Satoshi Okamoto Department of Physics, Columbia University Electronic Reconstruction in Correlated Electron Heterostructures: rds a general understanding of correlated electr at interface and surface Support: JSPS & DOE ER 46169 Refs.: .J.M., Nature 428, 630 (2004); PRB 70, 075101; 241104(R) (200 Collaborator: Andrew J. Millis cussion: H.Monien, M.Potthoff, G.Kotliar, A.Ohtomo, H.Hw
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Satoshi Okamoto Department of Physics, Columbia University Electronic Reconstruction in Correlated Electron Heterostructures: Towards a general understanding.
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Satoshi OkamotoDepartment of Physics, Columbia University
Electronic Reconstruction in Correlated Electron Heterostructures:
Towards a general understanding of correlated electrons at interface and surface
In this talk: Focus on“Charge leakage”, “Magnetic ordering”, “Metal/insulator”
Changes in local environment,interaction parameters,lower coordination #
Vacuum/other material
LiebschSchwiegerPotthoff
What are important effects?
General understanding of correlated-electron interface
Bulk: LaTiO3: Mott insulator with d 1
SrTiO3: band insulator with d 0
Dark field image Sr La
EELS results of “1 La-layer” heterostructure Fraction of Ti3+: d-electron density
Substantial Charge leakageDecay length ~1nm (2~3 unit cells)
Heterostructure is Metallic!!
[LaTiO3]n/[SrTiO3]m heterostructure
Ohtomo, Muller, Grazul, and Hwang, Nature 419, 378 (2002)
Transport property
Ideal playground and good starting point!
Distance from La-layer (nm)C
arrie
r de
nsi
ty (
cm-3)
La fraction
Ti3+ La
LaTiO3 & SrTiO3 have “almost the same lattice constant”
Key word of theoretical results: “Electronic reconstruction”
・ “ Spin & Orbital orderings” in Heterostructuresdiffer from bulk orderings
・ “ Edge” region ~ 3 unit-cell wide — Metallic!!
*Independent of detail of theory*
This talk:
1. Realistic model calculation for [LaTiO3]n/[SrTiO3]m-type
heterostructure (“Ohtomo-structure”) based on Hartree-Fock
2. Beyond Hartree-Fock effect by Dynamical-mean-field theory using simplified model heterostructure 2.1. Metallic interface and quasiparticle 2.2. Magnetic ordering (on-going work)
1. Realistic model calculation for [LaTiO3]n/[SrTiO3]m-type heterostructure (“Ohtomo-structure”)
(Results do not change: 5 < < 40)*Self-consistent screening*
“Ohtomo-structure”
2D Spin orderOrb. disorder
Spin FerroOrb. AF
Spin & orb.disorder
Bulk0 5 10 15
0.0
0.2
0.4
0.6
0.8
1.0
1 / n
U / t
Inve
rse
of “
La”
laye
r nu
mbe
r
Bulk limit
On-site Coulomb interaction
Hartree-Fock result for the Ground-state phase diagram
Key point: Different Spin & Orbital orderings than in the bulk“Electronic reconstruction”
-6 -4 -2 0 2 4 6
0.0
0.2
0.4
0.6
Orb
ital o
ccup
ancy
z
“2 La”, U/t = 10
yzxz
xy
2-dimensional orbital order(in-plane-translational symmetry)
Bulk: 3D AF orbital
yz
xz
La at z = 0.5
Details of phase diagram may depend on approximation method. Difference from bulk — generic
U’ = U 2J, J/U=1/9*
* Similar ratio is reported by photoemission
Spatial charge distribution
-8 -6 -4 -2 0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
1.0 # of "La" 8 7 4 3 2 1
C
harg
e de
nsity
z
Width of “Edge” region ~ 3 unit cells(robust to varying , U, and detail of theory)
Bulk like property at the center site at # of La (n) > 6
Compatible with the experiment?—mainly n < 6 studied.
U/t = 6
-8 -6 -4 -2 0 2 4 6 8
0.0
0.2
0.4
0.6
0.8
12 14 16 18 20 Exp.
U/t = 0 2 4 6 8 10
Cha
rge
dens
ity
z
-8 -6 -4 -2 0 2 4 6 8
0.0
0.1
0.2
0.3 12 14 16 18 20 Exp.
U/t = 0 2 4 6 8 10
Cha
rge
dens
ity
z
Comparison with experiments
“1 La”
“2 La”
Charge distribution width(Theory) > (experiments)
Theory with broadening
~
La at z = 0
La at z = 0.5
Distance from center of heterostructure
Sub-band structure and metallicity
F
Schematic view of sub-band structure
continuum
z kx,y
E
Wave function Dispersion in xy-direction
Total electron density (blue) and electron density from the partially-filled bands (red)
“6 La”U = 10t
“Edge region” is metallic!!
