these CE and FM regions form diagonal stripes,
is of the main interest for CMR effects. that alternate in a ...-FM-CE-FM-CE-... pattern along
onediagonal. The SOSstates23 arenotascomplexsince
Inotherwords,inthelimitx=1/2ithasbeenclearly
they have the same pattern of spins along the diagonal
shownbothexperimentallyandtheoreticallythatrobust
stripes, albeit spin rotated by 90o degrees. However,the
states of the CE form, displaying spin, charge, and or-
new state reported here mixes two very different states,
bital order, are very stable and dominant. In the other
the CE and FM states, in a combination that is stable
limit x = 0, there is also clear evidence of an A-type
inourcomputersimulationsonfiniteclusterseveninthe
AFM insulator with orbital order that is also very dom-
clean limit, namely without the need to have disorder or
inant. However, at intermediate hole densities between
strain effects to create such nanoscopic phase competi-
x=0 and 0.5 it is still not fully clear what kind of com-
tion. Itis this exoticcombinationofCEandFM regions
petitorstotheFMstatedoemergefromthemanyactive
that triggers the CMR effect in our computational stud-
degrees of freedom in manganites. The C E states24
1−x x ies once couplings are tuned to the vicinity of the metal-
are natural candidates, as assumed in Ref. 19, but the
insulator regime. As discussedin the conclusions section
recent prediction of a competing spin-orthogonal-stripe
ofthis manuscript,the presentresults createa varietyof
(SOS)statesatsmallelectron-latticecouplings,23 aswell
additional conceptual questions in manganites, and they
as the high degeneracy reported in Ref. 22 that was dis-
show that the level of complexity of these materials, at
cussed above, are clearly challenging our understanding
least within the model Hamiltonian framework and in
of this fundamental aspect of the CMR effect. Knowing
two-dimensionalgeometries,isfarlargerthananticipated
with accuracy the properties of the competing insulator
in previous investigations.
stateintheCMRregimewillcontributetounveilingwhy
this CMReffectoccursinthefirstplace,sinceitappears
that simply having a competition between a FM metal
II. MODEL AND THE MONTE CARLO
and any AFM insulator is not sufficient to have a CMR
TECHNIQUE
effect. The AFM insulator must have special properties
that are currently unknown.
In this manuscript, the standard two-orbital lattice
Motivatedbytheserecentdevelopments,inthepresent Hamiltonian for manganites will be studied using two-
publication the insulator found in the x = 1/4 CMR dimensionalfinite clusters. In the well-knownlimitofan
regime of Ref. 19 has been further investigated. To our infinite-Hundcoupling,themodelisexplicitlydefinedas:
H =− X taγγ′[cos(θi/2)cos(θi+a/2)+e−i(φi−φi+a)sin(θi/2)sin(θi+a/2)]d†iγσdi+aγ′σ+JAFXSi·Sj
iaγγ′σ hi,ji
+λX(Q1iρi+Q2iτxi+Q3iτzi)+(1/2)X(ΓQ21i+Q22i+Q23i), (1)
i i
where ta is the hopping amplitude for the e orbitals spring constants for breathing- and JT-modes. The rest
γγ′ g
γ = x2 − y2 and γ′ = 3z2 − r2 in the a direction, of the notation is standard. Throughout the numerical
J is the antiferromagnetic coupling between t spins simulations described below, cooperative lattice distor-
AF 2g
on neighboring sites i and j of a two-dimensional lat- tions are used, where the actual displacements uis for
tice, and the Q’s arethe various Jahn-Teller(JT) modes the oxygenatoms are the explicit variables in the Monte
(defined extensively in previous literature).2 λ is the di- Carlo sampling, instead of the linear combinations Qis
mensionless electron-lattice coupling constant, and τ = that make up the individual Jahn-Teller modes. Note
zi
Pσ(d†iaσdiaσ−d†ibσdibσ)arethepseudo-spinoperatorsde- that in our study both the t2g spins and the lattice de-
fined in Ref. 2. The last term represents the potential grees of freedom are considered classical for simplicity,
(elastic energy) for the distortions, with Γ the ratio of approximationwidely employed in previous efforts.2
3
The Monte Carlo technique used in these simulations of the values corresponding to several Monte Carlo ob-
has been extensively discussed in the past,2 and details servables when the model parameters are slightly mod-
will not be repeated here. The procedure to calculate ified, the overall convergence quality of the results can
conductances is also standard.25 However, an important also be monitored.
