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The metal-insulator transition of VO 2 revisited J.-P. Pouget Laboratoire de Physique des Solides, CNRS-UMR 8502, Université Paris-sud 91405 Orsay « Correlated electronic states in low dimensions » Orsay 16 et 17 juin 2008 Conférence en lhonneur de Pascal Lederer

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outline Electronic structure of metallic VO 2 Insulating ground states Role of the lattice in the metal-insulator transition of VO 2 General phase diagram of VO 2 and its substituants

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VO 2 : 1 st order metal-insulator transition at 340K Discovered nearly 50 years ago still the object of controversy! * *in fact the insulating ground state of VO 2 is non magnetic

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Bad metal in metallic phase: ρ ~T very short mean free path: ~V-V distance P.B. Allen et al PRB 48, 4359 (1993) metal insulator

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Metallic rutile phase cRcR ABAB (CFC) compact packing of hexagonal planes of oxygen atoms V located in one octahedral cavity out of two two sets of identical chains of VO 6 octahedra running along c R (related by 4 2 screw axis symmetry) A B

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e g: t 2g V-O σ* bonding bonding between V in the (1,1,0) plane (direct V-V bonding along c R :1D band?) bonding between V in the (1,-1,0) plane in the (0,0,1) plane V 3d orbitals in the xyz octahedral coordinate frame V-O π* bonding orbital located in the xy basis of the octahedron orbitals « perpendicular » to the triangular faces of the octaedron

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well splitted t 2g and e g bands V. Eyert Ann. Phys. (Leipzig) 11, 650 (2002) 3d yz and 3d xz : E g or π* bands of Goodenough 3d x²-y² : a 1g or t // (1D) band of Goodenough Is it relevant to the physics of metallic VO 2 ? LDA: 1d electron of the V 4+ fills the 3 t 2g bands t 2g egeg

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Electronic structure of metallic VO 2 LDA Single site DMFT Eg a 1g t 2g levels bandwidth~2eV: weakly reduced in DMFT calculations U LHB UHB Biermann et al PRL 94, 026404 (2005) Hubbard bands on both E g (π*) and a 1g (d // ) states no specificity of d // band!

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Fractional occupancy of t 2g orbitals orbital/occupancy LDA* single site DMFT* EFG measurements** x²-y² (d // ) f 1 0.36 0.42 0.41 yz (π*) f 2 0.32 0.29 0.26-0.28 xz (π*) f 3 0.32 0.29 0.33-0.31 *Biermann et al PRL 94, 026404 (2005) ** JPP thesis (1974): 51 V EFG measurements between 70°C and 320°C assuming that only the on site d electron contributes to the EFG: V XX = (2/7)e (1-3f 2 ) V YY = (2/7)e (1-3f 3 ) V ZZ = (2/7)e (1-3f 1 )

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VO 2 : a correlated metal? Total spin susceptiblity: N eff (E F )~10 states /eV, spin direction J.P. Pouget & H. Launois, Journal de Physique 37, C4-49 (1976) Density of state at E F : N(E F )~1.3*, 1.5**, 2*** state/eV, spin direction *LDA: Eyert Ann Phys. (Leipzig) 11, 650 (2002), **LDA: Korotin et al cond-mat/0301347 ***LDA and DMFT: Biermann et al PRL 94, 026404 (2005) Enhancement factor of χ Pauli : 5-8

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Sizeable charge fluctuations in the metallic state DMFT: quasiparticle band + lower (LHB) and upper (UHB) Hubbard bands LHB observed in photoemission spectra VO 2 close to a Mott-Hubbard transition? LHB Koethe et al PRL 97, 116402 (2006)

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Mott Hubbard transition for x increasing in Nb substitued VO 2 : V 1-X Nb X O 2 ? Nb isoelectronic of V but of larger size lattice parameters of the rutile phase strongly increase with x Very large increase of the spin susceptibility with x NMR in the metallic state show that this increase is homogeneous (no local effects) for x

