Orbital Structure and Magnetic Ordering in Layered Manganites: Universal Correlation and Its Mechanism

4 Pages·0.16 MB·English

by  S. Okamoto

Orbital Structure and Magnetic Ordering in Layered Manganites:
Universal Correlation and Its Mechanism
S. Okamoto, S. Ishihara, and S. Maekawa
Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan
(February 1, 2008)
Correlation between orbital structureand magnetic ordering in bilayered manganites is examined.
0
0 A level separation between the3d3z2−r2 and 3dx2−y2 orbitals in a Mn ion is calculated in theionic
model for a large number of the compounds. It is found that the relative stability of the orbitals
0
2 dominatesthemagnetictransitiontemperaturesaswellasthemagneticstructures. Amechanismof
the correlation between orbital and magnetism is investigated based on the theoretical model with
n
the two eg orbitals understrong electron correlation.
a
J
PACS numbers: 75.30.Vn, 75.30.Kz, 71.10.-w, 75.80.+q
7
1 Since the discovery of the colossal magnetoresistance relative stability of the orbitals and the magnetic tran-

(CMR), studies of manganites with cubic perovskite sition temperatures as well as the magnetic structures.
]
l structure have been renewed theoretically and experi- A mechanism of the correlation is investigated based on
e
mentally. Competition and cooperation between spin, the theoretical model with the e orbitals under strong
- g
r charge and orbital degrees of freedom as well as lat- electron correlation.
t
s tice cause the dramatic changes of transport and mag- We firstshow thatneither the tolerancefactornor the
.
t netic properties. Manganites with bilayered structure bond length governs T and the N´eel temperature T
a C N
m A2−2xB1+2xMn2O7, where A and B are trivalent and for the A-type AFM ordering [11]. The tolerance factor
divalent cations, respectively, are another class of CMR in the bilayered crystals is defined by t = (d +
- O(1)−A(1)
d materials [1,2]. Since an extremely large MR is observed d )/(2√2d )whered isabondlength
O(2)−A(2) Mn−O(3) A−B
n nearthe transitionfromparamagnetic(PM)insulatorto betweenAandBions. The positionofeachionisshown
o
ferromagnetic (FM) metal, it has been considered that in the inset of Fig. 1(a). Being based on the structural
c
[ several concepts proposed in the cubic compounds are data obtained by the neutron and x-ray diffraction ex-
applicable to the bilayered ones. periments[12–23],weevaluatet andthe bondlengthbe-
1
In cubic manganites, one of the key factors dominat- tween nearestneighboring (NN) Mn ions in the ab plane
v
1 ing the magnetic orderings is the tolerance factor [3]; a daMbn−Mn for a variety of compounds. TC and TN are
2 bending of a Mn-O-Mn bond decreases the hopping in- plotted as functions of t and dab in Figs. 1 (a) and
Mn−Mn
2 tegral of carriers. As a result, the ferromagnetic tran- (b), respectively. Almost all t’s are located in a narrow
01 sition temperature Tc decreases in the double exchange region where TC’s and TN’s are distributed randomly.
(DE) scenario. However, in bilayered manganites, the In addition, T is not correlated with dab , either.
0 C Mn−Mn
0 Mn-O-Mn bond angle is almost unchanged with chang- AlthoughTN increaseswithincreasingdaMbn−Mn,thiscor-
/ ing cations and carrier concentration, as shown later, in relationis opposite to that predictedby the DE scenario
t
a spiteofawidevarietyofthemagneticstructures. Various where, with increasing the bond length, the hopping in-
m key factorsdominating the magnetic ordering,whichare
tegral decreases and the FM interaction in the ab plane
- not included in the DE model, were experimentally sug- decreases. We conclude that the DE model, which in-
d
gested,e.g. the antiferromagnetic(AFM) superexchange
n cludes the change of the hopping integral caused by the
(SE) interaction[4], the locallattice distortion[5–8], the
o change ofthe bond angle/length,cannot explain TC(N).
c chargeandorbitaldegreesoffreedomandtheirorderings WealsoexaminedcorrelationsbetweenT andanum-
C(N)
: [9,10] and so on. However, systematics in their correla-
v berofotherquantities: thetorelancefactorevaluatedby
tions for a variety of compounds and their mechanisms
i the ionic radius, a Mn-O(3)-Mn bond angle, a Mn-O(1)-
X
still remain to be clarified.
