Electron correlation and Jahn-Teller interaction in manganese oxides

Naoto Nagaosa and Shuichi MurakamiDepartment of Applied Physics, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan

Hyun Cheol LeeAsia Pacific Center for Theoretical Physics, Seoul 151-742, Korea

~Received 19 September 1997!

The interplay between the electron repulsionU and the Jahn-Teller electron-phonon interactionELR isstudied with a large-d model for the ferromagnetic state of the manganese oxides. These two interactionscollaborate to induce local isospin~orbital! moments and to reduce the bandwidthB. The retardation effect ofthe Jahn-Teller phonon with frequencyV is especially important in reducingB, and a strongV dependenceoccurs even when the Coulomb interaction is dominant (U@ELR), as long asELR.V. The phonon spectrumconsists of two components: a temperature-independent sharp peak atv5V5V@(U14ELR)/U#1/2 and onecorresponding to the Kondo peak. A comparison of these results with experiments suggest thatV,ELR,U inthe metallic manganese oxides.@S0163-1829~98!51912-1#

Since the discovery of colossal magnetoresistance~CMR!the manganese oxidesR12xAxMnO3 ~R: rare earth metal ion,A: divalent metal ion! have been attracting intensiveinterest.1 One of the controversial issues is the role of theorbital degeneracy ofeg electrons. In the conventional mod-els of these compounds,2–5 the strong Hund’s coupling isconsidered to be of primary importance and the orbital de-generacy has been often neglected. Recently, however, Mil-lis et al.6,7 correctly pointed out that the Hund’s couplingalone is not enough to explain the CMR together with theinsulating temperature dependence of the resistivityr(T)above the ferromagnetic transition temperatureTc . The ad-ditional coupling that they present as important is the Jahn-Teller electron-phonon coupling, which lifts the double de-generacy ofeg orbitals and gives rise to the polaronic effect.The small polaron formation aboveTc leads to the insulatingr(T). In contrast with the insulating phase, the spin align-ment below Tc will enlarge the bandwidthB as B}cos(uij /2) ~u i j is the angle between the two neighboring

spinsSW i and SW j !, and the Jahn-Teller interaction enters theweak coupling regime. This scenario seems to be consistentwith the large isotope effect onTc by replacing O16 by O18.8

The strong correlation models of these compounds havealso been studied by several authors, especially for the un-doped cases.9,10 In these models the Jahn-Teller interaction isconsidered to be smaller than the Coulomb interactions, and

the effective Hamiltonian for the spin-orbital coupled systemand its phase diagram have been clarified. It is rather naturalto assume the strong electron correlation because the strongHund’s coupling originates from the strong electron-electroninteractions ~;5 eV!. However, it is not trivial that theelectron-electron interaction continues to be strong, even inthe effective Hamiltonian describing the low energy physicsafter the screening by oxygen orbitals and conduction elec-trons; and the controversial issue of whether or not theelectron-electron interaction and/or the Jahn-Teller couplingare in the strong coupling regime, remains.

Experimentally there are several anomalous featureswhich cannot be explained by the weak Jahn-Teller couplingdescribed above even in the low temperature ferromagneticphase belowTc :

~a! In the neutron scattering experiment no temperature-dependent phonon modes have been observed.11 The recentRaman scattering experiment also shows that the Jahn-Tellerphonons~especially their frequencies! are temperature inde-pendent and insensitive to the ferromagnetic transition atTc .12

~b! The photoemission spectra show a small discontinuityat the Fermi edge even atT!Tc , which suggests that someinteractions still remain strong there.13

~c! The optical conductivitys~v! at T!Tc is composedof two components, i.e., the narrow Drude peak~v

RAPID COMMUNICATIONS

PHYSICAL REVIEW BCONDENSED MATTER AND MATERIALS PHYSICS

THIRD SERIES, VOLUME 57, NUMBER 12 15 MARCH 1998-II

RAPID COMMUNICATIONS

Rapid Communications are intended for the accelerated publication of important new results and are therefore given priority treat-ment both in the editorial office and in production. A Rapid Communication inPhysical Review Bmay be no longer than four printedpages and must be accompanied by an abstract. Page proofs are sent to authors.

570163-1829/98/57~12!/6767~4!/$15.00 R6767 © 1998 The American Physical Society

,0.02 eV! and the broad incoherent component extendingup to v;1 eV.14 The Drude weight is very small, whichseems to be consistent with the photoemission spectra.

