Optical spectroscopic study of the interplay of spin and...

PHYSICAL REVIEW B 15 JANUARY 2000-IIVOLUME 61, NUMBER 4

Optical spectroscopic study of the interplay of spin and charge ina8-NaV2O5

A. Damascelli,* C. Presura, and D. van der MarelSolid State Physics Laboratory, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands

J. Jegoudez and A. RevcolevschiLaboratoire de Chimie des Solides, Universite´ de Paris-sud, Baˆtiment 414, F-91405 Orsay, France

~Received 1 June 1999!

We investigate the temperature-dependent optical properties ofa8-NaV2O5, in the energy range 4 meV–4eV. The symmetry of the system is discussed on the basis of infrared phonon spectra. By analyzing theoptically allowed phonons at temperatures below and above the phase transition, we conclude that a second-order change to a larger unit cell takes place below 34 K, with a fluctuation regime extending over a broadtemperature range. In the high-temperature undistorted phase, we find good agreement with the recentlyproposed centrosymmetric space groupPmmn. On the other hand, the detailed analysis of the electronicexcitations detected in the optical conductivity provides direct evidence for a charge disproportionated elec-tronic ground state, at least on a local scale: A consistent interpretation of both structural and optical conduc-tivity data requires an asymmetrical charge distribution on each rung, without any long-range order. We showthat, because of the locally broken symmetry, spin-flip excitations carry a finite electric dipole moment, whichis responsible for the detection of direct two-magnon optical absorption processes forEia. The charged-magnon model, developed to interpret the optical conductivity ofa8-NaV2O5, is described in detail, and itsrelevance to other strongly correlated electron systems, where the interplay of spin and charge plays a crucialrole in determining the low-energy electrodynamics, is discussed.

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I. INTRODUCTION

After many years of intensive experimental and theorcal work on CuGeO3 another inorganic compounda8-NaV2O5, has been attracting the attention of the scientcommunity working on low-dimensional spin systems,general, and on the spin-Peierls~SP! phenomenon, in particular. In fact, in 1996 the isotropic activated behavior of tmagnetic susceptibility was observed at low temperaturea8-NaV2O5.1 Moreover, superlattice reflections, with a latice modulation vectork5(p/a,p/b,p/2c), were found inan x-ray scattering experiment,2 and a spin gapD59.8 meVwas observed at the reciprocal-lattice point (2p/a,p/b,0),with inelastic neutron scattering.2 The SP picture was theproposed to explain the low-temperature properties ofcompound, with a SP transition temperatureTSP534 K.1 Inthis context,a8-NaV2O5 seemed to be a particularly inteesting material, as far as the full understanding of thephenomenon is concerned, because on the basis of the stural analysis reported by Carpy and Galy in 1975, sugging the noncentrosymmetric space groupP21mn,3 it wasassumed to be described by the one-dimensional~1D! spin-1/2 Heisenberg~HB! model better than CuGeO3. In fact, forthe latter compound it has been established bexperimentally4 and theoretically5 that the 2D character othe system cannot be neglected.

On the other hand, on the basis of experimental reslater obtained, the interpretation of the phase transitionstill controversial. First of all, the reduction ofTSP uponincreasing the intensity of an externally applied magnefield, expected for a true SP system, has not bobserved.6,7 For instance, Bu¨chneret al.,6 performing low-temperature magnetization and specific-heat measurem

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in magnetic field H<14 T, obtainedDTSP(H)5TSP(0)2TSP(14 T)'0.15 K, i.e., almost a factor of 7 smaller thawhat was expected on the basis of Cross and Fisher theo8,9

Furthermore, they claimed that the entropy reduction expmentally observed across the phase transition is considerlarger than the theoretical expectation for a 1D antiferromnetic ~AF! HB chain with only spin degrees of freedom anJ548 meV ~i.e., value obtained fora8-NaV2O5 from mag-netic susceptibility measurements1!. In order to explain thesefindings, degrees of freedom additional to those of the ssystem should be taken into account, contrary to the inpretation of a magnetically driven phase transition.6

This already quite puzzling picture became even mcomplicated when the crystal structure and the symmetrythe electronic configuration ofa8-NaV2O5, in the high-temperature phase, were investigated in detail. In fact, onbasis of recent x-ray-diffraction measurements10–12 it wasfound that the symmetry of this compound at room tempeture is better described by the centrosymmetric space grPmmn than by the originally proposedP21mn.3 On theother hand, the intensities and polarization dependence oelectronic excitations detected in optical conductivity specgave a direct evidence for a broken-parity electronic groustate,13,14 in apparently better agreement with the noncetrosymmetric space groupP21mn. The solution of this con-troversy and the assessment of the symmetry issue arfundamental importance for the understanding of the etronic and magnetic properties of the system and, ultimatof the phase transition.

In this paper we present a detail investigation of theterplay of spin and charge ina8-NaV2O5, on the basis ofoptical reflectivity and conductivity data in the energy ran

2535 ©2000 The American Physical Society

2536 PRB 61A. DAMASCELLI et al.

FIG. 1. High-temperature crystal structure ofa8-NaV2O5 :Pyramids around V ions and rows of Na ions are shown.

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4 meV–4 eV. We will start from the discussion of the rcently proposed high-temperature crystal structure,10–12 andits implications on the physics of this compound~Sec. II!. Agroup theoretical analysis for the two different space growill be reported and later compared to the experimentaobserved far-infrared phonon spectra. In Sec. III, we wpresent a theoretical model which we will use to interpqualitatively and quantitatively the optical conductivity spetra of a8-NaV2O5. In Sec. IV, we will analyze the data obtained with optical spectroscopy at different temperatuabove and belowTc534 K ~throughout the paper, we wilrefer to the transition temperature asTc because the interpretation of the phase transition is still controversial!. In particu-lar, we will concentrate on:

~i! Analysis of the temperature-dependent phonon specin relation to the x-ray-diffraction data, in order to: Firslearn more about the symmetry of the material in both hiand low-temperature phase. Second, detect signatures olattice distortion and study the character of the transition

~ii ! Detailed study of the very peculiar electronic amagnetic excitation spectra. In particular, we will show thwe could detect a low-frequency continuum of excitatiowhich, on the basis of the model developed in Sec. III,ascribe to ‘‘charged bimagnons,’’ i.e., direct two-magnoptical absorption processes.

Finally, in Sec. V, we will discuss the relevance of ofindings to the understanding of the nature of the phase tsition in a8-NaV2O5. In particular we will try to assesswhether the picture of a charge ordering transition, accopanied by the opening of a magnetic gap, is a valid altertive to the originally proposed SP description. In fact, thalternative interpretation has been recently put forward15–19

on the basis of temperature-dependent nuclear magnresonance~NMR! data.15

II. SYMMETRY PROBLEM IN a8-NaV2O5

The interpretation of the phase transition at 34 K as atransition1 is based on the noncentrosymmetric space gr

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P21mn originally proposed by Carpy and Galy for the higtemperature phase ofa8-NaV2O5.3 Following their crystal-lographic analysis, the structure can be constructed frdouble rows~parallel to theb axis! of edge sharing pyramidsone facing up and the other down, with respect to thea-bplane~see Fig. 1!. These double rows are connected by shing corners, yielding a planar material. The planesstacked along thec axis, with the Na in the channels of thpyramids. In thea-b plane~see Fig. 2!, we can then identifylinear chains of alternating V and O ions, oriented alongb axis. These chains are grouped into sets of two, forminladder, with the rungs oriented along thea axis. The rungsare formed by two V ions, one on each leg of the laddbridged by an O ion. The V-O distances along the rungsshorter than along the legs, implying a stronger bondalong the rung. In thea-b plane the ladders are shifted halfperiod along theb axis relative to their neighbors. The noncentrosymmetric space groupP21mn, allows for two in-

FIG. 2. Two-leg ladder structure formed by O and V ions in ta-b plane of a8-NaV2O5. Black dots are O ions; dark and lighgrey balls are V41 and V51 ions, respectively, arranged in 1Dchains in the space groupP21mn.

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PRB 61 2537OPTICAL SPECTROSCOPIC STUDY OF THE . . .

equivalent V positions in the asymmetric unit. These siwere interpreted as different valence states, V41 and V51,represented by dark and light grey balls, respectively, in F2. In this structure it is possible to identify well distinct 1magnetic V41 (S51/2) and nonmagnetic V51 (S50)chains running along theb axis of the crystal, and alternatineach other along thea axis. This configuration would beresponsible for the 1D character~Bonner-Fisher-type20! ofthe high-temperature susceptibility21 and for the SP transition, possibly involving dimerization within the V41 chains.1

Moreover, due to the details of the crystal structure, only oof the three t2g orbitals of V41, i.e., the dxy orbital, isoccupied.12 As a result, the insulating charactera8-NaV2O5, which has been observed in dc resistivmeasurements,22,23would be an obvious consequence of haing a 1D half filled Mott-Hubbard system.

However, recent redeterminations of the crysstructure10–12 showed that the symmetry ofa8-NaV2O5aboveTc is better described by the centrosymmetric spgroupPmmn: The suggested structure is very similar to tone proposed by Carpy and Galy3 ~a small difference is thesomewhat more symmetric eightfold coordination of Na10!.On the other hand, one major difference was found, namthe presence of a center of inversion in the unit cell. Tevidence is the very lowR(F) value of 0.015, and the facthat no lower value ofR(F) can be obtained when the x-raydiffraction data are refined omitting the inversion center.10 Inparticular, this result suggests the existence of only one kof V per unit cell, with an average valence for the V ions14.5. This finding has important implications for the undestanding of the electronic and magnetic propertiesa8-NaV2O5. In fact, we have now to think in the frameworof a quarater filled two-leg ladder system: It is still possibto recover an insulating ground state assuming that thdelectron, supplied by the two V ions forming a rung of tladder, is not attached to a particular V site but is sharedV-O-V molecular bonding orbital along the rung.12 In thisway, we are effectively back to a 1D spin system~a two-legladder with one spin per rung in a symmetrical positio!.However, in case of a SP phase transition, singlets wohave to be formed by electrons laying in molecularlike orbals and a rather complicated distortion pattern would havtake place: Not simply the dimerization within a real 1chain ~as, e.g., in CuGeO3), but a deformation of theplaquettes, formed by four V ions, within a ladder.

