Orbital Ordering, New Phases, and Stripe Formation in Doped Layered Nickelates

0.17 MB·English

by  Takashi Hotta

Orbital Ordering, New Phases, and Stripe Formation in Doped Layered Nickelates
Takashi Hotta1 and Elbio Dagotto2
1Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195, Japan
2National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32306
(Dated: February 2, 2008)
Ground-statepropertiesoflayerednickelatesareinvestigatedbasedontheorbital-degenerateHub-
4 bardmodelcoupledwithlatticedistortions,byusingnumericaltechniques. TheN´eelstatecomposed
0 ofspin S=1ionsisconfirmed intheundopedlimit x=0. Atx=1/2, novelantiferromagnetic states,
0 calledCE-andE-typephases,arefoundbyincreasingtheHund’scoupling. (3x2−r2/3y2−r2)-type
2 orbital ordering is predicted to occur in a checkerboard-typecharge-ordered state. At x=1/3, both
n Coulombicandphononicinteractionsarefoundtobeimportant,sincetheformerstabilizesthespin
a stripe, while thelatter leads tothe striped charge-order.
J
PACSnumbers: 75.30.Kz,75.50.Ee,71.10.Fd,75.47.Lx
9
2

] The existence and origin of “striped” structures con- the conflict between neighboring lattice distortions shar-
l tinues attracting considerable attention in the research ing the same oxygens. Then, static stripes stabilized by
e
- field of transition-metal oxides [1]. In a system with lattice distortions tends to occur along the diagonal di-
r
t dominantelectron-electronrepulsion,theWigner-crystal rection,asshowninthe stripes ofthe FM-phaseof man-
s
. state should be stabilized, but in real materials more ganites, stabilized by Jahn-Teller (JT) distortions [8].
t
a complicated non-uniform charge structures have been In simple terms, vertical or horizontal stripes in
m found. InNd-basedlightly-dopedcuprates,neutronscat- cuprates can be understood by the competition between
- tering experiments revealed incommensurate spin struc- Coulomb interaction and hole motion, while diagonal
d
tures [2] where antiferromagnetic (AFM) spin stripes stripes are better explainedas a consequence ofa robust
n
o are periodically separated by domain walls of holes. In electron-lattice coupling. However, a difficulty has been
c La2−xSrxCuO4, dynamical stripes are believed to exist foundfortheoreticalstudiesofstripeformationindoped
[ alongverticalorhorizontaldirections (Cu-O bonddirec- nickelates, since both Coulomb interaction and electron-

1 tion) [3]. In nickelates, the charge-ordered stripes are lattice coupling appear to be important. Since the Ni2+
v alongthe diagonaldirection[4]. Inmanganites,evidence ion has two electrons in the e orbitals, on-site Coulomb
g
6 forstripedcharge-orderingalsoalongthe diagonaldirec- interactions certainly play a crucial role to form spins
9 tion has been reported in the AFM phase for x>1/2 [5], S=1. When holes are doped, one electron is removed
5
while short-range diagonal stripe correlations have been and another remains in the e orbitals, indicating that
1 g
0 found in the ferromagnetic (FM) phase at x<1/2 [6]. the hole-doped site should become JT active. Then, in
4 In general, stripes can be classified into metallic or hole-doped nickelates both Coulombic and phononic in-
0 insulating. In La2−xSrxCuO4, the dynamical stripes ex- teractions could be of relevance, a fact not considered in
/
t hibit metallic properties, but they are easily pinned by previous theoretical investigations.