~d! The low temperature resistivityr(T) can be fitted by

r~T!5r01AT2, ~1!

whereA is a large constant of the order of 500mV cm/K2,15

again suggesting the strong electron correlation.~e! Contrary to the case of resistivity seen in~d!, the co-

efficient of T-linear specific heat is very small, withg52 mJ/K, which violates the Kadowaki-Woods law forthese compounds.16

Although~e! is difficult to reconcile with~a!–~d!, we con-sider the latter as evidence for strong coupling even atT!Tc . Because the spins are perfectly aligned atT!Tc , theonly remaining degrees of freedom are the orbital ones. Inthis paper, we study a larged model17 for the ferromagneticstate including both the electron-electron interactionU andthe Jahn-Teller couplingg. This method is the generalizationof Ref. 7 in two respects:~i! including the electron-electroninteraction,~ii ! including the quantum fluctuations. The latteris especially essential in describing the low temperatureFermi liquid state, which is described by the Kondo peak inthe larged limit.17 The strong electron-electron interactionwith the reasonable magnitude of the Jahn-Teller couplingexplains both the large isotope effect and~a!. Moreover, thefeatures of strong correlation seen in~b!–~d! are at leastconsistent with the largeU picture, although~e! still requiresfurther study, which we have not undertaken in this paper.

We start with the Hubbard-Holstein model for the ferro-magnetic state.

H52 (i , j ,a,a8

t i jaa8cia

† cj a81U(i

ni↑ni↓

2g(i

Qi~ni↑2ni↓!11

2 (i

S Pi2

M1MV2Qi

2D , ~2!

where the spin indicesa5↑,↓ correspond to the orbital de-grees of freedom as↑5dx22y2 and↓5d3z22r 2, and the realspins are assumed to be perfectly aligned. We consider onlyone Jahn-Teller displacement modeQi for each site, whilethere are two modesQ2i ,Q3i in the real perovskitestructure.18 However, this does not change the qualitativefeatures obtained below. The Jahn-Teller mode is assumed tobe an Einstein phonon with frequencyV. The transfer inte-

gral t i jaa8 depends on a pair of orbitals (a,a8) and the hop-

ping direction. These dependencies lead to the various lowlying orbital configurations, and thus they suppress the or-bital orderings in the ferromagnetic state. Actually, there isno experimental evidence for the orbital orderings, e.g., theanisotropies of the lattice constants and/or transport proper-ties in the ferromagnetic state. Then we assume that no or-bital ordering occurs down to the zero temperature due to thequantum fluctuations, and the transfer integral is assumed to

be diagonal for simplicity, i.e.,t i jaa85t i j daa8 . This means

that the ground state is a Fermi liquid with two degeneratebands at the Fermi energy. In order to describe this Fermiliquid state, we employ the large-d approach where Eq.~2! ismapped to the impurity Anderson model with a self-

consistent condition~Sec. II of Ref. 17!. This approach hasbeen applied to the manganese oxides in studying the Hund’scoupling by Furukawa,5 and in studying the additional Jahn-Teller coupling, by Milliset al.7 The action for the effectiveimpurity model is given by

S5E0

b

dtE0

b

dt8ca†~t!G0

21~t2t8!ca~t8!

1E0

b

dt@Un↑~t!n↓~t!2gQ~t!„n↑~t!2n↓~t!…#

1E0

b

dtM

2@„]tQ~t!…21V2Q~t!2#, ~3!

whereG021(t2t8) is the dynamical Weiss field representing

the influence from the surrounding sites. The self-consistency condition is that the on-site Green’s functionG( ivn)5„G0( ivn)212S( ivn)…21 should be equal to theHilbert transform of the density of statesD(«) as

G~ ivn!5E d«D~«!

ivn1m2S~ ivn!2«. ~4!

We take the Lorentzian density of statesD(«)5t/„p(«2

1t2)… because there is no need to solve the self-consistencyequation in this case, and the Weiss field is given by

G021~ ivn!5 ivn1m1 i t sgnvn . ~5!

The only unknown quantity is the chemical potentialm,which is determined by the electron number. The determina-tion of chemical potential requires some numerical calcula-tions, but our discussion below is not sensitive to the locationof the chemical potential. Then we takem as the parameter ofour model, and the problem is now completely reduced tothat of a single impurity. Now we introduce a Stratonovich-Hubbard~SH! field j~t! to represent the Coulomb interactionU. Then the action is given by

S5(vn

~ ivn1m1 i t sgnvn!ca†~vn!ca~vn!