improvement that has been employed here and in some Once the true ground state for a fixed model parame-
recent studies merits a more detailed discussion. In the tersetisidentified,the processisreversed,andthis time
present effort, the cool-down method has been used, a“heat-up”procedureiscarriedout,namelythe simula-
rather than a more standard procedure where a par- tion starts by using an initial configuration which is the
ticular temperature is chosen and the simulation is run properly converged last configuration of the cool-down
to converge to a particular set of equilibrium configura- process at the lowest temperature. These extra steps
tions. The latter has proved to be problematic in some increase the chances that true converged quantities are
casesbecausetheremaybecompetingmeta-stablestates obtained. The observables at each temperature are cal-
wheretheMonteCarlostatesbecome“trapped”andthis culated at this stage to a good accuracy by running the
prevents a proper convergence to the true ground state. simulation with an initial configuration borrowed from
The cool-down method, described below, is systematic the last configuration of the heat-up process. At this
and successful in achieving a better convergence to the stage in the simulation, 20,000 Monte Carlo steps for
groundstate,the only problembeing the increasedCPU thermalization were used, and another 100,000steps for
time required at the beginning of the process. measurements,wherethe measurementsare takenat ev-
The cool-down method starts by selecting a tempera- eryfiveMonteCarlosteps. AllthesevaluesoftheMonte
ture grid. Often a denser temperature grid is chosen at Carlo parameters and convergence process are kept the
low temperatures since it is in that range that metasta- sameforallthe resultsdiscussedbelow,unlessotherwise
bility problems are more likely to arise. The simulation is indicated explicitly.
starts in practice in our study at a very high tempera-
ture β = 3 with a completely random configuration of
the classicalspins andoxygenlattice degreesoffreedom. III. QUARTER DOPING, x=1/4
Asafirststep,thesystemisallowedtothermalizeforthe
first 10,000MC steps, and measurements are takendur-
Inthissection,theresultsobtainedusingaholedoping
ing the next 5,000 steps. Then, the temperature is low-
densityx=1/4arepresented. InFig.1,theMonteCarlo
eredto the next one in the temperature grid, e.g. β =4,
totalenergyvs. thesuperexchangecouplingJ isshown
AF
andthesimulationprocesscontinuesallthewaydownto
at a low temperature β = 200 for the case of a moder-
β =300,with5,000stepsforthermalizationandanother
ate electron-lattice coupling λ. In fact, this coupling has
5,000 steps for measurements at each temperature.
