Stabilization of A-type layered antiferromagnetic phase in LaMnO_3 by cooperative Jahn-Teller deformations

8 Pages·0.21 MB·English

by  M. Capone

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0
Stabilization of A-type Layered Antiferromagnetic Phase in
0
2 LaMnO by cooperative Jahn-Teller Deformations
3

n
a
J M. Capone1 , D. Feinberg2 and M. Grilli3

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1 1 IstitutoNazionalediFisicadellaMateriaandInternationalSchoolforAdvancedStudies(SISSA-ISAS),ViaBeirut2-4,34013
Trieste, Italy
] 2 Laboratoire d’Etudesdes Propri´et´es Electroniques des Solides, Centre National dela RechercheScientifique,associated with
l
e Universit´eJoseph Fourier, BP 166, 38042 Grenoble Cedex 9, France
- 3 IstitutoNazionalediFisicadellaMateriaandDipartimentodiFisica,Universit`adiRoma”LaSapienza”,PiazzaleAldoMoro
r
t 2, 00185 Roma, Italy
s
.
t Received: date/ Revised version: date
a
m
- Abstract. It is shown that the layered antiferromagnetic order in stoechiometric LaMnO3 cannot be un-
d derstood purely from electronic interactions. On the contrary, it mainly results from strong cooperative
n Jahn-Teller deformation. Thoseinvolveacompression of theMn−O octahedron along thec−axis(mode
o Q3 <0), while alternate Jahn-Teller deformations occur in the ab−plane (mode Q2). These deformations
c stabilize acertain typeof orbital ordering. Theresulting superexchangecouplings are calculated byexact
[
diagonalization, taking into account both eg and t2g orbitals. The main result is that antiferromagnetic
1 (ferromagnetic) coupling along the c−direction (ab−planes) can be understood only if the Jahn-Teller
v energyismuchlargerthanthesuperexchangecouplings,whichisconsistentwithexperiments.Thismech-
3 anismcontrastswiththatbasedonweakJahn-Tellercouplingwhichinsteadpredictselongation alongthe
4 c−axis (Q3 >0). The crucial role of the deformation anisotropy Q2 is also emphasized.
Q3
2
1 PACS. 71-70.E Jahn-Teller effect – 75-30.ET Exchangeand superexchangeinteractions
0
0
0 1 Introduction still lacking. Especially the dramatic dependence of all
/
t physical properties with very fine tuning of the chemi-
a
Perovskite oxides containing Mn ions have been the ob- cal composition requires a precise estimate of the various
m
jectofintenseinterestintherecentyears.Inspiteofbeing parameters,andclearidentificationofthedominantmech-
-
known for a very long time, these compounds have been anism for every doping. Surprisingly enough, such an un-
d
n reconsideredingreatdetailowingtotheircolossalmagne- derstanding is not yet reached in the insulating antiferro-
o toresistive properties. Starting from the ”parent” phases magnetLaMnO3,althoughitseemsessentialbeforequan-
c LaMnO (trivalent Mn) and CaMnO (tetravalent Mn), titatively studying the doped phases. This phase, when
3 3
v: substitutionaldopinghasrevealedanextremelyrichphase fully stoechiometric, presents a layered antiferromagnetic
i diagram.Understandingthisdiagramrequiresatleastthe order, with ferromagnetic couplings (F) in two directions
X
following ingredients : i) strong on-site Coulomb interac- andantiferromagnetic(AF)couplinginthe other[7].The
r tions;ii)the ”doubleexchange”mechanismdue tothe in- AFdirectionsareassociatedtoashorteningoftheMn O
a −
terplayofe electronitineracyandHund’s exchangewith bonds, leading to tetragonaldistortion, while in the F di-
g
the more localized t electron spins, which favours fer- rectionslongandshortbondsalternate,yieldingtheover-
2g
romagnetism [1,2,3]; iii) superexchange between t elec- all orthorhombic structure. In what follows, we shall ne-
2g
tronsaswellasbetweene electronsonneighbouringsites; glectthetiltingoftheMn O octahedraandconcentrate
g
−
iv) large electron-lattice interactions, in particular due to onlyontheMn O bondlengthdeformations.Thesecan
Jahn-Teller (JT) effect on Mn3+ ions [4,5]. All these ele- be understoodin−terms ofcooperativeJT effect. The cor-
ments are necessary to understand the interplay between responding lifting of e degeneracy can be viewed as an
g
spin, charge and orbital ordering. The latter lifts the de- orbitalordering,with occupied d orbitals pointing prefer-
generacy of the e orbitals by a cooperative Jahn-Teller entially in the directions of long Mn O bonds.
