F. Lechermann , L. Boehnke , D. Grieger and C. Piefke

F. Lechermann , L. Boehnke , D. Grieger and C. Piefke

Correlated Electron States in
Realistic Oxide Heterostructures
F. Lechermann1, L. Boehnke1, D. Grieger2 and C. Piefke1
1
I. Institut für Theoretische Physik, Universität Hamburg, Germany
2
International School for Advanced Studies (SISSA), Trieste, Italy
Phys. Rev. B 87, 241101(R) (2013), arXiv:1401.6105 (2014), JUROPA project hhh08
Prominent Oxide Heterostructures: LaTiO3/SrTiO3 and LaAlO3/SrTiO3
Metallicity and Ferromagnetism at the LAO/STO interface
The combination of metallic oxides within a heterostructure
architecture allows for the design of materials properties based
on the unique interface physics (for a recent review see
e.g. [1]). Especially interlacing different insulators (of band
or Mott kind), gives rise to an intriguing metallic 2DEG with
instabilities towards magnetic and superconducting order.
Here large supercell combinations of SrTiO3 (STO), a bulk
band insulator with LaTiO3 (LTO), a bulk Mott insulator and
LaAlO3 (STO), another bulk band insulator, are studied within
charge self-consistent DFT+DMFT. We find that structural
relaxations and strong correlations trigger orbital selectivity
in LTO/STO. Vacancy-induced correlations in LAO/STO
provide moreover an understanding of metallicity and intricate
ferromagnetism at the corresponding interface.
LDA results and construction of correlated subspace
ε-εF (eV)
1.0
• 80-atom supercells with
structural relaxation
0.5
0.0
xz/yz
xz/yz
xz/yz
xy
-0.5
• defect-free (DF) and oxygen
vacancy-hosting (VH) structure
M
• 25% vacancy-concentration
in Ti(12)O2 layer
DOS (1/eV)
• two electrons at Fermi level
for DF: polar-catastrophe
avoidance
x -y
0
Ti
2
(2)
Ti
0
(5)
2
1
0
Ti
(5)
Ti
(1)
-0.08
xy
x2− y2
xz/yz
correlated
subspace
for VH case:
strictly derived from
KS-Hamiltonian,
|e˜g i ∼ 0.55|z 2i±0.84|x2−y 2i
~140 meV
Ti
(345)
~155 meV
yz
xy
~320 meV
e~g
xz
yz
xy
• no ferromagnetic (Stoner) instability
in L(S)DA for both structural cases
2
1
charge self-consistent DFT+DMFT implementation [5]
A (1/eV)
VH-PM
40
20
νν
νν
2
-2
0
-1
1
-0.5
-0.1
0.1
M
~
eg
xy
ω (eV)
1 2
ω (eV)
0.1
3
4
increased peak heights at Fermi level, DF case shows no
ferromagnetic (FM) instability, lower eg -like Hubbard peak at
−1.2eV for VH case in agreement with photoemission.
kν
kνν 0
0.0
0
-0.5
M
Orbital Selectivity at the LTO/STO interface
Ti(1)
Ti(2)
Ti(3)
DF PM 0.02 0.16 0.02 0.16 0.07 0.05
PM 0.76 0.24 0.79 0.24 0.15 0.06
VH
↑ 0.41 0.19 0.43 0.19 0.07 0.03
FM
↓ 0.36 0.06 0.36 0.07 0.07 0.02
Ti(4)
Ti(5)
0.07 0.05 0.08 0.04
0.17 0.06 0.24 0.04
0.08 0.03 0.12 0.02
• mTi(12) ∼0.1 µB
Ti(1)
0.08 0.02 0.12 0.02
0
Ti
40
0.0
60
20
40
-0.1
0
40
30
-0.2
-1
β=160 eV (T=72.5K)
-1
β=80 eV (T=145.1K)
-0.3
-1
20
-0.4
40
2
4
6
8 10
2
iω
0
-2
0
-1
ω (eV)
nxz,yz=0.24
nxy =0.25 Z =0.62
xz,yz
Ti1
Z
=0.67
1 Zxz,yz=0.66
Zxy =0.61