beenfixedtoλ=1.3throughoutthex=1/4simulations
In general, the acceptance ratios of the Monte Carlo discussedbelow,sincepreviousinvestigationshaveshown
simulationsdeterioratesignificantlywithdecreasingtem- that this value is optimal for the existence of the CMR
perature. To avoid this problem, the acceptance ratios peak in the resistivity vs. temperature calculations.19
weremonitoredduringtheentiresimulationtomakesure Three states are here identified as ground states at
thattheydonotfallbelowa(lessthanidealbut)reason- this particular doping, including the new state that is
able rate (≈ %10) for the lowest temperature simulated the focus of the present publication. At small J , the
AF
in the calculation, β = 300. In practice it was observed groundstateis the well-knownFMmetalinducedbythe
that a Monte Carlo window of ∆ = 3/β is sufficient to double-exchange mechanism. At large J in the range
AF
guarantee that the acceptance ratios are still reasonable shown in Fig. 1, the ground state is a previously dis-
(here the Monte Carlo window denotes the amount by cussed C E -type charge-orderedinsulator.24 Figure
1/4 3/4
whichtheoxygencoordinates,andthetwoanglesneeded 1ofRef.19showsthatthisstatewaspreviouslybelieved
for each classical spin, are modified before the standard tobethemaincompetitortotheFMmetallicstate. The
Monte Carlo procedure is used to accept or reject the main surprise,and main result of this publication, arises
new configuration). at the intermediate superexchange couplings shown in
For a fixed set of model parameters (the electron- Fig.1. Inthis regime,anew intermediatestate isidenti-
latticecouplingλ,andthesuperexchangecouplingJ ), fied since the slope of the energy curve is different from
AF
the cool-down process alone might still not be sufficient the two extreme cases with purely FM or CE character-
for full convergence. However, many sets of these model istics. The intermediate state must actually be a com-
parameters were used in the Monte Carlo runs in paral- bination of the properties of the two neighboring states,
lel (employing several computer nodes) and in the end, since it is stable in a narrow region between them. In
by mere comparisonof energies, it was observed that of- fact, this novel state consists of diagonal FM regions lo-
tenforasubsetofthosemodelparametersaconvergence cated in real space sandwiched between CE-like zigzag
to the true ground state was found. The existence of a spin arrangements of a similar size in length, as shown
possible new ground state is then confirmed by compar- in Fig. 2. This new state will be called “FM-CE” in the
ing its energywith those ofthe neighboringstates in the present publication for simplicity. At low temperatures,
phase diagram. In short, by monitoring the smoothness the level crossing of the energies of the FM-CE state
4
-1.21
C E
1/4 3/4
-1.22
)
a
a
t 8x8
(-1.23
N 16x16
/
E
x=1/4
FM-CE
-1.24 λ=1.3
T=0.005
FM
-1.25
0.13 0.14 0.15 0.16 0.17 0.18
J /t
AF aa
FIG. 1. (color online) Monte Carlo total energy per site vs.
FIG. 2. (color online) Real-space representation of the new
JAF for the lowest temperature (β = 200) used in the cool-
intermediate state “FM-CE” found at x = 1/4. The ide-
down process, as described in the text. Results are obtained
alized spin configuration (left) is derived from the actual
using 8×8 and 16×16 lattices. Other parameters used are
Monte Carlo converged state (right) obtained at λ=1.3 and
asindicated. SymbolsrepresenttheactualMCresults. Error
bars are smaller than the size of the symbols (not shown). JAF=0.16. Thelocalchargedensitiesthataredepictedwith
circles intheidealized configuration are takenfrom thesame
Lines provideguides to the eye.
MonteCarlo snapshot shownontheright. Densitiesthatare
hni>0.65areshownwithlight-bluefilledcircles. Thelargest
circle displayed inthefigurecorresponds tohni≈1, showing
with the competitors FM and CE states signals a first- that the ferromagnetically oriented portion of the new state
order transition. While the FM regions do not display has approximately that density.
any strong indication of charge ordering, it is clear from
the Monte-Carlo snapshots that the CE-like regions are
charge-orderedwith a checkerboardpattern (see Fig. 2). ingstate.2,18,19 ThenewphaseFM-CEunveiledherehas
Note alsothatthe FMregionspresentanelectronic den- all the properties needed for the insulating state, but
sity close to 1 (i.e. x=0), namely close to the undoped in addition surprisingly it has a spin FM component.
