Review of Optical Properties of Materials

Review of optics

Absorption in semiconductors: qualitative discussion

Derivation of Optical Absorption Coefficient in Direct Semiconductors

Photons• When dealing with events on the atomic scale, it is often

best to regard light as composed of quasi- particlesPHOTONS

Photons are Quanta of lightElectromagnetic radiation is quantized

& occurs in finite "bundles" of energy

Photons– The energy of a single photon in terms of its frequency , or

wavelength is,

Eph = h = (hc)/

Maxwell – Electromagnetic Waves

Light as an Electromagnetic Wave

• Light as an electromagnetic wave is characterized by a combination of a time-varying electric field (E) & magnetic field (H) propagating through space.

• Maxwell’s equations give the result that both E & H satisfy the same wave equation:

2

22 2

1, ,E H E Hc t

Changes in the fields propagate through space with speed c.

Speed of Light, c• Frequency of oscillation, of the fields and their

wavelength, o in vacuum are related by;– c = o

• In any other medium the speed, v is given by;– v= c/n =

• n = refractive index of the medium • = wavelength in the medium

• And,• r = relative magnetic permeability of the medium • r = relative electric permittivity of the medium

rrn

The speed of light in a medium is related to the electric and magnetic properties of the medium,

and the speed of light can be expressed as

Electromagnetic SpectrumShorter wavelength

Longer wavelength

Larger Photon

Energy (eV)

Scattering

1- Refraction

2- Transmission

3a – Specular reflection

3b – Total internal reflection

3c – Diffused reflection

4 Dispersion –where different colors bend differently

41

3b

2

3a

3c

Incident light

“Semi-transparent” material

Interaction Between Light & Bulk MaterialMany different possible processes can occur!

Light when it travels in a

medium can be absorbed and

reemitted by every atom in its path.

Refraction, Reflection and Dispersion

Determined by refractive index; n

Small n

High n

n1 = refractive index of material 1

n2 = refractive index of material 2

Total Internal Reflection

n 2

i

n 1 > n2i

Incidentlight

t

Transmitted(refracted) light

Reflectedlight

k t

i>cc

TIRc

Evanescent wave

k i k r

(a) (b) (c)

Light wave travelling in a more dense medium strikes a less dense medium. Depending onthe incidence angle with respect to c, which is determined by the ratio of the refractiveindices, the wave may be transmitted (refracted) or reflected. (a) i < c (b) i = c (c) i> c and total internal reflection (TIR).

© 1999 S.O. Kasap, Optoelectronics (Prentice Hall)

Mechanism and Application of TIROptical fibre for communication

What sort of materials do you think are suitable for fibre optics cables?

Review of optical processes

Ene

rgy

Ground level

Excited level

E = h

The atom is at least partially in an excited state.

The atom is vibrating at frequency, .

Energy levels are everything in quantum mechanics.

Review of optical processes

Before After

Absorption

Stimulated emission

Spontaneous emission

• Recall: Semiconductor Bandgaps Eg are usually in the range: 0 < Eg < 3 eV

(up to 6 eV if diamond is included)

• Also, at equilibrium, at temperature T = 0, the valence band is full & the conduction band is empty.

• Now, consider what happens if electromagnetic radiation (“light”) is shined on the material.

• In the photon representation of this radiationIf hν Eg, some

electrons can be promoted to the conduction band leaving some holes in the valence band.

• Consider various types of spectra associated with this process:

• Absorption:– Looks at the number of absorbed photons (intensity) vs. photon frequency ω

• Reflection:– Looks at the number of reflected photons (intensity) vs. photon frequency ω

• Transmission:– Looks at the number of transmitted photons (intensity) vs. photon frequency ω

• Emission:– Looks at the number of emitted photons (intensity) vs. photon frequency ω

• Each of these types of spectra are

rich, complicated, & varied!• Understanding such spectra gives

huge amounts of informationabout electronic energy bands, vibrational properties, defects,

Appearance of insulator, metal and semiconductor

Appearance in terms of color depends on the interaction between thelight with the electronics configuration of the material.

Normally, High resistivity material: insulator transparent High conductivity material: metals metallic luster and opaque Semiconductors colored, opaque or transparent, color

depending on the band gap of the material For semiconductors the energy band diagram can explain the

appearance of the material in terms of luster and color.

Question: Why is Silicon Black and Shiny?

Answer.

Need to know, the energy gap of Si Egap = 1.2eV

Need to know visible light photon energy Evis ~ 1.8 – 3.1eV

Evis is larger than Silicon Egap All visible light will be absorbed Silicon appears black Why is Si shiny? Significant photon absorption occurs in silicon, because there are

a significant number of electrons in the conduction band. These electrons are delocalized. They scatter photons.