g
−
lattice deformationandleads totetragonalororthorhom- Several proposals have been made to explain layered
bic deformations of the cubic structure. antiferromagnetisminLaMnO .Goodenough[6]usedthe
3
Although Goodenough [6] provided long time ago a pictureof”semi-covalence”whereoxygenorbitalsplayan
qualitative understanding of the phase diagram of the essentialrole in overlappingempty d orbitals of Mn ions.
(La,Ca)MnO family, a full microscopic description is Thispicture,althoughusefulforqualitativepurposes,has
3
2 M. Capone et al.,:Layered AFordering from Jahn-Teller deformations in LaMnO3
notreceivedconfirmationbymicroscopiccalculationsand tenpercent,indicatingthatǫ>k T.Ontheotherhand,
B
doesnotallowtowritesimpleenoughmodels,forinstance neutron scattering measurements show that the local dis-
based ona Hamiltonian involving only metal orbital elec- torsions persist above the orthorhombic-cubic transition
trons and their basic interactions. A microscopic descrip- at750K [12].Thistemperatureonlymarksthedisappear-
tion requires to identify clearly the dominantinteractions ance of cooperative JT ordering, while distorted MnO
6
in the problem. In pioneering works, Kugel and Khom- octahedra still exist at higher temperatures. Photoemis-
skii [8], and Lacroix [9] (see also earlier work by Roth sion [13] measurements indicate that JT splittings are as
[10]), proposed that superexchange in the presence of e large as a few tenth of eV, comparable to the electronic
g
orbital degeneracy results in ferromagnetism and orbital hopping integrals between neighbouring sites. And opti-
ordering : Hund’s rule favours in this case different or- cal conductivity analysis [14] also shows evidence of large
bitals on neighbouring sites and ferromagnetic coupling. splittings.
Using a simplifiedmodelwith equalhoppingintegralsbe-
Intheseconditions,thedegenerateperturbationcalcu-
tween e orbitals leads to the same ordering along the
g lationofRef.[8]doesnotholdanymore.Inapreviouswork
three cubic lattice directions : the resulting structure is
wehavereconsideredtheproblemwihinperturbationthe-
an insulating ferromagnet, with ”antiferroorbital” order-
ory [15], making the opposite assumption, i.e. ǫ >> t2:
ing. However, taking properly into account the hopping U
This means that, given the crystal deformations, due to
integralsbetweendx2−y2 (denoted x) anddz2 (denoted z)
strong cooperative JT effect, the e orbitals split so as to
orbitals, Kugel and Khomskii [8] found the correct mag- g
give a certain type of orbital ordering. The orbitals sta-
netic structure.Startingwithdegeneratee orbitals,they
g
performedaperturbativecalculationin t and JH wheret, bilized at each sites are different from the one predicted
U U by pure superexchange. We have found that, depending
J andU arethe typicalhoppingintegral,the Hundcou-
H
on the values of the two JT modes Q and Q , different
pling and the on-site repulsion in the order.Based on the 2 3
magnetic ordering could be stabilized, among which the
weak electron-lattice coupling in the compound KCuF ,
3 layered ”FFA”. This ordering is always stabilized if the
they considered the JT couplings as a perturbation. As
Q mode is positive, e.g. for dilatation in the c-direction.
