Homopolar bond formation in ZnV 2 O 4 close to a metal-insulator transition
V. Pardo, S. Blanco-Canosa, F. Rivadulla, D.I. Khomskii, D. Baldomir, Hua Wu, J. Rivas
aa r X i v : . [ c ond - m a t . s t r- e l ] M a y Homopolar bond formation in ZnV O close to a metal-insulator transition V. Pardo,
1, 2, ∗ S. Blanco-Canosa, F. Rivadulla, D.I. Khomskii, D. Baldomir,
1, 2
Hua Wu, and J. Rivas Departamento de F´ısica Aplicada, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain Instituto de Investigaciones Tecnol´ogicas, Universidad de Santiago de Compostela, E-15782, Santiago de Compostela, Spain Departamento de Qu´ımica-F´ısica, Universidad de Santiago de Compostela, E-15782 Santiago de Compostela, Spain II. Physikalisches Institut, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, D-50937 K¨oln, Germany
Electronic structure calculations for spinel vanadate ZnV O show that partial electronic delocal-ization in this system leads to structural instabilities. These are a consequence of the proximity tothe itinerant-electron boundary, not being related to orbital ordering. We discuss how this mecha-nism naturally couples charge and lattice degrees of freedom in magnetic insulators close to such acrossover. For the case of ZnV O , this leads to the formation of V-V dimers along the [011] and[101] directions that readily accounts for the intriguing magnetic structure of ZnV O . PACS numbers: 71.27.+a;7.1.30.+h;75.25.+z
The transition between an antiferromagnetic (AF) in-sulator and a paramagnetic metal is associated to someof the most intriguing experimental observations in solidstate physics. Unconventional forms of superconductiv-ity, electronic phase separation, and a variety of non-Fermi liquid behavior are well-known examples. Al-though not completely understood, it is however becom-ing clear that a strong coupling between charge, spin,orbital and lattice degrees of freedom is an essential in-gredient to explain many of these phenomena. So, it isimportant to define the properties and conditions underwhich this coupling among different degrees of freedomtakes place. In a recent paper, Blanco-Canosa et al. [1].proposed that in single-valence systems, the transitionbetween localized and itinerant electron behavior takesplace through a transitional phase in which partial elec-tronic delocalization in the form of cation-clusters in anionic matrix occurs. In this situation, strong lattice in-stabilities can be anticipated, due to a purely electronicmechanism [2].In this paper we explore, using ab initio calculations,how the effect of a small (realistic) U/W ratio naturallycouples charge, spin and lattice degrees of freedom in amagnetic insulator close to the itinerant-electron limit.Particularly, we will discuss the case of ZnV O , whosemagnetic and orbital properties are still a matter of de-bate [3, 4, 5, 6]. We will show that, for small values of U,the most stable structure always consists of V-V dimersalong the [011] and [101] directions, that helps to under-stand the intriguing magnetic structure of the material.This is a dramatic example of a strong electron-latticecoupling due to the partial electronic delocalization inthe proximity of the itinerant electron limit.The electronic structure of the V ion (d ) in a tetrag-onally distorted octahedral environment has an orbitallydegenerate configuration. It is not yet clear what or-bital ordering, if any, could account for the experimen-tally found magnetic structure of the material [7]. Severalpictures have appeared recently [3, 4, 5, 8, 9], based ondifferent considerations, that predict various possible or- bital orderings. An “antiferro-orbital” picture has beenproposed [3], with the full occupation of a d xy orbital ateach site and with an alternation of d xz and d yz orbitalsalong the c-axis. Also, a “ferro-orbital” ordering of com-plex orbitals (d xz ± i d yz ) [4] and an “orbital-Peierls”ordering[8] with the orbitals in a d yz -d yz -d xz -d xz pat-tern along the tetragonally compressed c-axis were sug-gested. Considering a fully ab initio, all-electron picture,including spin-orbit effects, the most stable solution isan alternating orbital ordering along the [011] and [101]directions of the orbitals with an unquenched orbital an-gular momentum d xz ± i d yz [5], such that the orbitalangular momentum in every site is antiparallel to the spinmoment. However, most of these works rely on the as-sumption that U/W is large, a purely localized picture,which seems to contradict recent experimental findings[1]. In many of these works, the lattice symmetry foundin the existing experiments (space group I4 /amd) [10]was imposed as a rigid constraint. However, in general,electronic structure may have lower