a
m lattice effects and impurities. In La1.6−xNd0.4SrxCuO4, In this Letter, charge ordering in doped nickelates is
stripes along the bond-direction are pinned by lattice investigated based on the orbital-degenerate Hubbard
-
d distortions [1], but they are still metallic. Intuitively, model coupled to lattice distortions, using numerical
n vertical or horizontal stripes could be associated with techniques. After confirming the N´eel state composed
o the formation of “rivers of holes”, to prevent individual of S=1 spins at x=0, both cases x=1/2 and 1/3 will be
c
: charges from fighting against the AFM background [7]. analyzed. At x=1/2, novel AFM phases called CE- and
v Such stripes should be metallic, even if they are pinned, E-typehavebeenunveiled,whichareconsistentwithex-
i
X sincetheyareinducedbytheoptimizationofholemotion perimental results. Further including the JT-type coop-
r between nearest-neighbor Cu-sites via oxygens. erative distortion, (3x2 r2/3y2 r2)-type orbital order-
a However, in the diagonal stripes observed in mangan- ing is predicted. For x−=1/3, a−n incommensurate spin
ites and nickelates, charges are basically localized, indi- structure is induced by the Coulombic model, including
cating that such insulating stripes are not determined the level splitting between eg orbitals, but the charge
just by the optimization of the hole motion. In the FM stripedoesnotappear. Toreproducesimultaneouslyspin
state of manganites, the hole movement is already opti- and charge stripes, it is important to include the strong
mal and, naively, charges should not form stripes. Ob- couplingofeg electronsto lattice distortions,originating
viously, an additional effective local potential must be in the in-plane oxygen motions.
acting to confine electrons into stripes. If such a po- The model for nickelates includes three important in-
tential originates in lattice distortions, it is expected to gredients: The kinetic motion of e electrons, Coulomb
g
occuralongthebonddirectiontoavoidenergylossdueto interactionsamonge electrons,andelectron-latticecou-
g
2
3.0 the Hund’s rule coupling, the JT distortions are not
(a) x=0 U'=2(cid:13) (b)
active and it is possible to grasp the essential ground-
J=1(cid:13)
2.0 U=U'+2J(cid:13) U'-J U-D state properties using H. In Fig. 1(a), the Fourier
J'=J transform of spin correlations is shown vs. ∆, where
S(q) ((0p,0,0)) (c) S(q)=(1/N)Pi,jeiq·(i−j)hSizSjzi, with Siz= Pγ(d†iγ↑diγ↑
1.0 ((p,p/p)2,p/2) b−ed†ioγb↓dseiγr↓v)e/d2.forA∆s<ex3p,escutgegde,stainrgobthuastt t(hπe,πA)FpMeakphcaasne
0.0 is stabilized by sup∼er-exchange interactions. The rapid
0.0 2.0 4.0 6.0 8.0 decrease of S(π,π) for ∆>3 is understood by compar-
D
ing the energies for local∼triplet and singlet states, as
shown in Fig. 1(b). The ground-state properties change
FIG. 1: (a) Spin correlation S(q) vs. ∆ for x=0. (b) Two
kinds of local eg-electron arrangements for x=0. (c) AFM atU′−J=U−∆,leadingto∆=3J forthetransition. The
spin pattern theoretically determined for ∆<∼3. spinstructureatx=0isschematicallyshowninFig.1(c).
Let us turn our attention to the case x=1/2. The 8-
site tilted lattice is again used for the analysis, and the
plings between e electrons and distortions of the NiO
g 6 phase diagram Fig. 2(a) is obtained for ∆=0.5 [10]. In-
octahedra. Note that the electron-latticeterm is divided
creasing J, an interesting transformation from AFM to
intocouplingsfortheapicalandin-planeoxygenmotions.
