first-principles investigation of the cooperative jahn ...the cooperative jahn-teller distortion is...

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PHYSICAL REVIEW B, VOLUME 63, 224304

First-principles investigation of the cooperative Jahn-Teller effect for octahedrally coordinatedtransition-metal ions

C. A. Marianetti,1 D. Morgan,1 and G. Ceder1,21Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02

2Center for Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 0213~Received 2 November 2000; revised manuscript received 20 February 2001; published 22 May 2001!

Fundamental aspects of the cooperative Jahn-Teller effect are investigated using density functional theory inthe generalized gradient approximation. LiNiO2, LiMnO2, and LiCuO2 are chosen as candidate materials asthey possess small, intermediate, and large cooperative Jahn-Teller distortions, respectively. The cooperativedistortion is decomposed into the symmetrized-strain modes andk50 optical phonons, revealing that only theEg andA1g strain modes andEg andA1g k50 optical-phonon modes participate in the cooperative distortion.The first-principles results are then used to find values for the cooperative Jahn-Teller stabilization energy andthe electron-strain and electron-phonon coupling. It is found that the dominant source of anisotropy arises fromthe third-order elastic contributions, rather than second-order vibronic contributions. Additionally, the impor-tance of higher-order elastic coupling between theEg and A1g modes is identified, which effectively causesexpansion ofA1g-type modes and allows for a largerEg distortion. Finally, the strain anisotropy induced by theantiferromagnetically ordered states is shown to cause a significant difference in the cooperative Jahn-Tellerstabilization energy for the different orientations of the cooperative distortion.

DOI: 10.1103/PhysRevB.63.224304 PACS number~s!: 71.38.2k, 61.50.Ah, 71.20.Be

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I. INTRODUCTION

A. General background

The cooperative Jahn-Teller distortion is an energeticstability of the crystal structure with respect to certain modof distortion due to electronic degeneracy. In magneticides, the cooperative Jahn-Teller distortion is linked to maimportant phenomena. In LaMnO3, for example, the JahnTeller distortion plays a strong role in stabilizing theA-typeantiferromagnetic ordering.1 Jahn-Teller ordering has alsbeen invoked to understand charge ordering in thcompounds.2 Additionally, the Jahn-Teller distortion hapractical consequences in electrode materials for rechaable Li batteries. In LiMnO2 and LiNiO2, the transition-metal ions in the electrode material are cycled between JTeller active and inactive valence states during the chargand discharging of the battery. The resulting nonuniform vume and cell distortions can lead to rapid mechanical dedation of the electrode.3

In this paper, we study in detail the Jahn-Teller activityMn31, Ni31, and Cu31 in octahedral environments. In paticular, they are studied in the layered (R3̄m) and spinel-like(Fd3̄m) structures, which are observed in LiMnO2 andLiNiO2. Although the active octahedra in these ordered rosalts are edge sharing, the more general aspects of the reprobably also apply to the perovskite structures that hcorner-sharing octahedra.

Ligand field theory demonstrates that thed orbitals of atransition-metal ion in an octahedral complex will be spinto a set of threefold degeneratet2g states and twofold degenerateeg states~see Fig. 1!.

4 Both thet2g and theeg statesare split underEg-type octahedral distortions, commonly rferred to asQ2 andQ3 ~see Fig. 4!. All the systems investi-gated in this study have filled, or half-filled,t2g states sotheir splitting will not be considered hereafter. Splitting t

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eg states creates an electronic reduction in energy for mrials with partially occupiedeg states, allowing for the dis-tortion to operate until the increase in elastic energy haltsdistortion. Ni31, Mn31, and Cu31 are all susceptible toEg-type Jahn-Teller distortions. The Jahn-Teller active sconfiguration for each ion is pictured below~see Fig. 1!.Note that Mn31 is high spin while Ni31 and Cu31 are lowspin. In the high-spin state, splitting theeg states will allowMn31 to realize a decrease in energy, while Ni31 and Cu31

will not. In the low-spin configuration, Ni31 and Cu31 willrealize a decrease in energy while Mn31 will not. Strictlyspeaking, Cu31 is not Jahn-Teller active because theEg sym-metry is broken when the interelectronic interaction is acounted for.4 However, if the splitting due to the interelectronic interaction is small, this ion may still bsusceptible to a pseudo-Jahn-Teller effect and the energyels will be further split by the distortion. This is the case fLiCuO2, which is found to have a strong cooperative JahTeller distortion in experiment.

The previous discussion pertains to a system with discenergy levels, such as a molecule or a Jahn-Teller deHowever, the discussion above is completely applicablethe Jahn-Teller effect in solids with somewhat localized eltrons. In solids with octahedrally coordinated transition mals, a set of bonding (eg) and antibonding (eg* ) bands will be

FIG. 1. Jahn-Teller active spin states.

©2001 The American Physical Society04-1

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C. A. MARIANETTI, D. MORGAN, AND G. CEDER PHYSICAL REVIEW B63 224304

TABLE I. Space group and Wyckoff positions of the structures evalated in this study.

Structure Scho¨enflies International # Li Site Metal site O sit

Layered~rhombohedral! D3d5

R3̄m 166 b a 2c

Layered~monoclinic! C2h3 C2/m 12 2d 2a 4i

Spinel ~cubic! Oh7

Fd3̄m 227 16c 16d 32e

Spinel ~tetragonal! D4h19 141 /amd 141 8d 8c 16h

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formed, with the antibonding bands consisting primarilymetal orbitals and the bonding bands consisting primarilyoxygenp orbitals. Thet2g orbitals form nonbonding bandsTherefore, the valence electrons of transition-metal oxiwill occupy thet2g bands and theeg* bands. The cooperativdistortion of all the octahedra, in theQ2 or Q3 mode, willresult in the splitting of theeg* bands. However, there arseveral important differences in the solid. For the solidsvestigated in this study, the Jahn-Teller-active octaheshare edges and thus they do not distort independeWhether or not the octahedra share edges, a given octahwill feel a strain due to the distortion of the neighborinoctahedra. This is known as the electron-vibrational intertion.

Additionally, magnetic and qaudrupole interactions mexist between the Jahn-Teller centers. All of these intertions may cause the distortions of different octahedra toder and become cooperative.5 Above a critical temperaturethe entropic contribution to the free energy will overcomthe energetic benefit of ordering the octahedra and thetortion will become noncooperative. Additionally, the banwidth will create other differences between the isolated acooperative Jahn-Teller distortions, which are discussedSec. III B.

To study the Jahn-Teller distortion for these ions,chose to perform calculations for LiNiO2, LiMnO2, andLiCuO2 in the layered rhombohedral (R3̄m) structure and itsdistorted variant, monoclinic (C2/m), because of their im-

FIG. 2. Layered LiMO2(R3̄m). M -O bonds are designated witthick lines. In this structure, each~111! plane is occupied by asingle species. Every other~111! plane consists of oxygen, whilethe planes in between the oxygen planes alternate between mand Li. TheM -O octahedra share edges half with Li-O octaheand half with otherM -O octahedra. Corners of theM -O octahedraare only shared with Li-O octahedra.~Note that this is not a unitcell!.

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portance as cathode materials in rechargeable Li batteAdditionally, calculations are performed for the spinel-likcubic (Fd3̄m) structure and its distorted variant, tetragon(I41 /amd). A summary of the space groups and Wyckopositions for both structures is given in Table I. The geneconclusions of this study are similar for the layered and snel structures, indicating a rather small effect of structurethe properties of the distortion.