dxz nxz,yz=0.24
dyz nxy =0.26
dxy
nxz,yz=0.23
nxy =0.26 Zxz,yz=0.63
nxz,yz=0.20
nxy =0.32
xy
0
• five inequivalent Ti ions,
effective 3-band t2g correlated
subspace (U = 5eV, JH = 0.7eV)
ρ (1/eV)
1
• increased charge transfer
towards xy close to
interface in DFT+DMFT
• stronger pronounced xy QP
structure close to εF
→ orbital selectivity due
to relaxations and
strong correlations
NIC Symposium 2014, February 12-13 2014, Jülich, Germany
0
1
0
1
0
Ti2
Zxz,yz=0.66
Zxy =0.65
Zxy =0.60
nxz,yz=0.13
nxy =0.18 Z =0.75
xz,yz
Ti3
Zxz,yz=0.79
Zxy =0.79
nxz,yz=0.04
nxy =0.03 Z =0.91
xz,yz
Zxz,yz=0.97
Zxy =0.97
nxz,yz=0.02
nxy =0.01 Z =0.95
xz,yz
=0.99
1 Zxz,yz=0.99
nxz,yz=0.06
nxy =0.03
Zxy =0.95
xy
-1
nxz,yz=0.08
nxy =0.05
Zxy =0.91
Ti5
0
-2
nxz,yz=0.12
nxy =0.22
~
-0.3
PM
FM-up
eg
0.0 0.1 0.2 0.3 0.4 0.5
iω
• intricate double-exchange mechanism (with relevant
Hund’s exchange JH) responsible for FM state
• e˜g electrons less coherent with temperature,
stronger electron-electron scattering in PM phase
1
Summary
Charge self-consistent DFT+DMFT based on the combination of an accurate Kohn-Sham technique with advanced continuous-time quantum Monte-Carlo can cope with the correlated electronic structure of oxide heterostructures. Correlations ally with structural relaxations in driving orbital selectivity towards xy character in occupation and QP weight at the LTO/STO interface. Quantum-fluctuating ferromagnetism at the LAO/STO interface emerges from vacancyinduced key double-exchange processes in an correlated (e˜g , xy) subspace.
References
Zxy =0.74
Ti4
Z
xy
0.0 0.4 0.8 1.2 0.0 0.4 0.8 1.2
10
DFT+DMFT spectrum compared to LDA (black):
without (top) and with relaxations (bottom)
• without relaxations: white
bg, with relaxations: grey bg
-0.2
~
eg
• coexisting more localized e˜g level and more itinerant xy level
in interface Ti(12)O2 layer
• spin polarization most strong in xy orbital, while e˜g
dominantly occupied
β=40 eV (T=290.1K)
60
• 80-atom supercells
xy
0.05
-1
ω (eV)
• quasiparticle (QP) weight Z
recovers towards STO part,
xy minor lowered Z
0.05
β=20 eV (T=580.2K)
0
-10 -8 -6 -4 -2 0
n(r):occupied t2g bands
• effective 2-orbital Hubbard problem near quarter filling
-0.1
LDA
DFT+DMFT
(2)
• quantum
fluctuations
reduce the
FM moment
compared to
LDA+U [9]
-0.05
Im Σ(iω)
1000
20
(4,4)-2
0.0
-0.05
Im Σ(iω)
ρ (1/eV)
30
80
ρ (1/eV)
Local Ti(3d) fillings in DF and VH structure. DF: averaged xz, yz
and xy values. VH: (e˜g ,xy) on Ti(12) and (xz/yz,xy) on Ti(345).
(4,4)x2
10
X [A↑−A↓](k, ω)
Γ
0.01
• fully parallelized scheme, numerical cost: ∼50-100 processors for about 2 weeks per calculation
20
0
1
2
3
4
-2
ω (eV)
-1
0
1
PM-A(k, ω)
Local occupations and scattering properties in vacancy-hosting case
1
+ hHU i − Edc, with HU i Tr [Σ(iωn)G(iωn)
2
80
0
The k-resolved DFT+DMFT spectral function reveals correlated fermiology in PM and FM vacancy-hosting phase.