FM phases is found. This is natural, since at large J
Inlayerednickelates,allNiO octahedraaresignificantly
6 the system has a formal similarity with manganite mod-
elongatedalongthe c-axis,splitting thee orbitals. This
g els, where kinetic-energy gains lead to ferromagnetism,
splitting from apical oxygens should be included explic-
while at small J the magnetic energy dominates. How-
itly fromthe startand,then, the in-plane motionshould
ever, between the G-type AFM for J 0 and FM phase
be studied. The Hamiltonian H is given by ≈
for J U′, unexpected states appear which are mixtures
≈
of FM and AFM phases, due to the competition be-
H = − X taγγ′d†iγσdi+aγ′σ+∆X(nia−nib)/2 tween kinetic and magnetic energies. Typicalspin corre-
iaγγ′σ i lations S(q) are shown in Fig. 2(b). Note that peaks at
+ UXniγ↓niγ↑+J X d†iaσd†ibσ′diaσ′dibσ q=(π,0)and(π/2,π/2)indicate“C”and“E”typespin-
i,γ i,σ,σ′ structures, respectively (the notation is borrowed from
+ U′Xnianib+J′ X d†iγ↑d†iγ↓diγ′↓diγ′↑, (1) (tπhe/2M,πn/-o2x)iddeencootnetetxhte[C9]E).-tDypouebslterupcetaukrse,atfrqe=qu(eπn,t0l)yaonbd-
i i,γ6=γ′
servedinhalf-dopedmanganites[11]. Inhalf-dopednick-
where d (d ) is the annihilation operator for an elates,theCE-phaseisexpressedasamixtureoftype(I)
iaσ ibσ
eg-electron with spin σ in the dx2−y2 (d3z2−r2) orbital and (II) in Fig. 2(c), depending on the positions of the
at site i, n = d† d , n = n , a is the vec- S=1 and S=1/2 sites, although the “zigzag” FM chain
iγσ iγσ iγσ iγ Pσ iγσ
tor connecting nearest-neighbor sites, and ta is the structure is common for both types. The E-type phase
γγ′
nearest-neighbor hopping amplitude between γ- and γ′- is also depicted in Fig. 2(c). Note that the charge cor-
orbitals along the a-direction, given by tx = √3tx = relation always exhibits a peak at q=(π,π) (not shown
√3tx =3tx =3t/4 for the x-direction andaaty =−√3tayb= here), indicating the checkerboard-typecharge ordering.
−√3tyb=a3tyb=b3t/4 for the y-direction [9]. Haaereaftearb, t In experimental results, a peak at (π/2,π/2) in S(q)
ba bb
is the energy unit. In the second term, ∆(>0) is the has been reported [4], suggesting an AFM pair of S=1
levelsplittingbetweena-andb-orbitals. IntheCoulomb spins across the singly-occupied sites with holes. More-
interaction terms, U (U′) is the intra-orbital (inter- over,thecheckerboard-typechargeorderinghasbeenex-
orbital) Coulomb interaction, J is the inter-orbital ex- perimentallyobserved[4]. Thus,thespin-chargepatterns
changeinteraction,andJ′ isthe pair-hoppingamplitude ofCE(II)-andE-typeareconsistentwiththeexperimen-
betweendifferentorbitals. DuetotherelationsJ=J′and tal results. Our phase diagram has a robust region with
U=U′+J+J′,theindependentparametersareU′ andJ, a peak at (π/2,π/2), both for CE- and E-type phases,
with U′>J [9]. The calculationsareallcarriedoutusing although the CE-phase exhibits an extra peak at (π,0).
standard exact-diagonalization techniques. It is crucial Whether the E- or CE-phases are present in nickelates
to use this kind of unbiased methods for this first study canbe studied experimentally in the future by searching
that includes both Coulomb and lattice effects, even if forthis (π,0)peak. Note thatifdiffuse scatteringexper-
the technique restricts us to small N-site clusters. iments detect the AF correlation along the hole stripe,
First, consider the undoped case. The calculation is ashasbeenfoundatx=1/3[12],theCE(II)-typemaybe
done for an8-sitetilted cluster, equivalentin complexity theonlypossibility. Summarizing,the spin-chargestruc-
to a 16-site lattice for the single-band Hubbard model. tureinx=1/2experimentscanbeunderstoodwithinthe
Since at all sites the two orbitals are occupied due to Hamiltonian H by assuming a relatively large J.