B. Spinel-like and layered structures

The layered and spinel-like structures are both orderock salts~see Figs. 2 and 3!. The layered structure can benvisioned as two interpenetrating face-centered-cubictices, with one lattice consisting of oxygen and the othlattice consisting of alternating~111! planes of Li and metal.In the R3̄m space group the Li and the metal ions remafixed in the ideal rock salt positions, but the oxygen atohave a degree of freedom allowing the whole~111! oxygenplane to relax in thê111& direction. Physically, this relax-ation is caused by the broken symmetry induced by themetal ordering. Considering the effect on the individual otahedron, this planar relaxation corresponds to theT2g andtheT1g octahedral modes, commonly referred to asQ4 , Q5 ,Q6 , andQ19, Q20, Q21, respectively~see Fig. 4!. Becausethe Li-metal ordering breaks the cubic symmetry and cauthe oxygen atoms to relax, the point group of thea site isreduced fromOh to D3d upon ordering the rock-salt latticeIf a cooperativeQ3 distortion is imposed on this structurethe space group is reduced fromR3̄m to C2/m. However,

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FIG. 3. Spinel LiMO2(Fd3̄m). M -O bonds are designated witthick lines. As in the layered structure, corners of theM -O octahe-dra are only shared with Li-O octahedra. Also, theM -O octahedrashare edges half with Li-O octahedra and half with otherM -Ooctahedra. However, theM -O octahedra share different edges in tspinel and layered structures.~Note that this is not a unit cell!

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FIRST-PRINCIPLES INVESTIGATION OF THE . . . PHYSICAL REVIEW B 63 224304

this is still commonly referred to as the layered structurethe literature on lithium-metal oxides.

The spinel-like structure is so named due to the fact thais synethized by lithiating LiMn2O4, which has the spinecrystal structure (MgAl2O4), to form Li2Mn2O4. Eventhough the latter is not a true spinel, we shall simply referit as spinel, which is common in the literature on lithiummetal oxides. The spinel structure (Fd3̄m) can be envi-sioned as a face-centered-cubic oxygen lattice with an inpenetrating face-centered-cubic lattice having a particordering of Li and metal. As in the layered structure, tLi-metal ordering breaks the cubic symmetry and causesoxygen atoms to relax away from the ideal rock-salt potions, thus reducing the 16d site symmetry fromOh to D3d .

In both these structures, the metal-oxygen octahedra scorners with Jahn-Teller inactive Li-oxygen octahedEdges are shared half with Li-oxygen octahedra andwith other metal-oxygen octahedra. Edge sharing betwmetal-oxygen octahedra will be relevant because it prevthe octahedra from distorting independently. However,Q2 and Q3 components of the distortion on each metoxygen octahedra are independent of each other~see Sec.I C!. The metal-oxygen octahedra share different edges inspinel and layered structures. This difference will causecooperative distortion to be distinctly different in the twstructures, and it is most likely the reason why the JaTeller stabilization energy is greater in the layered structfor all three materials.

FIG. 4. Selected symmetrized octahedral modes. TheQ2 andQ3modes are Jahn-Teller active.

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C. Cooperative Jahn-Teller distortion

The cooperative Jahn-Teller distortion can be defined aregular ordering of local Jahn-Teller distortions. The mateals examined in this study are all found to be ferrodistortin experiment, and therefore we refer to the cooperatJahn-Teller distortion as the imposition of identicalEgmodes on each octahedron. It turns out that theEg latticestrains and theEg k50 optical phonons are the only modethat will induce equivalentEg distortions on each octahedron. The lattice strains are homogeneous deformationthe unit cell, while thek50 optical phonons are displacements of the basis atoms. These types of symmetrized mwill now be described, and the derivation of the modes usgroup theory will be discussed in Sec. III C.

TheEg lattice strains are listed in Table II for the layere(R3̄m) and spinel (Fd3̄m) structures, and from hereon thewill be referred to as theu2 andu3 strain modes. For spinelthe Eg strain modes are completely analogous to the octadral modes. For example, theu3 lattice strain in the cubiccell is compression in theX andY directions by an amount«and elongation in theZ direction by an amount 2«. It isinteresting to note that theEg strain modes induce only thEg octahedral modes on each octahedron. However,strain modes will also distort the lattice of transition meions, changing the distances and angles between the trtion metal ions. This will be shown to have important effecfor the antiferromagnetic interaction between the transitmetal ions. These statements also apply to the layered stures; however, it is slightly more complicated. Unlike spnel, theEg representation occurs twice in the representatof the strain for the rhombohedral lattice, and thereforelayered structure has two sets ofu2 andu3 lattice strains.

The Eg k50 optical phonon modes for the layere(R3̄m) and spinel (Fd3̄m) structures are shown in Table IIIand from hereon they will be referred to as thee2 and e3optical modes. Thee3 optical mode in spinel is shown as aexample in Fig. 5 and can be explained as follows. Consfixing all the metal and Li ions and inducing a positiveQ3octahedral distortion on all the metal-oxygen octahedra. Tsmall vectors indicate a displacement ofd while the largevectors indicate a displacement of 2d. Additionally, the grayarrows correspond to theQ3 octahedral distortion of the central metal-oxygen octahedron, whereas the black arrowsrespond to theQ3 octahedral distortions of the neighborin

ice.

TABLE II. Eg andA1g symmetrized strain modes for cubic~spinel! and rhombohedral~layered! lattices.

The Eg andA1g irreducible both occur twice in the representation of the strains for the hexagonal latt

Symmetrized strain modes

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TABLE III. Eg andA1g k50 symmetrized-phonon modes for the space groupsR3̄m andFd3̄m. Onlythe oxygen atoms participate in these modes. The position of each oxygen atom and the corresdisplacement vector is given for both space groups.

k50 Optical phononsPositions e1(A1g) e3(Eg) e2(Eg)

Fd3̄m~spinel!

~x,x,x! ~21,21,21! ~1,1,22! ~1,21,0!(x,2x1 14 ,2x1

14 ) ~21,1,1! ~1,21,2! ~1,1,0!

(2x1 14 ,x,2x114 ) ~1,21,1! ~21,1,2! ~21,21,0!

(2x1 14 ,2x114 ,x) ~1,1,21! ~21,21,22! ~21,1,0!

(2x,2x,2x) ~1,1,1! ~21,21,2! ~21,1,0!(2x,x1 14 ,x1

14 ) ~1,21,21! ~21,1,22! ~21,21,0!

(x1 14 ,2x,x114 ) ~21,1,21! ~1,21,22! ~1,1,0!

(x1 14 ,x114 ,2x) ~21,21,1! ~1,1,2! ~1,21,0!

R3̄m ~0,0,z! ~0,0,1! ~2,1,0! ~0,1,0!

~layered! (0,0,2z) ~0,0,21! ~22,21,0! ~0,21,0!

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metal-oxygen octahedra. Only the displacements of the ogen atoms belonging to the center octahedron are shoThe figure indicates that theQ3 octahedral distortion of thecentral octahedron is not affected by theQ3 distortions of theother octahedra. However, theQ3 distortion of the neighbor-ing octahedra will createQ4 , Q5 , Q6 , Q19, Q20, andQ21distortions in the central octahedron, which are theT2gmodes and theT1g modes~see Fig. 4!. In other words, theEgoptical modes induce localEg octahedral distortions in addition to local T2g and T1g distortions, in contrast to theEgstrain modes. However, the opticalEg modes do not distorthe transition metal lattice whereas theEg strains do. Itshould be noted that the perovskite structure does not hEg k50 optical modes.

As will be identified later in this study~see Sec. IV B!, theA1g modes play an important role in the cooperative disttion. TheA1g irreducible representation is the identity repr

FIG. 5. Thee3 k50 optical-phonon mode in the spinel structu

(Fd3̄m). Thee3 k50 optical phonon can be constructed by induing an equivalent octahedralQ3 distortion on everyM -O octahe-dron. The displacements in theZ direction correspond to 2d,whereas displacements in theX andY direction correspond to displacements ofd. The displacements are only labeled for the oxygatoms that belong to the centerM -O octahedron. Gray arrows correspond to displacements caused by theQ3 distortion of the centralM -O octahedron whereas the black arrows correspond to dispments caused byQ3 distortions of neighboring octahedra. This figure illustrates that theQ3 distortions of theM -O octahedra areindependent of one another.