o
1 Xn KS
∆N(k) =
G (iωn, k) P†(k) Σ(iωn) P(k) − (µ − µKS)1 Gbl(iωn, k)
β n
100
X
Γ
0.5
-0.1
0
-2 -1 0
2
1
0.0
-1.0
1
0
-3
kν
~
2
10
0.1
eg
xy
VH-FM
up
down
20
• Dynamical Mean-Field Theory (DMFT) using a hybridization continuous-time quantum Monte Carlo
solver (for a review see [6]) implemented in the TRIQS code [7, 8]
X
• charge self-consistency:
ρ(r) =
hr|Ψ i f (ε̃ )δ 0 + ∆N 0 (k) hΨ 0 |ri
0
3
0
0
i,m6=m ,σ
kν
Aloc (1/eV)
0
-0.1
ω (eV)
20
0.5
ω (eV)
3
0
kν
• correlated subspace: effective
3-orbital (U = 3.5eV, JH = 0.5eV)
for DF and effective 2-orbital
(U = 2.5eV, JH = 0.5eV) for VH
(12)
DF-PM
o
1 X n 0
†
†
†
†
U nimσ nim0σ̄ + U 00 nimσ nim0σ + JH dimσ dim0σ̄ dimσ̄ dim0σ + JH dimσ dimσ̄ dim0σ̄ dim0σ
nim↑nim↓ +
2
0
EDFT+DMFT = EDFT +
√
√
(4,4)- 2 × 2 LAO/STO
1.0
• Density Functional Theory (DFT) in Local Density Approximation (LDA):
mixed-basis pseudpotential electronic structure code
• Ti multi-orbital Hubbard Hamiltonian with full rotational invariance
• total energy:
4
DFT+DMFT spectral data in paramagnetic and ferromagnetic phase
P̄ R0 ∗0 (k)
0 ν m
(k)
εKS
kν ∆Nνν
3
• six electrons for VH: additional
electrons with mostly eg -like
character in Ti(12)O2 layer:
eg weight visible in bonding
charge density
-0.04
Ti
Rmm0
X
2
bonding charge
density:
(LDA)
(LDA)
(LDA)
ρb
=ρtot −ρatomic
0
Ti(2)
xz
R∗
R
P̄νm
(k) ΣR
mm0 (iωn ) P̄m0 ν 0 (k)
im
1
3 4
0
E-EF (eV)
0.04
40
H(int) = U
2
0.08
combine band theory and model Hamiltonians
by allowing for dominant local correlations
crystal Bloch basis: |kνi
Rk ∼ hRm|kνi
P̄mν
correlated subspace: |Rmi
X
(4)
(3)
LDA bands and local density of states
interface: projection formalism
Σνν 0 (k, iωn) =
Ti
Ti
xy
X
(5)
2
Ti
DFT+DMFT Approach: Materials Science meets Many-Body Theory
νν
2
z
xz
yz
xy
0
-1
z2
kνν 0
(3)
Ti
2
2
• five inequivalent Ti ions with
inplane differentiation
Right: LAO/STO,
photoemission [4],
t2g /eg dichotomy
0
X
Γ
vacancy-hosting
(1)
Middle: LTO/STO,
photoemission [3],
Hubbard peak at −1 eV
GRR
mm0 (iωn ) =
XM
Γ
(1)
Ti
eg
Ti
Left: LTO/STO,
structure [2],
arbitrary layerings
n
o
−1
R
P̄mν (k) [iωn + µ − HKS(k) − Σ(k, iωn)]
eg
defect-free
Experimental data on LTO/STO and LAO/STO interfaces
X
xy
xz/yz
2
3
4
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[4] G. Berner, M. Sing, H. Fujiwara, et al., Phys. Rev. Lett. 110, 247601 (2013)
[5] D. Grieger, C. Piefke, O. E. Peil, and F. Lechermann, Phys. Rev. B 86, 155121 (2012)
[6] E. Gull, A. J. Millis, A. I. Lichtenstein, et al., Rev. Mod. Phys. 83, 349 (2011)
[7] M. Ferrero and O. Parcollet, TRIQS: http://ipht.cea.fr/triqs
[8] L. Boehnke, H. Hafermann, M. Ferrero, F. Lechermann, and O. Parcollet, Phys. Rev. B 84, 075145 (2011)
[9] N. Pavlenko, T. Kopp, E. Y. Tsymbal, et al., Phys. Rev. B 85, 020407(R) (2012)