3
Considernowtheeffectofin-planeoxygenmotion(api- 10.0 1.0
(a)x=1/2 (b) U’=6, J=3
cal oxygenmotions havealready been included as an eg- 8.0 U=U'+2J(cid:13) FM 0.8 x=1/2 U’=8, J=6
lNteevir-emOl sibspolwnitdrtiitndtgier)ne.catAsiosnsu,mthiengextthraateolexcytgroenn-spmhoonvoenacloonugpltinhge J46..00 JD '==0J.(cid:13)5 CE E S(q) 00..46 CE
E
2.0 0.2
Heph = gX[−Q1i(nia+nib)+Q2iτxi+Q3iτzi] 0.0 G 0.0
i 0.0 2.0 4.0 6.0 8.0 10.0 (0,0) (p,0) (p,p) (p/2,p/2) (0,0)
U'
+ (k/2)X(βQ21i+Q22i+Q23i), (2) (c) CE-type (I) CE-type (II) E-type
i
where g is the electron-lattice coupling constant, Q is
1i
the breathing-mode distortion, Q and Q are, respec-
2i 3i
tively, the (x2 y2)- and (3z2 r2)-type JT distortions,
− −
τ = (d† d +d† d ), and τ = (d† d
xi Pσ iaσ ibσ ibσ iaσ zi Pσ iaσ iaσ −
d† d ). The second term is the quadratic potential
ibσ ibσ z
for adiabatic distortions, where k is the spring constant (d) (f)
t x
for the JT-mode and β is the spring-constant ratio for x 1
breathing- and JT-modes. From our experience in man- S=1 3x 2 -r2 d=0.5 } y
d=0.4
ganites, this ratio is here fixed to β=2 [13]. d=0.3 site 1
d=0.2
Since all oxygens are shared by adjacent NiO6 octa- 2 4 sites 2 & 4 d=0.1
hedra, the distortions are not independent. To consider S=1/2 1
smnuoectnhftescaossoihbpoleuerlatdotibvpeeereofffopretmcimt,biizonetdhp.rtihnHecoiLpwaleenv,cetzrho,esindOi-apigoroannctdaiilcsiezpaliatticoeins- al energy345...000 (1e)x=1/2 UJ3='=68(cid:13) Dl ==10..75(cid:13) -1 dd=d==00dd.0.==15.004..23}site 3 ztz x
and the optimization of all oxygen positions for 6- and Tot2.0 3y 2 -r2 -1 y
8-site clusters. In the actual calculations, Q , Q , and 0.0 0.1 0.2 0.3 0.4 0.5
1i 2i d (units of g/k)
Q are expressed by a single parameter d, for the shift
3i
of the O-ion coordinate. Note that the unit of d is g/k, FIG. 2: (a) Ground-state phase diagram at x=1/2. (b) S(q)
typically 0.1 0.3˚A. Then, the total energy is evaluated for the CE- and E-type phases, at the couplings indicated.
∼ (c) Spin and charge patterns for the CE- and E-typephases.
asa functionofdto findthe minimumenergystate. Re-
These are schematic views, since local charge-densities in
peatingthesecalculationsforseveraldistortionpatterns,
practice are not exactly 1 and 2. (d) Numerically obtained
it is possible to deduce the optimal state.
cooperative distortion pattern for an 8-site lattice at x=1/2.
After severaltrials, the optimaldistortionat x=1/2 is Black and open circles indicate Ni and O ions, respectively.
shown in Fig. 2(d). The diagonalization has been per- Opensymbolsindicate eg orbitalsin theoptimized state. (e)
formed at several values of d on the 8-site distorted lat- Total ground-state energy vs. d for x=1/2. (f) Orbital den-
tice and the minimum in the total energy is found at sities hτzii and hτxii for sites 1–4. See (d) for the site labels.
Optimized orbitals at d=0.3for sites 1 and 3are also shown.
d=0.3 (Fig. 2(e)). Here, the dimensionless coupling con-
stant λ is defined as λ=g/√kt. As mentioned above,
even without H , the checkerboard-type charge order-
eph
ing has been obtained, but the peak at q=(π,π) signifi- Now let us move to the case x=1/3. If the actual
cantlygrowsdue to theeffectoflatticedistortions. Note expected stripe structure at x=1/3 is faithfully consid-
that the distortion pattern in Fig. 2(d) is essentially the ered[4],itisnecessarytoanalyze,atleast,a6 6cluster.