symmetry. If thiswere the case, it would be difficult to detect if corre-sponding distortions are weak, as in the case of ZnV O (t g electrons involved). To take into account this pos-sibility, we carried out ab initio electronic structure cal-culations for ZnV O , valid for intermediate values ofthe Coulomb (Hubbard) repulsion U, and allowing forthe lower symmetry, to find the optimal electronic andlattice structure.We present here full-potential, all-electron, electronicstructure calculations based on the density functionaltheory (DFT), utilizing the APW+lo method [11], per-formed using the WIEN2k software [12]. For the struc-ture optimization, we used the GGA (generalized gra-dient approximation) in the Perdew-Burke-Ernzerhoff(PBE) scheme [13] as an exchange-correlation functional.The geometry optimization was carried out minimizingthe forces in the atoms and the total energy of the system,relaxing it from an initial trial state: unequal (dimerized)V-V distances along the [011] and [101] directions. Forthe electronic structure calculations we included strong FIG. 1: (Color online)Schematic representation of the “dimer-ized” structure resulting from our ab initio, all-electron cal-culations for realistic values of U in ZnV O . Bold (thin) linerepresents short (long) in-chain bonds. The magnetic struc-ture is indicated by arrows. correlations effects by means of the LDA+U scheme [14],where correlation effects are controlled by an effective U ( U eff = U − J ), U being the on-site Coulomb repulsionand J the Hund’s rule exchange constant. Spin-orbiteffects have been introduced as a second variation usingthe scalar relativistic approximation [15]. All our calcula-tions were fully converged with respect to the parametersused.ZnV O crystallizes in a distorted cubic spinel struc-ture, where the V atoms form a pyrochlore lattice ofcorner-sharing tetrahedra. Because of the geometry ofthe pyrochlore lattice, the AF interactions between theV atoms are highly frustrated, leading to small T N /Θ CW ratios.The material undergoes a magnetic transition anda structural transition to a low-temperature tetragonalphase (c/a <
1) below 50 K [10]. The lattice formed bythe V atoms can be described as built up by three V-Vchains running along the [110], [011] and [101] directions(we use below the cubic setting). The magnetic struc-ture, found by neutron diffraction [7], is AF along the[110] direction (within the ab plane), but along the [101]and [011] (off-plane) directions the spin ordering alter-nates, the V moments order ↑↑↓↓↑↑ (see Fig. 1).In the rest of the paper, for the sake of clarity,the structure derived from the tetragonal distortion de-scribed above will be called the “standard” structure,and the relaxed structure (see below) will be called the“dimerized” structure, to signal the formation of V-Vdimers along the chains.As we said before, the anomalous variation found in the
120 180 240 30010 CdV O E = 1.58 (2) eVMgV O E = 0.54 (4) eV MnV O E = 1.46 (1) eV
ZnV O E = 0.55 (2) eV ( c m ) Temperature (K)
FIG. 2: (Color online)Temperature dependence of the resis-tivity in the A V O series. Small differences in the resistiv-ity for MgV O and ZnV O could be due to the intergrainboundary scattering in the ceramic pellets (their room tem-perature resistivity differs only by ≃ pressure dependence of T N along the series A V O [1],has been interpreted as the consequence of a variation ofU/W, which first becomes progressively reduced by pres-sure, and then collapses close to the itinerant limit. Ac-cording to this, even though MgV O and ZnV O arestill semiconducting, a partial electronic delocalizationalong the V-V bonds leading to the formation of short-range cation clusters and to a lattice instability can beanticipated. In Fig. 2 we present the experimental re-sistivity curves for different vanadates in which the di-valent cation at the tetrahedral site has been used totune the V-V distance. The results show how the acti-vation energy decreases as the V-V distance does, andhence the metal-insulator transition is approached fromthe insulating side. On approaching this point, U/Wis reduced progressively [1] and hence partial delocaliza-tion can be anticipated along the V-V bonds. In or-der to check the possibility of a lattice instability dueto this electronic effect, we performed a structural op-timization, using the experimentally observed magneticstructure ( ↑↓↑↓↑↓ spins in [110] chains, ↑↑↓↓↑↑ in [101]and [011] directions). As a starting point, we selectedan artificially deformed structure that would give rise toV-V dimers (distances along the [101] and [011] chainsget short-long-short-long) and let the system relax untilforces on the atoms are smaller than 4 