Regarding the controversy about the x-ray structure demination fora8-NaV2O5 aboveTc , it is important to stressthat, even though it is now well established that the ‘‘moprobable’’ space group is the centrosymmetricPmmn~on thebasis of the statistics of the refinement!, deviations from cen-trosymmetry up to 0.029 Å are unlikely but cannotexcluded.10 Moreover, one has to note that x-ray-diffractiomeasurements are sensitive to the charge distribution ofelectrons and not of valence electrons. Therefore if a lobreaking of symmetry in the distribution of the Vd electronswere present on a local scale without the long-range orequired by the noncentrosymmetric space groupP21mn ~assuggested by the optical conductivity data we will presenSec. IV D!, it would not be detectable in an x-ray-diffractioexperiment: X-rays would just see the ‘‘space averaggiven by the centrosymmetric space groupPmmn.

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Group-theoretical analysis

An alternative way to assess the symmetry issue isinvestigate the Raman and infrared phonon spectra, coming the number of experimentally observed modes tonumber expected for the two different space groups onbasis of group theory. Obviously, a large number of modhas to be expected because each unit cell contains twomula units, corresponding to 48 degrees of freedom, whwill be reflected in 48 phonon branches.

In the space groupPmmnthe site groups in the unit celare:C2v

z for the 2 Na and the 2 O~1! atoms, andCsxz for the

4 V, the 4 O~2!, and the 4 O~3!. Following the nuclear sitegroup analysis method extended to crystals,24 the contribu-tion of each occupied site to the total irreducible represention of the crystal is

GNa1O(1)52@Ag1B1u1B2g1B2u1B3g1B3u#,

GV1O(2)1O(3)53@2Ag1Au1B1g12B1u12B2g1B2u

1B3g12B3u#.

Subtracting the silent modes (3Au) and the acoustic mode(B1u1B2u1B3u), the irreducible representation of the opcal vibrations, for the centrosymmetric space groupPmmn, is

G58Ag~aa,bb,cc!13B1g~ab!18B2g~ac!15B3g~bc!

17B1u~Eic!14B2u~Eib!17B3u~Eia!, ~1!

corresponding to 24 Raman (Ag ,B1g ,B2g ,B3g) and 18 in-frared (B1u ,B2u ,B3u) active modes.

In the analysis for the noncentrosymmetric space groP21mn, we have to consider only the site groupCs

yz for allthe atoms in the unit cell: 2 Na, 2 V~1!, 2 V~2!, 2 O~1!, 2O~2!, 2 O~3!, 2 O~4!, and 2 O~5!. Therefore the total irreducible representation is

83GCsyz583@2A11A21B112B2#.

Once again, subtracting the acoustic modesA11B11B2~there is no silent mode in this particular case!, the irreduc-ible representation of the optical vibrations, for the noncetrosymmetric space groupP21mn, is

G8515A1~aa,bb,cc;Eia!18A2~bc!17B1~ab;Eib!

115B2~ac;Eic!, ~2!

corresponding to 45 Raman (A1 ,A2 ,B1 ,B2) and 37 infrared(A1 ,B1 ,B2) active modes.

As a final result, we see that the number of optical vibtions, expected on the basis of group theory, is very differfor the two space groups. Moreover, whereas in the casPmmnthe phonons are exclusively Raman or infrared actifor P21mn, because of the lack of inversion symmetry, theis no more distinction betweengerade and ungeradeandcertain modes are, in principle~group theory does not saanything about intensities!, detectable with both techniques

III. CHARGED-MAGNON MODEL

In the context of the extensive work done, in the last feyears, on 2DS51/2 quantum AF and their 1D analogs, it ha

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been shown both experimentally and theoretically25–30 thatalso optical spectroscopy, besides elective techniquesRaman and neutron scattering, can be a very useful probstudying the spin dynamics in this systems. For examplethe 2D system La2CuO4 ~1D system Sr2CuO3) where two-magnon~two-spinon! excitations are in principle not optically active because of the presence of a center of inverthat inhibits the formation of any finite dipole momenphonon-assisted magnetic excitations were detected inmid-infrared region.25,28 These magnetoelastic absorptioprocesses are optically allowed because the phonon thervolved is effectively lowering the symmetry of the system

Let us now concentrate on a system where a breakinsymmetry is present because of charge ordering. We sthat, in this case, magnetic excitations are expected todirectly optically active and detectable in an optical expement ascharged bimagnons: Double spin-flip excitationscarrying a finite dipole moment. We will do that investigaing the spin-photon interaction in a single two-leg laddwith a charge disproportionated ground state. On the basthis model we will later interpret the optical conductivispectra ofa8-NaV2O5.

A. Model Hamiltonian

In this section we discuss the single two-leg ladderpicted in Fig. 3, wheret' and t i are the hopping parametefor the rungs and the legs, respectively,d' is the length of arung, and2D/2 and1D/2 are the on-site energies for thleft and the right leg of the ladder, respectively. We workquarter filling, i.e., one electron per rung. The total HamtonianHT of the system is

HT5H01H'1H i , ~3!

H05U(j

$njR↑njR↓1njL↑njL↓%1D(j

njC ~4!

H'5t'(j ,s

$L j ,s† Rj ,s1H.c.%, ~5!

H i5t i(j ,s

$Rj ,s† Rj 11,s1L j ,s

† L j 11,s1H.c.%, ~6!

FIG. 3. Sketch of the single two-leg ladder. Circles representionic sites having on-site energies2D/2 and1D/2 on the left andon the right leg of the ladder, respectively.t' andt i are the hoppingparameters for the rungs and the legs, respectively,d' is the lengthof a rung, andj is the rung index. Also indicated is the relatioassumed throughout the discussion, between the different enscales of the model.

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where L j ,s† (Rj ,s

† ) creates an electron with spins on theleft-hand ~right-hand! site of the j th rung, U is the on-siteHubbard repulsion,njR↑njR↓ (njL↑njL↓) is counting thedouble occupancies on the right-hand~left-hand! site, njC5(njR2njL)/2 is the charge displacement operator, andD isthe potential energy difference between the two sites. Csidering onlyH0 at quarter filling one immediately realizethat D is not only breaking the left-right symmetry of thsystem~for the symmetric ladderD50), but it is also intro-ducing a long-range charge ordering with one electronsite on the left leg of the ladder. Working in the limitt'@t i we can consider each rung as an independent polar mecule~with one electron! described by the HamiltonianH j 01H j' . The two solutions are lopsided bonding and anbonding wave functionsuL&5uuL&1vuR&, and uR&5uuR&2vuL&, with

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where ECT5AD214t'2 is the splitting between these tw

eigenstates. The excitation of the electron from the bondto the antibonding state is an optically active transition wa degree of charge transfer~CT! from the left to the right sitewhich is larger the bigger theD. It is possible to calculatethe integral of the real part of the optical conductivitys1(v)for this excitation, quantity that can be very useful in tanalysis of optical spectra. We start from the geneequation31

E0

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~En2Eg!U^nu(i

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, ~8!

whereV is the volume,qe is the electron charge andi is thesite index. As for a single rungRuxi uL&52d't' /ECT , weobtain the following expression, which is exact for one eletron on two coupled tight-binding orbitals:

ECT

s1~v!dv5pqe2Nd'

2 t'2 \22ECT

21 , ~9!

whereN is the volume density of the rungs. Comparing E~9! and the expression forECT with the area and the energposition of the CT peak observed in the experiment, weextract botht' andD.

In the ground state of the HamiltonianH01H' , eachelectron resides in auL& orbital, with some admixture ofuR&.Let us now introduce the coupling of the rungs along tlegs, consideringH i and a small fragment of the ladder witonly two rungs. We have now to take into account the sdegrees of freedom: If the two spins are parallel, the incsion of H i has no effect, due to the Pauli principle. If theform a S50 state, the ground state can gain some kineenergy along the legs by mixing in states where two eltrons reside on the same rung. We start from the singround state for the two rungs:

uL1L2&51

A2$uL1↑L2↓&2uL1↓L2↑&%, ~10!

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with eigenvalueEL1L252ECT , and we study the effect o

H i using a perturbation expansion int i /ECT , to the secondorder. However,H iuL1L2& does not coincide with any of thsinglet eigenstates of the system consisting of two electron one rung with the HamiltonianH j 01H j' . In order toobtain the exchange coupling constantJi between two spinson neighboring rungs, we have first to calculate the sespin-singlet eigenstates for the two electrons on one ruand then to evaluate the fractional parentages ofH iuL1L2&with respect to that set. The Hilbert subspace of two-electspin-singlet states is spanned by the vectors

uL jL j&5L j ,↑† L j ,↓

† u0&, ~11!

uL jRj&51

A2$L j ,↑

† Rj ,↓† 2L j ,↓

† Rj ,↑† %u0&, ~12!

uRjRj&5Rj ,↑† Rj ,↓

† u0&. ~13!

On this basis the Hamiltonian of thej th rung is

H j 01H j'5F U2D A2t' 0

A2t' 0 A2t'

0 A2t' U1DG . ~14!

A significant simplification of the problem occurs in thlargeU limit which, moreover, elucidates the physics of echange processes between a pair of dimers in an elegansimple way. We therefore take the limitU→`. The solu-tions for D50 are:32 The ~anti!symmetric linear combinations of uL jL j& and uRjRj& with energyU, and uL jRj& withenergyELR524t'

2 /U→0. The triplet states have energyFor a general value ofD, still obeyingD!U, the onlyrel-evanteigenstate for the calculation of the exchange procebetween neighboring dimers isuL jRj&, with energy ELR50. The other two states with energy of orderU are pro-jected out in this limit.