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sentation, and the distortions that project onto this represtation do not break the symmetry. TheA1g mode for theoctahedron, also known as theQ1 mode, is simply the uni-form expansion or contraction of all the bond lengths~seeFig. 4!. TheA1g strain andA1g k50 optical distortions willbe referred to asu1 ande1 , respectively~Tables II and III!.As explained previously, the~111! oxgen planes in the layered structure can relax in the^111& direction and this is thee1 k50 optical phonon. As for thee3 mode in spinel, thee1mode in spinel can be envisioned by inducingQ1 octahedraldistortions on each metal octahedron. Theu1 strain mode inspinel is simply a uniform expansion or contraction of tvolume. There are twoA1g strain modes in the layered structure, u1a andu1b . If a hexagonal coordinate system is cosidered, this corresponds to a uniform expansion or contion of thea andb lattice parameters, or a uniform expansioor contraction of thec lattice parameter.

We shall close this section with a short summary. Tcooperative Jahn-Teller distortion in the layered and spstructures, evaluated in this study, is some linear combtion of Eg and A1g strain modes andEg and A1g k50optical-phonon modes. TheEg strain modes andEg k50optical-phonon modes are the only two collective distortiothat will impose equivalentEg distortions on each JahnTeller active octahedron. Although theA1g mode exists inthe direct product of theEg irreducible representation, it isnot Jahn-Teller active in the sense that it is totally symmeand does not split theeg levels. However, it will be demon-strated that there is a strong coupling between theA1g andEg modes that causes theA1g modes to participate in thecooperative distortion.

D. Materials background

LiMnO2 can be synthesized in the orthorhombic,6 spinel,7

and layered8 type structures. In experiment, these systeare all found to be cooperatively Jahn-Teller distorted wthe orthorhombic structure being the ground state.6,9 Thesame result is found using density functional theory~DFT! inthe generalized gradient approximation, although the gro

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state was found to depend sensitively on the magnordering.10 Additionally, the bond lengths predicted usinDFT agree well with experiment.

LiNiO2 can be synthesized in the layered structure11

however, some small fraction of Ni atoms are found onlithium sites and the material has not yet been synthesstoichiometrically.12 Layered LiNiO2 displays a noncooperative Jahn-Teller distortion. X-ray diffraction data can be fitthe space groupR3̄m, which is an undistorted structure, sugesting that the Jahn-Teller distortion is not present. Hoever, extended x-ray absorption fine structure~EXAFS! datahas shown that the Ni-O octahedra are locally Jahn-Tedistorted, implying that the distortion is noncooperative13

This is consistent with the fact that the stabilization eneof the cooperative Jahn-Teller distortion is found to be vsmall in this study.

Layered LiCuO2 is a relatively new material which wafirst synthesized in 1992. Initially, LiCuO2 was incorrectlyassigned to the orthorhombic structure,14 until it was cor-rectly identified as being layered.15 The experimentally de-termined bond lengths of LiCuO2 indicate the presence of aextremely large Jahn-Teller distortion, which is also foundour calculations. As far as the authors are aware, none opublications regarding the initial synthesis of LiCuO2, norpublications addressing cathode performance16 orsuperconductivity17 acknowledge the presence of the larJahn-Teller distortion in this material.

II. OUTLINE

Previous studies have indicated that density functiotheory in the generalized gradient approximation can acrately model structural aspects of the cooperative Jahn-Tdistortion in LiMnO2 as compared to experiment.

10 In thisstudy, we perform first-principles calculations for LiMnO2,LiNiO2, and LiCuO2 for both the undistorted and coopertively Jahn-Teller-distorted systems. To our knowledthese are the firstab initio calculations that have been peformed for LiCuO2. The energy as a function of the JahTeller distortion is calculated using density functional theoin the generalized gradient approximation, yielding the coerative Jahn-Teller stabilization energy and the vibronic cpling for theEg strain and phonon modes. The radial depedence of the higher-order terms will be exploited to revthe fact that the higher-order elastic term is dominant othe higher-order vibronic term. These statements applyboth theEg strains and optical phonons.

Additionally, the role of A1g-type modes in the JahnTeller distortion are explored. It will be shown that the epansion ofA1g modes effectively allows for largerEg distor-tions. Finally, the coupling between the magnetic orderand the cooperative Jahn-Teller distortion is explored.

III. THEORY

A. The isolated Jahn-Teller center

In 1937, Jahn and Teller demonstrated that degeneelectronic states~excluding Kramers degeneracy! in a mol-ecule will always be unstable with respect to some distorti

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with the exception of the linear molecule.18 The distortion ofan octahedron can be described exactly by the 21 octahemodes19 ~see Fig. 4!. Van Vleck used perturbation theory texplicitly solve theE^ e Jahn-Teller problem, demonstratinthat the degenerateeg states in an octahedral complex will bsplit by theQ2 and Q3 octahedral modes.

20 In the simplestmodel, the Hamiltonian for an isolated octahedron or a JaTeller defect in an otherwise perfect crystal can be writtena sum of elastic and electronic energy~see Ref. 4 for a de-tailed derivation!. In the harmonic approximation, the elastenergy will be proportional to the square of the displacment, and the electronic energy can be found by perturbthe eg states with the vibronic interaction. The lower eigevalue of this simple Hamiltonian may be written in polcoordinates~equation 1! as follows~equation 2!:

Q35r cosf Q25r sinf ~1!

E5 12 kr21FEr, ~2!

wherek is an elastic constant andFE is the first-order vi-bronic coupling constant.

The energy can be plotted inQ2-Q3 space and the resuis a spherically symmetric potential known as the Mexichat potential. The minimum of this potential is a degenercircle in Q2-Q3 space, indicating that all linear combinationof Q2 and Q3 lying on the circle are degenerate. TheQ3mode is defined such that a positive tetragonal distortion ithe Z direction. It can be shown that theX andY orientationof the Q3 mode will correspond to a rotation of 2p/3 and4p/3 in Q2-Q3 space, respectively. The same can be shofor Q2 . Because of this degenerate circle inQ2-Q3 space,the system can move freely along the circumference betwdifferent amplitudes and orientations of the modes.

In order to attain a higher degree of accuracy, the elaenergy can be expanded to third order21 and the vibroniccoupling can be expanded to second order22 ~see Ref. 23 fora detailed derivation!. Diagnolizing this Hamiltonian and expanding to first order yields the following eigenvalue~equa-tion 3!:

E5 12 kr21FEr1GEr

2 cos~3f!1Br3 cos~3f!, ~3!

whereGE is the second-order vibronic coupling, andB is thethird-order force constant. Both the anharmonic elastic teand the second-order vibronic term end up contributingcos 3f term to the energy, with the elastic term going asr3

and the vibronic term going asr2. This potential is known asthe warped Mexican hat potential due to the fact thatspherical symmetry has been broken, reducing the poteto threefold symmetry. Intuitively, threefold symmetry mube retained due to the equivalence of the distortion in theX,Y, andZ orientations. Values ofB, GE,0 support minima atf50, 2p/3, and 4p/3, while B, GE.0 support minima atf5p/3, 2p/3, andp. Most systems are found to be distorted with a positiveQ3 distortion (f50).

19

Because both higher-order elastic and vibronic terms cate the same type of anisotropy inQ2-Q3 space, it is oftenambiguous as to what the dominant source of the anisotris, which is evident in the early literature. Opik and Pry

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were the first to account for the anisotropy in the potenand they used only third-order elastic contributions.21 In1958, approximately one year later, Liehr and Ballhauexplained the anisotropy with second-order vibronic copling disregarding the third-order elastic contributions.22 In-terestingly, the calculations they performed using crysfield theory yield a negative second-order vibronic couplinwhich stabilizes the negative distortion contrary to whatnormally observed in experiment. They concluded thatsecond-order vibronic coupling, as predicted by crystal fitheory, must be treated as a phenomenological constancause their calculations predict the wrong sign. Furtherculations by Ballhausen and Johansen using linear combtion of atomic orbitals methods were also deemedaccurate enough to calculate the second-order vibrocoupling.24 A later paper by Pryceet al. discusses bothhigher-order elastic and vibronic terms.25 First, they pointout that the third-order elastic term would support a positQ3 distortion in any reasonable potential model, due tofact that the twoM -O bonds are contracted twice as muchthe other four. Second, they reason that when the covaleis small, the second-order vibronic coupling may be negaor positive by a small difference, and that it would increawith increasing covalency. We address the issue of aniropy more quantitatively in Sec. IV C.