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Insulating phase: monoclinic M 1 tilted V-V pair V leaves the center of the octahedron: 1- V shifts towards a triangular face of the octahedron xz et yz orbitals (π* band) shift to higher energy 2- V pairing along c R : x²-y² levels split into bonding and anti-bonding states stabilization of the x²-y² bonding level with respect to π* levels Short V-O distance

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Driving force of the metal-insulator transition? The 1 st order metal- insulator transition induces a very large electronic redistribution between the t 2g orbitals Insulating non magnetic V-V paired M 1 ground state stabilized by: - a Peierls instability in the d // band ? - Mott-Hubbard charge localization effects? To differentiate more clearly these two processes let us look at alternative insulating phases stabilized in: Cr substitued VO 2 uniaxial stressed VO 2 The x²-y² bonding level of the V 4+ pair is occupied by 2 electrons of opposite spin: magnetic singlet (S=0)

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R-M 1 transition of VO 2 splitted into R-M 2 -T-M 1 transitions V 1-X Cr X O 2 J.P. Pouget et al PRB 10, 1801 (1974) VO 2 stressed along [110] R J.P. Pouget et al PRL 35, 873 (1975)

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M 2 insulating phase Zig-zag V chain along c V-V pair along c (site B) (site A) Zig –zag chains of (Mott-Hubbard) localized d 1 electrons

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In M 2 : Heisenberg chain with exchange interaction 2J~4t²/U~600K~50meV Zig-zag chain bandwidth: 4t~0.9eV (LDA calculation: V. Eyert Ann. Phys. (Leipzig)11, 650 (2002)) U~J/2t²~4eV U value used in DMFT calculations (Biermann et al) Zig-zag V 4+ (S=1/2) Heisenberg chain (site B) χ tot χ spin T M2M2 R T M2M2

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Crossover from M 2 to M 1 via T phase tilt of V pairs (V site A) Dimerization of the Heisenberg chains (V site B) 2J intradimer exchange integral on paired sites B Value of 2J intra (= spin gap) in the M 1 phase? J intra increases with the dimerization

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Energy levels in the M 1 phase Δρ Δρ dimer ΔσΔσ eigenstates of the 2 electrons Hubbard molecule (dimer) Only cluster DMFT is able to account for the opening of a gap Δρ at E F (LDA and single site DMFT fail) Δρ dimer ~2.5-2.8eV >Δρ~0.6eV (Koethe et al PRL 97,116402 (2006)) Δσ? S T S B AB

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Estimation of the spin gap Δσ in M 1 Shift of χ between the T phase of V 1-X Al X O 2 and M 1 phase of VO 2 51 V NMR line width broadening of site B in the T phase of stressed VO 2 :T 1 -1 effect for a singlet –triplet gap Δ: 1/T 1 ~exp-Δ/kT at 300K: (1/T 1 ) 1800bars =2 (1/T 1 ) 900bars If Δ=Δσ-Δs one gets for s=0 (M 1 phase) Δσ=2400K with Δ=0.6 3 K/bar 2J(M 1 )=Δσ >2100K G. Villeneuve et al J. Phys. C: Solid State Phys. 10, 3621 (1977) J.P. Pouget & H. Launois, Journal de Physique 37, C4-49 (1976) M2M2 T

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M1M1 T M2M2 J intra B (°K) + 270K 11.4 V YY A (KHz) The intradimer exchange integral J intra of the dimerized Heisenberg chain (site B) is a linear function of the lattice deformation measured by the 51 V EFG component V YY on site A For V YY = 125KHz (corresponding to V pairing in the M1 phase) one gets : J intra ~1150K or Δσ~2300K Site B Site A

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M 1 ground state Δσ~ 0.2eV<<Δρ is thus caracteristic of an electronic state where strong coulomb repulsions lead to a spin charge separation The M 1 ground state thus differs from a conventional Peierls ground state in a band structure of non interacting electrons where the lattice instability opens equal charge and spin gaps Δρ ~ Δσ

