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
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