limit, while the averagedensity of the CE region is close As shown in Fig. 3, the calculation of the density-of-
to x=1/2,rendering the overalldensity x=1/4. Thus, states shows that this new state has a gap at the Fermi
thisstatepresentsaspontaneousnanometer-scalecharge level, thus confirming its insulating character. This fig-
phaseseparationeveninthecleanlimithereinvestigated. ure shows the density-of-states obtained from a Monte
It is important to remark that very time consuming Carlo converged state as well as a for an idealized con-
simulations on a 16×16 lattice have also been carried figurationofspins, wherethe latteris shownbothbefore
out to investigate if the new phase is a spurious conse- and after broadening the delta-function peaks using a
quence of size effects on the 8×8 cluster. The 16×16 Lorentzian. In the idealized configuration, the density
simulations clearlyindicate that the new state is present of states was calculatedfor a fixed configurationof spins
in this lattice, actually on a wider range of the J cou- (see Fig.2), and the lattice degrees of freedom were not
AF
pling, suggesting that the bulk system will also display taken into account. Hence, this calculation effectively
the same novel FM-CE state found in the simulations corresponds to a zero electron-lattice coupling (λ = 0).
presented here. Note that for the 16 × 16 it was not This is interesting in the sense that it is known that
practical to use the same number of MC steps as used a finite electron-lattice coupling is needed for the pur-
for the 8×8 lattice. Instead, 1,000 thermalization and posesofCMR.WhiletheantiferromagneticcouplingJ
AF
2,000 measurement steps were used in the former. Also, appears sufficient for the stabilization of the spin struc-
for the three competing states that are considered here, ture, the electron-lattice coupling stabilizes the charge-
the spins were frozen to the ideal configurations during ordering characteristics. Thus, the present calculations
thesimulation,andthesimulationsstartedwiththecon- suggestthat both couplings playequally importantroles
vergedphononic configurationsthatwere borrowedfrom to generate the CMR resistivity peak.
the 8×8 lattice. The smoothness of the results at vari- Thestabilityofthesethreestateshasbeenstudiedvia
ous couplings indicate that a good convergencehas been calculationsoftheaveragecarrierdensitypersitehnivs.
reached in our study. the chemical potential µ. Figure 4 shows wide plateaus
From many previous theoretical and experimental in- at density hni = 0.75 varying µ, at three different cou-
vestigations, it is by now clear that the existence of a plings representative of the three different states, clearly
CMR relies on a competition between a FM metallic indicating that all these states are stable in the parame-
state and a charge, spin, and orbitally ordered insulat- terregionsstudiedinthis paper. Also,note thatthereis
5
40
T=0.005, λ=1.3 1.4
8x8
35 T=0, λ=0
)30 T=0, λ=0 (broadened) 1.2 λ=1.3
ts 1 β=200
i µ
n25 8x8
u
b. 20 x=1/4 n>0.8
ar J =0.16 <
ω) (15 AF 0.6 JAF=0.148
( 0.4 J =0.16
N10 AF
J =0.18
5 0.2 AF
0 0
-4 -3 -2 -1 0 1 2 3 4 -2 -1.5 -1 -0.5 0 0.5 1
ω/t µ/t
aa aa
FIG. 3. (color online) Density-of-states for the new FM-CE FIG.4. (coloronline)Averageelectronicdensityvs. chemical
stateshownforanidealized(T =0,λ=0,fixedspins)anda potential for the FM state, theintermediate new state “FM-
MonteCarloconvergedspinconfigurationsatdensityx=1/4 CE”,andtheC1/4E3/4 state,withparametersasindicatedin
(T =0.005, λ=1.3). The location of the chemical potential the figure. Note the wide plateau at hni = 0.75, suggesting
is indicated with µ and it is shown with dashed lines. The thattheintermediatestateisquitestable,similarlyasthetwo
density-of-states and the resistivity calculations (see Fig. 5) neighboring states FM and CE. In this calculation, 20,000
confirmthatthestateatλ=1.3isinsulating. Therestofthe Monte Carlo steps were used, where the initial 10,000 steps
parametersusedareindicatedinthefigure. Noticethatthere are discarded for warm-up purposes.
are states populated at the Fermi level for λ = 0, at least
afterbroadening,whiletherearenoneforλ=1.3, givingrise
to an insulating state. The sharp peak at ω = 0.0 in the IV. SPIN STRUCTURE FACTOR
λ = 0.0 case may correspond to localized states similarly as
previously reported in Ref. 19 in theCE-states context.