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Electronic parameters of the M 1 Hubbard dimer Spin gap value Δσ ~ 0.2 eV Δσ= [-U+ (U²+16t²) 1/2 ]/2 which leads to: 2t (Δσ Δρ intra ) 1/2 0.7eV 2t amounts to the splitting between bonding and anti-bonding quasiparticle states in DMFT (0.7eV) and cluster DMFT (0.9eV) calculations 2t is nearly twice smaller than the B-AB splitting found in LDA (~1.4eV) U Δρ intra -Δσ ~ 2.5eV (in the M 2 phase U estimated at ~4eV) For U/t ~ 7 double site occupation ~ 6% per dimer nearly no charge fluctuations no LHB seen in photoemission ground state wave function very close to the Heitler-London limit* *wave function expected for a spin-Peierls ground state The ground state of VO 2 is such that Δσ~7J (strong coupling limit) In weak coupling spin-Peierls systems Δσ

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Lattice effects the R to M 1 transformation (as well as R to M 2 or T transformations) involves: - the critical wave vectors q c of the « R » point star:{(1/2,0,1/2), (0,1/2,1/2)} - together, with a 2 components (η 1,η 2 ) irreductible representation for each q C : η i corresponds to the lattice deformation of the M 2 phase: formation of zig-zag V chain (site B) + V-V pairs (site A) the zig-zag displacements located are in the (1,1,0) R / (1,-1,0) R planes for i=1 / 2 M 2 : η 1 0, η 2 = 0 T: η 1 > η 2 0 M 1 : η 1 = η 2 0 The metal-insulator transition of VO 2 corresponds to a lattice instability at a single R point Is it a Peierls instability with formation of a charge density wave driven by the divergence of the electron-hole response function at a q c which leads to good nesting properties of the Fermi surface? Does the lattice dynamics exhibits a soft mode whose critical wave vector q c is connected to the band filling of VO 2 ? Or is there an incipient lattice instability of the rutile structure used to trig the metal-insulator transition?

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Evidences of soft lattice dynamics X-ray diffuse scattering experiments show the presence of {1,1,1} planes of « soft phonons » in rutile phase of (metallic)VO 2 (insulating) TiO 2 (R. Comès, P. Felix and JPP: 35 years old unpublished results) a R */2 c R */2 R critical point of VO 2 P critical point of NbO 2 Γ critical point of TiO 2 (incipient ferroelectricity of symmetry A 2U and 2x degenerate E U ) +(001) planes {u//c R } [001] [110] A2UA2U EUEU {u//[110]} smeared diffuse scattering c* R

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{1,1,1} planar soft phonon modes in VO 2 not related to the band filling (the diffuse scattering exists also in TiO 2 ) 2k F of the d // band does not appear to be a pertinent critical wave vector as expected for a Peierls transition but the incipient (001)-like diffuse lines could be the fingerprint of a 4k F instability (not critical) of fully occupied d // levels instability of VO 2 is triggerred by an incipient lattice instability of the rutile structure which tends to induce a V zig-zag shift* ferroelectric V shift along the [110] / [1-10] direction* (degenerate RI?) accounts for the polarisation of the diffuse scattering correlated V shifts along [111] direction give rise to the observed (111) X-ray diffuse scattering sheets *the zig-zag displacement destabilizes the π* orbitals a further stabilization of d// orbitals occurs via the formation of bonding levels achieved by V pairing between neighbouring [111] « chains » [111] [110] cRcR [1-10]

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phase diagram of substitued VO 2 R M1M1 0.03 x V 1-X M X O 2 0 dT MI /dx -12K/%V 3+ uniaxial stress // [110] R xV 5+ V 3+ M=Cr, Al, Fe VO 2+y VO 2-y F y M=Nb, Mo, W Oxydation of V 4+ Reduction of V 4+ M VO 2 dT MI /dx0 Sublatices AB