Colors of Semiconductors

I B G Y O R

Evis= 1.8eV 3.1eV

•If Photon Energy, Evis > Egap Photons will be absorbed

•If Photon Energy, Evis < Egap Photons will be transmitted

•If Photon Energy is in the range of Egap ;

•Those with higher energy than Egap will be absorbed.

•We see the color of the light being transmitted

•If all colors are transmitted = White

Why is glass transparent?

Glass is an insulator (huge band gap) The electrons find it hard to jump across a big energy gap:

Egap >> 5eV Egap >> E visible spectrum ~ 3.1- 1.8eV All colored photons are transmitted, no absorption, hence light transmission –

transparent. Defined transmission and absorption by Lambert’s law:

I = Io exp (- l) I = incident beam Io = transmitted beam = total linear absorption coefficient (m-1) = takes into account the loss of intensity from both scattering centers

and absorption centers. = approaching zero for pure insulator.

What happens during photon absorption process?

Photon interacts with the lattice

Photon interacts with defects

Photon interacts with valance electrons

Absorption – an important phenomenon in describing optical properties of semiconductors

Light, being a form of electromagnetic radiation,interacts with the electronic structure of atoms of amaterial.

The initial interaction is one of absorption; that is,the electrons of atoms on the surface of a materialwill absorb the energy of the colliding photons oflight and move to the higher-energy states.

The degree of absorption depends, among otherthings, on the number of free electrons capable ofreceiving this photon energy.

Absorption Process of Semiconductors

The interaction process is a characteristic of a photon and depends on the energy of the photon

Low-energy photons interact principally by ionization or excitation of the outer orbitals in solids’ atoms.

Light of low-energy photons (< 10 eV) is represented by infrared (IR), visible light, and ultraviolet (UV) in the electromagnetic spectrum.

High-energy protons (> 104 eV) such as x-rays (and gamma rays) scatter mainly elastically and are used for structure determination

The minimum photon energy required to excite and/or ionize the component atoms of a solid is called the absorption edge or threshold.

Absorption Process of SemiconductorsA

bsor

ptio

n co

effic

ient

(),

cm-1

Photon energy (eV)Absorption spectrum of a semiconductor.

Vis

Eg

~ v

is

Wavelength (m)

IRUV

Important region:

Valance-Conduction Absorption

h

Conduction band, EC

Valance band, EV

EgapEphoton

Process requires the lowest E of photon to initiate electron jumping (excitation)

• EC-EV = h

• EC-EV = Egap

• If h > Egap then transition happens•Electrons in the conduction band and excited.

Absorption

Types Direct and Indirect photon absorption For all absorption process there must be: Conservation of energy Conservation of momentum or the wavevector

The production of e-h pairs is very important for various electronics devices especially the photovoltaic and photodetectors devices.

The absorbed light can be transformed to current in these devices

Direct Band Gap

K (wave number)h

Conservation of E

h = EC(min) - Ev (max) = Egap

Conservation of wavevector

Kvmax + photon = kc

E

Direct vertical transition

Momentum of photon is negligible

Interband absorption in indirect gap semiconductors

Indirect-gap semiconductor: highest occupied and lowest unoccupied state have k≠0

Direct transitions possible for k0 strong direct interband absorption

occurs at E > Egap

Other possibility: momentum and energy can be conserved by photonabsorption and simultaneous absorption or emission of a phonon:

Indirect transitions possible with ‘assistance of a phonon’

Shown here are optically induced transitions

- during phonon emissiona phonon is generated in the process

- during phonon absorptiona phonon is generated in the process

Egap

Egap

Excitons

Excitons are combined electron-hole states:

A free electron and a free hole (empty electronic state in the valence band)exert Coulomb force on each other:

hydrogen-like bound states possible: excitonic states

eh

Coulomb force

n=3n=2n=1

E

k

Eb

Wave functions of electron and hole look similar to free electron and free hole

Note: exciton can move through crystal, i.e. not bound to specific atom!

