Hybrid Mott Band Metal InsulatorTransitions

Ansgar Liebsch

Institute for Solid State Physics

Research Center Julich

– p. 1

Band vs Mott Insulator → ‘Hybrid Insulators’

Band Insulator Mott Insulator

UHB LHB Val.

Valence Conduction LHB UHB

LHB Cond. UHB

– p. 2

OverviewMott transition in multi-orbital materials:

– Coulomb driven inter-orbital charge transfer– role of crystal field splitting– suppression of orbital fluctuations– subband filling / emptying → MIT

Dynamical mean field theory:– exact diagonalization for multiband materials

Ca2−xSrxRuO4 4d4 orbital selective Mott transition ?

V2O3 3d2 Hund vs. Ising exchange ?

LaTiO3 / SrTiO3 3d1 MIT in heterostructures ?3d1−x doping driven Mott transition ?

NaxCoO2 3d5+x topology of Fermi surface ?

– p. 3

Transition metal oxides: Ca2−xSrxRuO4

– p. 4

Ca2−xSrxRuO4: phase diagram

Maeno (PRL 2000 )

– p. 5

Ca2−xSrxRuO4:

0

0.5

1

1.5

-3 -2 -1 0 1

Den

sity

of S

tate

s (1

/eV

)

Energy (eV)

xy

xz,yz

coexisting narrow and wide bands: U/Wi ??

orbital selective Mott transitions ? (Anisimov et al)

crystal field splitting among t2g bands ! (Fang et al)

Sr → Ca: octahedral distortions

– p. 6

Dynamical mean field theoryG(iωn) =

∑

k

(

iωn + µ − H(k) − Σ(iωn))

−1

ij(t2g)

G0 = (G−1 + Σ)−1

exact diagonalization: solid → cluster = impurity + bath

G0,i(iωn) ≈ Gcl0,i(iωn) =

(

iωn + µ − εi −ns∑

k=4

|Vik|2

iωn − εk

)

−1

ε1

ε2

ε3

ε4

ε5

ε6

ε7

ε8

ε9

ε10

ε11

ε12

Vk

Vk

U

J

– p. 7

cluster HamiltonianHcl =

X

iσ

(εi − µ)niσ +X

kσ

εknkσ +X

ikσ

Vik[c+iσckσ + H.c.]

+X

i

Uni↑ni↓ +X

i<jσ≤σ′

(U ′ − Jδσσ′)niσnjσ′ −X

i6=j

J [ c+i↑

ci↓c+j′↓

cj↑ + c+i↑

c+i↓

cj↑cj↓]

cluster Green’s function:

Gcli (iωn) =

1

Z

X

νµ

|〈µ|c+iσ|ν〉|2

Eν − Eµ − iωn

[e−βEν + e−βEµ ]

=1

Z

X

ν

e−βEν

„

X

µ

|〈µ|c+iσ|ν〉|2

(Eν − Eµ) − iωn

+X

µ

|〈µ|ciσ|ν〉|2

(Eµ − Eν) − iωn

«

assumption: cluster self-energy ≈ solid self-energy:

Σcl = 1/Gcl0 − 1/Gcl ≈ Σ

Hcl sparse: Arnoldi algorithm ns = 12 N = 853776level spacing < 1 meV → finite size effects reducedED/DMFT for realistic t2g materials, Hund exchangecomplementary to QMC/DMFT Perroni, Ishida + A.L. PRB (2007)

– p. 8

Ca2−xSrxRuO4 nature of Mott transition ?Sr → Ca : nxy increases : crystal field ∆ Fang et al (2004)

0

0.5

1

1.5

-3 -2 -1 0 1

Den

sity

of S

tate

s (1

/eV

)

Energy (eV)

xy

xz,yz

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10n i

U (eV)

∆=0.5 eV 00.4 0.2

xy

xz,yz

dyn. correl.: nxy → 1 A.L.+ Ishida, cond-mat/0612539 PRL(2007)

suppression of orbital fluctuations:MIT in half-filled xz,yz band → no orbital selective MITfuture: role of octahedral distortions ? magnetic phases ?

– p. 9

Ca2RuO4 cluster spectra ∆ = 0.4 eV

0

1

-6 -4 -2 0 2

Den

sity

of S

tate

s (1

/eV

)

ω (eV)

(a) U=3.0 eV

0

1

2

-6 -4 -2 0 2

Den

sity

of S

tate

s (1

/eV

)

ω (eV)

(b) U=4.2 eV

0

1

2

-6 -4 -2 0 2

Den

sity

of S

tate

s (1

/eV

)

ω (eV)

(c) U=4.5 eV

gap between filled xy and xz,yz upper Hubbard band

– p. 10

V2O3 3d2 ED vs QMC

0

0.5

1

1.5

-1 0 1

Den

sity

of s

tate

s (1

/eV

)

ω (eV)

V2O3

ageg

0

0.1

0.2

0.3

0.4

0.5

3 4 5 6

ni

U (eV)

V2O3

T=230 Kna

ne

IsingHund

Keller et al, PRB (2004) A.L. (2007)

dynamical correlations: nag→ 0, MIT in half-filled e′g

suppression of orbital fluctuations

– p. 11

LaTiO3 3d1

0

1

2

-1 -0.5 0 0.5 1 1.5

Den

sity

of S

tate

s (1

/eV

)