We can now go back to the problem of two rungs wone electron per rung, and consider the hopping alonglegs treatingH i as a small perturbation with respect toH0

1H' . We proceed by calculating the corrections touL1L2&by allowing a finite hopping parametert i between the rungsUsing lowest-order perturbation theory, the ground-stateergy of a spin singlet is:

Eg,S5052ECT2^L1L2uH i1

H01H'1ECTH iuL1L2&.

~15!

We proceed by calculating the coefficients of fractional p

entage by projectingH iuL1L2& on the two-electron singleeigenstates of a single rung:

H iuL1L2&5A2t i (j 51,2

$u2uL jL j&1v2uRjRj&1A2uvuL jRj&%.

~16!

Calculating the second-order correction toEL1L2for the S

50 state, in the limit whereU→`, and realizing that the

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correction for theS51 spin configuration is zero, we obtaithe exchange coupling constantJi :

Ji58t i

2t'2

@D214t'2 #3/2

. ~17!

B. Interaction Hamiltonians and effective charges

Let us now consider three rungs of the ladder~see Fig. 4!to understand the role of spin-flip excitations in this systeBecause of the energy gainJi , in the ground state we havantiparallel alignment of the spins~Fig. 4, left-hand side!.The j th electron is described by the lopsided bonding wa

function uL j& and, because of the virtual hopping to thnearest-neighbor~NN! rungs, by states where two electronreside on the same rung. Working in the limitU→` thesestates have one electron in theuL& and one in theuR& state onthe same rung. Therefore the charge of thej th electron isasymmetrically distributed not only over thej th rung ~darkgray area in Fig. 4, left! but also on the NN rungs~light grayarea in Fig. 4, left!. However, in the latter case most of thcharge density is localized on the right leg of the ladderwe flip the spin of thej th electron~Fig. 4, right-hand side!no virtual hopping is possible any more to the NN runbecause of the Pauli principle. The electron charge distrition is now determined only by the lopsided wave functi

uL j& ~dark gray area in Fig. 4, right!. As a result, there is anet dipole displacement of the antiparallel spin configuratcompared to the parallel one: The spin-flip excitations caa finite dipole moment parallel to the rung direction (a axis!.This dipole moment can couple to the electric field of tincident radiation in an optical experiment, and resultwhat we refer to ascharged bimagnonexcitations. One mayremark, at this point, that only double spin flips canprobed in an optical experiment because of spin consetion. However, what we just discuss for a single spin flip cbe extended to the situation where two spin flips, on tdifferent rungs, are considered.

The coupling to light withEW iaW ~rung direction! can nowbe included using the dipole approximation. The only effeis to change the potential energy of theuR& states relative tothe uL& states. In other words, we have to replaceD with

FIG. 4. Pictorial description of the electric dipole moment asciated with a single spin flip in the asymmetrical two-leg ladderspins are antiparallel~left-hand side!, the charge of thej th electronis asymmetrically distributed not only over thej th rung but also onthe NN rungs~with opposite asymmetry!, because of the virtuahopping. If we flip its spin~right-hand side!, the j th electron isconfined to thej th rung, resulting in a reduced charge densitythe right leg of the ladder. Therefore there is a net dipole displament~along the rung direction! of the antiparallel spin configurationcompared to the parallel one.

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D1qed'Ea , whereEa is the component of the electric fielalong the rung. The coupling to the CT transition on trungs is given by the Hamiltonian

HCT5qed'EanC5qed'Ea(j

njC . ~18!

The Hamiltonian for the spin-photon coupling can be otained taking the Taylor expansion ofJi with respect toEa ,and retaining the term which is linear in the electric fieNoticing thatdEa5dD/qed' , we have

HS5qmd'EahS5qmd'Ea(j

SW j•SW j 11 , ~19!

where qm5(1/d')(]Ji /]Ea)5qe(]Ji /]D) is the effectivecharge involved in a double spin-flip transition, andhS

5( jSW j•SW j 11. In the limit U→`, the effective charge reduces to

qm5qe

3JiD

D214t'2

, ~20!

where Ji is given by Eq.~17!. One has to note that forsymmetrical ladder, whereD50, the effective chargeqmvanishes, and the charged magnon effect disappears: Thculiar behavior of the spin flips we just described is anclusive consequence of the broken left-right symmetry ofground state. On the other hand, the bonding-antibondtransition would still be optically active. However, thbonding-antibonding energy splitting would be determinonly by t' , i.e.,ECT8 52t' . At this point we like to speculateon the role played byD in determining not only the opticaactivity of the double spin-flip excitations but also the inslating ~metallic! nature of the system along the leg directioFor D50 and for strong on-site Coulomb repulsionU, theenergy cost to transfer one electron from one rung toneighboring one isECT8 52t' . Therefore the system is insulating for 4t i,ECT8 52t' and metallic in the opposite case.we now start from a metallic situation 4t i>ECT8 52t' ,switching on D we can realize the condition 4t i,ECT

5AD214t'2 which would result in a metal-insulator trans

tion and, at the same time, in optical activity for the chargbimagnons.

C. Spectral weights

In this section we will attempt to estimate what would bin an optical experiment, the relative spectral weights forCT and the charged bimagnon excitations of the single tleg ladder. In particular, we want to show that these quaties can be expressed in terms of properly defined correlafunctions. Considering the integral of the optical conductity, Eq. ~8!, and the interaction HamiltoniansHCT and HS ,we can write:

ECT

s1~v!dv5pqe

2d'2

\2V(nÞg

~En2Eg!u^nunCug&u2,

~21!

-

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ESs1~v!dv5

pqm2 d'

2

\2V(nÞg

~En2Eg!u^nuhSug&u2. ~22!

Let us first concentrate on the CT excitation which issimpler problem because, as we discussed in Sec. III Aeach rung we are dealing with one electron on two coup

tight-binding orbitalsuL& and uR&. Therefore only one en-ergy term (En2Eg)5ECT has to be considered in the prevous equation. We can then write

(nÞg

~En2Eg!u^nunCug&u2

5ECTH(n

u^nunCug&u22u^gunCug&u2J5ECT$^gunC

2 ug&2u^gunCug&u2%5ECT$^nC2 &2^nC&2%,

~23!

where we took into account that the expectation value ofnCover the ground state is a real number. From Eqs.~21! and~23!, we obtain

ECT

s1~v!dv5pqe

2d'2

4\2VECTgC~T!, ~24!

where we have introduced the correlation function

gC~T!54k^nC2 &2^nC&2lT . ~25!

For the ground state ofN independent rungs per unit volum

ug&5) j uL j&, we calculategC5N@N2(N21)4u2v22N(v2

2u2)2#. Because (v22u2)25124u2v2, it follows that gC5N(2uv)2. ThereforegC50 if the two sites on a rung arecompletely independent:u51 andv50, for t'50 andDÞ0. On the other hand,gC has its maximum valuegC5Nwhen the rungs form homopolar molecules:u5v51/A2, fort'Þ0 andD50. Noticing that 1/(2uv)2511(D/2t')2 onecan see that Eqs.~24! and ~9! give exactly the same result.

In performing a similar quantitative analysis for the spiflip excitations in the single two-leg ladder, we have to cosider the elementary excitations of the 1DS51/2 HB AF,system which does not have long-range magnetic ordeany temperature due to its 1D character. It has been shby Faddeev and Takhtajan33 that the true elementary excitations of the 1D HB AF are doublets ofS51/2 spinon exci-tations. Each spinon is described by the dispersion relati

e8~k!5p

2J sink, 0<k<p. ~26!

The two-spinon continuum, with total spinS51 or S50, isdefined bye8(q)5e8(k1)1e8(k2), where q5k11k2, andk1 andk2 are the momenta of the spinons. The lower bounary is found fork150 andk25q ~or the other way around!:e18(q)5(p/2)J sinq. The upper one for k15k25q/2:e28(q)5pJ sin(q/2).

In the present case, we are dealing with double spin flgenerated by the interaction HamiltonianHS . What is thenrelevant is the four-spinon continuum defined bye8(q)5( ie8(ki), whereq5( iki , and the indexi labels the four

b

-

n

nnsthnaalith

nT

be

idee

r

inf

oavrro-

ta-

lafo

statee

lu-

dif-dyeenin-

3v,

eli-

mir-

ityrs-

inin

ms.of

arfor

ck-At

d

ofity-

fde-le

forityes.

erot


spinons. Of this excitation spectrum we would actually proin an optical experiment only theq50 ~or equivalentlyq52p) states, with total spinS50. These states form a continuum atq50 extending frome18(0)50 to e28(0)52pJ. Infact, because Eq.~26! has its maximumemax8 5pJ/2 for k5p/2, four spinons with the samek5p/2 correspond to theq52p (q50) excited state with the highest possible eergy: e28(0)54emax8 52pJ.

If we now go back to Eq.~22!, we see that the evaluatioof the integrated optical conductivity for the spin excitatiois problematic because it requires a detailed analysis ofmatrix elements for the states within the four-spinon cotinuum, atq50. We can simplify the problem by means ofvery crude approximation and obtain a result which,though not rigorous, yet is meaningful for a comparison wthe experiment. Let us replace in Eq.~22! the quantity (En2Eg) with the average energy value of the four spinon cotinuum: ES5pJi . In this way, as in the case of the Cexcitations, we can write

ESs1~v!dv5

pqm2 d'

2

4\2VESgS~T!, ~27!

wheregS(T) is the spin-correlation function defined as

gS~T!54k^hS2&2^hS&

2lT . ~28!