×
same as that for half-doped manganites. This is quite However,suchalarge-sizeclusterwithorbitaldegeneracy
natural,sinceJTactiveandinactiveionsexistbipartitely cannotbe treatedexactly due to the exponential growth
alsoforhalf-dopednickelates. Then, due to this JT-type of the Hilbert space with cluster size. Then, a covering
distortionorbitalorderingforhalf-dopednickelatesispre- of the two-dimensional (2D) lattice using zigzag 6-sites
dicted, as schematically shown in Fig. 2(d). The shapes clusters is considered (Fig. 3(a)) by assuming a periodic
oforbitalsaredeterminedfromtheorbitaldensities, τ structure along the diagonal direction. The phase dia-
zi
and τ (Fig. 2(f)). The well-known alternate pathterni gram obtained by analyzing the zigzag 6-site cluster for
xi
of3xh2 ri2 and3y2 r2 orbitalsinhalf-dopedmanganites H is in Fig. 3(b). Typical spin and charge correlations
is den−oted by dash−ed lines. Increasing d, the shape of areinFigs.3(c)and(d),whereC(q)=(1/N)Pi,jeiq·(i−j)
orbitals deviates from 3x2−r2 and 3y2−r2, but it is still h(ni−hni)·(nj−hni)i, with ni=Pγniγ.
characterizedby the orbitalselongatingalongthex- and Since the momentum q is defined along the zigzag di-
y-directions (see insets of Fig. 2(f)). It would be very rection,thephaselabelledbyq=2π/3inFig.3(b)denotes
interesting to search for orbital ordering in half-doped an incommensurate AFM phase with the proper spin
nickelates,usingtheresonantX-rayscatteringtechnique. stripe structure. The phase labelled by q=π/3 indicates
4
(a) 8.0 bital ordering tends to appear in a hole pair separated
(b) x=1/3
by one site, the unit of the “bi-stripe” of manganites
6.0 UJ'==JU(cid:13)'+2J(cid:13) q=p /3 [5]. However, such a bi-stripe-type ordering contradicts
J4.0 D =0.5 thex=1/3stripedcharge-ordering,andthebi-stripe-type
q=2p /3(cid:13)
solution was found to be unstable in these calculations.
2.0 (stripes)
Onemayconsiderotherdistortionpatternswhichsatisfy
q=p
0.0 both (3x2 r2/3y2 r2)-type orbital and striped charge-
0.0 2.0 4.0 6.0 8.0 − −
U’(cid:13) ordering,butinsuchdistortionsnoenergyminimumwas
1.5 (c) x=1/3 U'=6(cid:13) J'=J(cid:13) 0.8 (d)x=1/3 U'=6(cid:13) obtained for d>0. After several trials, Fig. 3(e) has pro-
S(q)1.0 q=p q=U2=pU/3'+2J D q==0p.5/3 C(q)00..46 q=p UJD '===0JU.(cid:13)'5+2J(cid:13) tvuidrSeeusdmtomhfealramiyzoeinsrtegd,oppnotiiscmskiaeblllaesttseapstienh.,acvhearbgeee,nanddiscourbssietadl sbtarsuecd-
0.5 q=2p /3 on the e -orbital degenerate Hubbard model coupled
0.2 g
q=p /3 with lattice distortions. To understand the nicke-
q=0 q=0
0.0 0.0 late stripes, both Hund’s rule interaction and electron-
0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0
J J lattice coupling appear essentially important. At x=1/2,
(e)
2 4 6 0.6 (g)x=1/3 q=2p /3 (3x2 r2/3y2 r2)-type orbital ordering similar to that
U'=6(cid:13) − −
1 3 5 J=3 in half-doped manganites is predicted. Even FM phases
Total energy111111...468 (f) x=1/3 DlUJ==='=1306..75(cid:13)(cid:13) C(q)000...024 qq==0p /3 q=p Dl ==10..75(cid:13) cbsYpoaoulTnsilthdhdtiweiznbaiagdweutpathshl.tfaooayrrbsFsidloatiirzhsnceaxudinm=sksbp1iyMoo/nr3.cts,haM.enaTamt.tsrisHpcouaal.deltilaisyfa,oslJura.mplstTpteorrrodiairpnuntegleqadfutotiabhrodymenaPac,itarnaiironornlrdieei.vtrHye’s-.l
0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3
d (units of g/k) d (units of g/k) Areas Grant from the Ministry of Education, Culture,
Sports, Science, and Technology of Japan. E. D. is sup-
FIG. 3: (a) Zigzag 6-sites cluster covering the 2D lattice. ported by the NSF grants DMR-0122523and 0312333.