Fully-filled band
Partially-filled band
Hartree-Fock picture
Another example of“Electronic reconstruction”
SrTiO3SrTiO3LaTiO3
2. Beyond Hartree-Fock by Dynamical-mean-field theory (DMFT) 2.1. Metallic interface and quasiparticle 2.2. Magnetic ordering (on-going work)
Beyond Hartree-Fock effects by Dynamical-mean-field theory (DMFT)
i
iCoul
isiteonhop HHHH )()(
ii
isiteon nUnH )( Single-band Hubbard
ij
jihop cHddtH ..†
Impurity model corresponding to each layerKey assumption for DMFT: Self-energy zz’(k||,) is layer-diagonaland independent of in-plane momentum k||
Self-consistency
)(),( zimpzG
ij ij
j
ji
Laj
i
RR
ne
RR
eV
2
siteLa
2)(
2
1i
iiCoul nVH )()(
),(2
)( ||2||
2
kGkd
G lattzz
impz
1
||0||' )(ˆ)(ˆ),(ˆ kHkG latt
zz
'' )()( zzzzz
… …
HartreeCoulhop HHH :ˆ
0
LaTi
Sr
… …
bath
zth layer
Remaining problem: Solving many impurity models is computationally expensive.We need to solve many impurities.
z
x,y
Single-band heterostructure
+ self-consistency for charge distribution
Model
DMFT study 1: “Metallic interface” and quasiparticle band
2
2
1
10)(
nnn ii
i
2-site DMFT self-energy hascorrect behavior at two limits
& Reasonable behavior in between
Potthoff, PRB, 64, 165114 (2001)2-site DMFT
bath
bath-site
correlated-site
Self-energy
ni Im
in
nU /2
nZ 11~
“Mott” physics
Mass enhancement
Schematic view of “correct” in
Z: quasiparticle weight
We need impurity solver which becomes reasonable at high- and low-frequency regions
Strong coupling regime
0.0
0.10.0
0.10.0
0.10.0
0.10.0
0.10.0
0.10.0
0.1
-10 0 10 200.0
0.1
z = 7
z = 6
z = 5
z = 4
z = 3
z = 2
z = 1
z = 0
/t
A(z
,z;
)
0.0
0.10.0
0.1
0.0
0.1
z = 10
z = 9
z = 8
lowerHubbard
upperHubbard
coherentquasiparticle
empty
almosthalf-filled
U = 16t > bulk critical Uc
0.0
0.10.0
0.10.0
0.10.0
0.10.0
0.10.0
0.10.0
0.1
-10 0 10 200.0
0.1
z = 7
z = 6
z = 5
z = 4
z = 3
z = 2
z = 1
z = 0
/t
A(z
,z;
)
0.0
0.10.0
0.1
0.0
0.1
z = 10
z = 9
z = 8
)(Im1
);,(
lattzzGzzA
“10 La-layers” heteostructure (La at z=4.5,…,+4.5)
U = 6t < bulk critical Uc ~ 14.7t
coherent quasiparticle-banddominates the spectral weight
empty
almosthalf-filled
Results: Layer resolved spectral function
z = 10
“LaT
iO3”
reg
ion
“SrT
iO3”
reg
ion
Weak coupling regime
z = 0
z = 5
-10 -8 -6 -4 -2 0 2 4 6 8 100.0
0.2
0.4
0.6
0.8
1.0 ntot by HF
ntot
ncoh
ntot , n
coh
z
-10 0 10
/t
Charge density distribution: “Visualization of metallic region”
ntot: total charge densityncoh: quasiparticle (coherent) density
Center region is dominated by lower Hubbard band: insulatingEdge region, ~ 3 unit cell wide, isdominated by coherent quasiparticle:Metallic!!n > 6 needed for “insulating” centralLayersDMFT & Hartree-Fock give almostidentical charge distribution
0);,(2 zzAd
0
);,(2 zzAd coh
“10 La-layers”, U/t = 16
~
ntot
ncoh
Distance from center of heterostructure
Metallic interfaces
La-layers at z =4.5,…,+4.5
Cha
rge
dens
ity
“SrTiO3” “SrTiO3”“LaTiO3”
DMFT study 2: Magnetic ordering
Magnetic Phase Diagram for 1-orbital modelHartree-Fock result
Inve
rse
of “
La”
laye
r nu
mbe
r
Thin heterostructures show differentmagnetic orderings than in the bulk.Ferromagnetic and layer Ferri/AF (in-plane translation symmetry) orderingsat large U and small n region.
How is this phase diagram modifiedby beyond-Hartree-Fock effect?