One has to note that a nonzero value ofgS requires thatthe total HamiltonianHT and the interaction HamiltonianHSdo not commute with each other. This condition wouldrealized including, inHT or in hS , NNN interaction termswhich would, unfortunately, complicate the problem conserably. However, a finite value ofgS is obtained also in casof an AF broken symmetry of the ground state. We can thestimate an upper limit forgS starting from a long-range

Neel ordered state, i.e.,ug&5) j uL2 j ,↑L2 j 11,↓&. Over thisground state we can calculategS5N. On the other hand, foa random orientation of spins we would obtaingS50. Thereal spin configuration of the system is of course somethin between these two extreme cases and the value ogSdepends on the details of the many-body wave functionthe spins. In particular, it depends on the probability of hing fragments of three neighboring spins ordered antifemagnetically. ThereforegS is higher the lower is the temperature and reaches forT→0 its maximum value, whichwould be strictly smaller thangS5N in a truly 1D system. Inrelation to the interpretation of the optical conductivity daof a8-NaV2O5 within the charged magnon model, it is important to note that, in a dimerized singlet~or triplet! con-figuration of the single two-leg ladder, we havegS50.

In conclusion, the intensity of the spin fluctuations retive to the CT excitations in terms of effective charges,the single two-leg ladder system, is given by

ESs1~v!dv

ECT

s1~v!dv

5qm

2 gSES

qe2gCECT

. ~29!

e

-

e-

-

-

-

n

g

f--

-r

On the basis of the analysis presented above, we canthat the maximum limiting value for this quantity, in thlimit U→` and withJi given by Eq.~17!, is

ESs1~v!dv

ECT

s1~v!dv

<9p

32

D2Ji4

t i2t'

4. ~30!

IV. OPTICAL SPECTROSCOPY

We investigated the optical properties ofa8-NaV2O5 inthe frequency range going from 30 to 32 000 cm21. High-quality single crystals were grown by high-temperature sotion growth from a vanadate mixture flux.34 The crystals,with dimensions of 1, 3, and 0.3 mm along thea, b, andcaxes, respectively, were aligned by conventional Lauefraction, and mounted in a liquid-He flow cryostat to stuthe temperature dependence of the optical properties betw4 and 300 K. Reflectivity measurements, in near normalcidence configuration (u;10°), were performed on two dif-ferent Fourier transform spectrometers: A Bruker IFS 11in the frequency range 20–7000 cm21, and a Bomem DA3,between 6000 and 32 000 cm21. Polarized light was used, inorder to probe the optical response of the crystals along thaand b axes. The absolute reflectivity was obtained by cabrating the data acquired on the samples against a goldror, from low frequencies up to 15 000 cm21, and an alumi-num mirror for higher frequencies. The optical conductivwas calculated from the reflectivity data using KrameKronig relations.

Let us now, as an introduction to what we will discussdetail in the following sections, describe briefly the mafeatures of the optical spectra ofa8-NaV2O5, over the entirefrequency range we covered with our experimental systeIn Fig. 5 we present reflectivity and conductivity dataa8-NaV2O5 at 300 K forEia ~i.e.,' to the chain direction!and Eib ~i.e., i the chain direction! in the frequency range30–17 000 cm21. We can observe a very rich and peculiexcitation spectrum, characterized by strong anisotropydirection parallel and perpendicular to the V-O chains.

For Ei chain, the far-infrared region (v,800 cm21) ischaracterized by strong optical-phonon modes, with no baground conductivity, as expected for an ionic insulator.higher frequencies a strong electronic absorption~peaking at;9000 cm21) is directly observable in both reflectivity anconductivity. Moreover, in the mid-infrared region~between800 and 5000 cm21) we detected a very weak continuumexcitations, as shown by the finite value of the conductiv@see inset of Fig. 5~b# and by the reflectivity which is continuously raising in that frequency range@Fig. 5~a!#.

On the other hand, forE' chain, a weak broad band ooptical absorption, additional to the phonon modes, wastected in the far-infrared region. This is directly recognizabin the reflectivity spectrum@Fig. 5~a!#: Reducing the fre-quency of the incoming light, contrary to what observedEi chain, we can observe an overall increase in reflectivwhich masks the contribution due to the sharp phonon linAt the same time, the interference fringes, due to Fabry-P

it-

orn-

e

thth

t

th

bvicfon

e

seea

-arr

eas-

ase-

able

ofe

ew. 6.

yin-

225es-pen-rly,

ddesI.

ical

t

ns


resonances, which were dominating the optical reflectivbelow 140 cm21 for Ei chain, are almost completely suppressed forE' chain, indicating a stronger absorption fthe latter polarization. In the optical conductivity, a cotinuum, with a broad maximum in the far infrared@see insetof Fig. 5~b!#, extends from very low frequencies up to thelectronic excitation at;7500 cm21;1 eV, which is nowmuch more intense than the similar one detected forEichain. Moreover, along the direction perpendicular toV-O chains, we can observe an additional shoulder inconductivity spectrum, between 3000 and 5000 cm21.

Finally, for both polarizations, an absorption edge duethe onset of charge-transfer transitions was detected at;3eV ~not shown!, in agreement with Ref. 35.

A. Phonon spectrum: High-temperature phase

We will now concentrate on the phonon spectra ofhigh-temperature undistorted phase ofa8-NaV2O5. We willtry to assess the symmetry issue by comparing the numand the symmetry of the experimentally observed opticalbrations to the results obtained from the group-theoretanalysis for the two different space groups proposed,a8-NaV2O5, in the undistorted phase. In Fig. 6 we presethe reflectivity data forE perpendicular and parallel to thV-O chains~i.e., along thea andb axes, respectively!, up to1000 cm21, which covers the full phonon spectrum for thetwo crystal axes. In this respect, different is the case of thcaxis: Along this direction a phonon mode was detected;1000 cm21, as we will show in Sec. IV C while discussing the results reported in Fig. 11. In Fig. 6 the datashown forT5100 K, temperature which is low enough fo

FIG. 5. Optical spectra ofa8-NaV2O5 at 300 K forEia ~i.e.,'to the chain direction! and Eib ~i.e., i the chain direction!, in thefrequency range 30–17 000 cm21. Panels~a! and ~b! show reflec-tivity and optical conductivity, respectively. In the inset of panel~b!an enlarged view ofs1(v) from 40 to 3000 cm21 is presented.

y

ee

o

e

eri-alrt

t

e

the phonons to be sufficiently sharp and therefore moreily detectable. At the same time, it is far fromTc , so that wecan identify the phonon spectrum of the undistorted phwithout having any contribution from the one of the lowtemperature distorted phase.

In Fig. 6 three pronounced phonon modes are observfor Ei chain (vTO'177, 371, and 586 cm21, at 100 K!, andthree forE' chain (vTO'137, 256, and 518 cm21, at 100K!. It has to be mentioned that the feature at 213 cm21 alongthea axis, which looks like an antiresonance, is a leakagea c-axis mode36 due to the finite angle of incidence of thradiation on the sample (u;10°), and to the use ofp polar-ization ~therefore with a finiteEic component! to probe thea-axis optical response. Analyzing in detail the spectra fmore lines are observable, as shown in the insets of FigAlong the b axis a fourth phonon is present at 225 cm21:Being the sample a 300-mm-thick platelet characterized bsome transparency in this frequency region, Fabry-Perotterferences with a sinusoidal pattern are measurable. Atcm21 the pattern is not regular anymore, indicating the prence of an absorption process. From the temperature dedence, it can be assigned to a lattice vibration. Similathree more phonons are detected along thea axis at 90, 740,and 938 cm21.

In conclusion, sixa-axis and fourb-axis phonons weredetected ona8-NaV2O5, in the high-temperature undistortephase. The resonant frequencies of these vibrational moare summarized, for two different temperatures, in TableThese results compare better with the group-theoretanalysis for the centrosymmetric space groupPmmn@see Eq.~1!#, which gave seven and four vibrational modes forEia

FIG. 6. Reflectivity ofa8-NaV2O5 in the undistorted phase a100 K. The spectra are shown forE perpendicular and parallel tothe chain direction in panel~a! and ~b!, respectively. An enlargedview of the frequency regions, where very weak optical phonowere detected, is given in the insets.

n7

th

tnuo

tiveuleeeveo

ecthtrom

oti

iteif

igotenor

paoflsa

by

o

4 tomich

urefor-ing

aer-tri-rical

anym-hat

fre-ostof

nege ining

eitherdess ofhattornedase

be

ee inof

-el

60

ees,

-e

-ob-iesllic

ors,ex-ec-

of

e

2149


andEib, respectively, than with the analysis for the noncetrosymmetricP21mn. In fact, in the latter case 15 andphonons ought to be expected@see Eq.~2!#. Therefore inagreement with the x-ray structure redetermination,10–12alsothe optical investigation of the phonon spectra indicatescentrosymmetric space groupPmmnas the most probableone fora8-NaV2O5, at temperatures aboveTc .

However,P21mn is in principle not completely ruled ouas a possible space group on the basis of the optical-phospectra. In fact, group theory gives information only abothe number of phonons that one could expect but not abtheir resonant frequencies and, in particular, their effecstrength which, in the end, determines whether a moddetectable or not. The only really conclusive answer wohave been to detect more modes than allowed by symmfor the space groupPmmn. In the present situation, it may bpossible that the correct space group is still the nonctrosymmetricP21mn and that some of the phonons haescaped detection because the frequency is too low forsetup or due to a vanishing oscillator strength. In this respwe have to stress that, indeed, the oscillator strength ofadditional modes expected for the noncentrosymmeP21mn would have to be small because, as indicated frthe x-ray-diffraction analysis,10 if a deviation from cen-trosymmetry is present, it is smaller than 0.03 Å . We havealso been performing a lattice dynamical calculation for bspace groups which shows that, removing the center ofversion and changing the valence of the V ions~i.e., from theuniform V4.51 for all sites to an equal number of V41 andV51 sites!, the number of modes characterized by a finvalue of the oscillator strength does not increase, evenstrictly long-range charge ordering in 1D chains of V41 andof V51 ions is assumed.