In this section, the spin-structure factors of the vari-
ousstates discussedhere arecalculatedinorderto guide
future neutron scattering experiments applied to CMR
manganites. For this purpose, here a 24 × 24 lattice
is used, and the spins are fixed to the various C E
x 1−x
patterns as well as to the two new states discussed in
another stable region for hni=1.25 (the y-axis in Fig. 4 the previous two sections. First, the real-spacespin-spin
is normalized to hni = 2, the maximum density per site correlations were calculated for various spin pairs aver-
allowed by the two-orbital model), indicative of a possi- agedovertheentirelattice,(1/N)hS ·S i. Then,thespin
i j
ble particle-hole symmetry in the system. It is expected structurefactorS(k)wasobtainedbysimplyconsidering
that CMR should also reveal itself for systems that are the Fourier transform of the spin-spin correlations:
electron doped, but such a study is beyond the scope of
the present publication.
1
S(k)= XhSi·Sjiexpik·(ri−rj), (2)
N
i,j
The resistivity vs. temperature figures presented in
Ref. 19arehere reproducedfor the benefit ofthe reader. where r is the lattice location of the classical spin S .
i i
These curves display changes of over a couple of orders To better simulate the experimental situation, where
of magnitude near the ordering magnetic temperature, preferred directions in the spin order may not be stabi-
when parametrized as a function of the superexchange lizedduetotheformationofmultipledomains,thestates
coupling J . The location of the resistivity peak co- under discussion are symmetrized using various rotation
AF
incides with the Curie temperature T where the sys- andreflectionoperations. Forexample,forthesimpleE-
C
tem presents a spontaneous first-order transition to a phase, the zigzag chains canbe orientedeither along the
FM state with a finite magnetization. This first-order [11]or[1¯1]directions. Thesymmetryoperationtotrans-
transition is found for couplings very close to the phase formoneintotheotherinthiscasewouldbeaπ/2-degree
boundaryoftheFMmetallicstate(andarapidcrossover rotation either in the clockwise- or counter-clockwise di-
replaces the first-order transition as the distance to the rections. Inmorecomplicatedcases,suchasinthenewly
competing insulator increases). The details of these re- foundintermediatestates,therearetwoadditionalstates
sults have been already discussedin Ref. 19 and will not that can be obtained by first rotating the starting state
be repeatedhere. Butitis importantto remarkthatthe by π/2-degrees along the z-direction (perpendicular to
identificationofthenewFM-CEstateinthismanuscript the layers here investigated), and then taking the reflec-
doesnotalteratallthetransportCMRresultsofRef.19. tionoftheresultingstatealongthe[11]or[1¯1]directions.
6
103 fordopinglevelsx<0.5. Theirtentativeexplanationfor
J =0.12
AF their findings was expressed in terms of inhomogeneous
=0.14
electronic self-organization, where electron rich domain
=0.142
walls with short-range magnetic correlations are sepa-
=0.144
2 =0.146 rated from commensurate AF patches.
10 =0.148 Inourpresentresultsanalternativeexplanationtothe
2) =0.15 results of Ye et al.17 can be envisioned. The notorious
e =0.16
h/ peak that they observed along the (0,0)−(π,π) direc-
(
ρ tion might very well be due to the existence of the new
1 state reported here that lies in parameter space in be-
10
tween the FM and C E states, in the phase diagram
x 1−x
varying J . It is clear from our S(k) calculations for
AF
the intermediate state that, just like as in the neutron-
scatteringresultsofYeetal.,thepeaklocatedinbetween
0
10 8x8 the (0,0) and (π,π) peaks has a tendency to move for-
λ=1.3, x=1/4 wardinthe(π,π)directionastheholedopingisincreased
from x= 1/4 to x= 1/2. While it was quite reasonable
toexplaintheneutronscatteringresultswithshort-range
0 0.02 0.04 0.06 0.08 0.1
incommensurate orderings as was done in Ref. 17 using
T/t
aa inhomogeneousstates,therelevantphysicsmightalsobe
explainedvia long-rangeorderingusing the intermediate
FIG.5. (color online)Resistivityvs. temperatureatx=1/4
states found in the present manuscript. Further work
doping and moderate electron-lattice coupling (λ = 1.3)
is needed to better clarify this important aspect of the
parametrized as a function of the superexchange coupling
interpretation of the neutron results.