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Main features of the general phase diagram Substituants reducing V 4+ in V 3+ : destabilize insulating M 1 * with respect to metallic R formation of V 3+ costs U: the energy gain in the formation of V 4+ -V 4+ Heitler-London pairs is lost dT MI /dx -1200K per V 4+ -V 4+ pair broken Assuming that the energy gain ΔU is a BCS like condensation energy of a spin-Peierls ground state: ΔU=N(E F )Δσ²/2 One gets: ΔU1000K per V 4+ - V 4+ pair (i.e. per V 2 O 4 formula unit of M 1 ) with Δσ~0.2eV and N(E F )=2x2states per eV, spin direction and V 2 O 4 f.u. *For large x, the M 1 long range order is destroyed, but the local V-V pairing remains (R. Comès et al Acta Cryst. A30, 55 (1974))

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Main features of the general phase diagram Substituants reducing V 4+ in V 5+ : destabilize insulating M 1 with respect to new insulating T and M 2 phases but leaves unchanged metal-insulator transition: dT MI /dx0 below R: the totally paired M 1 phase is replaced by the half paired M 2 phase formation of V 5+ looses also the pairing energy gain but does not kill the zig-zag instability (also present in TiO 2 !) as a consequence the M 2 phase is favored uniaxial stress along [110] induces zig-zag V displacements along [1-10] Note the non symmetric phase diagram with respect to electron and hole « doping » of VO 2 !

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Comparison of VO 2 and BaVS 3 Both are d 1 V systems where the t 2g orbitals are partly filled (but there is a stronger V-X hybridation for X=S than for X=O) BaVS 3 undergoes at 70K a 2 nd order Peierls M-I transition driven by a 2k F CDW instability in the 1D d // band responsible of the conducting properties at T MI tetramerization of V chains without charge redistribution among the t 2g s (Fagot et al PRL90,196403 (2003)) VO 2 undergoes at 340K a 1 st order M-I transition accompanied by a large charge redistribution among the t 2g s Structural instability towards the formation of zig-zag V shifts in metallic VO 2 destabilizes the π* levels and thus induces a charge redistribution in favor of the d // levels The pairing (dimerization) provides a further gain of energy by putting the d // levels into a singlet bonding state* *M 1 phase exhibits a spin-Peierls like ground state This mechanism differs of the Peierls-like V pairing scenario proposed by Goodenough!

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acknowledgements During the thesis work H. Launois P. Lederer T.M. Rice R. Comès J. Friedel Renew of interest from recent DMFT calculations A. Georges S. Biermann A. Poteryaev J.M. Tomczak

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Supplementary material

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Main messages Electron-electron interactions are important in VO 2 - in metallic VO 2 : important charge fluctuations (Hubbard bands) Mott-Hubbard like localization occurs when the lattice expands (Nb substitution) - in insulating VO 2 : spin-charge decoupling ground state described by Heitler-London wave function The 1 ST order metal-insulator transition is accompanied by a large redistribution of charge between d orbitals. for achieving this proccess an incipient lattice instability of the rutile structure is used. It stabilizes a spin-Peierls like ground state with V 4+ (S=1/2) pairing The asymmetric features of the general phase diagram of substitued VO 2 must be more clearly explained!

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LDA metallic

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T=0 Spectral function half filling full frustration X.Zhang M. Rozenberg G. Kotliar (PRL 1993) ω/D metallic VO 2 : single site DMFT D~2eV zig-zag de V phase M 2 D~0.9eV

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LDA phase métallique Rphase isolante M 1

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Structure électronique de la phase isolante M1 LDA Pas de gap au niveau de Fermi! Eg { a 1g B AB Niveaux a 1g séparés en états: liants (B) et antiliants (AB) par lappariement des V Mais recouvrement avec le bas des états Eg (structure de semi- métal)

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Cluster DMFT Gap entre a 1g (B) et E g Structure électronique de la phase isolante M1 Eg a 1g Single site DMFT Pas de gap à E F Eg a 1g LHB UHB U B AB LHB UHB Stabilise états a 1g

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LDA: Phase M 2 paires V 1 zig-zag V 2

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