mRy/a.u. (smallenough to consider the system is relaxed). The systemrelaxes to a structure away from the “standard” one,forming chains with an alternation of short-long V-V dis-tances (Fig. 1). The same relaxed structure is obtainedstarting from different articially deformed structures, in-cluding the one with the chains having short-intermedite-long-intermediate V-V distances along the [101] and [011]chains (the structure proposed in Ref. 8). This shows wehave found a stable (its energy is lower than the “stan-dard” one by 30 meV/V for U eff = 0) and reproduciblestructure.The reason why one needs to displace the V atomsfrom their “standard” positions is because the latter is alocal minimum for the system, as our calculations show(very small forces on the atoms). However, if we movethe system slightly out of that structural local minimum,a different local minimum can be found, the one withdimerization shown in Fig. 1 with a smaller total energyand hence more stable.For carrying out this computational experiment, oneneeds to reduce the symmetry of the compound from the“standard” I4 /amd to the P4 U eff . For this we needed to find the energyminimum for the “standard” structure. The V :d ionsin the tetragonal environment have an orbitally degen-erate configuration, with the d xy orbital fully occupiedand the additional electron in a d xz -d yz doublet. Wetried different possible orbital orderings, and the energyminimum, for the “standard” structure, once spin-orbitcoupling is included, is found for the second electron inan l z = ± eff (see Fig. 3). Forsmall values of U eff (below about 2.2 eV) the “dimer-ized” structure is more stable than the “standard” one.Such small values of U eff would be consistent withour experimental findings of the system being close tothe itinerant electron limit. It is remarkable that inthe “dimerized” structure, even a GGA calculation isenough to open a tiny gap in the density of states( ∼ xz (d yz ) electrons to fill up a bonding Standard(cid:13) Dimerized(cid:13) E ne r g y ( m e V / V ) (cid:13) U(cid:13) eff(cid:13) (eV)(cid:13)
FIG. 3: (Color online)Total energies (in meV/V) of the twostructures as a function of U eff . For small values of U eff the“dimerized” structure proposed in this work is more stable.Note that, for any U eff , we take as zero the energy of themost stable structure. molecular orbital along the [101] ([011]) direction (seeFig. 1). Then the bonding-antibonding splitting opensthe gap and stabilizes this “dimerized” structure, whichdoes not need occurrence of an orbital ordering (see be-low). This is contrary to the case of isostructural spinelMgTi O , where the structure along the off-plane chainsis short/intermediate/long/intermediate, where no bandgap opens within the GGA scheme [16].For U eff = 1.7 eV, the “dimerized” structure is morestable than the “standard” one by 5 meV/V. For a moredelocalized case (smaller values of U eff compared to thebandwidth W= 2 eV), the “dimerized” structure is al-ways the lowest-energy solution. However, if we increasethe electron-electron on-site interaction U, i.e. drivingthe material towards the strongly localized limit, themost stable situation is the “standard” structure. So,from our calculations we conclude that as U/W is re-duced on approaching the itinerant behavior, a strongcoupling between charge and lattice degrees of freedomtakes place, determining the low-temperature propertiesof the system. The “dimerized” structure is consistentwith the experimental evidences of lattice instabilities,and the fact that it is stable for small values of U eff con-firms our hypotheses about its origin. It is remarkablethat no special constraints have been introduced in thecalculations, so the specific results obtained for ZnV O may be a manifestation of a general trend which mayexist also in other systems close to a localized-itinerantcrossover [17].In Table I we can see the V-V distances in the “dimer-ized” structure and we can compare them with the othermembers of the series. The shortest V-V distance is be- TABLE I: V-V distances (in ˚A) along the different directionsin the “standard” (experimental values from Ref. 7) and the“dimerized” (our calculations) structure of ZnV O and alsofor the other members of the series AV O in the cubic phase(our experimental results).in-plane off-plane short off-plane long“standard” 2.98 2.97 2.97“dimerized” 2.98 2.92 3.01CdV O O O low the critical distance for electron itineracy ( ∼ -V bondacross a shared octahedral edge in an oxide [18]. This is inagreement with the experimental prediction of the forma-tion of V-V molecular orbitals close to a metal-insulatortransition in ZnV O [1], and is similar to the case ofMgTi O , where a tetramerization