JAF. A canonical CMR is observed at this doping. The ar-
rows indicate the Curie temperature for ferromagnetism at
each valueof JAF. This result is reproduced from Ref. 19.
V. CONCLUSIONS AND SUMMARY
All these states could appear in real crystals since they In this manuscript, the region of the phase diagram
are all energy degenerate, and the average S(k) calcu- wherepreviousinvestigations19 unveiledaclearCMRef-
latedbyaveragingoverthemcouldbedirectlycontrasted fectincomputersimulationshasbeenrevisited,withem-
against the results of neutron scattering experiments. phasis on the detailed properties of the insulating state
InFigure6,thespinstructurefactorsforvariousstates that competes with the double-exchange induced FM
are shown in the first quadrant of the Brillouin zone. metalto generatesucha CMR effect. In agreementwith
For the previously studied CxE1−x states, the dominant thosepreviousinvestigations,this competinginsulatoris
peak at (π/2,π/2) evolves into other peaks along the confirmed to have “CE” characteristics with an associ-
(π,0)−(0,π) direction as x is varied.22 Thus, if these ated magnetic peak in the spin structure factor S(k) at
CxE1−x states are searched for with neutron scattering, wave-vector (π/2,π/2) for hole density x = 1/4. How-
thediagonal(π,0)−(0,π)mustshowmoreintensitythan ever, this insulating state was found to have an unex-
the opposite one (0,0)−(π,π). pected and far more complex structure, since, in addi-
The new states FM-CE found here display interesting tionto CE regions,italso containsFM regions,with the
behaviorthatmeritsfurtherdiscussion. Thesestatesnot combination having diagonal stripe characteristics that
only have peaks along the (π,0) − (0,π) direction, in- alternate from CE to FM. This exotic structure gener-
duced by the CE component, but also along the other atesextrapeaksinS(k) dueto the FMcomponent. The
diagonal (0,0)−(π,π) direction (although not with the standardx=1/4CEstate,withoutFMregions,isstabi-
same intensity along the two diagonals). This is to be lizeduponfurtherincreasingthesuperexchangecoupling
expected, since the new states possess properties inher- betweenthe t spins. Inotherwords,hereitis reported
2g
ited from both the FM state and the C E states. that at x = 1/4 and in a narrow region of parameter
x 1−x
While the peaks along the (π,0) − (0,π) state are at- space, there is a new unexpected phase between the FM
tributed to the C E characteristics, the peaks along metal and the CE insulator, which is a mixture of those
x 1−x
the(0,0)−(π,π)couldsimilarlybeattributedtotheFM two states. It appears that this new state is the com-
characteristics (with (0,0) being the characteristic mo- petitor of the FM metallic phase that causes the CMR
mentumofalong-rangeuniformFMstate). Theseresults effect in the Monte Carlo computer simulations. Note
are actually similar to those reported by Ye et al. us- that the present results are obtained in the clean limit,
ingneutronscatteringexperimentsappliedtosingle-layer i.e. without quenched disorder so this new state with
manganites.17 Those experimental results were the first mixed characteristics is intrinsic of the standard model
to identify the existenceofpeaksalongthe (0,0)−(π,π) formanganites(althoughitappearsinanarrowregionof
direction in a single-layer manganite Pr Ca MnO , parameterspace). Also note thatallthe previousresults
1−x 1+x 4
7
FIG. 6. (color online) The spin structure factors for the various states at the densities indicated in the figure using a 24×24
lattice. For each case, various rotation and reflection operations are used to symmetrize the states, as explained in the text,
for better comparison with potential neutron scattering results.