Eb is the excitonbinding energy =

energy released uponexciton formation, or

energy required forexciton breakup

Excitonic absorption

Light can excite an electron from the valence band and generate an excitonat energies slightly below the bandgap

see absorption at Ephot = Egap – Eb (absorption slightly below Egap)

Exciton binding energy on the order of a few meVThermal energy at room temperature: kT ~ 25 meV

exciton rapidly dissociates at room temperature absorption lines broaden / disappear for higher temperatures

E

k

Eb

Optical transitions related to dopant atoms

Ga: 3 valence electrons

Si: 4 valence electrons

As: 5 valence electrons

Donor levels

Substitute Si atom with As atom (impurity atom in the Si lattice): weakly bound extra valence electron

Low T: donors neutral, electron weakly boundlow energy light can excite donor electron in to conduciton band

Binding energy Ed similar to kT at room temperature (‘RT’):At room temperature the bound electron is quickly released impurity mostly ionized at RT : Arsenic is a donor in Si

At RT such transitions are typically too broad to observe

Low T

RT

Acceptor levels

Substitute Si atom with Ga atom : empty electronic state just above the Si valence band: at finite temperature, Si valence electron may fill acceptor level location of unoccupied valence state (hole) can orbit the charged Ga dopant

Binding energy Ea similar to kT at room temperature (‘RT’):At room temperature the hole can leave the dopant, producing a ‘free charge’

‘hole’ = available electron state

Infrared absorption due to dopants

Dopant binding energies low: donor level related absorptions invisible at RT,but observable at low temperatures

Example: direct valence band → acceptor level absorption in boron doped Si

Transition at ~40 meV absorption at 30 m : infrared

Dopant related transitions

Possible dopant related transitions:

Typically visible at low T, but not clearly observable at RT

Free carrier absorption

At RT, predominant dopant related absorption is free carrier absorptionin which a photon excites an electron into a higher lying state

Example: p-type semiconductors: filled states in the conduction band:optical transitions possible at Ephot < Egap !

Free electrons: absorption typically indirect phonon-assisted transition

Free holes can make direct transitionsfrom the heavy-hole band to the light-hole band

holes cause stronger free carrier absorption than electrons

Free carrier absorption

Free electron absorption can be described by the Drude model

Dopant levels in semiconductors range from ~1014 - 1018 /cm3

which is ~108 – 104 lower than free electron densities in metals

Plasma frequency of doped semiconductors 104 - 103 lower than of metals: IR

3

2

3

2

2

2

)(",1)('

ppr

pr

2

2

2

2

)(")(p

p

ccc

At frequencies above plasma frequency, εr is complex and is described by

Electron FCA up for lower energies

Free hole absorption less well defined

Derivation of Optical Absorption Coefficient in Direct Semiconductors

Chuang Ch. 9

Outline of derivation

•Absorption Coefficient:

•Examples: lasers, solar cells, etc.

•Fermi’s Golden rulepoly-Si Solar Cells

( )( , ) ( , ) zoI z I z e

( )

•Absorption Coefficient & Simplifications

•Time-dependent perturbation

•Direct-gap net absorption rate

Fermi’s Golden Rule

2'2 ( ( ) ( ))i f fi f i f iW H E E E E

Absorption Emission

Ei

EiEf

Ef

Direct-Gap Net Absorption Rate

22 2 ' 1v c

vc cv c v v ck k

πR H δ(Ε Ε )f fV

Ec

Ev

k

E

Assumptions:kv = kc = k

Undoped, low excitationfv = 1, fc = 0

22 2 'cv c vk

π H δ(Ε Ε )V

Rabs

2 2

*2vh

kEm

2 2

*2c ge

kE Em

Absorption Coefficient

•How to find H’cv?

( )/absR

P2

2 2

2 2 '( / 2) cv c v

kr o o

π H δ(Ε Ε )n c A V

)(ˆ2

1),(2

rVAepm

trHo

pem

eArHo

o ˆˆ2

)('

rdψrH'ψ H v*ccv

3)('

22

2

2 ˆ cv c vkr o o

πe e p δ(Ε Ε )n c m V

( )

Momentum matrix element

(no. of photons absorbed per second per unit volume)

(no. of injected photons per second per unit area)

More Practical Form

22

2ˆ( ) ( )cv J g

r o o

πe e p N En c m

11

22

e

gocv m

Emp

2 2 22 3

32

2ˆ22

cv gkr o o r

πe V k e p δ(E ) kn c m V m

( )

2 2 22 3

32

2ˆ22

cv gr o o r

πe k e p δ(E )d kn c m m

3 / 22

2 1/ 22 2 2

21ˆ( )2

rcv k g k k

r o o

mπe e p δ(E )dn c m

•Using E-k (dispersion) relationship:

3 / 2

2 2

21( )2

rJ k k

mN

Conclusions

Yu, Cardona: p. 260

Absorption Coefficient at 5K

Red: calculation at 300K

•Example: InSb•Eg = 0.17eV

•Not 100% accurate•Parabolic band approximation•nr depends on wavelength•Exciton absorption below bandgap

•Different for 2D,1D,0D•Density of States