It has to be mentioned that many independent investtions of the Raman and infrared phonon spectra,a8-NaV2O5, were reported.13,35–43Of course, there was noimmediately a perfect agreement between all the differdata. As a matter of fact, in early experimental studieslattice vibrations,35,37the optical spectra there presented weinterpreted as an evidence for the noncentrosymmetric sgroup P21mn. At this stage, we believe that the resultsour investigation performed on high quality single crystagive a very complete picture of the infrared vibrationmodes ofa8-NaV2O5, in the undistorted phase, forEia andEib. Moreover, our results are completely confirmedthose recently obtained by Smirnov and co-workers36,38~whocould measure also thec-axis phonon spectrum! on samplesof different origin.

Before moving on to the phonon spectra characteristicthe low-temperature distorted phase ofa8-NaV2O5, we

TABLE I. Resonant frequencies of the optical vibrational modcharacteristic of the undistorted phase ofa8-NaV2O5, obtained byfitting the optical phonons with Fano line shapes.

Pol. T ~K! vTO (cm21)

Eia 4 91.12 139.15 255.49 517.30 740.62 938.100 89.67 136.81 255.82 517.74 740.49 938.

Eib 4 177.46 224.67 373.91 587.20100 177.18 225.20 371.02 585.88

-

e

ontuteisdtry

n-

urt,e

ic

hn-

a

a-n

tf

ece

,l

f

would like to discuss the temperature dependence, from300 K, of the lattice vibrations characteristic of the uniforphase. We are here referring to those optical modes whare observable at any temperatures aboveTc and thereforereflect the symmetry of the system in the high-temperatphase. In order to extract, from the experimental data, inmation about their parameters, we had to fit the data usFano line shapes for the phonon peaks.44 In fact, most ofthem, in particular along thea axis, are characterized bystrong asymmetry indicating an interaction with the undlaying low-frequency continuum. In this case the symmecal Lorentz line shape was not suitable and the asymmetFano profile had to be used: The latter model containsadditional parameter which takes care of the degree of asmetry of an excitation line, due to the coupling between tdiscrete state and an underlaying continuum of states.

The results for the percentage change in resonantquency, the damping, and the oscillator strength of the mintense optical phonons of the undistorted phasea8-NaV2O5, are plotted in Fig. 7, between 4 and 300 K. Ocan observe that some of the modes show a sudden chantheir parameters between 40 and 32 K, i.e., upon reducthe temperature belowTc534 K. In particular, the oscillatorstrength of the phonons at 371 and 586 cm21, along thebaxis, decreases across the phase transition, suggestinga transfer of spectral weight to zone boundary folded moactivated by the phase transition or to electronic degreefreedom. We will see later, in the course of the paper, tfor the chain direction the reduction of the phonon oscillastrength will be compensated mainly by the strength gaiby the zone boundary folded modes activated by the phtransition. On the other hand, forE' chain, the change inspectral weight of the electronic excitations will appear toparticularly important.

B. Phonon spectrum: Low-temperature phase

Upon cooling the sample belowTc534 K, significantchanges occur in the optical phonon spectra as one can sFig. 8, where we present the reflectivity spectraa8-NaV2O5 measured, forE' chain ~a! and Ei chain ~b!,below ~4 K! and just above~40 K! the phase transition. Contrary to the case of CuGeO3, where we could clearly observin reflectivity only a single, very weak additionaphonon,46–48 ten new lines are detected forE' chain (vTO'101, 127, 147, 199, 362, 374, 410, 450, 717, and 9cm21), and seven forEi chain (vTO'102, 128, 234, 362,410, 718, and 960 cm21), some of them showing a quitlarge oscillator strength. The stronger intensity of the linwith respect to the case of CuGeO3, suggests larger latticedistortions ina8-NaV2O5, on going through the phase transition or, alternatively, a larger electronic polarizability. Thresonant frequencies~at T54 K! of all the detected zoneboundary folded modes are summarized in Table II. Theservation of many identical or almost identical frequencfor both axes, which is an important information for a fuunderstanding of the structural distortion, is an intrinsproperty of the low-temperature phase. Experimental errlike polarization leakage or sample misalignment, arecluded from the well-defined anisotropy of the phonon sptrum of the undistorted phase. A more detailed discussion

s

e

reneste

g.

ncIda

ahan

a-re

Wemthef thetry

pan-

is

ed

he

r

w a

toeo

-e

e

by

ry

600


this point will be presented in Sec. IV C, in relation to threflectivity data reported in Fig. 11.

An enlarged view of few of the frequency regions whethe zone-boundary folded modes were detected, is giveFig. 9. We can observe the gradual growing of the thmodes, whose oscillator strength increases the more theperature is reduced belowTc534 K. Moreover, the reflec-tivity data acquired forE' chain @panel ~a! in Fig. 8 andpanels~a! and ~b! in Fig. 9# show that also the underlayincontinuum as a considerable temperature dependencefact, the reflectivity is decreasing, over the entire frequerange, upon reducing the temperature from 40 to 4 K.particular, in Fig. 9~a!, we can see that very pronounceinterference fringes, due to Fabry-Perot resonances,present at 4 K for frequencies lower than 136 cm21, indicat-ing a particularly strong reduction of the absorption in thfrequency region. This effect is, in our opinion, related to topening of the magnetic gap in the excitation spectrum,will be discussed in great detail in Sec. IV D.

FIG. 7. Resonant frequency shift, damping, and oscillastrength plotted vsT, for the most intense optical phonons of thundistorted phase. We are here referring to modes which areservable at any temperatures aboveTc and therefore reflect the symmetry of a8-NaV2O5 in the high-temperature phase. These modare still well detectable belowTc ; the reduction of their oscillatorstrength across the phase transition marks the symmetry changto the lattice distortion.

inem-

Inyn

re

ted

Of all the detected folded modes, for experimental resons ~i.e., higher sensitivity of the detector and therefohigher accuracy of the results in this frequency range!, wecarefully investigated the ones at 718 (Eia,b) and 960 cm21

(Eia), for several temperatures between 4 and 40 K.fitted these phonons to a Fano profile because the 718-c21

modes show an asymmetrical line shape. The results ofpercentage change in resonant frequency and damping, onormalized oscillator strength, and of the Fano asymmeparameter are plotted versus temperature in the differentels of Fig. 10. We use here an asymmetry parameterQ ~rad!redefined in such a way that the larger the value ofQ, thestronger is the asymmetry of the line: A Lorentz line shapeeventually recovered forQ50.49

Let us now start by commenting on the results obtainfor the oscillator strength: In Fig. 10~d! we see thatS has asimilar behavior for the three different lines. However, t960-cm21 peak vanishes atTc whereas the two 718-cm21

modes have still a finite intensity atT540 K and, as a matteof fact, disappear only forT.60270 K. At the same time,the line shape of the 960 cm21 mode is perfectly Lorentzianat all temperatures, whereas the two other phonons shoconsistently increasing asymmetry forT.32 K. Also the

r

b-

s

due

FIG. 8. Reflectivity spectra ofa8-NaV2O5 measured forE'

chain ~a! andEi chain ~b! below ~4 K! and just above~40 K! thephase transition. Along both axes, new phonon lines activatedthe phase transition are observable forT,Tc .

TABLE II. Resonant frequencies, at 4 K, of the zone-boundafolded modes detected ona8-NaV2O5 in the low-temperature~LT!phase, withE' chain andEi chain.

Pol. ~LT! vTO (cm21)

Eia 101 127 147 199 362 374 410 450 717 9Eib 102 128 234 362 410 718 96

rden

an

rr,btstu

ec-

ednsi-

if-

hease

dif-theeho-

ity

nt

to

tre-n a

e

l

onr

nah

m


increase in resonant frequency@Fig. 10~a!# and the reductionof the damping@Fig. 10~c!# are much more pronounced fothe two 718-cm21 modes. From these results we concluthat the second-order character of the phase transitionicely shown by the behavior ofS for the 960-cm21 foldedmode. On the other hand, pretransitional fluctuations mfest themselves in the finite intensity of the 718-cm21 modesaboveTc , and in the extremely large value ofQ andg forthese lines (g;35 cm21 at 40 K!. In other words, the latticedistortion is already taking place in the system, at tempeture much higher thanTc , but with a short-range charactewhereas a coherent long-range distortion is realized onlylow the phase-transition temperature. In this respect, iworth mentioning that similar pretransitional fluctuationbelow 70 K, have been observed also in the course of a s

FIG. 9. Detailed temperature dependence of some of the zboundary folded modes observed in the reflectivity spectra, foT,Tc , with E' chain @~a! and ~b!# andEi chain @~c! and ~d!#.

FIG. 10. Temperature dependence of the change in resofrequency~a! and damping~c!, of the normalized oscillator strengt~d!, and of the Fano asymmetry parameter~b!, for the foldedphonons observed below the phase transition at 717 and 960 c21

with Eia, and at 718 cm21 with Eib.

is

i-

a-

e-is,dy

of the propagation of ultrasonic waves along the chain dirtion of a8-NaV2O5,50 and, very recently, in x-ray diffusescattering measurements.51

C. Symmetry of the lattice distortion

We saw in the previous section that many of the foldzone-boundary phonon modes, activated by the phase tration, show the same resonant frequency forEia and Eib~see Table II!, even though they are characterized by a dferent oscillator strength along the two different axes~e.g.,S;0.021 forEia andS;0.014 forEib, for the 718-cm21

modes at 4 K!. This is quite a surprising result because tphonon spectrum of the undistorted high-temperature phhas a well defined anisotropy, at any temperature, withferent resonant frequencies for vibrations polarized alonga andb axes~see Table I!. Similar anisotropy should then bpresent below the phase transition if the system is still ortrhombic, even for the folded modes.

