Utilizing the short wavelength of X-ray to study low-energy local excitations

q-dependence of the spectral weights and dispersions

Wei Ku (BNL & SUNY Stony Brook)

Acknowledgement

Ben Larson & Jon Tischler (ORNL)

Chi-Cheng Lee & Hung-Chung Hsueh (BNL & Tamkang U. Taiwan)

Ken Finkelstein (CHESS, Cornell)

Paul Zschack (UNICAT-APS & UIUC)

Oscar Restrepo & Adolfo Eguiluz (UT-Knoxville & ORNL)

Peter Abbamonte, James P. Reed & Serban Smadici (UIUC)

Chen-Lin Yeh (BNL & Tamkang U. Taiwan)

Tim Graber (U. of Chicago)

Abhay Shukla (Universit ´e Pierre et Marie Curie)

Jean-Pascal Rueff (Synchrotron SOLIEL)

sq , 1

VS

q , 2Im

G G

q

G ,

Non-Resonant Inelastic X-Ray Scattering (NIXS)

[hkl]

sample

Detector

Spherically BentAnalyzer Crystal

q

UNI-CAT ID-337.59 keVE ~ 1.1 eVIo ~ 5•1012 Hz

UNI-CAT ID-33CHESS C-LineE ~ 0.3 eVIo ~ 1011 Hz

100

100

100110

110

110

111

IXS Measurement Directions

Sq , 2V Im

G ,

G

q

G ,

Absolute Response CalculationsAbsolute IXS Measurements

New in-gap features at large q! What are these excitation? Strong angular dependence? (100) != (111) Difference between NiO & CoO?

25

20

15

10

5

0s(

q,

)

(eV

-1n

m-3

)302520151050

²E (eV)

CoO (1.1 eV Res.)

(111) (100)

2.0 A-1

7.0 A-1

25

20

15

10

5

0

s(q

,

) (e

V-1

nm

-3)

302520151050

²E (eV)

NiO (1.1 eV Res.)

(111) (100)

2.0 A-1

7.0 A-1

Strong Within-Mott-Gap Excitations at Large q

NiO CoO

q = 2/Å q = 2/Å

q = 7/Aq = 7/A

Charge Excitations in NoO and CoOSmall momentum transfer cases

Linear response within time-dependent density functional theory LDA+U approximation greatly improves the gap and line shape Work well at small q in absolute unit

0 10 20 30 40Energy (eV)

0.0

0.2

0.4

0.6

0.8

1.0

-Imv

0 5 10 15Energy (eV)

0.0

0.2

0.4

0.6 expLDALDA+U

NiO, q ~ 0.7 /Å

Charge Excitations in NoO at Large q

Large q excitations local d-d excitation (dipole forbidden) Strong angular dependence and nodal directions ?

0.014

0.012

0.010

0.008

0.006

0.004

0.002

0.000

s(q,

) (

eV

-1 Å

-3)

161412108642

²E (eV)

NiO (111) and (100)

q(111) = 3.5 A-1 (Meas.)

q(100) = 3.5 A-1 (Meas.)

q(100) = 3.5 A-1 (0.35 eV Res.)

q(111) = 3.5 A-1 (0.35 eV Res.)

q(111) = 3.5 A-1 (Calc.)

q(100) = 3.5 A-1 (Calc.)

B. Larson et. al, Phys. Rev. Lett. 99, 026401 (2007)

25

20

15

10

5

0

s(q

,)

(eV

-1n

m-3

)

302520151050²E (eV)

CoO (1.1 eV Res.)

(111) (100) 2.0 A

-1

7.0 A-1

25

20

15

10

5

0

s(q

,)

(eV

-1n

m-3

)

302520151050²E (eV)

NiO (1.1 eV Res.)

(111) (100) 2.0 A

-1

7.0 A-1

25

20

15

10

5

0

s(q

,)

(eV

-1n

m-3

)

302520151050²E (eV)

CoO LSDA+U = 8 eV

2.0 A-1

(111) 1.9 A

-1 (100) 7.0 A

-1 (111)

7.0 A-1

(100)

25

20

15

10

5

0

s(q

,)

(eV

-1n

m-3

)

302520151050²E (eV)

NiO LSDA+U = 8 eV

2.0 A-1

(111) 2.0 A

-1 (100)

7.0 A-1

(111)

7.1 A-1

(100)

Large-q only excitations local d-d excitation (dipole forbidden) Strong local interaction needed for correct energy How about the strong angular dependence and nodal directions ?

Linear Response & LDA+U Approximation

L

L

L

Time dependent density functional theory with LDA+U approximation:

local Fockp-h attraction

local Hartree

Hartreelong-range screening

d.c. LDAxcf w

1 21 1 2 2 , ; 1 2 1 2 ,; ; ;m m mn m n n n

mnm n

t t M L t t t t M

x xx x 1 *, 1 1m m m mM x x x

C.-C. Lee, H.-C. Hsueh, and Wei Ku, to be published

Real-Space Picture of Local Excitons

d x 5

eg x 2

t2g x 3

eg x 2

a1g x 1

e’g x 2

2 21 3ge z r 2 22 3ge x y

1 1 3 1 3 2ge yz zx xy

1 : 2ga yz zx xy

2 1 3 1 3 2ge yz zx xy

F of NiO

F of CoO

ˆ

1

ˆ

1

, ; 1,1 ;1 ,1;1 1 1 1i ie ew L w

q x q xq q

L

Energy-resolvedWannier orbitals

X-ray sees this

particle

hole

EF

e’g

eg

Local Excitations in NoO and CoOPoint group symmetry and new selection rules

Local point group symmetry nodal directions new selection rules

Anisotropy of Local Excitations

Nodal direction point group symmetry Lack of [100] node in CoO weak symmetry breaking