1E0

b

dtFD2

Uj~t!21

M

2@„]tQ~t!…21V2Q~t!2#

1z~t!Q~t!2„Dj~t!1gQ~t!…„n↑~t!2n↓~t!…G , ~6!

whereD5(t21m2)/t andz~t! is the test field to measure thephonon correlation function. At this stage the electron iscoupled with the linear combination of the SH fieldj and thephononQ as h(t)[j(t)1(g/D)Q(t). Then the effectiveaction can be derived as

S5(vn

@~ivn1m1it sgnvn!ca†~vn!ca~vn!

2Dh~ivn!„n↑~2 ivn!2n↓~2 ivn!…#

1(vn

1

2M ~vn21V2!

F2z~ ivn!z~2 ivn!12D2

U

RAPID COMMUNICATIONS

R6768 57NAOTO NAGAOSA, SHUICHI MURAKAMI, AND HYUN CHEOL LEE

3M ~vn21V2!h~ ivn!h~2 ivn!1

2gD

U

3„z~ ivn!h~2 ivn!1h~ ivn!z~2 ivn!…G , ~7!

where V5VAUeff /U. The effective interactionUeff is thesum of the Coulomb repulsionU and the lattice relaxationenergy ELR5g2/(2MV2) as Ueff5U14ELR . The phononGreen’s functiond( ivn)5^Q( ivn)Q(2 ivn)& is given by

d~ ivn!51

M ~vn21V2!

14S gD/U

M ~vn21V2!

D 2

xh~ ivn!, ~8!

wherexh( ivn)5^h( ivn)h(2 ivn)& is the orbital suscepti-bility of the electronic system. The first term is the usualphonon Green’s function with the renormalized frequencyV,which is higher than the bareV. This V does not depend onthe electron response function and hence is temperature in-dependent. WhenU!ELR , V is much larger thanV and thefirst term of Eq.~8! is irrelevant in the phonon frequencyregion v;V;0.1 eV. In the opposite limitU@ELR , therenormalization of the phonon frequency is small. The sec-ond term depends on the temperature~Kohn anomaly! andshows the characteristic line shape with the broadening.From the above consideration together with the experimentalfact that no temperature dependence of the phonon frequencyhas been observed, we conclude that the Coulomb interactionU is larger than the lattice relaxation energyELR .

Now let us study the electronic system in detail. Follow-ing Sec. III C of Ref. 19, we can obtain the solution of theintegral equation for the electron Green’s function in thepresence of the time-dependent fieldh~t!, and hence for theeffective action. The result is

Seff5(vn

D2

U

vn21V2

vn21V2 h~ ivn!h~2 ivn!

2E0

b

dt2D

p Fh~t!tan21 h~t!21

2ln„11h~t!2

…G1

1

p2 E0

b

dtE0

b

dt8P1

t2t8h~t!

dh~t8!

dt8

31

h~t!22h~t8!2 ln11h~t!2

11h~t8!2 , ~9!

where P denotes the principal value. Equation~9! differsfrom Eqs. ~3.40!–~3.43! of Ref. 19 by the frequency-dependent coefficient ofh( ivn)h(2 ivn) due to the retarda-tion effect of the phonons. A similar model without the Cou-lomb repulsionU has been studied by Yu and Anderson,20

which corresponds to the limitU→0, V→`, V2U5finite inEq. ~9!. First, we consider the saddle point solution for thehfield, assuming the static configurationh(t)5h0 . Then theaction for the static solution becomes

S05bF D2

Ueffh0

222D

p S h0tan21h021

2ln~11h0

2! D G . ~10!

WhenUeff /(pD),1, there is a single minimum at the origin,and the double minima appear forUeff /(pD).1. We are in-terested in the limit of strong correlationUeff /(pD)@1, andin this limit the tunneling events~instantons! between thetwo minima ath56h0>6(Ueff/2D) play an essential rolein producing the Kondo peak at the Fermi level. The thirdterm in Eq. ~9! gives the nonlocal interaction between theinstantons along the time axis, which corresponds to theJzterm when it is mapped to the Kondo problem. The fugacityz of the instanton, which corresponds toJ' , can be esti-mated following Sec. IV B of Ref. 19. Lett0 be the width ofthe instanton, i.e., hopping time, and the action for a singleinstanton is given by

Sinst~t0!5(vn

2g2D2vn2

U2M ~vn21V2!V2

h inst~ ivn!h inst~2 ivn!

1 16 t0Ueff2ln t0. ~11!