a result, orbital and magnetic ordering result from su- 3
But in the real case Q < 0, FFA order is realized only
perexchange (SE) only: Intraorbital SE dominates in the 3
if the in-plane alternate Q mode is sufficiently large and
c-direction(definedasthez-axis),leadingtoAFcoupling, 2
overcomesthecontraryeffectofQ .Lookingatstructural
whileinterorbitalSEdominatesintheab-directions,yield- 3
numbers, one checks that this is actually the case. Nev-
ing F coupling. Occupied orbitals are dominantly dz2−x2 ertheless the system is close to the point where the FFA
and dz2−y2, therefore, as Kugel and Khomskii remark, order becomes unstable towards FFF. This results in the
for Cu2+ in KCuF (hole orbital), JT coupling implies
3 F exchange (along the ab-plane) being larger than the
a shortening as experimentally observed (c/a < 1). How-
AFexchange(alongthec-axis).Thisfeaturehasbeenob-
ever,forMn3+ ionswithoneelectroninthee levels,they
g tainedfrominelasticneutronscattering[16],anditcannot
correctly point out that repulsion between metal and an-
beexplainedbytheKugel-Khomskiimodel,whichobtains
ion orbitals, together with JT coupling, would trigger a
on the contrary that the F superexchange is of order JH
lengthening of the c-axis (c/a>1), in contradiction with U
times the AF one, thus much smaller.
the actual structure. In a recent work, Feiner and Oles
[11] reconsidered Kugel and Khomskii’s model, including Theinterplaybetweenlatticedistortionsandmagnetism
both Hund’s coupling betwen eg and t2g orbitals and the has also been investigated from ab-initio calculations of
antiferromagnetic superexchange interaction between t2g the electronic structures [17,18,19]. All conclude with a
spins (equal to 3 in the ground state). Their results con- prominent role of those distortions to stabilize the actual
2
firms those ofRef.[8]: They find the correctlayeredstruc- magnetic order. In particular, Solovyev et al. [18] have
ture (whichthey callMOFFA),but onlyif thedz2 orbital found that the c-axis exchange is antiferromagnetic only
has lower energy than the dx2−y2 one, contrarily to what iftheJTdistortionissufficientlylarge.ForLaMnO3with
happensforelectron-likeorbitals(caseofLaMnO ).This its very large distortionthey obtainthe layeredantiferro-
3
contradictionsetsthelimitsoftheKugel-Khomskiimodel magnetic structure, but it is close to the border between
for LaMnO . We believe that the JT effect, on the con- FFA and FFF phases. Very recent Monte Carlo calcu-
3
trary,hastobe consideredfromtheverybeginninginthe lations have also demonstrated the relevance of the JT
model. interaction in stabilizing the FFA magnetic order [20].
Essentially, the assumption that the eg degeneracy is In the present work, we reconsider the problem, be-
lifted principally by superexchange may be justified in yond any perturbation theory, by exact diagonalizations
KCuF3,butis definitely notcorrectinLaMnO3.Infact, onpairsofMn3+ sites.Thetwoeg orbitalsareconsidered
this couldholdonlyifthetypicalJTsplitting ǫwasmuch togetherwiththequantum 3-spinsduetotheelectronsin
smallerthanthesuperexchangesplitting,oforder t2.The the t levels. Our conclusio2ns confirm the essential role
U 2g
latter (related to the magnetic transition temperatures) of JT deformations, especially the Q mode, to stabilize
2
beingoftheorderofafewmeV,theformerismuchlarger. the layered AF order. They also demonstrate that it is
Although there is no precise evaluation of this quantity, essentialtoincludeHund’scouplingwitht orbitals,and
2g
thisissupportedbyexperiment:Ontheonehand,thede- thattheroleoftheintrinsict AFexchangeistoslightly
2g
formations of Mn-O bonds is extremely large, more than stabilize the FFA order with respect to the FFF one.
M. Capone et al.,:Layered AFordering from Jahn-Teller deformations in LaMnO3 3
2 The Model l s), the m bonds lying in the z direction and the s,l
−
ones in the x,y planes.
