funding: office of basic energy sciences, u.s. department of energy #s de-fg02-06er46285 & de-...
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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
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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)
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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
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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
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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
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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
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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)
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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
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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]
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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.
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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
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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
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Full response
All processes:
• Exciton
• Interband
• Plasmon
• Core levels
• Compton scattering
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Isolation of the exciton
Truncated at 16.5 eV
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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
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Wannier Functions
M(x) = a*2s(x) a2p(x)
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Wannier functions
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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; )
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Off-diagonal terms – true imaging
J. A. Golovchenko, Phys. Rev. Lett. 46, 1454 (1981)
W. Schülke, Phys. Lett. A 83, 451 (1981)