Mn bond length, Mn-O bond lengths, a lattice spacing
r In this letter, we study the correlation between mag-
a between NN bilayers, lattice constants and the valence-
neticorderingandorbitalstructureinbilayeredmangan-
bond sum for a Mn ion. However, there are not clear
ites. The two eg orbitals, i.e. the 3d3z2−r2 and 3dx2−y2 correlations between these parameters and T .
orbitals in a Mn3+ ion split in the crystalline field of the C(N)
Let us focus on the correlation between T and
C(N)
bilayered structure and one of them is occupied by an
a relative stability of the e orbitals. We employ the
g
electron. It is known that the occupied orbital controls
ionic model to examine the electronic energy-levelstruc-
the anisotropy of the magnetic interaction as well as its
tures. This model may be justified by the following
strength. The level separation between the orbitals is
considerations [24]: (1) the manganites at x = 0 are
calculated in the ionic model for a large number of the
classified as charge-transfer type insulators in which the
compounds. We find a universalcorrelationbetween the
ionic model provides a good starting point [25]. (2) The
1
240 (a) V (b) V 240 V
O(2) F F F
200 A(2) DE E 200 E
T (K) 11268000 M OAO (((n311)))O DCEWKNTSPCBXWMGUAL U X T OAPGBNWWMJLCSCKDD T (K) 11268000 TDD NKSCCXLWWBJU GA
40 O N I O N 40 x2- y2 N P M I 3z2- r2
0 HQR R Q H HF E Q R
0
0.97 0.98 0.99 1.00 3.84 3.87
-0.1 0.0 0.1 0.2 0.3
t dab (Å) D V (eV)
Mn-Mn
FIG. 1. TC and TN as functions of (a) t and (b) FIG. 2. TC and TN as functions of ∆V. Filled and open
daMbn−Mn. Filled and open circles indicate TC and TN, re- circlesindicateTC andTN,respectively. ∆V’sarecalculated
spectively. t and daMbn−Mn are obtained from the structural for the same compounds in Fig. 1. Note that in the region
data in the following compounds: A: La1.4Sr1.6Mn2O7 (s) with large positive (negative) ∆V, the 3d3z2−r2 (3dx2−y2)
[12], B: La1.3Sr1.7Mn2O7 (p) [12], C: La1.2Sr1.8Mn2O7 (s) orbital is occupied by an electron.
[12], D: La1.1Sr1.9Mn2O7 (s) [12], E: La1.04Sr1.96Mn2O7 (s)
[12],F:LaSr2Mn2O7 (s)[12],G:La1.4Sr1.6Mn2O7(p)[13],H:
is the unit vector in the z(x) axis. The Ewaldmethod is
Pr1.4Ca1.3Ba0.3Mn2O7 (p)[14],I:Nd1.4Ca1.6Mn2O7 (p)[14],
J: La1.4Sr1.6Mn2O7 (p) [15], K: La1.2Sr1.8Mn2O7 (p) [16], L: used for the lattice summation. Zi’s for Mn and O ions
La1.2Sr1.8Mn2O7 (s) [17], M: La1.2(Sr0.8 Ca0.2)1.8Mn2O7 (s) and a cation at A site are chosen to be 3+x, −2 and
[17], N: La1.2(Sr0.7 Ca0.3)1.8Mn2O7 (s) [17], O: La1.2(Sr0.6 (8 2x)/3, respectively. The difference of the potentials
−
Ca0.4)1.8Mn2O7 (p) [17], P: (La0.8Nd0.2)1.2Sr1.8Mn2O7 (s)
[17], Q: (La0.6Nd0.4)1.2Sr1.8Mn2O7 (s) [17], R: Sm1.2Sr1.8 ∆V =V3z2−r2 Vx2−y2, (4)
−
Mn2O7 (p) [18], S: La1.2Sr1.4Ca0.4Mn2O7 (p) [19], T:
NdSr2Mn2O7 (p) [20], U: Nd1.1Sr1.9Mn2O7 (p) [20], V: represents the relative stability of the orbitals; with in-
LaSr2Mn2O7 (p) [21], W: LaSr1.6Ca0.4Mn2O7 (p) [22], X: creasing ∆V, the energy level of the 3d3z2−r2 orbital for
NdSr2 Mn2O7 (s) [22] where (s) and (p) indicate the sin- an electron relatively decreases.
gleandpolycrystallinesamples,respectively. Theinsetof(a) T and T are plotted as functions of ∆V in Fig. 2
N C
shows a schematic pictureof thebilayered structure.
wherethestructuraldataatroomtemperatureareused.