¿From the discussion of the preceding section it is clear Since in the present work we will not attempt to per-
that the basic physical ingredients requiredfor a satisfac- formany energyminimization by including the elastic in-
tory description of the Manganites should involve both teractions due to the lattice, we here disregard these en-
Coulomb and lattice (namely JT) interactions. Accord- ergytermsbytreatingtheJTdeformationsQ=(Q2,Q3)
ingly we consider the following model asexternalfieldsimposedbyalatticeorderinginvolvinga
much higher energy scale than the magnetic ones. There-
H =Ht+HH +HUU′ +HJ +HJT (1) fore in the following the various magnetic couplings will
be determined in terms of assigned lattice deformations.
with This viewpoint, which already guided us in the perturba-
tiveanalysisofthestabilityofFFAantiferromagnetismin
Ht = − X taαα′c†iασci+aα′σ theundopedLMO [15]is definitely justifiedbythe exper-
iaαα′σ imental observation that the JT energy splitting is much
HH = −JH X c†iασsσσ′ciασ′ largWeretehxaanctallyl mdiaaggnoentailcizceouthpelinHgasm. iltonian in Eq. (1) for
iασσ′
asystemoftwositeswithopenboundaryconditions.The
  two sites are located either on the same xy plane or on
× Si+ X ciα′σ˜sσ˜σ˜′ciα′σ˜′ adjacentplanesandthe suitablehoppingmatrixelements
 α′6=ασ˜σ˜′  betweenthevariousorbitalshavebeenconsideredaccord-
HUU′ = UX(cid:16)c†iα↑ciα↑(cid:17)(cid:16)c†iα↓ciα↓(cid:17) ingTtoheexJpTreesnseiorgnyss(p2l)i.ttingǫ=g Q2+Q2andthedefor-
iα p 2 3
mationanisotropyratior Q /Q aregivenexternalpa-
+ U′ X (cid:16)c†iασciασ(cid:17)(cid:16)c†iα′σ′ciα′σ′(cid:17) rameters and are fixed for≡any2diag3onalizationprocedure.
iα6=α′σσ′ Once the ground state is found, the effective exchange
coupling between the total spins on the two sites can be
H = J S S
J tX i· j determined. Specifically, since the Hamiltonian conserves
hiji
the total spin of the two-site cluster, we determine the
HJT = gX(cid:16)c†iαστ(3)αα′ciα′σQ3i+c†iαστ(2)αα′ciα′σQ2i(cid:17), ground states with total spin ST = 4,MST = 4 and
S = 3,M = 3. Then the magnetic coupling is given
i T ST
by the energy difference E(S = 4,M = 4) E(S =
The first term represents the kinetic energy with the 3,M =3)=2J.OncethemTagneticcSoTuplings−(andpTar-
ST
electrons in the Manganese 3dx2−y2 (α = x) or 3d3z2−r2 ticularlytheirsign)alongthevariouslatticedirectionsare
(α=z) orbitals hopping from site i to the nearest neigh- found, the resulting magnetic phase is also determined.
bor (nn) site i + a in the a lattice direction. Here s is
the vector of Pauli matrices for spins and τ the vector of
Pauli matrices for orbital pseudospins in the x,z basis.
3 Results
Specifically,forastandardchoiceofthe phasesfortheor-
bital wavefunctions,the hopping betweenthe x and the z
orbitals are given by In order to gain insight from the physical processes un-
derlying the intersite magnetic couplings, we first carry
tx,y = 3t; tx,y = t; out a comparison between the results of the perturbative
xx zz − analysis of the superexchange interactions (see Ref. [15])
tx = √3t; ty =√3t
xz − xz and the exact numerical calculations. The perturbative
tzzz = −4t tzxx =tzxz =0. (2) analysisnotonlywasperformedassumingverylargelocal
Coulomb interactions (U,U′ and J much larger than t),
H
Together with the Hund coupling givenby HH the ki- but the additional assumption was made that the JT en-
neticenergygivesrisetotheusual“double-exchange”itin- ergy splitting ǫ greatly exceeds the typicalsuperexchange