The last term can be written in terms of the derivativeh(t)5dh inst(t)/dt, which is the localized function around thecenter of the instanton (t50) with the width of the order oft0 and the integral ofh(t) over t is 2h0 . We assumeh(t)5(h0 /t0)e2utu/t0. Then the summation overvn can beeasily done and we obtain

Sinst~t0!54ELR

V2t0f ~Vt0!1

1

6t0Ueff2 ln t0 , ~12!

where f (a)5a(a12)/„4(11a)2…. Minimizing Sinst(t0)

with respect tot0 , we obtaint0 ast0>6/Ueff for Ueff@6 Vand t0>3/Ueff1A(3/Ueff)

216ELR /(V2Ueff) for Ueff!6V.Introducing the dimensionless variablesx5ELR /V and y5U/ELR , the reduction factorR for the instanton fugacityz5h0

21e2R is given by

R52xA114y21, for y@1/max~x,x2!, ~13!

R5A11 23 x2~41y!, for y!1/max~x,x2!. ~14!

The main conclusion here is that whenx.1, i.e., ELR.V,the reduction factorR is proportional tox even in the limitU@ELR . This is because the overlap of the phonon wavefunction enters the tunneling matrix element even when theCoulomb interaction is dominating.

In Sec. IV B of Ref. 19 the fugacity of the instantons andthe long range interaction between them can be mapped tothe anisotropic Kondo model with the Kondo couplingsgiven by

N~0!J'>D

Ueffe2R, N~0!Jz>

D

Ueff, ~15!

whereN(0) is the density of states for the Kondo problemand the reduction factorR is estimated to be Eqs.~13! and~14!. The scaling equations forJ' andJz are given as21

2dJ'

d ln L5JzJ'N~0!, 2

dJz

d ln L5J'

2 N~0!, ~16!

whereL is the energy cutoff. In the limit of largeR(@1),i.e., J'!Jz , we approximateJz as a constant in the firstequation of~16!, and we obtain

RAPID COMMUNICATIONS

57 R6769ELECTRON CORRELATION AND JAHN-TELLER . . .

J'~L!5J'S L0

L D JzN~0!

, ~17!

where L0 is the original cutoff of the order oft. Thetemperature T0 at which J'>Jz is given as T0>L0 exp@2R/JzN(0)#. Scaling fromT0 to the Kondo tem-peratureTK , whereJzN(0);J'N(0);1 can be neglectedwhenR@1, we obtain the estimate for the Kondo tempera-ture as

TK;t expF2RUeff

D G . ~18!

This Kondo singlet state corresponds to the Fermi liquid inthe original large-d system~Sec. VII of Ref. 17!. Here themomentum dependence of the self-energy is neglected, andthe effect of the correlation is represented by the wave-function renormalizationZ given by

Z5S 12]S~v!

]v D 21

, ~19!

which gives the residue of the quasiparticle.Z is related tothe mass enhancement asm* /m5Z21. In the large-d limitthe physical quantities can be calculated in terms of the localelectron Green’s functionG( ivn) in Eq. ~4!, because thevertex corrections can be neglected~see Sec. IV of Ref. 17!.The spectral function of the electron has a so-called Kondopeak near the Fermi energy, which corresponds to the coher-ent motion of the electrons. The Kondo temperature in Eq.~18! gives the width of the Kondo peak and hence the effec-tive bandwidthB, i.e., the effective mass enhancement isestimated asZ215m* /m5t/TK;exp@RUeff /D#. This strongmass enhancement manifests itself in the physical quantitiesas follows~see Sec. VII of Ref. 17!. The orbital susceptibil-ity xh( ivn→v1id! has a peak atv50 with the peak heightof the order ofh0

2/TK and with the width of the order ofTK .The specific heat coefficientg should be proportional toB21,while the coefficientA of T2 in Eq. ~1! scales asA}B22. Theferromagnetic interaction between spins is due to the doubleexchange mechanism,2–4 the strength of which is determinedby the kinetic energy gain when the spins are perfectlyaligned. Hence the ferromagnetic transition temperatureTc isexpected to be proportional toB, and we expect a largeisotope effect whenR.1. In other words, the large isotope

effect does not rule out the large Coulomb interactionU.The edge of the photoemission spectrum at the Fermi energyand the Drude weight are reduced by the factorm/m* . Asmentioned above, many experiments except for those of spe-cific heat seem to suggest that the manganese oxides in theirferromagnetic state belong to the strongly correlated region.Then it is natural that the large isotope effect is observed.The bandwidth is rather difficult to pin down accurately, buta rough estimate fromA in Eq. ~1! is B;0.1 eV, which iscomparable toV. This suggests that the suppression is notextremely large. Another experimental fact is that thetemperature-independent Jahn-Teller phonon is observed atthe frequency near the original one,12 which means thatU islarger or at least of the order ofELR . Then we conclude thatthe relative magnitude of the energy scales isV,ELR,U forthe ferromagnetic phase in the manganese oxides. The smallspecific heat coefficient necessitates the physical mecha-nisms which have not been included in our model. One pos-sibility is that the intersite process, e.g., the orbital singletformation, and another possibility is that the proximate quan-tum criticality is affecting the Fermi liquid behavior.