Broad shades are drawn by considering experimental er-
rors. It is clearly shown that both T and T are
ionic property is predominant between bilayers. (3) The C N
correlated with ∆V; T increases with decreasing ∆V
energy-level structure given by band-structure calcula- N
and there is an optimal value of ∆V( 0.08 eV) for
tions shows the same tendency with those by the ionic
∼
T . We estimate the strength of the correlation be-
model [6,26]. The energy levels of the e orbitals split C
g
tween T and ∆V by using the correlation coefficient:
due to the electrostatic potential and one of the orbitals N
is occupied by an electron in a Mn3+ ion. r = N1 Pl (TNl − TN)(∆Vl − ∆V)/(σTNσ∆V) where l
indicates a sample and N is the number of samples.
Byusingalargenumberofthestructuraldata[12–23],
∆V (T ) and σ (σ ) are the mean value and the
we calculate the Madelung potential for a hole in the N ∆V TN
standard deviation of ∆V (T ), respectively. We ob-
3d3z2−r2 and 3dx2−y2 orbitals at site j defined by N
tain r = 0.89 0.11 for single crystal samples and
− ±
1 r = 0.15 0.04 for all samples including polycrys-
V3z2−r2 = 2(cid:8)V(~rj +rdzˆ)+V(~rj −rdzˆ)(cid:9), (1) tals. −One m±ight think that the TC(N) v.s. ∆V curve
in Fig. 2 just reflects the relation between T and x
C(N)
and
in La Sr Mn O (LSMO) [12,7]. However, when
2−2x 1+2x 2 7
we pay attention to T ’s for samples with the same x
Vx2−y2 =V(~rj +rdxˆ), (2) (e.g., the samples C aCn(dNK) -R), we notice that the corre-
lationremains. The correlationbetweenT and ∆V ex-
respectively [27]. Here, V(~r ) is given by N
j
plains that between T and dab shownin Fig. 1(b),
N Mn−Mn
Z e2 since ∆V is a decreasing function of dab in the re-
V(~rj)=X ~r i ~r , (3) gion, 3.84˚A<dab <3.88˚A. Mn−Mn
j i Mn−Mn
i6=j | − |
In Fig. 3, we present the magnetic phase diagram at
withapointchargeZ eatsiteiandtheposition~r ofthe T = 0 as a function of ∆V and x. The structural
i i
site. r (=0.42˚A) is the radius of a Mn 3d orbital where data at room temperature are used. Symbols connected
d
itsradialchargedensitybecomesmaximum[28]andzˆ(xˆ) by dotted lines correspond to a series of LSMO with
2
0.5
0.25
A 3z2- r2 R
0.20 G I 0.4
FM
0.15 Q M U FM
V) 0.10 J B P LC W 0.3
V (e 0.05 NK S D E X V Dt/00.2
D 0.00 T
-0.05 H F 0.1 A-AFM
x2- y2 A-AFM
-0.10
0.0
0.55 0.60 0.65 0.70 0.75
0.30 0.35 0.40 0.45 0.50
x x
FIG.4. The calculated magnetic phase diagram at T = 0
FIG. 3. The magnetic phase diagram at T = 0 as a
as a function of ∆ and x. Note that in the region with large
function of ∆V and x. Filled, open and crossed squares
indicate the FM, A-type AFM and PM samples, respec- positive(negative) ∆,the3d3z2−r2 (3dx2−y2)orbitalisoccu-
pied by an electron.