erancy of the eg electrons. The (strong) on-site Coulomb energy scale of order t2/U. In this way the ground state
interactions, are represented by the intraorbital repulsion can safely be assumed to be formed by just one singly
U and by the interorbital U′ =U −2JH term. occupied eg level. Accordingly the exact numerical cal-
For simplicity here and in the following we will not culations to be compared with the analytic results have
distinguish between the Hund exchange energy between been performed for Q < 0 and t = 0.2eV, U = 8eV,
3
electronsintheeg andt2g orbitals.Theantiferromagnetic JH = 1.2eV, ǫ = 0.4eV, and Jt = 0. Fig. 1 reports the
superexchange coupling between neighboring t2g spins is superexchange interactions both in the planar and inter-
consideredwithHJ,whiletheJTinteractionbetweenthe planardirectionsobtainedwithboththeperturbativeand
eg electrons and the (cooperative) lattice deformation is the exact-diagonalization analysis. As it is apparent, the
given by the last term HJT. The Jahn-Teller modes are perturbative Jxy and Jz display the same qualitative be-
definedintermsoftheshort(s),medium(m)andlong(l) havior as in the exact calculation. This confirms that, at
Mn O bonds by Q =√2(l s) and Q = 2/3(2m leastinthe ǫ t2/U limit,asubstantialpartofthe mag-
− 2 − 3 p − ≫
4 M. Capone et al.,:Layered AFordering from Jahn-Teller deformations in LaMnO3
Fig. 1. Magnetic couplings for Q3 <0, t=0.2eV, U =8eV, Fig.2.MagneticcouplingsJxy(solidline)andJz (dottedline)
JH = 1.2eV, ǫ = 0.4eV, and Jt = 0 as function of the defor- vs. the deformation ratio r = Q2/|Q3| for negative values of
mation anisotropy ratio from exact diagonalization and per- Q3, for t = 0.14eV, U = 6eV, ǫ = 0.3eV, Jt = 2.1meV and
turbationtheory.Dashedline:Jxy from exactdiagonalization; JH =1.2eV (JH =0.9eV) in the upper(lower) panel.
Dot-dashedline:Jz from exactdiagonalization; Solid line:Jxy
from perturbation theory; Dotted line: Jz from perturbation
theory. Negative (resp. positive) values indicate ferro (resp.
antiferro) magnetic interactions.
netic effective interactions is generated by the superex-
change processes due to the hopping of electrons lying
in the lower e level on the same or on different nearest
g
neighbor e orbitals. On the other hand, the quantitative
g
comparison indicates that the range of stability for the
FFA phase (i.e. J < 0 and J > 0) is modified. In fact
xy z
a positiveJ togetherwith a negativeJ areobtainedin
z xy
the exact calculation on a somewhat larger range of lat-
ticedeformationanisotropies(Q /Q 2.5).Inorderto
2 3
| |≥
establisha tighterconnectionbetweenthe experimentally
determinedJ’s and the observeddeformations,andto in-
vestigatethe roleofthe variousinteractionsinthe model,
a more systematic analysis is required. Assuming the JT
interaction to be relevant in stabilizing the FFA phase,
weinvestigatethe behaviorofthe exchangeconstantsJ
xy
(denoted ”intraplane”) and J (denoted ”interplane”) in
z
terms of ǫ and the deformation ratio r. Fig. 3. Magnetic couplings (see Fig. 2) vs. the deformation
Figs. 2 and 3 report Jxy and Jz as functions of r for ratio r = Q2/|Q3| for negative values of Q3, for t = 0.14eV,
| |
the Q3 < 0 case (the one relevant for LMO) at a large U = 6eV, ǫ = 0.01eV, Jt = 2.1meV and JH = 1.2eV (JH =
(ǫ 2t) and at a small (ǫ < 0.1t) value of the JT split- 0.9eV) in the upper(lower) panel.