B. Jahn-Teller effect in a crystal

Thus far, the energetics have been considered for anlated octahedron. The Jahn-Teller effect in a crystal ismore complicated as theeg electrons now form energy bandand not discrete energy levels. Only a small number of hisymmetryk-pointswill be degenerate and therefore the JahTeller effect in the solid is largely a pseudo-Jahn-Tellerfect. The formal problem of the Jahn-Teller effect felectronic states in crystals may be addressed in the sway Jahn and Teller addressed instability in molecules. Bman analyzed the direct products of the irreducible represtation of two different space groups in order to find the actphonon modes for those given space groups.26,27 Thus, theactive phonons can be found for each high-symmetryk-pointin the first Brillouin zone. As pointed out by Englman23 andCracknell,28 this formalism is severely limited by the facthat the vast majority ofk-pointsin the first Brillouin are nothigh-symmetry points and therefore little can be concludfrom group theory. A more practical treatment involves treing the Jahn-Teller effect in the solid as an array of loJahn-Teller centers having several different types of intetions: magnetic, quadrupole, and electronic-vibratiointeractions.5 All three of these interactions may cause obital ordering in the solid. Additionally, if there is significanelectron hopping, or bandwidth, this must be includedsome manner.

Kristofel’ points out that when considering electrophonon coupling in crystals, the bandwidth mustconsidered.29 When considering wide bands, only select phnon modes may be active, while in the case of narrow baphonons from the entire spectrum may be active. The appriate limit for a given system will depend on the relati

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difference between the strength of the vibronic coupling athe bandwidth. The materials analyzed in this studynarrow-band materials and phonons from the entire spectwill be active. However, we only address the ferrodistorticooperative distortion that by definition only involvesk50type distortions.

Even in narrow-band systems, finite bandwidth will haan appreciable effect. In a crystal the cooperative Jahn-Tedistortion will split energy bands that are only degeneratea few k-points, not discrete energy levels as in moleculeThe end result is that the change in electronic energy afunction of the cooperative distortion will be nonlinear fosmall distortion. This occurs for two reasons. First, banhaving significant width will not split uniformly or linearlyas a function of the distortion, which is essentially a resultthe pseudo-Jahn-Teller effect. Simplified models have shothe electronic energy to be proportional tor2 for the initialstages of the distortion.30,31 Second, even if the bands spllinearly and uniformly at eachk-point as a function of thedistortion, band overlap will cause the change in the eltronic energy to be nonlinear. In this study, we find fLiMnO2 that the electronic term is approximately quadrafor small distortions and transitions to linear behavior onthe bands no longer overlap~see Sec. IV C!. It should benoted that in this context the electronic term representstotal change in electronic energy due to the splitting ofeg* bands.

Kanamori was one of the first to model Jahn-Teller crytals using the phonon formalism.32 The Hamiltonian is writ-ten in terms of the phonon modes and strain modes. Abronic coupling coefficient is assumed for each phonmode and for theEg strain modes,commonly referred to asthe electron-phonon coupling and the electron-strain copling, respectively. He also considers the ferrodistortive cain which only theEg strains participate, demonstrating ththe potential in the space of theEg strains has the samfunctional form as the warped Mexican hat potential@seeequation ~3!#. He does not considerEg k50 opticalphonons, however, the potential of theEg k50 opticalphonons will also have the same functional form as equa~3!. Therefore, there will be a vibronic coupling for both thEg strain modes and theEg k50 optical-phonon mode. Inthe present study, it is found that the vibronic coupling is tsame for theEg strain modes andEg k50 optical phononmodes~see Sec. IV C!. However, when both the strains anthe optical phonons are considered, the situation is comcated by the interaction between them.

C. Symmetrizing the crystal distortions

An arbitrary crystal distortion can be described by spefying the 3n atomic coordinates, wheren is the number ofatoms in the primitive unit cell. However, taking the proplinear combinations will yield displacement vectors thtransform as the irreducible representations of the spgroup. These symmetrized displacement modes can reabe found using standard group-theoretical techniques andspace group of the structure. As long as a given irreducrepresentation only appears once in the representation o

4-6

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tructure


TABLE IV. Relative energies and structural parameters calculated in this study. All calculationsperformed ferromagnetically. The zero for the energy is chosen to be the undistorted layered structugiven compound. Experimental values are included in parentheses wherever available. Both LiNiO2 andLiCuO2 have small negative distortions, and in some cases they are degenerate with the undistorted sor metastable by 1 meV.

MaterialStructure

typeSpacegroup Mode

Magneticordering

Energy~meV!

M -O bondlengths~Å! ~4!

M -O bondlengths~Å! ~2!

Volume~Å3!

Q1~Å!

Q3~Å!

LiNiO2 Layered R3̄m F 0 1.98(1.97)11

1.98~1.97!

34.4~34.0!

0.00 0.01

LiNiO2 Layered C2/m 1Q3 F 211 1.91(1.91)12

2.14~2.07!

34.6 0.04 0.27

LiNiO2 Layered C2/m 2Q3 F 0 2.00 1.95 34.5 0.0120.05LiNiO2 Spinel Fd3̄m F 1 1.98 1.98 34.5 0.00 0.00

LiNiO2 Spinel I41 /amd 1Q3 F 23 1.92 2.12 34.5 0.03 0.24LiNiO2 Spinel I41 /amd 2Q3 F 2 1.99 1.96 34.4 0.0120.02LiMnO2 Layered R3̄m F 0 2.04 2.04 37.4 0.00 0.01

LiMnO2 Layered C2/m 1Q3 F 2215 1.94(1.92)7

2.40~2.31!

38.5~37.0!

0.12 0.54

LiMnO2 Layered C2/m 2Q3 F 271 2.14 1.91 37.5 0.0420.26LiMnO2 Spinel Fd3̄m F 3 2.05 2.05 37.6 0.00 0.01

LiMnO2 Spinel I41 /amd 1Q3 F 2208 1.94(1.94)6

2.40~2.29!

38.5~36.9!

0.12 0.53

LiMnO2 Spinel I41 /amd 2Q3 F 267 2.13 1.91 37.6 0.0420.25LiCuO2 Layered R3̄m F 2205 2.03 2.03 35.9 0.02 0.01

LiCuO2 Layered R3̄m NSP 0 2.02 2.02 35.9 0.00 0.01

LiCuO2 Layered C2/m 1Q3 NSP 2341 1.86(1.84)14

2.95~2.76!

38.9~37.7!

0.46 1.29

LiCuO2 Layered C2/m 2Q3 NSP 1 2.07 1.94 35.9 0.0120.13LiCuO2 Spinel Fd3̄m NSP 0 2.02 2.02 36.0 0.00 0.01

LiCuO2 Spinel I41 /amd 1Q3 NSP 2290 1.87 2.87 38.5 0.42 1.18LiCuO2 Spinel I41 /amd 2Q3 NSP 21 2.05 1.97 35.9 0.0120.09

noib

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group, the corresponding symmetrized modes are alsomal modes or phonon modes of the crystal. For irreducrepresentations that appear more than once, the dynammatrix must be used to find what linear combination of tgiven symmetry modes is to be used to construct the phoA computer code is now publicly available that will find thsymmetrized displacement modes using group theoretechniques.33 This code was used to find the symmetrizstrain modes andk50 phonon modes for the spinel anlayered structures in this study.