tively. ∆V’s for filled and open circles are obtained from
the data below TC and TN, respectively. Symbols connected
by dotted lines indicate a series of LSMO. Bold arrows show
ion, the two e orbitals are introduced and the t elec-
δ(∆V) = ∆V(T < T )− ∆V(T > T ). ∆V’s are g 2g
C(N) C(N) trons are treated as a localized spin with S = 3/2. The
calculated for the same compounds in Fig. 1. Note that in
first two terms in Eq. (6) correspond to the so-called t-
the region with large positive (negative) ∆V, the 3d3z2−r2
(3dx2−y2) orbital is occupied by an electron. and J-terms in the tJ model, respectively, with the two
e orbitals under strong electron correlation. The third
g
and fourth terms describe the Hund coupling between
x=0.3 0.5 [12]. In addition, ∆V’s calculatedby using e and t spins and the AFM SE interaction between
− g 2g
thedatabelowTC(N) inLSMOarealsoplotted. Boldar- t2g spins, respectively. The splitting of the energy levels
rowsindicate the changeof the Madelung potential with between 3d3z2−r2 and 3dx2−y2 orbitals is represented by
changing temperature from T >TC(N) to T <TC(N): thelastterm: Hz =−∆PiTiz wherethepseudospinop-
δ(∆V) ∆V(T <TC(N)) ∆V(T >TC(N)). (5) erator is given by T~i = 21Pγγ′σd†iγσ~σγγ′diγ′σ with diγσ
≡ − being the annihilation operator of an e electron at site
g
Wefindthatthemagneticstructuresaregovernedby∆V i with spin σ and orbital γ. The +( ) eigenstate of T
iz
−
and x; the FM (A-type AFM) phase is locatedin the re- corresponds to the state where the 3d3z2−r2 (3dx2−y2)
gionwithsmaller(larger)xandmoderate(smaller)∆V. orbital is occupied by an electron. The anisotropies of
Let us focus on δ(∆V) in LSMO. δ(∆V)’s are negative the hopping integral and the SE interactions due to the
at x =0.3 and 0.35. The absolute value of δ(∆V) grad- layered structure are considered. The explicit expres-
ually decreases with increasing x and δ(∆V) becomes a sion and derivation of the Hamiltonian are presented in
small positive value at x = 0.4. Below T , ∆V seems Refs. [27] and [29].
C
to approach to the optimal value of ∆V 0.08 where The calculated magnetic phase diagram at T = 0 is
∼
T becomes maximum as seen in Fig. 2. On the other presented in Fig. 4 [30,31] where the mean field approx-
C
hand, δ(∆V)’s are negative at x=0.45 and 0.48 where imation is adopted. We note that the phase diagram
the A-type AFM structure appears. The orbital struc- derived in this approximation explains that in the cu-
ture and its stability in the FM phase have been studied bic manganites [32]. The characteristic features shown
by measuring the striction in Ref. [10]. The difference in Fig. 3 are well reproduced by the present theory; the
of the Mn-O bond lengths between PM and FM states A-type AFM phase appears in the region with higher x
was reported in Ref. [7]. These experimental results are andsmaller∆thanthatoftheFMone. Therangeofthe
consistent with the present results of δ(∆V) in Fig. 3. horizontal axis in Fig. 4 is larger than that in Fig. 3 by
Now we theoretically investigate a mechanism of the about 0.25. This discrepancy may be attributed to the
correlationbetweenmagnetic orderingandorbitalstruc- neglectoftheorbitalfluctuation[33]. However,thechar-
ture. WestartwiththefollowingHamiltonian[27,29,30]: acteristicsofthephasediagramareinsensitivetothepa-
rameters in the model. In the FM (A-type AFM) phase,
= + + + + . (6)
H Ht HJ HH HAF Hz the orbitals are uniformly aligned with 0 < θ < 0.72π
(0.72π < θ < π) where θ describes the orbital state as
Instead of the bilayered structure, the simple tetragonal
latticeconsistingofMnionsisconsidered. Inthismodel, |θi =cos(2θ)|3d3z2−r2i−sin(2θ)|3dx2−y2i. The present re-
sultssuggestthatadimensionalityoftheFMinteraction
the magnetic structure with FM and AFM alignments
iscontrolledbytheorbitalstructure;intheA-typeAFM
perpendicularandparalleltothecaxis,respectively,cor-
phase, the FM ordering in the ab plane is caused by the
responds to the A-type AFM structure [11]. In each Mn
3
DEinteraction,whiletheAFMinthecdirectionisbythe [10] T. Kimuraet al.,Phys.Rev. Lett.81, 5920 (1998).
AFMSE.Whenthe3dx2−y2 orbitalisstabilized,theDE [11] ThefollowingtwolayeredAFMstructureshavebeenob-
interaction in the ab plane (c direction) becomes strong served [12,13]: AFM-I consisting of the FM spin order-
(weak) and the A-type AFM phase appears [34]. A mix- ingintheplaneandtheAFM(FM)onealongthecaxis
within (between NN)bilayers, and AFM-IIconsisting of
ing of the orbitals is essential in the FM phase where
the FM in the plane and the FM (AFM) along the c
the FM interaction overcomes the AFM SE one in the
axiswithin(betweenNN)bilayers.Sincetheintrabilayer
three directions. We note that, in Fig. 4, the FM phase
magnetic coupling is much larger than the interbilayer
appears not around ∆ = 0 but in a region of ∆ > 0,
one [35], we term theAFM-I structure theA-typeAFM
since the anisotropy in the hopping integral due to the
one and regard theAFM-II as the FM.
layeredstructurestabilizesthe3dx2−y2 orbitalmorethan [12] M. Kubotaet al., cond-mat/9902288.