On-site interaction
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
layerFerri/AF
Ferro
in-plane (3D) AF
layerAF
Para
1/n
U/t
How difficult is dealing with in-plane symmetry breaking?
DMFT Self-consistency equations
),(2
)( ||2||
2
kGkd
G lattzz
impz
1
2
1
||' ),(ˆ
N
lattzz
t
t
t
tt
t
kG
)(|| zzkz Va
N: total layer #
Without in-plane symmetry breaking
),(2
)( ||)(),(2||
2
)(
kGkd
G lattBzABzA
impBzA
1
||
2||
1||
||
||2
||1
||' ),(ˆ
NBk
Bk
Bk
kNA
kA
kA
lattzz
t
tt
tt
t
t
tt
tt
t
kG
With in-plane symmetry breaking
)()()( BzAzBzA Va dispersion plane-in:||k
Coulomb ranged-long from potential:zV
We have to invert at least twice larger matrix at each momentum k|| and frequency,thus time consuming. This talk: Only in-plane-symmetric phases
Numerical methods with finite # of bath-orbital are weak dealing with thermodynamics.
We want to do finite temperature calculation as well.
0.0 0.2 0.4 0.6 0.8 1.00.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
QMC
2-site DMFT
U = 2
U = 4
U = 2
U = 4
T C
n
Trial: Magnetic phase diagram of single-band Hubbard modelon an infinite-dimensional FCC lattice
Energy diagramimpurity part of 2-site DMFT(U=4, n=0.5)
Hubbard-Stratonovich transformation (complete square) against two terms:
}exp{],[)()(exp0 intSxDnUnd
0
†22 )()}()(){(2)()(4
1cixcUxd
USint
Semiclassical approximation: keep (il=0, zero Matsubara frequency) (spin field), saddle-point approximation for x (charge field) at given . very slow spin-fluctuation dominates
1/ ]2/)()([)( xiiaediG nTV
nimp
12/)()(Tr xiiaUTxi n
2/)()(lnTr4
1 22 xiiaTxU
V n
)()()( 1nimpnn iGiai
ain: Weiss fieldDMFT self-consistency equation
Spin-field charge-field
0 5 10 15 200.0
0.2
0.4
0.6
0.8
1.0
Semiclassical QMC Hartree-Fock Strong coupling
T/t
U/t
Examples of magnetic phase diagram by semiclassical DMFT
2D square lattice (half filling)
Hubbard model on various lattice S.O., Fuhrmann, Comanac, & A.J.M., cond-mat/0502067.
Néel temp.
Semiclassical approximation gives excellent agreement with QMC.n=1: charge fluctuation is suppressed
0.0 0.2 0.4 0.6 0.8 1.00.00
0.05
0.10
0.15HF (101)
TC
QMC T
C
TN
Semiclassical T
C
TN
T
n
0.90 0.95 1.000.00
0.05
0.10
T
n
3D face-centered-cubic (FCC) lattice
Curie temp.
QMC data: Ulmke, Eur. Phys. J. B 1, 301 (1998).
Néel temp.for layer AF
Examples of magnetic phase diagram by semiclassical DMFT
TC is higher than QMC by a factor ~2, better agreement than HF and 2-site DMFT.Correct n-dependence Charge fluctuationassociated with spin fluctuationCorrect phase, AF phase at n~1, is obtained.
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
layerFerri/AF
Ferro
in-plane (3D) AF
layerAF
Para
1/n
U/t
Hartree-Fock resultMagnetic Phase Diagram
Inve
rse
of “
La”
-lay
er n
umbe
r
On-site interaction
Magnetic moment appears at “interface region.”In-plane symmetry breaking: now in progress.
Layer Ferri/AF phases are washed away. Ferromagnetic ordering appears at large U region, and prevails to large n.
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
?3D AF
FerroPara
1/n
U/t-10 -8 -6 -4 -2 0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
ntot
ncoh
ntot , n
coh
z
Inve
rse
of L
a-la
yer
num
ber
On-site interaction Distance from center of heterostructure
Main results of DMFT study on “1-orbital heterostructure”:Different orderings in heterostructures than in the bulk
Metallic interface, ~ 3 unit cells wide.
“SrTiO3” “SrTiO3”“LaTiO3”
Future problemIn-plane symmetry breaking: in progress
DMFT study on the realistic three-band model How spin & orbital orderings are modified?
Combination between DMFT & 1st principle calculation Effect of lattice distortion; largeness of , orbital stability
Material dependence d2, d3, d4,…systems, various combinations between them and with others
SummaryModel calculation for [LaTiO3]n/[SrTiO3]m-type heterostructure
Key word: “Electronic reconstruction”Key results independent of details of theory: Thin heterostructures show different orderings than in the bulk. Interface region between Mott/band insulators (~3 unit cells wide) becomes always metallic.