The representation for thek50 phonons can be decomposed into

@A1g~O!1Eg~O!1T1g~O!12T2g~O!1A2u~Li, M ,O!

1Eu~Li, M ,O!12T1u~Li, M ,O!1T2u~Li, M ,O!#

for spinel (Fd3̄m) and

@A1g~O!1Eg~O!1A2u~Li, M ,O!1Eu~Li, M ,O!#

for layered (R3̄m). The representation for the strain modcan be decomposed into (A1g1Eg1T2g) for a cubic latticeand into (2A1g12Eg) for a rhombohedral lattice. OnlyEg

22430

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and A1g strain andEg and A1g k50 optical-phonon modesparticipate in the cooperative distortion in the systems evaated in this study.

IV. RESULTS

A. Total energy calculations

First-principles calculations using density functiontheory in the generalized gradient approximation have deonstrated the ability to model the structural aspects ofcooperative Jahn-Teller distortion for LiNiO2, LiMnO2, andLiCuO2. All energies were calculated with the Vienna Abinitio Simulation Package~VASP!.34,35VASP solves the Kohn-Sham equations using ultrasoft pseudopotentials.36,37 Aplane-wave basis set with a cutoff energy of 600 eV wchosen. The results compiled in Table IV were calculausing a minimum of 116 irreduciblek-points for the primi-tive layered unit cell~4 atoms! and 29 irreduciblek-pointsfor the primitive spinel unit cell~16 atoms!. Higher k-pointmeshes, 334 irreduciblek-points for the primitive layeredcell and 72 irreduciblek-points for the primitive spinel cell,were tested and the total energy differed by less than 1 m

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in both cases. The tetrahedron method with Blochl corrtions was used to integrate the band energy for all the resin Table IV. All other calculations in this paper were calclated with a lower-k-point mesh, eight irreduciblek-pointsfor the primitive spinel cell, and band integrations were pformed using the method of Methfessel-Paxton with a smeing width of 200 meV. These changes were made in ordereduce calculation time and introduce errors of appromately 5–10 meV in the total energy, which is insignificafor the results of this study. Calculations were performedthe spinel and layered structures for the undistorted strucand the corresponding positive and negative amplitudethe cooperativeQ3 distortion. Additionally, the energy wacalculated as a function of the cooperativeQ3 distortion forall three materials with ferromagnetic spin polarization. Twas performed by perturbing the undistorted structureusing the method of steepest descent to relax to the mmum. The energies, bond lengths, magnitude of the octadral Q3 distortion, and volume are shown in Table IV witthe experimental values included in parentheses where aable.

As shown, the calculations agree well with experiment.all cases, the short bonds of the octahedron are predictewithin 1% of experiment, while the long bonds are overpdicted in all three cases with an error of 3–7% as compato experiment. LiNiO2, LiMnO2, and LiCuO2 serve as inter-esting examples of study because they possess small, imediate, and large Jahn-Teller distortions, respectivwhere the size is associated with the magnitude ofQ3 or thestabilization energy. The values of the cooperative JaTeller stabilization energies~the energy of the cooperativeldistorted structure minus the energy of the undistorted stture! for these materials immediately yield insight into eperimental observations. Unless otherwise indicated, theromagnetic, layered energy values will be discussedsimplicity as all the results obey the same general trends.energies will be given per formula unit. The stabilizatioenergy for LiMnO2 is 2215 meV while the stabilization energy for LiNiO2 is only 211 meV. These values are consitent with the experimental observations of a strong cooptive distortion in LiMnO2 and a noncooperative distortionroom temperature in LiNiO2.

We also calculated the energy as a function of the coerative distortion. As noted in Sec. I C, the cooperative dtortion consists ofA1g and Eg strain modes andA1g andEg k50 optical phonon modes.This is indeed found in thecalculation as the only nonzero displacement projectionsthe A1g and Eg strain modes and the A1g and Egk50 opticalphonons. In order to best represent the data, we plotenergy as a function of theQ3 octahedral mode, which wilreflect the contributions of both theu3 strain ande3 opticalmodes. The curve for LiCuO2 is calculated entirely low spinand will be specifically discussed below. As shown in Fig.the distortion is twice as large in terms ofQ3 and 20 timesmore energetically stable in LiMnO2 as compared to LiNiO2,whereas in LiCuO2 it is roughly twice as large and 1.5 timemore stable than in LiMnO2. Several interesting featureshould be noted in this plot. First, the derivatives of all tcurves are close to zero for the undistorted state. This wo

22430

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not be expected if the electronic term were linear. Howevthe electronic term is quadratic for small distortions becathese systems are solids with finite bandwidths~see Sec.III B !. Second, for a given material, the cooperative JaTeller stabilization energy in the spinel structure is alwaless than that of the layered structure. The difference is rasmall for intermediate distortions, as in LiMnO2, while itincreases for large distortions, as in LiCuO2. Presumably,this difference is an elastic effect that arises due to the mner in which the octahedra are shared, although it isexactly clear why the layered structure is energetically mfavorable. Third, all three systems are highly anisotropic.noted in Sec. III B, there are two sources of anisotropy whseparately considering the potential of theEg strains or theEg k50 optical phonons: vibronic anisotropy and elastic aisotropy. Both terms have a cos 3f dependence but the elastic term goes asr3 while the electronic term goes asr2. Asnoted in the Sec. III A, it is not immediately evident whicterm is causing the asymmetry. It is shown below thatdominant source of this anisotropy is the anharmonic elaterm, both for the strain and optical phonons~see Sec. IV C!.Lastly, a particularly interesting feature is that LiCuO2 be-gins with a very small slope, quite similar to that of LiNiO2,but the distortion continues to grow far larger than thatLiMnO2. This is due to the fact that LiCuO2 has a smalllinear vibronic coupling but it is elastically much softer thaLiNiO2 and LiMnO2, partially because of the presence ofextra electron in theeg* band.

The electronic density of states~DOS! are plotted for lay-ered LiMnO2 and LiNiO2 in the undistorted and distortestructures~see Figs. 7 and 8!. The energy scales are shiftesuch that the center of gravity of theeg* bands is referencedto zero. The cooperative distortion clearly splits the band

As stated in Sec. I A, Cu31 does not have degenerateegelectrons once the interelectronic interaction is taken iaccount. Within density functional theory, all of the higsymmetryk-points will display degeneracies as prescribed

FIG. 6. Energy vs OctahedralQ3 mode. All energies are peformula unit. These results were obtained by starting from thedistorted structure and using the method of steepest descent into determine the trajectory of the distortion. The energy is plotteda function of theQ3 octahedral distortion, which reflects contributions of both theu3 strain mode and thee3 k50 optical-phononmode. In all cases, the undistorted energy is referred to zero.layered structure is designated asL while spinel is designated asS.Only LiMnO2 has a significant stabilization energy in the negatdistortion.

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the appropriate irreducible representations of the spgroup, and the interelectronic interaction will not break tsymmetry. The concern for this is that, in reality, the electrcorrelation might significantly split theeg* bands and greatlyweaken the Jahn-Teller effect. However, our calculatiocompare well with experiment. Therefore, it is reasonableassume that the GGA is accurate enough to elucidatebehavior of LiCuO2 reported in this work.

LiCuO2 is also more complicated than the other two cabecause high-spin LiCuO2 is not susceptible to the JahnTeller distortion, nor the pseudo-Jahn-Teller distortiowhereas low-spin LiCuO2 is susceptible. We shall discuslayered LiCuO2 but spinel LiCuO2 displays the same qualitative behavior. In the undistorted structure, the high-sstate is 205 meV lower in energy than the low-spin state~seeFig. 9!. This means that the stable undistorted structurehigh spin and is not Jahn-Teller active. At a value ofQ350.3 Å, the system transits from high spin to low spin. Tdistorted state is 136 meV lower in energy than the undtorted, high-spin state, and 341 meV lower than the untorted, low-spin state. It is expected that the distortionobserved to be cooperative in experiment considering this stable by 136 meV with respect to the undistorted hispin state.