ω (eV)

LaTiO3

ag

eg 0

0.1

0.2

0.3

0.4

0.5

3 4 5

ni

U (eV)

LaTiO3

na

ne

Pavarini et al, PRL (2004) A.L. (2007)

dynamical correlations: ne → 0, MIT in half-filled ag

suppression of orbital fluctuations

– p. 12

quasi-particle spectra: LaTiO3 V2O3

Pavarini et al (2006) Poteryaev et al, cond-mat/0701263

excitation gap:

LaTiO3: LHB ag → empty eg bandsV2O3: LHB eg → empty ag band

– p. 13

Ca2RuO4 V2O3 LaTiO3

nσ ≈ (0.8, 0.6, 0.6) nσ ≈ (0.2, 0.4, 0.4) nσ ≈ (0.3, 0.1, 0.1)→ (1.0, 0.5, 0.5) → (0.0, 0.5, 0.5) → (0.5, 0.0, 0.0)

a

e

a

e e

Ca RuO 2 4

n=4

ea

a

ee

V O2 3

n=2

a e

a a

e

n=1

LaTiO3 – p. 14

La1−xSrxTiO3 3d1−x doping driven MIT

0

0.1

0.2

0.3

0.4

0.5

3 4 5 6

ni

U (eV)

LaTiO3

na

ne

n=1.0

n=0.90

n=0.95

A.L. (2007)

LaTiO3 / SrTiO3 heterostructures: La / Sr interdiffusion:3d0.95 inhibits eg → ag charge transfer: no MIT

thin LaTiO3 layers on SrTiO3:cubic symmetry shifts Uc upwards → no MIT at UTi

– p. 15

Na0.3CoO2: topology of Fermi surface ?

t2g bands 0.4 ag holes 0.3 e′g holes D.J. Singh, PRB (2000)

-1.5

-1

-0.5

0

0.5

Ene

rgy

(eV

)

Γ M K Γ

a a

aa

e1

e2

e2

e2 e2

e1 hole pockets not seen in ARPES ??

– p. 16

Na0.3CoO2 QMC vs ED / DMFTCoulomb driven inter-orbital charge transfer for J = U/4

0.7

0.8

0.9

1

0 1 2 3 4 5

nm

U (eV)

T=10 meV

ag

eg ED

QMC

dynamical correlations: stabilization of eg hole pocketsQMC: Ishida, Johannes + A.L. PRL (2005) ED: Perroni, Ishida + A.L. PRB (2007)

– p. 17

Na0.3CoO2 LDA + U

0.7

0.8

0.9

1

0 1 2 3

ni

U (eV)

J=U/4

J=0 ag

eg (b) LDA+U

Ishida, Johannes + A.L. PRL (2005)

∆εa = 2/3(ne − na)(U − 5J)

∆εe = −1/3(ne − na)(U − 5J) → J=0.9 eV: Uc=4.5 eV

subtle balance:J vs U, shape of DOS, dynamical vs static correlations

– p. 18

Na0.3CoO2 QMC + crystal field∆ = εa − εe: eg bands shifted down J=0

role of H(k) ! Marianetti, Haule, Kotliar, cond-mat/0612606

– p. 19

Na0.3CoO2 ED vs QMC same H(k)

13

13.5

14

14.5

0 5 10 15 20 25

Re

Σi (i

ωn)

(e

V)

iωn (eV)

ag

eg

µ = 14.41 eV

Na0.3CoO2 U=3 eV J=0

β = 50 na=0.735 ne=0.957

A.L. (2007) Marianetti, Haule, Kotliar, cond-mat/0612606

– p. 20

Na0.3CoO2 ED

0.7

0.8

0.9

1

0 1 2 3 4 5 6

ni

U (eV)

na

ne

J=U/5

J=0

H(k) Zhou et al, PRL (2005) A.L. (2007)

– p. 21

BaVS3

J = U/4 vs J = U/7 Lechermann et al, PRL (2005)

local Coulomb correlations can reduce / enhanceorbital fluctuations !

– p. 22

LaTiO3 vs BaVS3: both 3d1

0

1

2

-1 -0.5 0 0.5 1 1.5

Den

sity

of S

tate

s (1

/eV

)

ω (eV)

LaTiO3

ag

eg

opposite charge transfer: importance of density of states!– p. 23

Na0.3CoO2 quasi-particle bands vs ARPES

Perroni, Ishida + A.L. PRB (2007) Qian et al PRL (2006)

∼ 30 % band narrowing: 1.5 → 1.0 eV

– p. 24

V2O3 ED: Hund vs Ising exchange

-2

-1

0

0 1 2 3 4 5

Im Σ

i (iω

n)

ωn

Σa

Σe

V2O3

T=20 meV

Hund

Ising

-2

-1

0

0 1 2 3 4 5

Re Σ

i (iω

n)

ωn

Σa

Σe

V2O3

T=20 meV

Hund

Ising

A.L. (2007)

Ising: nearly localized eg:modifies (i) ag self-energy (ii) intra-eg scattering(see: two-band model !)

– p. 25