Black circles denote Ni ions, and dashed lines indicate hole
positions. (b) Phase diagram at x=1/3. Each phase is char-
acterized by the momentum that shows a peak in S(q). (c)
S(q) and (d) C(q) vs. J for U′=6 and ∆=0.5. (e) Coopera-
tivedistortionpatternforthezigzag6-sitesclusteratx=1/3. [1] J. M. Tranquada et al., Nature 375, 561 (1995); H. A.
(f) Total ground-state energy and (g) C(q) vs. d for x=1/3. Mook et al.,Nature 395, 80 (1998).
[2] S. -W. Cheong et al., Phys. Rev. Lett. 67, 1791 (1991);
T.E.Masonetal.,ibid.68,1414(1992);T.R.Thurston
a spin spiral state, which will eventually turn to the FM et al.,Phys.Rev. B 46, 9128 (1992).
phase in the thermodynamic limit. Thus, the spin stripe [3] Forx<0.02,thespin-glassphaseexhibitsadiagonalspin
phaseappearsbetweenthecommensurateAFMandFM- modulation. See M. Matsuda et al., Phys. Rev. B 65,
like phases, similar to the case of x=1/2. However, as 134515 (2003).
[4] J.M.Tranquadaetal.,Phys.Rev.Lett.73,1003(1994);
seen in Fig. 3(d), C(q) in the spin stripe phase does not
V. Sachan et al., Phys. Rev. B 51, 12742 (1995); H.
show the striped charge structure (q=2π/3). Rather, bi-
Yoshizawa et al., Phys. Rev. B 61, R854 (2000); R. Ka-
partite charge ordering characterized by a peak at q=π jimoto et al.,Phys. Rev.B 67, 014511 (2003).
still remains. Namely, the Hamiltonian H can explain [5] S. Mori et al.,Nature 392, 473 (1998).
thespinstripe,butdoesnotreproducethestripedcharge [6] P. Dai et al., Phys. Rev. Lett. 85, 2553 (2000); C. P.
ordering at x=1/3, indicating the importance of H . Adams et al.,ibid. 85, 3954 (2000).
eph
ConsidernowtheeffectofH forx=1/3. Aftereval- [7] V.J.Emeryetal.,Phys.Rev.B56,6120(1997);C.Buh-
eph leretal.,Phys.Rev.Lett.84,2690(2000);A.L.Malvezzi
uating total ground-state energies for several kinds of
and E. Dagotto, Phys.Rev. B 63, 140409 (2001).
distortions, the pattern in Fig. 3(e) has been found to
[8] T. Hotta et al.,Phys. Rev.Lett. 86, 4922 (2001).
provide the optimal state at x=1/3. This type of distor- [9] E. Dagotto et al., Phys.Rep. 344, 1 (2001).
tioninduces aspatialmodulationofthe levelsplittingas [10] Since ∆ of nickelates is half of that of cuprates, from
δ1/2=δ2=δ3= δ4/2=δ5=δ6 [14], where δi is the level thelattice constantsfor CuO6 andNiO6 octahedra, itis
− −
splitting causedbythe in-plane oxygenmotions,andthe reasonable toselect ∆/t=0.5.
site numbers are in Fig. 3(e). The minimum energy is [11] Y. Murakami et al., Phys.Rev.Lett. 80, 1932 (1998).
[12] A. T. Boothroyd et al., Phys. Rev. Lett. 91, 257201
foundatd=0.1(Fig.3(f)). Themodulationoflevelsplit-
(2003).
ting stabilizes the striped charge ordering characterized
[13] T. Hotta et al.,Phys. Rev.B 60, R15009 (1999).
by a q=2π/3 peak in C(q) (Fig. 3(g)).
[14] This breathing-mode modulation is consistent with ex-
Notethat(3x2 r2/3y2 r2)-typeorbitalorderingdoes perimentalresults.SeeJ.M.Tranquadaetal.,Phys.Rev.
− −
not occur in Fig. 3(e). Phenomenologically, such or- B 52, 3581 (1995).