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