To further check these findings, we performed reflectivmeasurements on a single crystal ofa8-NaV2O5, at the fixedtemperature of 6 K, rotating the polarization of the incidelight of an anglef ~see Fig. 11!. The sample was aligned insuch a way to have the electric field of the light parallelthea axis of the crystal, forf50° and 180°, and parallel tothe b axis, for f590°. For these polarizations of the lighthe obtained results are, obviously, identical to those psented in Fig. 8. However in Fig. 11 the data are shown idifferent frequency range, extending up to 1050 cm21. Wecan then observe at;1000 cm21 along thea axis, whichagain was probed withp-polarized light, the antiresonancdue to the leakage of ac-axis phonon.36 Turning the polar-ization from f50° to f590° in stepsDf510°, we ob-serve a gradual decrease of thea-axis contribution and, at thesame time, an increase of theb-axis contribution to the totareflectivity. From 90° to 180°~not shown!, the behavior isthe reverse and the reflectivity spectra measured atf590°1nDf are just identical to those acquired atf590°2nDf. The spectra satisfy the following relations:

R~45°!1R~245°!5R~0°!1R~90°!, ~31!

R~f!5R~0°!cos2~f!1R~90°!sin2~f!. ~32!

e-

nt

FIG. 11. Reflectivity ofa8-NaV2O5, at 6 K, for several orien-tations of the electric field of the light in thea-b plane~as shown inthe sketch, withDf510°). The valuef50° corresponds toE'

chain, andf590° to Ei chain.

a

reEele

thecc

olun

eV

dete.e

isn

tslew-tys-

iva-

m:na-a-erst,of

pa-ay

uldis-c-o

ary

talthet

po-uchentstwoal-ng

nd-

oil-ge-

ing-larared

000

res

c-tweion

at

dne

:

en

e-


This suggests that the system, even in the distorted phhas the optical axes along the crystal directionsa, b, andc@on the contrary, for a triclinic or a monoclinic structuthere would be no natural choice of the optical axes and~32! would not be satisfied#. However, the low-temperaturfolded mode at 718 cm21 can be observed for all possiborientations of the electric field of the incident radiation~seeFig. 11!. Because identical resonance frequencies, alonga and theb axes of the crystal, were observed for six of tfolded zone-boundary modes, we cannot believe in an adental coincidence.

A possible explanation for the observed effect is as flows. In the low-temperature phase a nonmagnetic grostate is realized, and true singlets are present ina8-NaV2O5.The most probable situation is that the singlets are formwithin the ladders, over the plaquettes defined by fourions. In this case, as shown in Fig. 12, within one ladthere is an alternation of ‘‘empty’’ plaquettes and plaquetcontaining a singlet~represented by a gray ellipsoid in Fig12!. Let us now assume that in adjacent ladders the chargarranged in an oblique charge ordering pattern~OCOP!, asshown in the left-hand side of Fig. 12. The OCOP is constent with the doubling of the lattice constant observed, alothea axis, in low-temperature x-ray scattering experimen2

Moreover, two different kinds of V ions are identifiabwhich could possibly explain the results obtained in lotemperature NMR experiments15 ~for a more detailed discussion see Sec. V!. In fact, there are V ions adjacent to empplaquettes~white circles! and V ions adjacent to plaquettecontaining a singlet~black circles!, arranged in zigzag pat

FIG. 12. Possible charge ordering pattern ina8-NaV2O5, belowTc . The grey ellipsoids represent the electronic charge involvethe formation of singlets, within the ladders, over every secoplaquette formed by four V ions. On adjacent ladders, the chargassumed to define an oblique charge ordering pattern~OCOP; see,e.g., left-hand side!. Two different kinds of V ions are identifiableV ions adjacent to empty plaquettes~white circles! and V ionsadjacent to plaquettes containing a singlet~black circles!, arrangedin zigzag patterns on each ladder. If domains with different oritation for the OCOP are present in the system~left- and right-handsides!, the zigzag ordering of the different kind of V ions is dstroyed in correspondence of a domain wall~dashed line!.

se,

q.

he

i-

-d

d,

rs

is

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-

terns on each ladder. Note that in our picture the nonequlence of plaquettes along theb axis, indicated by the grayellipsoids in Fig. 12, is really a chicken-and-egg probleEither the singlet formation occurs first, causing an altertion alongb of nonequivalent plaquettes, which, in combintion with the OCOP alonga gives rise to a zigzag chargordering. Alternatively, the zigzag charge order occurs fiwhich, in combination with OCOP causes the alternationnonequivalent plaquettes alongb. The resulting chargemodulations are only small deviations from average occution, which we represented graphically by the different grshades.

At this stage, the low-temperature phonon spectra shostill be characterized by a well defined anisotropy, with dtinct eigenfrequencies for different polarizations of the eletric field of the incoming radiation. However, there is nreason for the polarization of the folded zone-boundmodes to be precisely along thea or theb axes. In fact, theyare polarized with a finite angle with respect to the crysaxes~e.g., along directions parallel and perpendicular todiagonal long-range singlet pattern!. If domains are presenin the system, with opposite orientation for the OCOP~left-and right-hand sides of Fig. 12!, each domain will still becharacterized by a phonon spectrum with a well definedlarization, as just discussed. However, in presence of sdomains the phonon spectra measured in the experimcorrespond to an average over many domains with thepossible orientations randomly distributed. As a result,most no anisotropy would be found for light polarized aloa andb axes.

A short-wavelength alternation of descending and asceing OCOP domains~also a 2a32b supercell! correspondsclosely to the structure recently reported by Lu¨deckeet al.45

A similar structure was also considered by Riera and Pblanc as one of the many possible realizations of chardensity wave in this system.19

D. Optical conductivity

Very interesting information and a deeper understandof the symmetry ofa8-NaV2O5 are obtained from the detailed analysis of the optical conductivity data, in particuwhen the electronic excitations are considered and compto the result relative to CuGeO3 ~see Fig. 13!. On the lattercompound we observed sharp phonon lines below 1cm21,46–48 multiphonon absorption at;1500 cm21, veryweak @note the low values ofs1(v) in the inset of Fig.13~a!# phonon-assisted Cud-d transitions at ;14 000cm21,52 and the onset of the Cu-O charge-transfer~CT! ex-citations at;27 000 cm21. On a8-NaV2O5, besides thephonon lines in the far-infrared region, we detected featuthat are completely absent in CuGeO3: A strong absorptionpeak at;8000 cm21 and, in particular forE' chain, alow-frequency continuum of excitations@see inset of Fig.13~b!#. Our goal is to interpret the complete excitation spetrum for a8-NaV2O5, trying, in this way, to learn more abouthe system on a microscopic level. In order to do that,have first to identify possible candidates for the excitatprocesses observed in the experimental data.

On the basis of intensity considerations, the peak;8000 cm21;1 eV, in the optical conductivity of

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a8-NaV2O5, has to be ascribed to an optically allowed ecitation. Contrary to the case of the absorption detecte; 14 000 cm21 on CuGeO3, it cannot be interpreted just aa d-d exciton on the transition-metal ion~i.e., V in the caseof a8-NaV2O5). In fact, this kind of transition is in principleoptically forbidden and only weakly allowed in the presenof a strong parity-broken crystal field or electron-phoncoupling. We have now to recall one of the main conquences of the structural analysis which suggested as aprobable space group, fora8-NaV2O5 in the undistortedphase, the centrosymmetricPmmn: The existence of onlyone kind of V per unit cell, with an average valence of14.5.Taking into account that the hopping parameter alongrungst' , is approximately a factor of 2 larger than the oalong the legs of the laddert i ,53,54 all the other hoppingamplitudes being much smaller~including the nn txy),a8-NaV2O5 can be regarded as a quarter filled two-leg lader system, with one Vd electron per rung shared by twoions in a molecular bonding orbital.12 In this context, themost obvious candidate for the strong optical absorptionserved at 1 eV is the on-rung bonding-antibonding transitiwith a characteristic energy given by 2t' . However, on thebasis of LMTO band-structure calculation53 and exact diago-nalization calculation applied to finite-size clusters,54 t' wasestimated to be;0.3020.35 eV, which gives a value for thbonding-antibonding transition lower than the experimetally observed 1 eV. This energy mismatch, and the strdependence of the intensity of the 1 eV peak on the poization of the light, are important pieces of information awill be discussed further in Sec. IV E.

FIG. 13. Optical conductivity, at 300 K, of CuGeO3 ~a! anda8-NaV2O5 ~b!, for E parallel and perpendicular to the Cu-O orthe V-O chains. Inset of panel a: enlarged view ofs1(v) ofCuGeO3, from 800 to 30 000 cm21. Inset of panel~b!: enlargedview of s1(v) of a8-NaV2O5, from 40 to 3000 cm21.

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Let us now turn to the infrared continuum detectedE' chain@see inset of Fig. 13~b!#. We already showed, discussing the low-temperature reflectivity data~see Figs. 8 and9!, that this broad band of optical absorption has a considable temperature dependence: The overall reflectivity wdecreasing upon reducing the temperature and, forT,Tc ,Fabry-Perot resonances appear at frequencies lower thancm21, indicating a particularly strong reduction of the opcal absorption in that frequency region. These effectscome even more evident when we consider the lotemperature optical conductivity calculated from treflectivity data via Kramers-Kronig~KK ! transformations~see Fig. 14!. However, when we perform this kind of calculation we will obtain unphysical oscillations in the opticconductivity at frequencies where reflectivity is dominatby interference fringes. In addition, because KK transformtions are integral equations one could expect some effec~e.g., a wrong estimate of the background conductivity! alsoat frequencies other than those where interference frindirectly show up. More in detail, the concern is not that tinfrared continuum detected forE' chain could be an arti-fact due to a pathological KK analysis~in fact, while discuss-ing Figs. 5, 8, and 9, we showed that the continuum isrectly observable in reflectivity!, but rather that the absolutvalue ofs1(v) could be wrong. In order to verify the reliability of the KK analysis, not only in relation to this poinbut also to the effect of different low- and high-frequenextrapolations~which are needed to perform the KK integraover frequencies ranging from zero to1`), we comparedthe KK results tos1(v) we obtained with alternative methods:

FIG. 14. Far-infrared optical conductivity ofa8-NaV2O5, forE' chain~a! andEi chain~b!. Along both axes new phonon linesactivated by the phase transition, are present forT,Tc534 K.Moreover, forE' chain, the temperature dependence of the lofrequency continuum is observable.