∼
ting respectively. Different values of the Hund coupling
J areconsidered. OnecanfirstobservefromFig.2that
H
inthelarge-ǫcasetheincreaseoftheHundcouplingshifts competing with the generated F interaction
downwards both the intraplane and the interplane mag-
C
neticcouplings.Thisoutcomecanberationalizedinterms JF xy,z . (4)
of perturbatively generated superexchange processes pro- xy,z ≈−U +ǫ (5/2)JH
−
viding AF effective couplings of the form
Thenumericalcoefficients A,B,andC stemfromthe dif-
A B ferent hopping matrix elements between the different or-
JAF xy,z + xy,z (3)
xy,z ≈ U +(3/2)J U +ǫ bitals in the different directions [15]. Specifically, while
H
M. Capone et al.,:Layered AFordering from Jahn-Teller deformations in LaMnO3 5
the A’s are related to the hopping processes between two
nearest-neighbourlower-lyinge orbitals,theB andC co-
g
efficients are due to hoppings between one low-lying and
onehigherJT-splitorbitals(thisiswhythecorresponding
denominators involve ǫ). The A,B and C coefficients are
independent from the Coulomb interactions, which only
determinetheenergiesofthevirtualintermediatestatesin
thesuperexchangeprocesses.Theaboveschematicexpres-
sions clearly show that, when J is increased for a fixed
H
ǫ t, the F spin configurationbecomes more favourable,
≫
sincethe Fcouplingbecome stronger,while the AFinter-
action weakens. We remark that purely electronic models
such as in Refs.[8,11] make use of degenerate perturba-
tion theory. Then the orbital splitting is of order of the
exchange couplings J and therefore those models become
invalid if ǫ > J, which is the case in LaMnO . More se-
3
riously, the orbital order resulting from purely electronic
interactionsisatodds withthatobtainedfromthe actual
Jahn-Teller distortions, showing that those distortions do
notresultfromanorbitalorderingofelectronicorigin,but
are on the contrary the mere source of orbital ordering.
Fig. 4. Magnetic couplings vs. the JT splitting energy ǫ for
Another quite generic effect, which can be interpreted
t = 0.14eV, U = 6eV, Jt = 2.1meV for Q2/Q3 = −3 (upper
in terms of perturbatively generated superexchange pro- panel) and Q2/Q3 =3 (lower panel). Solid line: Jxy for JH =
cessesisthetendencyofJz toacquireaF(oratleastaless 1.2eV; Dotted line: Jz for JH = 1.2eV; Dashed line: Jxy for
AF) character at low values of Q2/|Q3| (this can also be JH =0.9eV; Dot-dashed line: Jxy for JH =0.9eV.
accompanied by an upturn of J for r tending to zero).
z
| |
This occurs because for Q < 0, the lowest e level pro-
3 g
gressively loses its 3d3z2−r2 component: By schematically is not so transparent. However, the effect of JH favoring
writingthelowerandtheuppere statesas a x +η z
g ferromagnetism is still present.
| i∝| i | i
and b η x + z respectively,ηvanisheswith r 0.
| i∝− | i | i | |→ An important difference betweenthe results in Figs.2
Now,thesuperexchangealongzisdrivenbytheinterplane
and 3 is that the FFA phase is generically obtained in a
hopping, which is only allowedbetween 3d3z2−r2 orbitals. broad range of parameters when ǫ t. In particular, for
Furthermore one can see [15] that the ferromagnetic su- ≫
rather realistic values of J 5t 1eV the deformation
perexchange arises from a b hoppings, which are H ∼ ≈
| i → | i ratios required to generate negative (i.e. F) couplings in
of order η, while the antiferromagnetic coupling is mostly
thexy planesandpositiveonesinthezdirectionarequite
generatedfromintraorbital a a hopping(theAterm
in Eq.(3). Since this latter|isio→f or|dierη2, itis quite natu- reasonable|r|∼2−3.Thesamedoesnotholdinthe case
ofsmallJTsplitting,whereJ andJ havethesamesign
ral that in the low-r region,as η decreases,the superex- xy z
| | (FFF).Thereforeafirstresultisthatasizableǫisneeded
changealongz isferromagneticandvanisheswithη.This
in order to obtain both the FFA phase and reasonable
ferromagnetic tendency is, however, contrasted (and ac-
lattice distortion ratios Q /Q .
tually overcome in Figs. 2 and 3) by the independent AF 2 3
This result is also confirmed by the calculation of J
superexchange J between the t spins, which becomes xy
t 2g
and J as a function of ǫ, at a fixed value of the defor-
relatively more important. Of course, when r increases, z
the η2 terms in the hopping become relevant,|t|he intraor- mationanisotropyratio r. Figs. 4 and 5 report the values
of J and J for r = 3 and r = 3 respectively. While
bital a a hopping starts to dominate and Jz even- xy z −
| i → | i the positiver caseis genericforperovskitematerialswith
tually becomes (more) positive (i.e. AF).
