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### The metal-insulator transition of VO2revisited

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 l’honneur de Pascal Lederer

outline

- Electronic structure of metallic VO2
- Insulating ground states
- Role of the lattice in the metal-insulator transition of VO2
- General phase diagram of VO2 and its substituants

VO2: 1st order metal-insulator transition at 340K

*

Discovered nearly 50 years ago

still the object of controversy!

*in fact the insulating ground state

of VO2 is non magnetic

Bad metal

insulator

metal

in metallic phase: ρ ~T

very short mean free path: ~V-V distance

P.B. Allen et al PRB 48, 4359 (1993)

A

B

cR

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 VO6 octahedra running along cR

(related by 42 screw axis symmetry)

V 3d orbitals in the xyz octahedral coordinate frame

eg:

V-O

σ* bonding

orbital located in the xy basis of the octahedron

bonding between V in the (1,1,0) plane (direct V-V bondingalong cR :1D band?)

t2g

V-O

π* bonding

orbitals « perpendicular » to the triangular faces of the octaedron

bonding between V in the (1,-1,0) plane in the (0,0,1) plane

well splittedt2gand egbands

3dx²-y²: a1g or t// (1D) band of Goodenough

Is it relevant to the physics of metallic VO2?

t2g

3dyzand 3dxz: Egor π* bandsofGoodenough

1d electron of the V4+

fills the 3 t2g bands

eg

V. Eyert Ann. Phys. (Leipzig)

11, 650 (2002)

Electronic structure of metallic VO2

LDA

Single site DMFT

UHB

LHB

U

t2g levels

bandwidth~2eV: weakly reduced in DMFT calculations

a1g

Eg

Hubbard bandson both Eg (π*)

and a1g (d//) states

no specificity of d// band!

Biermann et al PRL 94, 026404 (2005)

Fractional occupancy of t2g orbitals

orbital/occupancy LDA* single site DMFT* EFG measurements**

x²-y² (d//) f1 0.36 0.42 0.41

yz (π*) f2 0.32 0.29 0.26-0.28

xz (π*) f3 0.32 0.29 0.33-0.31

*Biermann et al PRL 94, 026404 (2005)

** JPP thesis (1974): 51V EFG measurements between 70°C and 320°C

assuming that only the on site d electron contributes to the EFG:

VXX = (2/7)e<r-3> (1-3f2)

VYY = (2/7)e<r-3> (1-3f3)

VZZ = (2/7)e<r-3> (1-3f1)

VO2: a correlated metal?

- Total spin susceptiblity:

Neff (EF)~10 states /eV, spin direction

J.P. Pouget& H. Launois, Journal de Physique 37, C4-49 (1976)

- Density of state at EF:

N(EF)~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 χPauli: 5-8

Sizeable charge fluctuations in the metallic state

- DMFT: quasiparticle band + lower (LHB) and upper (UHB) Hubbard bands
- LHB observed in photoemission spectra
- VO2 close to a Mott-Hubbard transition?

LHB

Koethe et al PRL 97, 116402 (2006)

Mott Hubbard transition for x increasing inNb substitued VO2: V1-XNbXO2?

- 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<xC

magnetism becomes more localized when x increases (Curis Weiss behavior of χspin for x large)

- beyond xC ~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)

Insulating phase: monoclinic M1

Short V-O distance

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 cR :

x²-y² levels split into bonding and anti-bonding states

stabilization of the x²-y² bonding level with respect to π* levels

The x²-y² bonding level of the V4+ pair is occupied by 2 electrons of

opposite spin: magnetic singlet (S=0)

Driving force of the metal-insulator transition?- The 1st order metal- insulator transition induces a very large electronic redistribution between the t2g orbitals
- Insulating non magnetic V-V paired M1 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 VO2

uniaxial stressedVO2

R-M1 transition of VO2 splitted into R-M2-T-M1transitions

V1-XCrXO2

J.P. Pouget et al PRB 10,

1801 (1974)

VO2 stressed along [110]R

J.P. Pouget et al PRL 35,

873 (1975)

(site A)

(site B)

Zig-zag V chain

along c

V-V pair

along c

Zig –zag chains of (Mott-Hubbard) localized d1 electrons

Zig-zag V4+ (S=1/2) Heisenberg chain (site B)

χspin

χtot

M2

T

R

T

M2

In M2: 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)

Crossover from M2 toM1via T phase

Dimerization of the Heisenberg chains (V site B)

tilt of V pairs (V site A)

2J intradimer exchange integral

on paired sites B

Jintra increases with the dimerization

Value of 2Jintra (= spin gap) in the M1 phase?

