Satoshi OkamotoDepartment of Physics, Columbia University

Electronic Reconstruction in Correlated Electron Heterostructures:

Towards a general understanding of correlated electrons at interface and surface

Support: JSPS & DOE ER 46169

Journal Refs.: S.O. & A.J.M., Nature 428, 630 (2004); PRB 70, 075101; 241104(R) (2004).

Collaborator: Andrew J. Millis

Discussion: H.Monien, M.Potthoff, G.Kotliar, A.Ohtomo, H.Hwang

Interface Science including Correlated-Electron Systems(ex. High-Tc cuprates, CMR manganites…)

Important for Application: spin valve, Josephson junction… Experiment (surface* sensitive): ARPES, STM,…

Tunneling magnetoresistance (TMR)(La,Sr)MnO3/SrTiO3/(La,Sr)MnO3

2

2

1

2

)(

)()0(TMR

P

P

H

HH

High TMR ratio (high polarization P ~ 90%) is only achieved at very low T <<bulk TC~370K

Photoemission experiment on CaVO3 and SrVO3

Maiti et al., Europhys. Lett. 55, 246 (2001).Bowen et al., Appl. Phys. Lett. 82, 233 (2003).

Dependent on photon energySurface bulk

Interface phases look different from bulk…

SrTiO3

(La,Sr)MnO3

(La,Sr)MnO3

SrTiO3

(La,Sr)MnO3

(La,Sr)MnO3

curr

ent

H

(0) (H)

*Surface= interface between material & vacuum

Theory: LiebschSchwieger, Potthoff

Key question: What is electronic phase? FM/AFM/SC, Metal/insulator,…(contrast: usual surface science; what is lattice reconstruction)

V

Charge leakage Atomic rearrangement/interdiffusion

Strain fields

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”)

Model

Ti d-electron (electronically active) ・ tight-binding t2g (xy, xz, yz) bands ・ Strong on-site interaction ・ Long-range repulsion

Extra “+1 charge” on La site (La3+ vs Sr2+) ・ Potential for d-electron

Neutrality condition ・ # of Ti d-electron=# of La ion

LaTi

Sr

n = 2z

x,y

… …

ij ij

j

ji

Laj

i

RR

ne

RR

eV

2

siteLa

2)(

2

1

baabba

baba

baba

aaa

isiteon

ddddJnnU

nnJUnnUH

††

,

)(

'

'

iii

Coul nVH )()(

Hamiltonian for Ti t2g electron

t2g bands

Electrostatic potential

On-site Coulomb

,,

†)( ..aij

jaiaaij

ahop cHddtH

Kimura et al. PRB 51, 11049 (1995)

Attraction by La ions

d-d repulsion

t ~ 0.3eV

U from ~2 to ~ 6 eV

Okimoto et al., PRB 51, 9581 (1995)Mizokawa & Fujimori, PRB 51, 12880 (1995)

SrTiO3: Almost ferroelectric(q 0, T 0) >> 102

We need (l 1Å, T 300K)This work: =15

(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)

>>TC

Energy unit: variance of non-interacting DOS

QMC data: Ulmke, Eur. Phys. J. B 1, 301 (1998).

QMC

2-site DMFT

Ei

E0

3

2

1

0 0

12

3

45i = 6

…

DMFT study 2: Magnetic ordering

Alternative: semiclassical approximationHasegawa, JPSJ 49, 178; 963 (1980).S.O., Fuhrmann, Comanac, & A.J.M.,

cond-mat/0502067.

22

4

1mUnUn

spin:

charge:

nnm

nnKey point:

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

S.O., Fuhrmann, Comanac, & A.J.M., cond-mat/0502067.

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.

0.0

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 = 0

/t

z = 1

z = 2

z = 3

z = 4

z = 5

z = 6

Lay

er-r

esol

ved

Spe

ctra

l fun

ctio

n

“LaT

iO3”

“SrT

iO3”

reg

ion

-8 -6 -4 -2 0 2 4 6 80.0

0.2

0.4

0.6

0.8

1.0

m

n

n, m

z

“4 La-layers”, U/t=18, T/t=0.1“SrTiO3” “SrTiO3”“LaTiO3”

Para

Ferro

DMFT result

?

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.