the lattice elongated in the z direction (c/a>1), the lat-
As far as the superexchange along the planes is con-
ter choice is more pertinent to the case of the undoped
cerned,at small r this is instead dominated by the large
LMO, where c/a<1. As already discussed by Kugel and
| |
hopping between 3dx2−y2 orbitals, which favor the a Khomskii[8]foradifferentmodelandasconfirmedbythe
| i →
a hopping and, consequently produces an AF magnetic
perturbativeanalysisofRef.[15],theJTdeformationand
| i
coupling. On the contrary, for large r, orbital ordering
the superexchange interactions cooperate when Q > 0
| | 3
implies that the main superexchange contribution comes
like in KCuF so that it is not surprising that for all val-
3
from hopping between different orbitals, thus favouring
ues of J the FFA is realized over a much broader range
H
ferromagnetism [8].
of ǫ. On the other hand, for Q < 0, Fig. 4 shows that
3
All the above arguments are obviously only valid as the conditions for a FFA phase, J < 0 and J > 0,
xy z
long as the conditions for the perturbation theory nearly are only realized for a smaller range of ǫ values. In par-
hold.Ontheotherhand,thesimpleperturbativeapproach ticular a sizeable minimum value of ǫ is required to have
between non-degenerate states breaks down when ǫ an AF coupling along z, while exceedingly large values of
t2/U as in Fig. 3 and the interpretation of the resul≈ts ǫ (of order J ) produce an AF coupling also along the
H
6 M. Capone et al.,:Layered AFordering from Jahn-Teller deformations in LaMnO3
planes. Both the minimum and the maximum values of ǫ
forobtainingtheFFAphaseincreaseuponincreasingJ .
H
However, the maximum value of ǫ increases more rapidly
and the overalleffect is that, increasing J , the available
H
range in ǫ to obtain an FFA phase is enlarged. Again the
behavior displayed in the exact calculations reported in 3
Fig. 4 can easily be interpreted in terms of the pertur-
2
bative superexchangeprocessesschematicallyrepresented
in Eqs. (3) and (4). First of all these expressions at once 1
accountfor the increasing behaviorof the couplings upon
0
increasing ǫ: While only the interorbital part of J (the 0 1 2
AF
contributionproportionaltoB)decreasesuponincreasing
ǫ, the whole ferromagnetic part in Eq. (4) is suppressed
when ǫ grows, so that the total coupling, although ferro-
magneticatsmallJTenergysplitting,eventuallyvanishes
and becomes positive.
Moreover it turns out that, for r > 2 3 the hop-
| | −
pings generate smaller A,B,C coefficients in the z direc-
tion. This accounts for the more rapid rise of J when ǫ
z
is increased. Finally, along the same line of the discus-
sion of Fig. 2, one can easily observe that an increasing Fig. 5. Zero-temperature phase diagram for t=0.14eV, U =
JH strenghtenstheferromagneticcomponentandweakens 6eV, Jt =2.1meV and JH =1.2eV.
the antiferromagnetic one, thus rationalizing the generic
tendency of all curves to be shifted downwards when J
H
grows.
Besides the above specific findings, the occurrence of
the various magnetic phases can be cast in a phase dia-
gram at zero temperature illustrating the stability region
of these phases in terms of the JT energy splitting and
the deformation ratio. In the light of their richer com-
plexity and of the present interest in the Manganites, we
here consider in greater detail the case of Q < 0 of rel-
3
evance for the undoped LMO, while the Q > 0 case is
3
only described in the inset of Fig. 5. Figs. 5 and 6 report
the phase diagram for two different values of the Hund
coupling. Both phase diagrams display the same qualita-
tive features. In particular, at moderate and large values
of ǫ a N´eel AAA phase is found for weak planar distor-
tions (small r). As seen in the discussion of Fig. 2, in the
very-small-r region, J is naturally positive, while the
xy
superexchange between e levels along z, although ferro-
g
magnetic, is small so that the direct superexchange be-
tween t spins may easily dominate and gives rise to the
2g
AAA phase (see Figs. 2 and 3). As it can be also be seen
Fig. 6. Zero-temperature phase diagram for t=0.14eV, U =
from Fig. 1, it can be checked that the AAA phase is
replaced by the so-called C-like antiferromagnetic AAF 6eV, Jt =2.1meV and JH =1.0eV.