3d3z2−r2. Itisworthtomentionthechangeoftheorbital [13] D. N. Argyriou et al., Phys.Rev.B 59, 8695 (1999).
structure associatedwith the magneticordering: By uti- [14] P. Laffez et al.,J. Appl.Phys. 80, 5850 (1996).
lizingthemeanfiledapproximationatfinitetemperature, [15] E.-O. Chi et al.,Phys.Rev. B60, 12 867 (1999).
we compare the orbital structures above and below the [16] D. N. Argyriou et al., Phys.Rev.Lett. 78, 1568 (1997).
magnetic transition temperatures. It is found that (1) [17] T. Akimoto et al.,Phys. Rev.B 59, R14 153 (1999).
[18] P. D. Battle et al.,J. Appl.Phys. 83, 6379 (1998).
there is an optimal mixing of the orbitals for the FM
[19] C. H.Shen et al., J. Appl.Phys. 86, 2178 (1999).
state and the orbital structure tends to approachto this
[20] P. D. Battle et al.,Phys.Rev. B 54, 15 967 (1996).
structurebelow TC and(2)the 3dx2−y2 orbitalstructure [21] R.Seshadrietal.,SolidStateCommun.101,453(1997).
is stabilized below T . The theoretical results are con-
N [22] T. Akimoto et al.,(unpublished).
sistentwith δ(∆V)’s showninFig. 3by consideringthat
[23] For the samples G, J, R, T and U where thetwo chemi-
the changeof ∆V associatedwith the magnetic ordering cally distinct phasesare observed,thestructural dataof
is caused by that of the orbital structure. the major phase are adopted.
Insummary,weexaminecorrelationbetweenmagnetic [24] Y. Ohtaet al., Phys.Rev.B 43, 2968 (1991).
ordering and orbital structure in bilayered manganites. [25] T. Arima et al.,Phys. Rev B 48, 17 006 (1993).
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g
calculatingthe Madelungpotentialsinalargenumberof [27] S. Ishihara et al., J. Phys. Soc. Jpn. 66, 2965 (1997).
[28] J.C.SlaterinQuantumTheoryofAtomicStructure(Mc-
the bilayeredcompounds. We findthatthe A-type AFM
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3d3z2−r2 and 3dx2−y2 orbitals for the FM ordering. A [30] In the Hamiltonian, we consider the SE interactions
rtheleaotrioynweixthpltahinestwaomeegchoarbniitsamlsoufnthdeerusntirvoenrgsaellceoctrrroelnatcioorn- J1a,b2(c) between eg electrons, the AFM SE one JAabF(c) be-
between orbital and magnetism. tweent2g spinsandthehoppingintegralt0ab(c) ofeg elec-
tronswherethesubscriptab(c)indicatestheinteraction
TheauthorswouldliketothankY.Moritomo,T.Aki-
intheabplane(thecdirection)[27,29]. Thesevaluesare
moto, Y. Tokura, T. Kimura, Y. Endoh, K. Hirota, chosen as Jab /tab =0.25 (0.075), Jab /tab =0.002 and
1(2) 0 AF 0
M. Kubota and G. Khaliullin for their valuable discus-
sions. This work was supported by CREST, NEDO and ta0b/tc0 =pJ1a,b2/J1c,2 = pJAabF/JAcF = 1.5 in the numeri-
cal calculation. tab is represented as t for simplicity.
Grant-in-Aid for Scientific Research Priority Area from 0 0
[31] The phase diagram at T = 0 for the bilayered mangan-
theMinistryofEducation,ScienceandCultureofJapan.
ites has also been studied in theHartree-Fock theory by
S.O.acknowledgesthefinancialsupportofJSPS.Partof
R. Maezono and N. Nagaosa, cond-mat/9904427.
thenumericalcalculationwasperformedintheHITACS- [32] S.Okamotoetal.,Phys.Rev.B61,(2000),andY.Endoh
3800/380superconputingfacilitiesinIMR,TohokuUniv. et al., Phys.Rev.Lett. 82, 4328 (1999).
[33] S. Ishihara et al., Phys.Rev.B 56, 686 (1997).
[34] R. Maezono et al., Phys.Rev.B 57, R13993 (1998).
[35] H.Fujiokaet al.,J.Phys.Chem.Solids 60,1165(1999).
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4