In summary, we have studied a model of manganese ox-ides in the ferromagnetic state taking into account both theCoulomb repulsionU and the Jahn-Teller interactionELR .These two interactions collaborate to induce the local orbitalmoments. In the strong coupling case, i.e.,Ueff5U14ELR@t ~t: transfer integral!, the overlap of the phonon wavefunctions (;e2ELR /V) enters the tunneling amplitude be-tween the two minima for the orbital moment. The phononspectral function consists of two parts, i.e., the sharp peak atthe renormalized frequencyV5VAUeff /U and the broadpeak with the width of the order of the bandwidth. Theseresults reconcile the large isotope effect onTc with the ap-parent temperature-independent phonon spectrum assumingU.ELR.V.

The authors would like to thank Y. Tokura, A. Millis, J.Ye, G. Kotliar, Q. Si, H. Kuwahara, S. Maekawa, H. Fuku-yama, K. Miyake, S. Ishihara, T. Okuda, K. Yamamoto, andY. Endoh for valuable discussions. This work was supportedby COE and Priority Areas Grants from the Ministry of Edu-cation, Science and Culture of Japan. Part of this work hasbeen done in the APCTP workshop and the Aspen Center forPhysics, and we acknowledge their hospitality.

1R. M. Kusterset al., Physica B155, 362~1989!; K. Chaharaet al., Appl.Phys. Lett.63, 1990~1993!; R. von Helmoltet al., Phys. Rev. Lett.71,2331~1993!; S. Jimet al., Science264, 413~1994!; Y. Tokuraet al., J.Phys. Soc. Jpn.63, 3931 ~1994!; H. Y. Hwanget al., Phys. Rev. Lett.75, 914 ~1995!.

2C. Zener, Phys. Rev.82, 403 ~1951!.3P. W. Anderson and H. Hasegawa, Phys. Rev.100, 675 ~1955!.4P.-G. de Gennes, Phys. Rev.118, 141 ~1960!.5N. Furukawa, J. Phys. Soc. Jpn.63, 3214~1994!.6A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett.74,

5144 ~1995!.7A. J. Millis, R. Mueller, and B. I. Shraiman, Phys. Rev. B54, 5389

~1996!; 54, 5405~1996!.8Guo-meng Zhao, K. Conder, H. Keller, and K. A. Muller, Nature~Lon-

don! 381, 676 ~1996!.9K. I. Kugel’ and D. I. Khomskii, JETP Lett.15, 446 ~1972!.

10S. Ishihara, J. Inoue, and S. Maekawa, Physica C263, 130 ~1996!.11M. C. Martin, G. Shirane, Y. Endoh, K. Hirota, Y. Moritomo, and Y.

Tokura, Phys. Rev. B53, 14 285~1996!.12K. Yamamoto and Y. Tokura~private communication!.13D. D. Sarmaet al., Phys. Rev. B53, 6873~1996!.14Y. Okimoto et al., Phys. Rev. Lett.75, 109 ~1995!.15A. Urushibaraet al., Phys. Rev. B51, 14 103~1995!.16B. F. Woodfield, M. L. Wilson, and J. M. Byers, Phys. Rev.

Lett. 78, 3201 ~1997!; T. Okuda and Y. Tokura~private communica-tion!.

17For a review see, A. Georgeset al., Rev. Mod. Phys.68, 13 ~1996!.18J. Kanamori, J. Appl. Phys.31, 14S~1960!.19D. R. Hamann, Phys. Rev.2, 1373~1970!.20C. C. Yu and P. W. Anderson, Phys. Rev. B29, 6165~1984!.21P. W. Anderson, G. Yuval, and D. R. Hamann, Phys. Rev. B1, 4464

~1970!.

RAPID COMMUNICATIONS

R6770 57NAOTO NAGAOSA, SHUICHI MURAKAMI, AND HYUN CHEOL LEE