# Presentation: 934104

## 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)

## Non-Resonant Inelastic X-Ray Scattering (NIXS)

**Absolute Response Calculations**

**Absolute IXS Measurements**

## Strong Within-Mott-Gap Excitations at Large q

-   Strong angular dependence?  (100) != (111)
-   Difference between NiO & CoO?

- Strong Within-Mott-Gap Excitations at Large *q*

- NiO

- CoO

*q* = 2/Å

*q* = 2/Å

*q* = 7/A

*q* = 7/A

## Charge Excitations in NoO and CoOSmall 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

- NiO,*  q* ~ 0.7 /Å

## Charge Excitations in NoO at Large q

- Charge Excitations in NoO at Large *q*

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

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

##   Large-q only excitations  local d-d excitation (dipole forbidden)

-   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

- Time dependent density functional theory with LDA+*U* approximation:

- local Fock
- p-h attraction

- local Hartree

- Hartree
- long-range screening

- d.c.

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

## Real-Space Picture of Local Excitons

- Energy-resolvedWannier orbitals

-  X-ray sees this

- particle

- hole

## Local Excitations in NoO and CoOPoint group symmetry and new selection rules

- 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 CoOSensitive probe of weak symmetry breaking

- 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 pair
- p-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

- 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

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

## Matrix Element and Structure in q-space

- Matrix Element and Structure in *q-*space

- real-space

*q*-space

## Effective Two-Particle Hopping

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

- Define effective two particle kinetic kernel T via

- local

- local

- Propagation of exciton

- 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 for
- Landau continuum
- 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* )

*L**ph,hp *( *R*, *t* ) * 4

## 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* )

*L**ph,hp *( *R*, *t* ) (scaled by *R*2)

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

## Propagation of Frenkel Excitons

*t* ( *fs *)

*L**ph,hp *( *R*, *t* ) (scaled by *R*2)

- (0, 0, 1) * 12
- (0, 0, 2) * 22

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

## Conclusion

-   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: ( *a**0*, *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 ...)

## 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**

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