Funding: Office of Basic Energy Sciences, U.S. Department of Energy #s DE-FG02-06ER46285 & DE-AC02-98-CH10886

Dynamical reconstruction of the valence exciton in LiFPeter AbbamonteUniversity of Illinois

Collaborators

Wei Ku (BNL)

Tim Graber (APS) James Reed (UIUC) Serban Smadici (UIUC)Young Il Joe (UIUC)

Chen Lin Yeh (Tamking University)

Scattering:

First (second?) principles

Abhay Shukla (U. Marie et Pierre Curie), Jean-Pascale Rueff (Soliel)

Excitons: Frenkel vs. Wannier

Frenkel (Xe, Organics, …) Wannier (Si, Ge, Cu2O, …)J. Frenkel, Phys. Rev., 37, 17 (1931) G. H. Wannier, Phys. Rev., 52 191 (1937)

conduction band

valence band

Alkali Halides: Intermediate case?

Discovery of excitons in alkali halidesHilsch, R., & Pohl, R. W., Über die ersten ultravioletten Eigenfrequenzen einiger einfacher kristalle, Z. Physik 48, 384-396 (1928)

Marginal case btwn. Frenkel and Wannier Mott, N. F., Conduction in polar crystals. II. The conduction band and ultra-violet absorption of alkali halide crystals, Trans. Faraday Soc. 34, 500-506 (1938)

Electron transfer modelOverhauser, A. W., Multiplet structure of excitons in ionic crystals, Phys. Rev. 101, 1702-1712 (1956)

“Excitation” modelDexter, D. L., Exciton models in alkali halides, Phys. Rev. 108, 707-712 (1957)

It’s all just WannierHopfield, J. J., & Worlock, J. M., Two-quantum absorption spectrum in KI and CsI, Phys. Rev. 137, A1455-A1464 (1965)

GW correction / Solve Bethe-Saltpeter eqn.Rohlfing, M., & Louie, S. G., Electron-hole excitations in semiconductors and insulators, Phys. Rev. Lett. 81, 2312-2315 (1998)

No data on (time-dependent) structure of excitons

Inelastic x-ray (or n+ or e–) scattering

Couple light to electrons

(Lorentz force law)

Be sure to second-quantize to get photons

Multiply out to get interactions

Do perturbation theory (1st Born approximation)

Turns out to be a Green’s function

2 1ˆ( , ) | ( ) | 0 ( ) Im ( , )n

n

S n n

k k k

Inelastic x-ray (or n+ or e–) scattering

• density-density Green’s function

• density propagator

• susceptibility

c(k,w) :

Describes how disturbances in electron density travel about the medium.

(0,0)

(x,t)Causality

Frenkel vs. Wannier in IXS

Wannier’s Excitonic Basis:

Excitons come from diagonalizing

For Frenkel exciton, dominated by one term:

• Frenkel exciton keeps its size / shape through its life.

•Wannier changes.

• IXS determines which description – independent of H

( ) , | | 0,iH e H

K R

R

K R R β β

00 ( ) , | | 0,0iH e H K R

R

K R R

| , R R βR. Knox, Theory of Excitons (Academic

Press, NY, 1963)

Results

F 2p Li 2s

• Data from APS 15-ID• F 2p to Li 2s• Only see longitudinal exciton• Singlet-triplet splitting << g spin

state unimportant

Phase problem / “arrow of time”

Cannot invert with only Im[(k, )]

• c(x,t) = 0 for t < 0

• Raw spectra do not really describe dynamics – no causal information

• Must assign an arrow of time to the problem. Permits retrieval of c(x,t) – view dynamics explicitly.

Re[w]

Im[w]

Problem #1

Problem #1:

Im[c(k,w)] must be defined on infinite w interval for continuous time interval

Solution:

Extrapolate.

Side effects:

• c(x,t) defined on continuous (infinitely narrow) time intervals.

• Wmax plays role of pulse width.

Problem #2

Problem #2:

Discrete points violate causality

Im[c(k,w)] must be defined on continuous w interval. Periodicity incompatible with causality.

Solution:

Analytic continuation (interpolate)

Side effects:

• c(x,t) defined forever. Vanishes for t < 0.

• Repeats with period T = 2p/Dw = 13.8 femtoseconds

• Dw plays role of rep rate

Nyquist’s theorem

wt

f(t)

wN = 2 wmax

too small aliasing

- wmax wmax

Nyquist frequency

DtN = p/wmax = 20.7 as

DxN = p/qmax = 0.635 Å

|f( )|2

Full response

All processes:

• Exciton

• Interband

• Plasmon

• Core levels

• Compton scattering

Isolation of the exciton

Truncated at 16.5 eV

Result

Very close to Frenkel limit

• Delocalized over 3a

• Periodic internal structure a/3

• 283 as period

• Time-independent – very close to Frenkel limit

• Should be able to describe as single pair of Wannier functions

Wannier Functions

M(x) = a*2s(x) a2p(x)

Wannier functions

Conclusions

• Exciton in LiF is a Frenkel exciton, despite old controversy

• No contradiction between CT and Frenkel – relative motion between e- and h+ is quenched

• NSLSII: Focus on momentum variable (imaging)

Problem (k1,k2; ) (k,k; )

Off-diagonal terms – true imaging

J. A. Golovchenko, Phys. Rev. Lett. 46, 1454 (1981)

W. Schülke, Phys. Lett. A 83, 451 (1981)