Energy levels in the M1 phase

AB

Δρdimer

Δρ

B

S

eigenstates of the 2 electrons Hubbard molecule (dimer)

Δρdimer

T

Δσ

Only cluster DMFT is able to account for

the opening of a gap Δρat EF

(LDA and single site DMFT fail)

Δρdimer~2.5-2.8eV >Δρ~0.6eV

(Koethe et al PRL 97,116402 (2006))

Δσ?

S

Estimation of the spin gap Δσ in M1

2J(M1)=Δσ >2100K

- Shift of χbetween the T phase ofV1-XAlXO2 and M1 phase of VO2
- 51V NMR line width broadening of site B in the T phase of stressed VO2 :T1-1 effect

for a singlet –triplet gap Δ: 1/T1~exp-Δ/kT

at 300K: (1/T1)1800bars=2 (1/T1)900bars

If Δ=Δσ-Δ’s one gets for s=0 (M1phase)

Δσ=2400K with Δ’=0.63 K/bar

M2

G. Villeneuve et al

J. Phys. C: Solid State

Phys. 10, 3621 (1977)

T

J.P. Pouget& H. Launois, Journal de Physique 37, C4-49 (1976)

The intradimer exchange integral Jintra of the dimerized Heisenberg chain

(site B) is a linear function of the lattice deformation measured by the 51V EFG component VYY on site A

M1

Site B

T

M2

Site A

JintraB(°K) + 270K ≈ 11.4 VYYA (KHz)

For VYY= 125KHz (corresponding to V pairing in the M1 phase) one

gets : Jintra~1150K or Δσ~2300K

M1 ground state

Δσ~ 0.2eV<<Δρ is thus caracteristic of an electronic state where strong coulomb repulsions lead to a spin charge separation

The M1 ground state thus differs from a conventional Peierls ground state in a band structure of non interacting electronswhere the lattice instability opens equal charge and spin gaps Δρ ~ Δσ

Electronic parameters of the M1 Hubbarddimer

- Spin gap value Δσ ~ 0.2 eV

Δσ= [-U+ (U²+16t²)1/2]/2

which leads to:

2t ≈ (Δσ Δρintra)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 ≈ Δρintra-Δσ ~ 2.5eV

(in the M2 phaseU 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 VO2 is such that Δσ~7J (strong coupling limit)

In weak coupling spin-Peierls systems Δσ<J

Lattice effects

- the R to M1 transformation (as well as R to M2 or T transformations)involves:

- the critical wave vectors qc of the « R » point star:{(1/2,0,1/2) , (0,1/2,1/2)}

-together, with a 2 components (η1,η2) irreductible representation for each qC:

ηi corresponds to the lattice deformation of the M2 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

M2: η1≠0, η2= 0 T: η1> η2 ≠0 M1: η1= η2 ≠0

- The metal-insulator transition of VO2 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 qc which leads to good nesting properties of the Fermi surface?

- Does the lattice dynamics exhibits a soft mode whose critical wave vector qc is connected to the band filling of VO2 ?
- Or is there an incipient lattice instability of the rutile structure used to trig the metal-insulator transition?