phase in the J = 0 case. At small-to-intermediate values
t
ofǫ,aprogressiveincreaseof r drivesthesystemtowards
| | A-type antiferromagnetism FFA experimentally observed
the phase AAF. In this phase J keeps its AF character,
xy
in undoped LMO.
while the negative superexchange between e levels along
g
z is small, but no longer is overcome by J . At larger val- At smallvalues of the JT splitting,the phase diagram
t
ues of ǫ the AAF phase is not present, but the intimate isprominentlyoccupiedbyaFFFphase.Inthis latterre-
nature of the AAA phase changes upon increasing r. In gard,fromthecomparisonofFigs.5and6,theimportant
particular while at low r the AF along z is determ| i|ned observation can be done that the FFF phase at low and
by J , at larger r, the|su|perexchange between e levels moderate ǫ’s is greatly stabilized by the increase of the
t g
along z is itself A| F| and therefore the t2g superexchange Hund coupling JH, as previously expected.
contributes,butitisnotstrictlynecessarytotheAFcou- Within the present exact numerical treatment of the
pling along z.On the other hand, a further increase of r modelinEq.(1)itisalsopossibletoattemptat“precise”
| |
promotes a F coupling along the planes and leads to the estimatesofJ andofJ .Asanexample,wereporthere
xy z
M. Capone et al.,:Layered AFordering from Jahn-Teller deformations in LaMnO3 7
a realistic sets of parameters (among many others) pro- imposes them as external parameters of the calculation.
viding the values J = 0.83 meV and J = 0.58 meV Actually we do not believe that suchdeformationscanbe
xy z
−
experimentally observedwith inelastic neutron scattering easily determined by microscopic models, which should
[16]. Assuming Q /Q = 3.2, a value largely confirmed incorporate complex effects such as long-range Coulomb
2 3
| |
by many groups [12], we take t = 0.124eV, U = 5.81eV, interactions, cation and anion size and tilts of the MnO
6
J = 1.2eV, J = 2.1 meV, and ǫ = 0.325eV. The quite octaedra.On the other hand realistic deformations as ob-
H t
reasonable values of the model parameters needed to re- tained from experiments can easily be imposed and the
produce the measured magnetic couplings is an indirect consequent local electronic structure can be determined
test of the validity of the considered model. We empha- exactly: orbital ordering results essentially from coopera-
size that the ”anomalous” trend J > J is correctly tive Jahn-Teller deformations.
xy z
| | | |
reproduced, and that our fit is relatively flexible concern- Moreover,andquiteimportantlyforaquantitativede-
ing parameters U, JH or t, provided ǫ is large enough. termination of the magnetic coupling and of the stability
ofthemagneticphases,weherealsotakeintoaccountthe
electronic Coulombrepulsion. This interactionis perforce
4 Conclusions largerthantheJT interactionandcontributestoits insu-
lating behavior as well as to the numerical values of the
Inthispaperwepresentedtheresultsofcalculationsbased exchange couplings.
ontheexactdiagonalizationofamodelaimingtodescribe Finally we showed that using reasonable parameters
the stoechiometric LaMnO3. The model includes strong theexperimentalvaluesofthemagneticcouplingscaneas-
local Coulomb interactions as well as a JT coupling be- ily be reproduced. Of course precise estimates depend on
tween the electrons and the Q2 and Q3 lattice deforma- theknowledgeofthevariouscouplingsenteringthemodel,
tions. which are not always available neither from experiments
Despite the smallness of our cluster, we believe that nor from reliable first principle calculations.However cal-
our determination of the magnetic couplings not only is culating the magnetic couplings for various parameters
qualitatively,butalsoquantitativelysignificant.Thisisso and matching the numerical results with the experimen-
because, in the presently considered undoped LMO, the tally obtained values provides useful connections between
coherentchargemobility is negligible due to the large on- theinvolvedparametersandsetlimitstothepoorlyknown
site Coulomb repulsions and to the substantial JT defor- physical quantities.
mations. As a consequence the magnetic interactions do
notarise,e.g.,fromFermisurfaceinstabilitiesorothercol-
lectiveeffects,butareratherdeterminedbyshort-distance
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