1. 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

2. 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

3. 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

4. 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

5. 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

6. 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

7. 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

8. 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!

9. 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 )

10. 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

11. 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)

12. 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<x C magnetism becomes more localized when x increases (Curis Weiss behavior of χ spin for x large) beyond x C ~0.2: electronic conductivity becomes activated electronic charges become localized local effects (induced by the disorder) become relevant near the metal-insulator transition metal-insulator transition with x due to combined effect of correlations and disorder concept of strongly correlated Fermi glass (P. Lederer)

14. 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

15. 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)

16. 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)

17. 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

18. 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

19. 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

20. 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

21. 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

22. 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

23. 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

24. 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<J

25. 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?

26. 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

27. {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]

28. 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

29. 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))

30. 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 !

31. 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!

32. 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

33. Supplementary material

34. 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!

35. LDA metallic

36. 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

37. LDA phase métallique Rphase isolante M 1

38. 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)

39. 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

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