of the Ti chains hasbeen observed[16].As mentioned above, for the “standard” structure it ispossible to stabilize different orbital orderings. Amongthem, the most stable one has an unquenched orbital an-gular momentum of about 0.7 µ B per V site antiparallelto the spin moment at each site. However, if we an-alyze the electronic structure of the “dimerized” struc-ture, such an orbital ordering is not found. In fact, wedo not observe any orbital ordering. The occupations ofthe levels d xz and d yz are almost identical but the or-bital angular momenta are fairly small (about 0.1 µ B ).If we use a basis set with the real combination of orbitals:d xz ± d yz , they are also equally populated. Hence, thereis no trace of orbital ordering left once dimerization ofthe V chains occurs. The results we present here do notdepend on a particular orbital ordering. The dimeriza-tion of the structure is caused by a spin-lattice couplingwithout orbital ordering being involved, the only neces-sary ingredient is the collapse of U/W in the vicinity ofa metal-insulator transition.In all our calculations, we have assumed that the mag-netic structure is the one obtained experimentally [7].But we have also carried out the calculations for differ-ent magnetic orderings to try to discern the values of thedifferent exchange constants, assuming there exist an in-plane coupling (J in ) and an out-of-plane coupling (J out ).We have used the total energies obtained for various mag-netic couplings, with U eff = 1.7 eV, fitting them to aHeisenberg model (H= - P i,j J ij S i S j ) and estimating themagnetic coupling. Different results are obtained in the“dimerized” and “standard” structures (the latter withan orbital ordering with l z = ± out can be subdivided in the coupling forthe short J s and for the long bonds, J l . The in-plane cou-pling remains constant and is strongly AF: J in = -16 meVfor both structures, mainly due to the singly occupied d xy orbital. Changes occur in the out-of-plane coupling:J out = -12 meV for the “standard” structure (highly frus-trated effective AF coupling). For the “dimerized” struc-ture, J s = 10 meV and J l = -3 meV. The exchange in shortbonds is FM, and AF in the long ones. This dimerizedstructure removes the magnetic frustration and explainsthe experimentally observed off-plane ↑↑↓↓↑↑ magneticstructure (see Fig. 1). Our work resolves the difficultyin obtaining the correct magnetic structure met in otherapproaches [3, 4, 9].The structure we have obtained is only stable in a limitclose to itineracy, when U is comparable to W , and henceour conclusions can be applied to ZnV O and not tomore localized members of the series, like MnV O andCdV O , because they are far from the metal-insulatortransition. It is the closeness to the transition what leadsto the formation of V-V dimers giving rise to the appear-ance of molecular orbitals with the subsequent chargedelocalization in a dimer in a material that is still semi-conducting.Summarizing, ab initio calculations show that ho-mopolar V-V bonding occurs in some members of theA V O series. The appearance of this effect is deter-mined by a considerable reduction of U/W, as it has beenproposed to occur close to the itinerant-electron limit.Our results prove that charge and lattice degrees of free-dom couple strongly in magnetic insulators that approachthe itinerant-electron limit. A possible physical pictureexplaining why there occurs dimerization in the ↑↑↓↓↑↑ chains close to the itinerant regime is that in this case theexchange interaction resembles double exchange. Short-ening of ↑↑ bonds leads to a larger hopping and to a gainin double exchange energy, stabilizing this spin order-ing. In contrast, reduced hopping in longer V-V bondsweakens this tendency, allowing for ↑↓ ordering in suchbonds. The unusual properties of many localized-electronsystems that are close to the itinerant crossover shouldbe revisited on the light of the results presented in thiswork.The authors thank the CESGA (Centro de Supercom-putacion de Galicia) for the computing facilities and theMinisterio de Educaci´on y Ciencia (MEC) for the finan-cial support through the projects MAT2006/10027 andHA2006-0119 and also the Xunta de Galicia through theproject PXIB20919PR. F.R. also acknowledges MEC forsupport under program Ram´on y Cajal. The work ofD.Kh. and H.W. was supported by DFG via the projectSFB 608, and by the European Project COMEPHS. ∗ Electronic address: [email protected][1] S. Blanco-Canosa, F. Rivadulla, V. Pardo, D. Baldomir,J. S. Zhou, M. Garc´ıa-Hern´andez, M. A. L´opez-Quintela,J. Rivas, and J. B. Goodenough, Phys. Rev. Lett. ,187201 (2007). [2] Y. Horibe, M. Shingu, K. 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