15. optical processes and excitons
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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 PresentationTRANSCRIPT
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.
+ phonon emission (Stokes process)– phonon absorption (anti-Stokes)
2-phonon creation
XPS
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.
Kr at 20K
Lowest atomic transition of Kr = 9.99eV.In crystal it’s 10.17eV.Eg = 11.7eV → Eex = 1.5eV
The translational states of Frenkel excitons are Bloch functions.
Consider a linear crystal of N non-interacting atoms. Ground state of crystal is
1 2 1g N Nu u u u uj = ground state of jth atom.
If only 1 atom, say j , is excited: 1 2 1 1 1j j j j N Nu u u v u u u (N-fold degenerated)
In the presence of interaction, φj is no longer an eigenstate.For the case of nearest neighbor interaction T :
1 1j j j jH T j = 1, …, N
Consider the ansatz i k j a
jk je
i k j ajk
j
e HH 1 1ji k j a
jj jTe
i k ai k j a
j
ik ajT ee e 2 cos kT ka
ψk is an eigenstate with eigenvalue 2 coskE T ka
Periodic B.C. → 2 skN a
, , 12 2N Ns
Alkali Halides
The negative halogens have lower excitation levels
→ (Frenkel) excitons are localized around them.
Pure AH crystals are transparent (Eg ~ 10 eV)
→ strong excitonic absorption in the UV range.
Prominent doublet structure for NaBr ( iso-electronic with Kr )
Splitting caused by spin-orbit coupling.
Molecular Crystals
Molecular binding >> van der Waal binding → Frenkel excitons
Excitations of molecules become excitons in crystal ( with energy shifts ).
Davydov splitting introduces more structure in crystal (Prob 7).
Weakly Bound (Mott-Wannier) Excitons
Bound states of e-h pair interacting via Coulomb potential 2eUr
4
2 2 22n geE E
n
are
where1 1 1
e hm m n = 1, 2, 3, …
Cu2O at 77Kabsorption peaksEg = 2.17eV = 17,508 cm–1
For Cu2O, agreement with experiment is excellent except for n = 1 transition.
Empirical shift for data fit gives
12
80017,508cmn
With ε = 10, this gives μ = 0.7 m.
Exciton Condensation into Electron-Hole Drops (EHD)Ge: gEe VB e CB h VB ~1ns exciton ~8 s
For sufficiently high exciton conc. ( e.g., 1013 cm−3 at 2K ), an EHD is formed.
→ τ ~ 40 µs ( ~ 600 µs in strained Ge )
Within EHD, excitons dissolve into metallic degenerate gas of e & h.
Ge at 3.04KFE @ 714 meV : Doppler broadened.EHD @ 709 meV : Fermi gas n = 21017 cm−3.
EHD obs. by e-h recomb. lumin.
Unstrained Si
Raman Effect in Crystals
k k K
k k K
1st order Raman effect (1 phonon )
Cause: strain-dependence of electronic polarizability α. 2
0 1 2u u u = phonon amplitudeLet
0 cosu t u t 0 cosE t E t
Induced dipole: 1p t u t E t 1 0 0 cos cosu E t t
1 0 0 cos cosu E t t StokesAnti-Stokes
21n nI u K K 1n K
21n nI u K K n K
App.C:
→ 1
nII n
K
K
/ Bk Te 0 0T
1st order Ramanλinc = 5145AK 0
GaP at 20K.ωLO = 404 cm−1. ωTO = 366 cm−1 .1st order: Largest doublet.2nd order: the rest.
Si
Electron Spectroscopy with X-RaysXPS = X-ray Photoemission SpectroscopyUPS = Ultra-violet Photoemission Spectroscopy
Monochromatic radiation on sample : KE of photoelectrons analyzed.→ DOS of VB (resolution ~ 10meV)
Only e up to ~ 50A below surface can escape.
Ag: εF = 0
5s
4d
Excitations from deeper levels are often accompanied by plasmons.
E.g., for Si, 2p pk ~99.2eV is replicated at 117eV (1 plasmon) and at 134.7eV (2 plasmons).
ωp 18eV.
Energy Loss of Fast Particles in a Solid
Energy loss of charged particles measures Im( 1/ε ).
14 t
E DPPower dissipation density by dielectric loss:
EM wave: e e14
R Ri t i tEe i Ee
P
2 cos co14
s sint t tE
218
E
Particle of charge e & velocity v : , et
t
Dr v
r
Isotropic medium: , , , D k k E k
1
4,
, Re Re ,,
i t i tDe i D e
kk k
kP
2 1 1sin cos1 ,4
sint t tD
k
2 1 1sin cos s1 n, i,4
t tD t
k kP 2 11 ,
8D
k
22
,1 ,8
D
kk 21, , Im
81D
k kP
Energy loss function =
1Im,
k
2
02 1Im lnk ve
v
P