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~i! High-frequency ellipsometry~0.8–4.5 eV!.55

~ii ! Inversion of the Fresnel formulas in regions wherecould measure both reflectivity and transmission.55

~iii ! From the parameters of a detailed fit of the reflectity data. The fit is based on a routine which allows usreproduce also the interference fringes due to multiple refltions within a thin sample.

As a result, we could conclude thats1(v) calculated viaKK transformation is reliable from very low frequencies uto 17 000 cm21, with an accuracy in the infrared of.1%~except in regions where interference fringes were detect!.Above this frequency~we actually measured the reflectivitup to 32 000 cm21), our s1(v) starts to show a dependencon the high-frequency extrapolation and therefore islonger completely reliable.

In Fig. 14~b! we can observe, forEi chain, only thesharpening of the phonon lines and the appearance ofzone-boundary folded modes. On the other hand, forE'chain we can observe, in addition to the folded modes beTc , first the increase of the continuum intensity upon coolthe sample from 300 to 40 K and, subsequently, a reducof it from 40 to 4 K. In the frequency region below the 13cm21 phonon, whereas a reduction is observable in gofrom 300 to 40 K, not much can be said about the absoconductivity of the 4 K data, because of the interferenfringes.

In order to evaluate the low-temperature values of cductivity more precisely, we can fit the interference fringdirectly in reflectivity. In fact, fitting the period and the amplitude of them, we have enough parameters to calcuright away the real and imaginary part of the index of refration and, eventually, the dynamical conductivitys1. The soobtained values are plotted versus the temperature in15~a!. The data for 100, 200, and 300 K were obtainedKramers-Kronig transformations, whereas those for 4 andK from the fitting of the interferences in the range 105–1cm21. The 40-K point, as at that temperature very shallfringes start to be observable, was calculated with both mods, which show a rather good agreement in this case. Aresults we can observe@see Fig. 15~a!# a pronounced reduction of s1, across the phase transition, corresponding toopening of a gap in the optical conductivity.

It is difficult to evaluate precisely the gap size becausethe phonon at 139 cm21 @inset of Fig. 15~a!#. However, wecan estimate a lower limit for the gap of;136 cm21 (1763 meV!. The important information is that the value of thgap in the optical conductivity is approximately twice thspin gap value observed in inelastic neutron-scatteexperiments2 and magnetic susceptibility measurements56

for T,34 K. This finding indicates that the infrared broaband of optical absorption has to be related to electrodegrees of freedom: Because the range of frequency ccides with the low-energy-scale spin excitations, the mlikely candidates for this continuum are excitations involvitwo spin flips~as already proposed in the early stage ofinvestigation ofa8-NaV2O5).37

In relation to what we just discussed, it is interestingnote the effect of the opening of a gap, in the optical coductivity, on the phonon located at 139 cm21. This can beclearly identified in the inset of Fig. 15~a!, where an enlargedview of s1(v) at low frequency is given, for 100 and 4 K

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At 100 K this mode is characterized by a very pronouncanomaly of its line shape. As a matter of fact, fitting thphonon with a Fano line shape we obtained for the asymtry parameter the valueQ521.7 rad, indicating a strongcoupling to the continuum. However at 4 K, once the gapopen and the intensity of the underlying continuum hascreased, a sharp and rather symmetrical line shape is reered.

An additional information we can extract from the coductivity spectra is the value of the integrated intensity, aits temperature dependence, for the infrared broad banoptical absorption detected forE' chain. The oscillatorstrength, obtained by integratings1(v) ~with phonons sub-tracted! up to ;800 cm21, is displayed in Fig. 15~b!. Itincreases upon cooling down the sample from room tempture to Tc , and rapidly decreases forT,Tc . The possiblemeaning of this behavior will be further discussed in the nsection.

As a last remark, it is important to mention that lowtemperature measurements of the dielectric constant, inmicrowave frequency range, were recently performeda8-NaV2O5 by different groups.57,58 The reported resultsshow no anomaly, across the phase transition, for thepart of the dielectric constant along theb axis. On the otherhand, along thea axis, a pronounced decrease belowTc wasfound. The observed behavior, as a matter of fact, is in vgood agreement with the temperature dependence of thecillator strength of the low-frequency continuum present,

FIG. 15. Panel~a!: Temperature dependence of the optical coductivity at v5115 cm21, for E' chain. The values are obtainefrom Kramers-Kronig analysis, forT.Tc , and by fitting the inter-ference fringes in reflectivity, forT,Tc . The inset shows an enlarged view ofs1(v), from 110 to 190 cm21. Panel~b!: Oscillatorstrength of the low-frequency electronic continuum, forE' chain,plotted versus temperature.

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the optical spectra, along thea axis @Fig. 15~b!#. This com-parison is meaningful because the contribution of all possoptical excitations to the static dielectric constant is jgiven by their oscillator strengths. The very good agreemof the different results proves, as already suggested byauthors of Refs. 57 and 58, that the change in the very lfrequency dielectric constant is related to electronic~mag-netic! degrees of freedom.

E. Charged bimagnons

In the previous section we identified possible candidafor the 1-eV excitation~on-rung bonding-antibonding transtion!, and for the low-frequency continuum detecteda8-NaV2O5 for E' chain ~double spin flips!. However, wealso stressed that, for a symmetrical quarter filled two-ladder system, the bonding-antibonding transition wouldat 2t';0.7 eV and not at the experimentally observed vaof 1 eV. Moreover, also intensity and polarization (E'chain! of the continuum cannot be understood assumingcomplete equivalence of the V sites required by the spgroupPmmn. In fact, because of the high symmetry, no dpole moment perpendicular to the V-O chains would besociated with a double spin-flip process. Therefore no optactivity involving magnetic degrees of freedom woulddetectable. Not even phonon-assisted spin excitations conered by Lorenzana and Sawatzky26,27,29~where the involvedphonon is effectively lowering the symmetry of the syste!could be a possible explanation for the observed spectra:energy is too low for this kind of processes which, on tother hand, are probably responsible for the weak minfrared continuum in theEi chain spectra@see Fig. 13~b!and, in particular, its inset#.

A possible solution is to assume a charge disproportiation on each rung of the ladders without, however, any pticular long-range charge ordering~see Fig. 16!. The latter is,in fact, most probably excluded by the x-ray-diffraction rsults and by the phonon spectra of the undistorted phase.worth mentioning that the assumption of charge disproptionation has been confirmed by calculations of the optconductivity by exact-diagonalization technique on finisize clusters.59 Only in this way it was possible to reproducthe energy position of the 1 eV peak, and the anisotrobetweena and b axes ~see Fig. 13!. We can now try tointerpret the optical data ona8-NaV2O5 in the framework ofthe charged-magnon model developed in Sec. III.

Let us start from the analysis of the absorption band aeV. We can model thej th rung, formed by two V ions, withthe HamiltonianH j 01H j' , as in Eqs.~4! and~5!, where thepotential energy difference between the two sitesD has therole of a charge disproportionation parameter: For the smetric ladderD50. The splitting between the two bondin

and antibonding eigenstatesuL& and uR&, given by ECT

5AD214t'2 , corresponds to the photonenergyof the optical

absorption. We see that, ifDÞ0, the experimental value of 1eV can be reproduced.

A second crucial piece of information is provided by tintensityof the absorption. We can calculate the integraintensity of the 1-eV peak, in the conductivity data, and thtake advantage of its functional expression derived in Sec@Eq. ~9!#. This way we obtain from the spectraut'u'0.3 eV,

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value which is in very good agreement with those obtainfrom linear muffin-tin orbital band structure calculation,53

and exact diagonalization calculation applied to finite-sclusters.54 Combining this number withECT51 eV, we ob-tain D'0.8 eV. The corresponding two eigenstates ha90% and 10% character on either side of the rung. Therethe valence of the two V ions is 4.1 and 4.9, respectiveand the optical transition at 1 eV is essentially a charge trafer ~CT! excitation from the occupied V 3d state at one sideof the rung to the empty 3d state at the opposite side of thsame rung.

Let us now turn to the infrared continuum detectedE' chain. The presence oftwo V states (uL& and uR&) perspin, along with the broken left-right parity of the grounstate on each rung, gives rise to the fascinating behaviothe spin flips described in detail in Sec. III for the quartfilled single two-leg ladder, and in a pictorial way foa8-NaV2O5, in Fig. 16: If in a small fragment of the laddeeach spin resides in auL& orbital, with some admixture ofuR&, the inclusion of the coupling between the rungs, viaHamiltonianH i @Eq. ~6!#, has no effect when the spins aparallel, due to the Pauli principle. If the spins form anS50 state, the ground state gains some kinetic energy athe legs by mixing in states where two electrons reside onsame rung. In the limitU→`, these states have one electrin the uL& and one in theuR& state on the same rung. Asresult, there is a net dipole displacement of the singlet s

FIG. 16. Cartoon of the electric dipole moment associated wa single spin flip ina8-NaV2O5, when a local breaking of symmetry in the charge distribution on each rung is assumed. If thneighboring antiparallel spins are laying on the same leg ofladder ~top!, the charge of thej th electron is asymmetrically distributed not only over thej th rung but also on the NN rungs~withopposite asymmetry!, because of the virtual hoppingt i . If we flipits spin~bottom!, the j th electron is confined to thej th rung, result-ing in a reduced charge density on the right leg of the laddTherefore there is a net dipole displacement~along the rung direc-tion! of the antiparallel spin configuration compared to the paraone.

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compared to the triplet state:The spin-flip excitations carry afinite electric dipole moment along the rung direction.