B. Decomposing the cooperative distortion

The actual trajectory of the distortion has been calculausing the method of steepest descent, so the distortion caprojected onto all the symmetrized strain modes andk50optical phonons. As noted above, the only nonzero pro

FIG. 7. Electronic DOS for distorted and undistorted layerLiMnO2. The units are per formula unit. The center of gravitythe eg* bands is referred to zero in both cases.

FIG. 8. Electronic DOS for distorted and undistorted layerLiNiO2. The units are per formula unit. The center of gravity of teg* bands is referred to zero in both cases.

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tions were theA1g and Eg strain modes and theA1g andEg k50 optical phonons. First, the octahedralQ3 distortionis decomposed into contributions of theu3 strain mode ande3 k50 optical mode~see Fig. 10!. In this plot, theQ3 am-plitude resulting from theu3 strain mode is found by multi-plying the normalized amplitude of theu3 mode by a factorof a(2&)21, wherea is the cubic lattice parameter, and thQ3 amplitude resulting from thee3 mode is found by multi-plying the normalized amplitude of thee3 mode by a factorof one half. The two individual contributions are plotted asfunction of the total octahedralQ3 distortion. Only the re-sults for LiCuO2 are shown as the decomposition is neaidentical for LiMnO2 over the range of its distortion (Q350 – 0.55 Å). At Q3 values less than 0.5 Å, theQ3 distor-tion is created largely by the lattice strains. However, assize of the distortion increases, the optical-phonon modegins to play a larger role.

A similar analysis can be performed to show the relatcontributions of theu1 strain mode ande1 optical mode tothe Q1 octahedral mode. TheQ1 octahedral mode has beeplotted as a function of the magnitude of theQ3 octahedral

FIG. 9. Energy vs octahedralQ3 mode for high-spin and low-spin LiCuO2. The high-spin state is lower in energy than the lospin state in the undistorted structure. However, the low-spin sis Jahn-Teller active and becomes stable over the high-spin stathe distorted structure.

FIG. 10. The magnitude of the strain and optical contributioof the octahedralQ3 mode are plotted vs the total magnitude of toctahedralQ3 mode in LiCuO2. As shown, theu3 strain modeaccounts for a larger portion of the octahedralQ3 distortion.

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distortion for LiCuO2 ~see Fig. 11!. Additionally, the totalQ1 distortion has been decomposed into contributions ofu1 strain mode ande1 k50 optical-phonon mode, by multiplying the normalized modes by a factor ofa(2&)21 andone half, respectively. The curves for LiMnO2 and LiNiO2are similar over the ranges of their respective distortions,therefore are not shown. TheA1g distortions begin to operatwhen the magnitude ofQ3 is approximately 0.2 Å, andscales with the size of theQ3 distortion being very small inLiNiO2 and very large in LiCuO2. As shown, there is only aslightly larger contribution from the optical mode for smaand intermediate distortions, corresponding to LiNiO2 andLiMnO2, with the optical mode becoming more dominantlarge distortions. TheA1g strain mode is simply a uniformvolume expansion, and the resulting volume expansion aciated with the cooperativeQ3 distortion in the different sys-tems is shown in Table V. The negativeQ3 distortions alsoinduce a positiveA1g distortion. However, the negativeQ3distortions are all small and therefore the resultingA1g dis-tortions are small.

The A1g distortion plays an obvious role in the coopertive distortion. Recall that the octahedralQ1 distortion is auniform expansion or contraction of the octahedron. Whthe positiveQ1 distortion is superimposed with theQ3 dis-tortion, it cancels the inward motion of the oxygen atoms areinforces the outward motion of the oxygen atoms. Givthe proper linear combination,A6Q112)Q3 , two atoms

FIG. 11. The amplitude of theQ1 octahedral distortion is plottedas a function of theQ3 octahedral mode in LiCuO2. The Q1 octa-hedral distortion is further broken down into contributions from tu1 strain mode and thee1 k50 optical phonon.

TABLE V. Percent volume expansion in going from the undtorted structure to the positive amplitude of the cooperativeQ3distortion.

Material Structure%Volume

change

LiCuO2 Layered 8.4LiCuO2 Spinel 6.9LiMnO2 Layered 2.9LiMnO2 Spinel 2.4LiNiO2 Layered 0.6LiNiO2 Spinel 0.1

22430

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move outwards and four atoms remain stationary. Forample, if theA1g distortion were removed from the distortestructure of LiCuO2, the short bonds would measure 1.65as opposed to 1.87 Å. The results imply that theA1g modesare associated with elastic anharmonicity through highorder elastic coupling betweenA1g andEg ~i.e., terms that goasA1gEg

2 andA1g2 Eg). As noted above,Q1 does not operate

until Q3 is approximately 0.2 Å, whereupon it becomes eergetically favorable to expend energy to expand the octadra in order to reduce the elastic energy associated withcontracted bonds from theQ3 distortion. This makes sensphysically considering that a typical interatomic potentwill increase rapidly after the bond length is contracted snificantly. This hypothesis is also supported by the fact tthe positiveQ1 distortion is induced by both the positive annegative mode of theQ3 distortion. TheQ1 distortion is veryimportant in LiCuO2, effectively softeningQ3 and allowingthe system to distort to a much higher degree.

Previous studies have considered the role of theA1g dis-tortion. The Hamiltonian can have linear terms inA1g-typedistortions as theA1g irreducible representation is containewithin the symmetrical product of theEg representation.

4

However, this will simply shift theeg levels by a constanindependent of the Jahn-Teller distortion becauseA1g is to-tally symmetric. Van Vleck acknowledged this when he fisolved theE^ e problem, and he states that the undistortstate can be considered as equilibrated with respect toQ1 .

20

Liehr and Ballhausen used crystal field theory to perfocalculations for the Jahn-Teller effect in octahedcomplexes.22 They included first-order and higher-order copling to Q1 when expanding the matrix elements of thHamiltonian. A linear term inQ1 is needed in their generaHamiltonian to describe the relative expansion or contractof different transition metals due to the ligand field, indepedent of the Jahn-Teller distortion. In our calculations, thereno need to consider terms linear inA1g because the structurhas already been equilibrated with respect toA1g distortions.Although their model does account for higher-order couplbetweenQ1 andQ3 , they make no mention of the implications of this in their paper.

More recently, Atanasovet al.have considered the role oA1g distortions in partially lithiated spinel LixMnO2 usingdensity functional calculations in clusters.38 Their calcula-tions predict that, given the Mn7O14

2 cluster equilibrated withrespect toA1g type distortions, a negativeQ1 distortion willbe induced by theQ3 distortion, which is exactly opposite towhat we have found in this study. It is not completely clewhy their results are qualitatively different than ours, but this most likely due to approximations introduced in their eletronic structure method. This is supported by the fact tbond lengths predicted by our calculations are more accuas compared to experiment. The authors claim that theviation from experiment may be due to localization beimore pronounced in cluster calculations. Ultimately, the oservation ofQ1 increasing with theQ3 distortion makesmore sense and the same trend is observed in LiCuO2 wherethe effect is much more pronounced.

