1. Optical Properties 8 Interaction of light with matter
Classical and quantum models to describe
refraction, reflection, and absorption of solids
Generation of light by electrons in solids

2. Review of Chapter 4 Index of refraction without absorption:
Index of refraction with absorption:
Absorption constant, :
Relationship between r and 
(4.34) and (8.1)
(4.35) and (8.2)
(4.36) and (8.3)
(4.41) and (8.4)
General task is to calculate a(w) from a specific absorption mechanism, and hence to determine its effect on other properties affected by the index of refraction.

3. Contents of this chapter
Classical wave solution for the problem of reflection at the boundary between two regions with different dielectric constant and absorption index.
Major sources of absorption in solids.
Recombination of excited carriers: photoconductivity and luminescence
Applications
② min. E → max wavelength
③ excitons
⑤ Free electron transition; both k and E change. both phonon and photon involved called indirect transition

4. Reflection x I R T
The continuity requirement is that electric and magnetic components (being tangential components) must be conserved across the interface.

5. Optical reflection Incident wave Reflected wave Transmitted wave yx y z
Incident wave
Reflected wave
Transmitted wave
(1) Energy flow in the EM filed is given by E x H, so that flow of energy in the reflected wave must have opposite direction – inserting minus sign in EyR
(2) Third Maxwell equation:
(8.5)
(8.6)
(8.7)
(8.8)
(8.11)
(8.12)
(8.9)
(8.10)

6. Optical reflection I T R A A″ A x (8.15)x A I A″ A T R
(8.15)
(for an interface between vacuum and a material without absorption)
(8.16)
(for an interface between vacuum and a material with absorption index G)
(8.17)

7. Antireflection coating1 2
glass
air
anti-reflection
coating ① ②
(assume that absorption is negligible)
(a) Region 0:
(8.18)
(8.19)
(b) Region 1:
(8.20)
(8.21)
(c) Region 2:
(8.22)

8. Antireflection coating
From the conservation of the tangential components Ey and Hz at each interface,
(8.23)
Where, (1) and (2) can be constructively or destructively interfered depending on the r and thickness of the film.
For r1=1, r2=2, and r3=3, R=25%
With a film thickness of d=l1/4, R=2%,

9. Light Electron Analogies (Revisit Textbook 68 page)
V=V0 I R T
Reflection/Transmission
V=0
V=V0 I R T
Reflection/Transmission
When electron wave is passing to a region of lower velocity – no phase change
When electron wave is passing to a region of higher velocity – phase change

10. Light Electron Analogies
V=0
V=V0 I R T
Reflection/Transmission I R T
Reflection/Transmission
E > V0
V=0
E < V0 I R
Reflection only I R
Reflection only

11. Light Electron Analogies
Reflection/Interference/Transmission V3 V2
Reflection/Interference/Transmission
E > V2
Reflection/Interference/Transmission
Thin film
→ attenuation in thin film V0
Reflection/Interference/Attenuation/Transmission
E<V0
tunneling

12. Summary of absorption processes
In order of decreasing energy of transition ① Electron transition from the valence band to higher-lying conduction bands. =105106 cm-1 ② Electron transition from the valence band to the lowest-lying conduction band with a minimum required energy of the forbidden band gap. The magnitude and variation with energy of the absorption constant depends on whether it is direct or indirect transition. ③ Optical transition producing bound electron-hole pair (excitons), requiring less energy than to produce a free electron-hole pair by the system ④ Imperfections (to or from a band between discrete levels) ⑤ Free carriers (classical ): a transition to higher energy states within the same band or to higher bands; involves the absorption of both photons and phonons because both E an k must be change in the transition ⑥ Lattice vibrations (Reststrahlen absorption)

13. Absorption processes in solids
Figure 8.3 Characteristic types of optical transitions shown both for the flat band model and the E vs. k plot. (1) Excitation from the valence band to higher lying conduction band, (2) excitation across the band gap, (3) exciton formation, (4) excitation from imperfections, (5) free carrier excitation

14. Band to band transition
(8.26)
<Direct transition>
Direct band gap: energy difference between the conduction band and balance band extrema occurring at the same value of k
(8.24)
 increases rapidly with photon energy larger than EG to values in the range of 105106 cm-1.
Need to explain the Kphoton
Need to explain the reason why the alphs is proportional to the sqrt of energy coming from the density of state
Holds only for a small range of photon energies greater than EG
Most semiconductors (not Si) are direct band gap materials.

15. Band to band transition
Indirect transition (Si): the extrema of conduction and valence bands occur at different points in k space.
+ : absorption
 : emission
phonon emission
(8.30)
(8.29)
2nd-order process
absorption
(8.31)
spontaneous
stimulated emission
(major role in laser)

16. Band to band transition
Optical absorption properties of indirect band gap materials; Ge and Si Ge Si
EGi=0.74 eV corresponding to a conduction band minimum at the 111 zone face
A direct band gap of 0.90 eV at k=0
EGi=1.17 eV in the 100 direction about 85% of the way to the zone face
A direct band gap of 2.5 eV at k=0
FIG Energy bands E(k) vs k near the conduction and valence band extrema including spin-orbit splitting energies for Ge and Si. Energies given are for 0°K

17. Band to band transition
Fig. 8.8 and 8.7
FIG Analysis of absorption spectra of Ge and Si with a one-phonon model to show the presence of indirect transitions. (After G. G. MacFarlane and V. Roberts, Phys. Rev. 97, 1714 (1955); 98, 1865 (1955).)
FIG Dependence of absorption constant on photon energy for Ge. (After W. C. Dash and R. Newman, Phys. Rev. 99, 1151 (1955).)