Evidences of soft lattice dynamics

{u//[110]}

[110]

- X-ray diffuse scattering experiments show the presence of {1,1,1} planes of « soft phonons » in rutile phase of

(metallic)VO2 (insulating) TiO2

[001]

smeared diffuse

scattering ┴ c*R

cR*/2

+(001) planes

{u//cR}

R critical point of VO2

Γ critical point of TiO2

(incipient ferroelectricity

of symmetry A2Uand

2x degenerate EU)

Pcritical point of NbO2

aR*/2

EU

aR*/2

A2U

(R. Comès, P. Felix and JPP: 35 years old unpublished results)

{1,1,1} planarsoft phonon modes in VO2

- not related to the band filling (the diffuse scattering exists also in TiO2)
- 2kF 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 4kF instability (not critical) of fully occupied d// levels

- instability of VO2 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

[110]

[111]

cR

[1-10]

correlatedV 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// orbitalsoccurs via the formation of bonding levels achieved by V pairing between neighbouring [111] « chains »

phase diagram of substitued VO2

Sublatices A≡B

Sublatices A≠B

dTMI/dx≈0

R

dTMI/dx ≈ -12K/%V3+

M1

xV5+

x

V3+

0.03

0

Reduction of V4+

Oxydation of V4+

VO2

M

V1-XMXO2

M=Cr, Al,Fe

M=Nb, Mo, W

VO2+y

VO2-yFy

uniaxial stress // [110]R

Main features of the general phase diagram

- Substituants reducing V4+ in V3+ : destabilize insulating M1* with respect to metallic R

formation ofV3+ costs U: the energy gain in the formation of V4+-V4+ Heitler-London pairs is lost

dTMI/dx ≈ -1200K per V4+-V4+ pair broken

Assuming that the energy gain ΔU is a BCS like condensation energy

of a spin-Peierls ground state:

ΔU=N(EF)Δσ²/2

One gets: ΔU≈1000K per V4+ - V4+ pair (i.e. perV2O4 formula unitof M1)

with Δσ~0.2eV and N(EF)=2x2states per eV, spin direction and V2O4 f.u.

*For large x, the M1 long range order is destroyed, but the local V-V pairing remains

(R. Comès et al Acta Cryst. A30, 55 (1974))

Main features of the general phase diagram

- Substituants reducing V4+ in V5+ : destabilize insulating M1 with respect to new insulating T and M2 phases

butleaves unchangedmetal-insulator transition: dTMI/dx≈0

below R: the totally paired M1 phase is replaced by the half paired M2 phase

formation of V5+ looses also thepairing energy gain but does not kill

the zig-zag instability (also present in TiO2!)

as a consequence the M2 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 VO2!

Comparison of VO2and BaVS3

- Both are d1 V systems where the t2g orbitals are partly filled

(but there is a stronger V-X hybridation for X=S than for X=O)

- BaVS3 undergoes at 70K a 2nd orderPeierls M-I transition driven by a 2kF CDW instability in the 1D d// band responsible of the conducting properties

at TMItetramerization of V chainswithout charge redistribution among the t2g’s

(Fagot et al PRL90,196403 (2003))

- VO2 undergoes at 340K a 1st order M-I transition accompanied by a large charge redistribution among the t2g’s

Structuralinstability towards the formation of zig-zag V shifts in metallic VO2 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*

*M1 phase exhibits a spin-Peierls like ground state

This mechanism differs of the Peierls-like V pairing scenario proposed by Goodenough!

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

Main messages

- Electron-electron interactions are important in VO2

- in metallic VO2: important charge fluctuations (Hubbard bands)

Mott-Hubbard like localization occurs when the lattice expands (Nb substitution)

- in insulating VO2: spin-charge decoupling

ground state described by Heitler-London wave function

- The 1ST order metal-insulator transitionis 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 V4+ (S=1/2) pairing

- The asymmetric features of the general phase diagram of substitued VO2 must be more clearly explained!

LDA

metallic VO2: single site DMFT

T=0 Spectral function half filling full frustrationD~2eV

zig-zag de V phase M2

D~0.9eV

ω/D

X.Zhang M. Rozenberg G. Kotliar (PRL 1993)

Structure électronique de la phase isolante M1

LDA

LDA

AB

B

a1g

Niveaux a1g séparés en états:

liants (B) et antiliants (AB)

par l’appariement des V

Mais recouvrement avec le bas des états Eg (structure de semi-métal)

{

Eg

Pas de gap au niveau de Fermi!

Structure électronique de la phase isolante M1

Single site DMFT

Cluster DMFT

UHB

a1g

B

Eg

LHB

AB

UHB

U

LHB

a1g

Eg

Stabilise états a1g

Gap entre a1g(B) et Eg

Pas de gap à EF

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