presentedinRef. 19 with regardsto transportandCMR analysis must be carried out only using clusters where
effects are unchanged and confirmed, the only modifica- the state under discussion fits properly, otherwise frus-
tionbeing the characterizationof the insulating compet- tration effects will complicate the study. The present
ingstatethatisfoundtobeamixtureFM-CEasopposed status of computer simulations of models of manganites
to just a CE-like state. prevents such an analysis to be carried out because of
There are a few caveats that the reader should be the rapid growth of the computer effort as N4, with N
alerted about with regards to the new results presented the number of sites, due to the fermionic diagonaliza-
in this paper. First, states as complex as the one un- tions needed. Thus, the present efforts are restricted
veiledherewithaFM-CEmixtureinanalternatingdiag- to two lattice size only, but it is certainly reassuring to
onal striped pattern are difficult to analyze with regards find out that in both of these two clusters the FM-CE
to finite size effects. The reason is that a size scaling state is stable. Second, the study carried out here has
8
been performed using two-dimensional clusters, since a hinted toward an even higher degree of complexity with
three-dimensional study with the new complex state fit- similar patterns that are repeated in a variety of ways
ting properlyinside the studied lattice cannotbe carried along, e.g. one of the diagonals.22,23 The present results
outatpresentagainduetotherapidgrowthoftheeffort add a new layer of complexity since now the diagonal
withthenumberofsites. Then,whileourworkinprinci- stripes alternate between having FM and CE character-
ple does apply to single-layermanganites, it is unclear if istics. This new state far from being a pathology ap-
they can be extrapolated to three dimensions assuming parently is the key element to induce the famous CMR
a stacked arrangement of the new state. For this reason effect, at least in the computer simulations using finite
is that our predictions for neutron scattering have not clusters reportedhere andin Refs. 18 and19, and in the
been comparedagainstavailableneutronscatteringdata clean limit. Thus, a new avenue may have been opened
for the three-dimensional manganites, such as LCMO, in the study of manganites that hopefully will locate us
but it was contrasted only against single-layer neutron closer to understanding these materials and the associ-
results. Third, in this and related efforts it is often as- atedCMReffect. Thepresentefforthasrevealedanovel
sumed that screening effects are sufficient to suppress unexpectedstate that wasnotenvisionedbefore, since it
long-range Coulombic forces. This is usually a reason- emergesfroma nontrivialcompetition ofseveraltenden-
able assumption widely used before in model Hamiltoni- cies.
ans for manganites,cuprates, and other transitionmetal
oxides. However, the new FM-CE state has nanoscopic
regions where the electronic densities seem to be differ-
ent, i.e. the FM and CE regions do not have the same
VI. ACKNOWLEDGMENT
averageelectronicdensity. Thus,anotherissuethatmust
be studied in future efforts is the stability of the present
results against long-range Coulomb repulsion.26 This work was supported by the National Science
Withthecaveatspreviouslydescribed,the resultspre- Foundation under grant DMR-11-04386. The com-
sentedheresuggestadegreeofcomplexityinthestudyof putational effort was supported in part by the Na-
manganite models that is well beyond previous expecta- tional Science Foundation through Teragrid resources
tions. ThemuchstudiedCEstateatx=1/2withsimul- under grant number TG-DMR110033. The compu-
taneousspin/charge/orbitalorderinzigzagarrangements tations were performed on Kraken (a Cray XT5)
wasalreadyconsideredratherexoticinthecomplexoxide at the National Institute for Computational Sciences
context. The recent theoretical efforts unveiling highly- (http://www.nics.tennessee.edu/). This research
degenerate states involving diagonalstripes already sug- usedtheSPFcomputerprogramandsoftwaretoolkitde-
gested that this simple picture may be incomplete and veloped at ORNL (http://www.ornl.gov/~gz1/spf/).
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