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In order to better understand the energetics of each pnon and strain mode, it is instructive to calculate the eneof the cooperative distortion while considering theEg andA1g strain and phonon modes separately~see Fig. 12!. Sepa-rate calculations were performed for the energy as a funcof the u3 strain mode ande3 optical mode, clearly revealingthe energetic importance of theA1g modes. As shown in Fig12 for spinel LiMnO2, allowing only theu3 strain modereproduces the steepest descent data extremely well fortortions less thanQ350.3 Å. Thee3 optical mode is muchless energetically favorable than theu3 strain mode. In thisstudy, it was found that both of these modes have the svibronic coupling, so the observed differences must beto differences in the elastic terms~see Sec. IV C!. This de-composition explains why theEg strain mode dominates thcooperative distortion. In order to clearly illustrate the effeof the A1g mode, a calculation can be performed as a fution of theu3 strain mode, while allowing theu1 strain modeto relax at every step. The presence of theu1 strain modecauses the minimum to increase by 0.15 Å and the stabiltion energy to increase by roughly 50 meV. When theu1strain mode is allowed to operate, the results are much clto the steepest descent results with the final differences bdue to the absence of thee3 and e1 distortions. The sameanalysis can be performed for LiCuO2 ~see Fig. 13!. As forLiMnO2, the cooperative distortion consisting solely ofu3lattice strains is only accurate for small distortions. Thesence of theA1g modes greatly reduces the size of the dtortion. Allowing the u1 strain mode to operate greatly reduces the energy of the distortion. However, there is stisignificant departure from the steepest descent curve at ldistortions indicating that thee3 and e1 optical modes aresignificant in representing the energetics for large distortio

C. Vibronic coupling of the strain modesand the optical modes

In this section, we shall discuss specific calculations texplain the differences between the distortions in the diff

FIG. 12. Energy as a function of the octahedralQ3 distortion inspinel LiMnO2. Four different cases are considered. The steedescent~SD! calculation represents the true trajectory of the disttion ~same as calculation in Fig. 6!. Theu3 strain and thee3 opticalcurves show the energy as a function of the respective mode.u31u1 strain curve is obtained by taking theu3 strain curve andallowing theu1 strain mode to relax at every point along the pa~this is equivalent to relaxing the volume at every point!.

22430

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ent materials, the source of anisotropy in theEg strain andEg k50 optical phonon potentials, the differences in straand optical modes, and determine the vibronic couplingthe Eg strain andEg k50 optical phonon. Separate calcultions are performed for the energy as a function of theu3strain distortion and the energy as a function of thee3 opticaldistortion, while not allowing anyA1g distortions to operateWe shall only consider the spinel structure for these calcutions because theEg irreducible representation appears ononce in the representation of the strains, whereas in theered structure it appears twice. This simplifies the analyand we expect the same conclusions to hold in both sttures.

First, we examine the splitting of theeg* bands. We cal-culate the change in electronic energy by evaluating equa~4!, taking the limits of integration from the lowesteg* stateto the center of gravity of theeg* bands, as a function of thedistortion.

E5E «•D~«!d«. ~4!Because the band energies cannot be referenced absofor different distortions, we must assume that the centergravity of theeg* states remains unchanged. Therefore, osplitting of theeg* bands is taken into account when calclating the change in electronic energy. As indicated in SIII B, the change in electronic energy for a cooperative dtortion in a solid with finite bandwidth is expected to vanonlinearly for small distortions. The change in electronenergy is plotted as a function of the positiveu3 strain dis-tortion for all three materials~see Fig. 14!. As before, theamplitude of the strain mode is multiplied by a factora(2&)21 such that the amplitude of the mode is equal toinduced amplitude of theQ3 octahedral mode. This will al-low us to directly compare the strain modes and optimodes below. Once thet2g bands and theeg* bands began tooverlap, no larger distortions were considered. As shownthree curves begin nonlinearly and transition to appromately linear dependence. The nonlinear region can be arately represented by a quadratic term, with the rms ebeing less than 3 meV for all three cases. The first-orvibronic coupling of theEg strain mode can be taken as thslope of the linear portion of the curve. The vibronic co

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FIG. 13. Energy as a function of the octahedralQ3 distortion inspinel LiCuO2. See Fig. 12 for further description.

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plings of the Eg strain mode for LiMnO2, LiNiO2, andLiCuO2 are21.70,21.16, and20.84 eV/Å, respectively.

Up until this point, only the vibronic coupling of theu3strain mode has been considered. It is also useful to calcuthe change in electronic energy as a function of thee3 opticalmode. This was performed for LiMnO2 and the result isnearly identical to the case of theu3 strain mode, with allpoints in the nonlinear region being practically equivaleand all points in the linear region being within 15 meVeach other on a scale of 700 meV.This indicates that thevibronic coupling of the u3 strain mode and e3 optical modeare practically the same. This suggests that, to a high degrof accuracy, the change in electronic energy is only a fution of the octahedralQ3 distortion. It is also instructive tocalculate the change in electronic energy for a cooperadistortion that is not ferrodistortive. This is performed binducing a distortion in which three of the four metal-oxygoctahedra, in the primitive spinel unit cell, have octahedQ3 distortions in different directions but identical in magntude. This distortion is some linear combination ofk50 op-tical phonons, and we characterize the amplitude ofmode by the magnitude of theQ3 octahedral distortion. Theresult is very similar to theu3 strain mode, with the nonlin-ear region being practically equivalent and the linear reghaving roughly the same slope but shifted by approxima230 meV. This further illustrates that the change of eletronic energy in the cooperative distortion can be appromately considered to be a function only of theQ3 octahedraldistortion. If the electrons were completely localized, thwould be expected, but with the absence of the initial nlinear behavior. However, as shown in Fig. 7 LiMnO2 has abandwidth of roughly 1 eV. The above calculations weonly performed for LiMnO2; however, the results are probably similar in LiNiO2. Also, these arguments probably hofor small distortions in LiCuO2.

The curves shown in Fig. 6 are now better understoLiMnO2 has a much larger distortion than LiNiO2 becausethe vibronic coupling in LiMnO2 is much larger than forLiNiO2. Obviously, differences in elastic terms will alscontribute to the differences between the two materials.terestingly, LiCuO2 has the smallest vibronic coupling, buthe largest distortion and stabilization energy. This mustdue to the elastic terms being much smaller in LiCuO2 thanin LiMnO2 and LiNiO2. This reasoning is supported by th

FIG. 14. Electronic energy change vsu3 strain distortion. Thechange in electronic energy is calculated from the splitting of theeg*peaks in the DOS.

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fact that there are two electrons in the antibondingeg* bandsin LiCuO2 whereas there is only one in the other two marials. It is well known that filling an antibonding band tendto decrease the elastic constants.

In order to determine the dominant source of anisotroin the Eg strain potential, we will use two different calculations. First, we will use the same approach as above byculating the change in electronic energy for positive anegative amplitudes of theu3 strain distortions for LiMnO2.The results for the positive and the negative amplitudevery similar, with the negative amplitude providing a slightlarger gain in energy. However, the calculation of the toenergy indicates that the positive amplitude has a lower tenergy~see Fig. 12!, which must be caused by the elastcontribution. The anisotropic terms can be more accuraquantified as follows. Consider the function defined in eqtion ~5!:

f ~r!5 12 @E~r,0!2E~r,p!#. ~5!

This will eliminate all symmetric terms from the Hamitonian, leaving only the second-order vibronic coupling athe third-order elastic term. This function can then be fit tosecond- and third-order polynomial, revealing each term.the u3 mode, this gives a cubic term of22165 eV/Å

3 and aquadratic term of 154 eV/Å2. Considering an amplitude o0.5 Å, the third-order elastic term is roughly seven timlarger than the second-order vibronic term. The same fit wperformed for thee3 mode yielding a cubic term of21888eV/Å3 and a quadratic term of 210 eV/Å2, which behaves inthe same fashion. Additionally, the fit predicts the secoorder vibronic term to stabilize the negative amplitude of tu3 distortion and the third-order elastic term to stabilize tpositive amplitude of theu3 distortion, which was also seeabove when calculating the electronic energy. Therefore,can conclude that, for these materials, the stability ofpositive Q3 distortion is due to the dominant anharmonelastic coupling. The same trends were found for LiNiO2 andLiCuO2.

