15. Optical Processes and Excitons Optical Reflectance Kramers-Kronig Relations Example: Conductivity of Collisionless Electron Gas Electronic Interband Transitions Excitons Frenkel Excitons Alkali Halides Molecular Crystals Weakly Bound (Mott-Wannier) Excitons Exciton Condensation into Electron-Hole Drops (Ehd) Raman Effect in Crystals Electron Spectroscopy with X-Rays Energy Loss of Fast Particles in a Solid
15. Optical Processes and Excitons. Optical Reflectance Kramers-Kronig Relations Example: Conductivity of Collisionless Electron Gas Electronic Interband Transitions Excitons Frenkel Excitons Alkali Halides Molecular Crystals Weakly Bound (Mott-Wannier) Excitons - PowerPoint PPT Presentation
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15. Optical Processes and Excitons
Optical ReflectanceKramers-Kronig Relations
Example: Conductivity of Collisionless Electron Gas
Electronic Interband Transitions
ExcitonsFrenkel Excitons
Alkali Halides
Molecular Crystals
Weakly Bound (Mott-Wannier) Excitons
Exciton Condensation into Electron-Hole Drops (Ehd)
Raman Effect in CrystalsElectron Spectroscopy with X-Rays
Energy Loss of Fast Particles in a Solid
Optical Processes
Raman scattering: Brillouin scattering for acoustic phonons. Polariton scattering for optical phonons.
k γ << G for γ in IR to UV regions.→ Only ε(ω) = ε(ω,0) need be considered.
Theoretically, all responses of solid to EM fields are known if ε(ω,K) is known.
ε is not directly measurable.Some measurable quantities: R, n, K, …
Optical Reflectance
Reflectivity coefficient
E reflr
E inc ie
Consider the reflection of light at normal incidence on a single crystal.
Let n(ω) be the refractive index and K(ω) be the extinction coefficient.
→ 11
n iKrn iK
see Prob.3
n iK N Complex refractive index
Let 0 expinc i t E E k r
exptrans i n i K t E k r exp expK i tn r rk k
Reflectance
2
2
E reflR
E incl 2r 2 (easily measured)
θ is difficult to measure but can be calculated via the Kramer-Kronig relation.
i → 2 2n K 2nK
Kramers-Kronig RelationsRe α(ω) KKR → Im α(ω) α = linear response
x F
22
2 j j jj
d d Fxdt dt M
Equation of motion:(driven damped uncoupled oscillators)
jj
x x
Fourier transform: 2
i tdf t e f
i tf d t e f t
Linear response: t
x t dt t t F t
→
2 2j j j
j
Fi x
M
→ 2 2jj j jj j
Fx x
M i
2 2j
j
j j
fi
1
jj
fM
2 2
2 22 2
j j
jj
j
j
f i
Let α be the dielectric polarizability χ so that P = χ E. 2 2
22 j j j
d d ne Epdt dt m
j jj j
nex P p → →2
jnefm
Conditions on α for satisfying the Kronig-Kramer relation:
• All poles of α(ω) are in the lower complex ω plane.
• C d ω α /ω = 0 if C = infinite semicircle in the upper-half complex ω plane.
It suffices to have α → 0 as |ω | → .
• α(ω) is even and α(ω) is odd w.r.t. real ω.
Example: Conductivity of Collisionless Electron GasFor a free e-gas with no collisions (ωj = 0 ):
1
m i
0 1 1 i
m
2
1m
m
2 2 2 2
1 1s s sP ds P ds
s m s
2
1m
KKR
41
PE
4 nexE
24 ne
Consider the Ampere-Maxwell eq. 4t
c
DH J
Treating the e-gas as a pure dielectric:
ct
DH
Fourier components:
4i i E E → 14
i
2i i n e 2n e i
m
pole at ω = 0
Treating the e-gas as a pure metal: 4t
c
EH E
→ 4t t
D EE
2
2
41 nem
→ 2 24 ne
m
Electronic Interband Transitions
R & Iabs seemingly featureless.
Selection rule c v k k
allows transitions k B.Z.
→ Not much info can be obtained from them?
Saving graces:Modulation spectroscopy: dnR/dxn, where x = λ, E, T, P, σ, …
0c v k k kCritical points where
provide sharp features in dnR/dxn which can be easily calculated by pseudo-potential method (accuracy 0.1eV)
dR/dλ
Electroreflectance: d3R/dE3
R
Excitons
Non-defect optical features below EG → e-h pairs (excitons).
Frenkel excitonMott-Wannier exciton
Properties:• Can be found in all non-metals.• For indirect band gap materials, excitons near direct gaps may be unstable.• All excitons are ultimately unstable against recombination.• Exciton complexes (e.g., biexcitons) are possible.
0c v k k kExciton can be formed if e & h have the same vg , i.e. at any critical points
GaAs at 21KI = I0 exp(–α x)
Eex = 3.4meV
3 ways to measure Eex :• Optical absorption.• Recombination luminescence.• Photo-ionization of excitons (high conc of excitons required).
Frenkel Excitons
Frenkel exciton: e,h excited states of same atom; moves by hopping.E.g., inert gas crystals.