B. Larson et. al, Phys. Rev. Lett. 99, 026401 (2007)

Local Excitations in NoO and CoOSensitive probe of weak symmetry breaking

Lost of nodal directions : extremely sensitive to weak symmetry breaking Visualization of symmetry breaking via Wannier functions

NiO CoO

NiO CoO

CoO NiO

Formation of Frenkel Excitons in Local Picture

p1 h1

p1 h1

same pairp-h attraction

+

p1 h1

p1 h1

Hybridization of Frenkel Excitons in Wannier basis

local Fock

+

p1 h1

p2 h2

local Hartree

+

p1 h1

p1 h1

Tightly-Bound Excitons in Charge Transfer Insulators:case study of LiF

P. Abbamonte et. al., to be published

Tightly bound exciton

Charge transfer insulator

p-h in different atoms

Frenkel or Wannier exciton ?

Inelastic X-ray scattering

Structured spectral weight

Clear dispersion at large q !

observe fs dynamics

20

15

10

5

0

-5

-10

½ x

y z

x

yz

½ ½

Excitons in LiF as a Frenkel Exciton in a “Super Atom”

q = 0~1.5 Intensity divided by 2.60 1 2 3 4

161412

10

x

y

z

3 3 5

x

yz

½

Matrix Element and Structure in q-space

real-space

q-space

Effective Two-Particle Hopping

1 21 1 2 2 , ; 1 2 1 2 ,; ; ;m m mn m n n n

mnm n

t t M L t t t t M

x xx x 1 *, 1 1m m m mM x x x

C-L Yeh, H.-C. Hsueh, and Wei Ku, to be published

Define effective two particle kinetic kernel T vialocal

local

Propagation of exciton L L

L

L

local

local

T gives hopping of p-h pair in real space dispersion in q-space

Effective Two-Particle Hopping in LiF

C-L Yeh, H.-C. Hsueh, and Wei Ku, to be published

T() is complex and strongly -dependent to fully account for1. Landau continuum2. Lower mobility with stronger p-h binding

Re{T()} Im{T()}

(0,.5,.5) (0,.5,.5)

(0,0,1) (0,0,1)

within the continuum fast decay

NN hopping dominant cos() like dispersion

Time Evaluation of Charge Fluctuation in LiF

at the source of perturbation well defined averaged frequency steady decay in time

t ( fs )

Lph

,hp (

R, t

) *

4(0, 0, 0) * 22Lph,hp ( q , )

(

eV)

q (reciprocal lattice unit)

Propagation of Frenkel Excitons

(0, 0.5, 0.5) * 0.72

(0, 1, 1) * 1.42

(0, 1.5, 1.5) * 2.12

t ( fs )

Lph

,hp (

R, t

) (

scal

ed b

y R

2 )

along the (011) “direct” path efficient propagation steady group velocity

t ( fs )

Lph

,hp (

R, t

) (

scal

ed b

y R

2 )(0, 0, 1) * 12

(0, 0, 2) * 22

along the (001) “indirect” path velocity decreases interference due to multiple scattering

Propagation of Frenkel Excitons

Non-resonant IXS measurement vs. theory in absolute unit Non-resonant inelastic scattering at large q

sub-atomic spatial resolution beyond dipole selection Strong anisotropy & nodal directions of spectral weight at large q

direct access to spatial distribution of underlying orbital local point group symmetry new selection rules Clearer signature of dispersion at large q

propagation of excitations in space and time good (space, time) resolution: ( a0, fs )

Theory of local dynamics based on 1st-principles Wannier function real-space picture of local excitons and their propagation visualization of particle holes pairs and their nodal directions suitable for charge-transfer & more itinerant systems applicable for exciton decay near surfaces and in nano-systems Potential applications in correlated materials (orbiton, polaron, phason …)

Conclusion

The Correspondence is Less Direct With Resonant Emission X-ray Spectroscopy (REXS) And Cluster Calculations

REXS Observes A Range of Gap Excitations In NiO and CoO

0.05

0.04

0.03

0.02

0.01

0.00

s(q,

)

( e

V -

1 Å

-3)

3.5eV3.02.52.01.5²E (eV)

NiO

q = 7 (Å-1

)

Along (001) Along (111)

~1.3 eV

IXS

0.025

0.020

0.015

0.010

0.005

0.000

s(q,

)

(eV

-1 Å

-3)

3.0eV2.52.01.51.00.5

E (eV)

CoO<111>

q = 7 A-1

~1.4 eV

IXS

IXS Peak Positions

REXS

Butorin et al., PRB 54, 4405 (1996)

The Non-Resonant X-Ray Scattering Observations Are Similar to Spin-Polarized Resonant-Exchange Electron Scattering In NiO and CoO

Fromme et al., PRB 75, 693 (1995)

NiOC-SPEELS

0.05

0.04

0.03

0.02

0.01

0.00

s(q,

)

( e

V -

1 Å

-3)

3.5eV3.02.52.01.5²E (eV)

NiO

q = 7 (Å-1

)

Along (001) Along (111)

~1.3 eV

IXS

0.025

0.020

0.015

0.010

0.005

0.000

s(q,

)

(e

V -

1 Å

-3)

3.0eV2.52.01.51.00.5

E (eV)

CoO<111>

q = 7 A-1

~1.4 eV

IXS