D. Strain anisotropy induced by antiferromagnetic ordering

Thus far, the present study has only addressed structwith ferromagnetic spin polarization. As stated in Sec.the Eg strain modes will distort the metal lattice. In the udistorted structure for both layered and spinel, a given trsition metal will have six metal nearest neighbors. Theu3strain mode will break the set of six nearest-neighbors intset of four and two. In order to illustrate this, the Mn-Mbond lengths for the six nearest neighbors are plotted vethe magnitude of the octahedralQ3 distortion for the steepesdescent calculation of the cooperative distortion in spiLiMnO2 ~see Fig. 15!. The positiveu3 distortion, which isdefined inZ orientation~Table II!, creates four long metalmetal bonds~bonds 0–2, 0–3, 0–4, and 0–5 in Fig. 16! andtwo short metal-metal bonds~bonds 0–1 and 0–6!, while thenegative distortion creates two long metal-metal bon~bonds 0–1 and 0–6! and four short metal-metal bond~bonds 0–2, 0–3, 0–4, and 0–5!. This is opposite to how thecooperative distortion effects the metal-ligand bonds. T

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magnitude of the slope of the two-bond curve is roughlytimes that of the four-bond curve. The exchange interacwill be a function of the metal bond lengths, so the magneenergy will clearly change as a function of the distortion.

The antiferromagnetic state introduces an additional coplexity as compared to the ferromagnetic case. The magnground state of both spinel and layered LiMnO2 is antiferro-magnetic, and both systems are frustrated. As shown in16, a given metal atom can have four antiferromagnebonds~bonds 0–2, 0–3, 0–4, and 0–5! and two ferromag-netic bonds~bonds 0–1 and 0–6!. The antiferromagnetic ordering reduces the symmetry, causing theZ orientation of theu3 strain to be inequivalent to theX andY orientations of theu3 strain. For both the positive and negative cooperativetortions, it is found that the lower-energy states correspto a deformation in which the short metal bonds are antiromagnetic. The positiveu3 distortion creates one set oshort bonds, so for the particular antiferromagnetic orderin Fig. 16 theX andY orientations ofu3 will result in shortantiferromagnetic bonds~bonds 0–2, 0–5 and 0–3, 0–4, rspectively! and theZ orientation ofu3 will result in shortferromagnetic bonds~bonds 0–1 and 0–6!. The negativeu3distortion creates four short bonds, so theZ orientation ofu3will result in four short antiferromagnetic bonds~bonds 0–2,0–3, 0–4, and 0–5! while theX andY orientations ofu3 will

FIG. 15. Percent change in Mn-Mn bonds in spinel LiMnO2 forthe cooperativeQ3 distortion ~steepest-descent calculation!. Asshown, a positive cooperativeQ3 distortion will create four longMn-Mn bonds and two short Mn-Mn bonds due to theu3 straincomponent of the distortion, while the opposite occurs for the netive cooperativeQ3 distortion. Note that this is the opposite of hothe cooperativeQ3 distortion effects the Mn-O bond lengths.

FIG. 16. The figure shows the frustrated antiferromagneticdering present in the spinel structure. EachM atom has four anti-ferromagneticM-M bonds and two ferromagneticM-M bonds.

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only create two short antiferromagnetic bonds~bonds 0–3,0–4 and 0–2, 0–5, respectively! and two short ferromagneticbonds~bonds 0–1, 0–6 and 0–1, 0–6!. The energy was cal-culated for the positive and negative amplitudes of the coerativeQ3 distortion for theX, Y, andZ orientations of thecooperative distortion in spinel LiMnO2 ~see Table VI!. Theantiferromagnetic ordering creates approximately a 70 mdifference in energy for the different orientations of the potive cooperative distortion and a 90 meV difference for tdifferent orientations of the negative cooperative distortioThe same trends were observed in layered LiMnO2. There-fore, there is a significant coupling between the magneordering and theEg strain modes, which makes certain orentations of the cooperative distortions energetically favable.

V. CONCLUSIONS

Density functional calculations, in the generalized graent approximation, have been used to elucidate severalportant aspects of the cooperative Jahn-Teller distortionoctahedrally coordinated transition-metal ions. Our calcutions accurately predict the structural characteristics ofcooperative Jahn-Teller distortion in spinel and layerLiNiO2, LiMnO2, and LiCuO2, as compared to experimenThe predicted metal-oxygen bond lengths were in reasonagreement with experiment, having errors ranging fro1–7 %. The stabilization energy of the ferromagnetic cooerativeQ3 distortion was found to be211, 2215, and2341meV in layered LiNiO2, LiMnO2, and LiCuO2, respectively,and24, 2211, and2290 meV in spinel LiNiO2, LiMnO2,and LiCuO2, respectively~see Table IV!. Additionally, itwas found that the Jahn-Teller distortion will induce a sptransition in LiCuO2, causing the initial high-spin state ttransition to low spin~see Fig. 9!.

The total cooperative distortion is decomposed into

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TABLE VI. Relative energies of antiferromagnetic LiMnO2 indifferent orientations of the cooperativeQ3 distortion. All energiesare referenced to the undistorted, antiferromagnetic calculationshown, the antiferromagnetism breaks the symmetry and causepositive X and Y orientations of the cooperative distortion to blowest in energy. The lowest energy orientations are those whproduce a maximal number of short antiferromagnetic bonds.

Orientation Q3 ~Å! Energy~meV! Metal-metal bonds

2 short AF bondsX, Y 0.52 2263 2 long AF bonds

2 long F bonds

Z 0.53 2194 2 shortF bonds4 long AF bonds

2 short AF bonds2X,2Y 0.24 272 2 shortF bonds

2 long AF bonds

2Z 0.24 2161 4 short AF bonds2 long F bonds

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symmetrized lattice strains and the symmetrizedk50 opticalphonons, and it is found that only theEg and A1g strainmodes andEg and A1g k50 optical-phonon modes particpate in the cooperative distortion. TheEg strain modes andEg k50 optical phonons both impart octahedralEg distor-tions. It is found that theu3 strain mode is responsible for thmajority of the octahedralQ3 distortion for small and inter-mediateQ3 distortions while the contributions of thee3 k50 optical phonon become more important for large disttions. TheA1g-type modes cause all the metal-oxygen bolengths to increase, canceling the contraction of the meoxygen bond lengths induced by theEg distortions. TheA1g-type modes begin to operate at roughlyQ350.2 Å,whereupon it becomes energetically favorable to expendergy to expand the octahedron in order to reduce the enassociated with excessively short bond lengths~see Fig. 11!.This effect is a higher-order elastic interaction betweenA1g andEg modes. The importance of this effect dependsthe magnitude of theQ3 distortion, being negligible inLiNiO2, intermediate in LiMnO2, and extremely importanin LiCuO2.

The vibronic coupling of theu3 strain mode was found tobe 21.70, 21.16, and20.84 eV/Å for spinel LiMnO2,LiNiO2, and LiCuO2, respectively, by examining the spliting of theEg peaks in the DOS~see Fig. 14!. Additionally,it was shown that the electronic energy varies quadraticwith the distortion while theEg bands are overlapping antransits to a linear relationship thereafter. It was also fou

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that the change in electronic energy calculated from the sting of the DOS in LiMnO2, as a function of the distortionwas essentially identical for theu3 strain mode and thee3k50 optical phonon mode, and very similar for a nonferrdistortive distortion. This indicates that the vibronic coplings for theu3 strain mode and thee3 k50 optical phononmode are practically the same, and that the change in etronic energy is approximately a function only of the magtude of theQ3 octahedral distortion for a cooperative distotion in LiMnO2. Also, it was shown that the stronanisotropy favoring the positive amplitude of theu3 ande3modes is elastic in origin, not vibronic.

Finally, it is found that symmetry breaking of the antiferomagnetically ordered state in LiMnO2 can create roughly70 meV differences in the stabilization energy for differeorientations of the cooperative distortion~i.e., X, Y, or Z!.This derives from the fact that the cooperative distortion snificantly changes the metal-metal bond lengths, thus chaing the exchange interaction.

ACKNOWLEDGMENTS

We would like to thank Professor Michael Kaplan fohelpful discussions. Also, we would like to thank Eric Wfor carefully reviewing the entire manuscript. Finally, wwould like to thank the Department of Energy for supportithis work ~Contract No. DE-F602-96ER45571!

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