From the expression for the exchange-coupling conscalculated in the limitU→` @Eq. ~17!#, and using the valuesof t' and D calculated from the optical data andut iu'0.2eV,60 we obtainJi'30 meV. It is significant that this valueis comparable to those reported on the basis of magnsusceptibility measurements, which range from 38 tomeV ~see Refs. 56 and 1, respectively!. Moreover, for theeffective charge of the bimagnon excitation process@Eq.~20!#, we obtainqm /qe50.07. We can also compare the eperimental intensity of the spin fluctuations relative to tCT excitations to the maximum limiting value calculated fthis quantity@Eq. ~30!#. For the parameters obtained from thconductivity spectra, the maximum relative intensity;0.0014. The experimental value is;0.0008, atT5300 K,in good agreement with the numerical estimate.

We can now interpret the temperature dependence ofspin-fluctuation continuum shown in Fig. 15~b!: The oscilla-tor strength increases upon cooling the sample from rotemperature toTc , and rapidly decreases forT,Tc . Withinthe charged-magnon model, the increase marks an incrin short-range AF correlations of the chains: In fact, thetensity of the spin fluctuations is proportional to the spin-scorrelation functiongS(T) defined in Eq.~28!. Below Tc ,NN spin-singlet correlations become dominant;gS is sup-pressed, along with the intensity of the spin fluctuations.

In conclusion, by a detailed analysis of the optical coductivity, we provided a direct evidence for a local breakiof symmetry in the electronic charge distribution over trungs of the ladders ina8-NaV2O5, in the high-temperatureundistorted phase. Moreover, we showed that a direct tmagnon optical absorption process is responsible forlow-frequency continuum observed perpendicularly toV-O chains.

V. DISCUSSION

As we discussed in the course of this paper, the interptation of the phase transition ina8-NaV2O5 is still contro-versial. On the one hand, the SP picture was suggestepreliminary magnetic susceptibility1 and inelastic neutronscattering measurements.2 On the other, x-ray-diffraction redeterminations of the crystal structure of the compoundthe high-temperature undistorted phase,10–12showed that thedimerization within well defined 1D magnetic chains is nvery probable ina8-NaV2O5. Moreover, the magnetic fielddependence ofTc , expected for a true SP system on the baof Cross and Fisher theory,8,9 has not been observed.6,7

Additional information came recently from NMRexperiments:15 AboveTc , only one kind of V site~magnetic!was clearly detected, whereas no signature of nonmagnV51 was found. However, another set of magnetic V siwhich are invisible in the NMR spectrum could not be rulout.15 On the other hand, below the phase transition, tinequivalent sets of V sites were detected and assigneV41 and V51. These results were then interpreted as andication of the charge-ordering nature of the phase transitin a8-NaV2O5.15 Subsequently, many theoretical studihave been addressing this possibility.16–19The main point tobe explained is why a charge ordering would automatica

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result in the opening of a spin gap~with the same transitiontemperature!. Among all the models, the more interestinseem to be those were a zigzag charge ordering on theders is realized. In this case, the ordering would immediaresult in a nonmagnetic ground state, due to the formationspin singlets on NN V sites belonging to two differeladders16 or, alternatively, to the modulation ofJi within asingle ladder.18

In the discussion of our spectra of the optical conductity, we showed that the infrared continuum, observed ppendicularly to the V-O chains direction, could be attributto direct optical absorption of double spin-flip excitationHowever, in order to have optical activity for these procesand, at the same time, to explain the energy position andpolarization dependence of the on-rung bonding-antibondtransition at 1 eV, we had to assume a charge disproportation between the two V sites on a rung. This assumptseems to contrast the results of the x-ray-diffractianalysis10–12 and of NMR measurements.15 As far as theformer is concerned, it is obvious that if the charge dispportionation does not have a long-range character it wonot be detectable in an x-ray-diffraction experiment, bwould still result in optical activity of charged bimagnoexcitations. Regarding the disagreement with the NMresults,15 we saw that the existence of another kind of V sitcould not be completely excluded in the NMR spectrum15

Furthermore, in comparing the optical to the NMR dathere could also be a problem with the different time scaof the two experimental techniques~i.e., ‘‘fast’’ and ‘‘slow’’for optics and NMR, respectively!. Therefore if there arecharge fluctuations in the system, as one would expect fcharge-ordering phase transition, the experimental rescould be different.

From the analysis of the optical conductivity, we cannreally say which is the more appropriate picture for the phtransition ina8-NaV2O5. We saw that, indeed, the electronand/or magnetic degrees of freedom are playing a leadrole. This is shown, e.g., by the strong temperature depdence of the infrared spin fluctuation spectrum which,turn, determines the anomalous behavior of the microwdielectric constant.57,58 However, the reduction of the oscillator strength of the infrared continuum forT,Tc , which wediscussed as a consequence of NN spin-singlet correlatbecoming dominant belowTc , could also result from a zig-zag charge ordering on the ladders. In this case, as forperfectly symmetrical quarter filled ladder, the optical actity of double spin flips would simply vanish.

In our opinion, for a better understanding of the phatransition, the pretransitional fluctuations observed with otical spectroscopy, ultrasonic wave propagation,50 and x-raydiffuse scattering,51 have to be taken into account. Theprobably reflect the high degree of disorder present insystem above the phase transition, which is also shownthe local and ‘‘disordered’’ charge disproportionation necsary to interpret the optical conductivity. In this context,charge ordering transition, where more degrees of freedthan just those of the spins are involved, seems plausiblwould also explain the experimentally observed entropyduction, which appeared to be larger than what expecteda 1D AF HB chain.6

Also the degeneracy of the zone-boundary folded mod

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activated by the phase transition, could be an important piof information. However, at this stage of the investigationthe electronic and magnetic properties ofa8-NaV2O5, veryimportant insights could come from the x-ray-diffraction determination of the low-temperature structure. Probably, onat that point it will be possible to fit all the different resulttogether and come up with the final interpretation of thphase transition.

VI. CONCLUSIONS

In this paper we have been discussing the high- and lotemperature optical properties ofa8-NaV2O5, in the energyrange 4 meV–4 eV. We studied the symmetry of the marial, in the high-temperature undistorted phase, comparthe different crystal structures proposed on the basis of x-diffraction3,10–12 to our infrared phonon spectra. We founthat the system is better described by the centrosymmespace groupPmmn, in agreement with the recent structurredetermination.10–12On the other hand, by a detailed analysis of the electronic excitations detected in the optical coductivity, we provided direct evidence for a charged disprportionated electronic ground state, at least on a local scA consistent interpretation of both structural and optical coductivity data requires an asymmetrical charge distribution each rung, without any long-range order. We showed thbecause of the locally broken symmetry, spin-flip excitatiocarry a finite electric dipole moment, which is responsibfor the detection of charged bimagnons in the optical sptrum for E' chain, i.e., direct two-magnon optical absorption processes. By analyzing the optically allowed phonoat various temperatures below and above the phase tration, we concluded that a second-order change to a larunit cell takes place below 34 K, with a fluctuation regimextending over a very broad temperature range.

The charged-magnon model we have been discussingthis paper has been developed explicitly to interpret the pculiar optical conductivity spectra ofa8-NaV2O5. However,our opinion is that the importance of this model goes beyothe understanding of the excitation spectrum of just this pticular system. We expect this model~obviously in a moregeneral and rigorous form! to be relevant to many of thestrongly correlated electron systems, where the interplayspin and charge plays a crucial role in determining the loenergy electrodynamics.

At this point we would like to speculate that a specificase where charged magnon excitations could be observethe stripe phase of copper oxides superconductors61 and of

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other strongly correlated oxides.62 In fact, even though themagnetism is quasi-2D in these systems, in the stripe phasymmetry-breaking quasi-1D charge ordering takes plwhich could result in optical activity for the charged manons. For instance, in the optical conductivity spectraLa1.67Sr0.33NiO4 below the charge ordering transition,63 it ispossible to observe some residual conductivity inside theergy gap, whose nature has not been completely undersso far. Without further experimental and theoretical invegations it is of course not possible to make a strong claHowever, the reported features are suggestive of chamagnon excitations. In fact, for the value ofJ obtained fromthe two-magnon Raman scattering on this material,64 directtwo-magnon optical absorption is expected in the sameergy range~i.e., from zero to;0.2 eV! where the residuastructure has been experimentally observed.63

Pushing the speculation to its limit, one could imagthat even the local breaking of symmetry introduced bsingle impurity in an AF could result in optical activity fopure spin-flip excitations. One particular system wheresituation is probably realized is Zn-substituted YBa2Cu3O6,where Zn is introduced in the CuO2 planes. In the cupratparent compounds direct bimagnon optical absorption isallowed due to the inversion symmetry; only phonon-assismagnetic excitations are optically active, whenever thevolved phonon is lowering the symmetry of the system.fact, phonon assisted bimagnon excitations in the minfrared region were detected on both YBa2Cu3O6 andYBa2Cu2.85Zn0.15O6.65 But on the Zn substituted system itpossible to observe, in the optical conductivity spectra, vous features with frequencies coincident with those of pdouble spin-flip excitations and therefore most probablycribable to direct charged bimagnon absorption processe65

ACKNOWLEDGMENTS

We gratefully acknowledge M. V. Mostovoy, D. I. Khomskii, T. T. M. Palstra, and G. A. Sawatzky for stimulatindiscussions. We thank D. Smirnov, J. Leotin, M. Gru¨ninger,P. H. M. van Loosdrecht, and M. J. Rice for many usecomments, and C. Bos, A. Meetsma, and J. L. de Boerassistance. Andrea Damascelli is grateful to M. Picchiand B. Topı´ for their unlimited cooperation. This investigtion was supported by the Netherlands Foundation for Fdamental Research on Matter~FOM! with financial aid fromthe Nederlandse Organisatie voor Wetenschappelijk Onzoek ~NWO!.

-.

*Present address: Laboratory for Advanced Materials, McCulloBuilding, Stanford University, 476 Lomita Mall, Stanford, Cafornia 94305-4045. Email address: [email protected]

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