18. Degenerate semiconductors
Degenerate means EF lies in a conduction band.
Non-degenerate : Fermi lies in band gap. E EC k EG EF kF A B
Lowest direct transition energy
Increasing impurity
Density of free electrons
where E=(EF-Ec), the height of the Fermi level above the bottom of the conduction band

19. Excitons (Wannier Excitons)
Bound electron-hole pairs
Neutral in charge; diffuse
Localized entity
Bube 책 참고하여 정리 n=1은 왜 관찰이 안되는지?
EH: the ground state energy of the isolated hydrogen atom (-13.5 eV)
The Wannier-Mott excitons lying just below the conduction band and an example of absorption spectrum of Cu2O at 77 K.

20. Excitons Binding energy or dissociation energy
where EH is the ground state energy of the isolated hydrogen atom ( eV) and M*r is a reduce mass
Exciton binding energy is a strong function of dielectric constant ε
An exciton can diffuse. The total energy of an exciton is
For an indirect excitons, the optical absorption corresponds to exciton bands.
For tightly bound excitons such as in ionic and molecular materials with low dielectric constants, excitons are localized to atoms (anions) or molecules.
(8.32)
Dissociation rate of an exciton is low at low temperature and high at high temperature.
(8.33)

21. Imperfection Absorption
Localized state arising from this type

22. (transition only for small range of k)
Imperfections
Transition 4 in Fig. 8.3.
Imperfections form localized states (localized energy levels) indicated by a short line on a flat band diagram.
Shallow imperfections (small energy difference between the imperfection energy levels and either conduction or valence band edge)
large extension → uncertainly in  x is large
→ k is small from the uncertainty principle
E to make free carrier
Deep imperfections
strong binding to carrier
carrier is strongly localized
is small
is large k E
Deep imperfection
(transition for all range of k) x k
shallow
(transition only for small range of k)

23. Imperfections Imperfection absorption: ①, ②, ③ CB
(③: as in the case of impurity )
Transition ① and ②: absorption consists of a threshold followed by a region of higher absorption.
Transition ③: a narrow peaked band with maximum at the energy separation between the two levels. CB VB ① ② ③
Types of imperfections
Point imperfections: vacancy or misplaced atoms
Impurities
Larger structural defects: dislocations, imperfection complexes, grain boundaries, and surfaces
Absorption constant for the transitions of ① and ②
=SoptNI ; (8.34)
where Sopt is the absorption cross section (10-15 to cm2)and NI the density of imperfections (1014 to 1018 cm-3)

24. Free carriers Transition 5 in Fig. 8.3
Indirect process involving both a photon and a phonon
Classical model : 
Quantum picture?
Transition to higher energy state k E
2nd order process

25. Free carrier absorption
Classical
(8.35)
(8.36) ↑
damping term
(8.37), (8.38)

26. Free carrier absorption①
(8.39) ②
“Skin effect” ③ ④
semi-conductor optical free carrier absorption

27. Free Carrier absorption
Quantum Model : indirect transition
Classical Model

28. Plasma Resonance Absorption
Optical transition due to collective action of the carriers
Plasma resonance frequency
Consider a collection of free electrons to be displayed a distance  by an electric field E,
The equation of motion is
(8.40)
(8.46)
(8.41)
(8.42)
(8.43)

29. Plasma Resonance Absorption
It is the frequency at which an undamped plasma of free electrons can oscillate as a whole.
For large ,
 = + and r is real if   p
Transparent above this region
Normal waves propagates
 =  and r is complex if   p
No wave propagates
Plasmon: quanta of energy associated with the oscillations electric charge involved in plasma resonance with energy of
Surface plasmon: characteristic of charge oscillation occurring at the surface of a metal. The possibility of coupling between optical photons and surface plasmon makes possible a number of interesting applications in holography, planer light source, enhanced Raman scattering, submicron electronic circuitry.
Reflectivity
Fig. 8.14

30. Polarization of bound electrons
Bonding : charged particles attached to nucleii by spring
The force FE exerted on an electron of charge q by the electric field
The restoring force by bond
The equation of motion including a damping constant
숙제: derive the last two equations in this slide.

31. Interaction of light with matter
r2-2-1 
wo (lo)
Resonance frequency n l
숙제 2. Sell Meier 식을 유도하시요.
Sell-Meier equation

32. Photoelectronic effects
Photoelectronic effects; effects related to iight emission or light detection related. We frequently use photonics or optoelectronics for the description of the phenomena.
Results of absorption
If electrons are excited to conduction band, the free carrier density increases to increase the conductivity; Photoconductivity
When the excited electrons give up their energy when they return to their initial state in the form of photons, light is emitted; luminescence
Fate of a free electron-hole pair formed by photoexcitation
Captured by localized imperfections
Recombine with each other directly or at localized imperfections
Pass out of the material at one end without being replaced at the other

33. Recombination process
Intrinsic recombination: pure intrinsic materials with little imperfections
At thermal equilibrium
Every excitation makes a hole and excited electron pair.
 is the recombination rate in cm3 sec-1

34. Recombination process
Small light case
(Low f)
Large signal case (high f)
Luminescence emission intensity or rate of recombination under the assumption that all of the recombination processes are radiative.
n is the photoconductivity lifetime

35. Recombination processes

36. Recombination process
Recombination coefficient and capture cross section
If an electron moves to the volume, then captured.

37. Recombination process
What happens to the energy upon recombination?

38. Recombination process
Rate of excitation
Rate of recombination
lifetime

39. Optical spectra Absorption Excitation
PC (photoconductivity) :Change of conductivity by photon
Luminescence

40. Review Exciton Imperfection Recombination Recombination
Rate of excitation
Rate of recombination
lifetime

41. Review
Absorption
Photoconductivity

42. Review
Luminescence