Lecture 21

Optical properties

Incoming light Reflected light

Transmitted light

Absorbed lightHeat

Light impinging onto an object (material) can be absorbed, reflected, or transmitted.

If the medium is transparent (no absorption):

Reflectivity (R) = Reflected fraction Incident fraction Transmitivity (T) = Transmitted fraction

Incident fraction

1 RT

In more detail:

Reflection: the incident and exit angle with respect to the normal to the surface are identical.

Transmission: the refractive index change from one material to the next determines the change in direction from outside to inside the medium.

Scattering: on a rough surface, locally the surface normal varies, resulting in a broad macroscopic distribution of “reflected” light called scattering.

Absorption: the incoming light partially penetrates the material, transfers energy to electron and/or lattice excitations. These in turn may relax back to the ground state by emitting light and or phonons.

1n

2n

vcn Refractive index n is the ratio of the vacuum speed of light c to the speed of light in the medium v. Frequency is the constant.

2211 sinsin nn

Transmission

vcn

The amplitude of the reflected light depends on the polarization of the light and the dielectric properties of the material. The components of E parallel and perpendicular to the plane of incidence.

2112

2112||

2211

221121 coscos

coscoscoscoscoscos

nnnnr

nnnnr

Reflection

A finite fraction of the light intensity dI is absorbed over a small distance dx into the material. The absorption coefficient is a material property.

In a semiconductor is proportional to the frequency of visible light and a material property . means that there is no absorption.

)()( )(oo

xxo xIIeIxIxI o

Absorption

cf4

0

Windows Mirrors

X-rays for detectors

xxxo exAeIxI o )()( )(

2112

2112||

2211

221121 coscos

coscoscoscoscoscos

nnnnr

nnnnr

vcn 2211 sinsin nn

The parameters T, R, and A (for absorption) along with n and are optical material constants.

1 RT When there is no absorption A=0 is zero.

Transmission

Reflection

Absorption

Depth profiling of shallow water

Reflection data for Al, Ag, Au, and Cu

Band diagram for Al

Silicon Aluminum Silver Copper

Density of states per energy

Energy

Fermi level

Au

2.3 eV

Ag

4.0 eV

Cu

2.0 eV

Energy from 3d levels

2 eV

4 eV

Al

more absorption

Background: Light moving in z-direction with electrical polarization in x-direction

xo

xx Et

Et

Ez

c

2

2

2

22

)(cznti

ox eEE

oo

iin2

ˆ 2

From Maxwell’s equations

“trial” solution (note: complex)

Complex index of refraction

o

o

ni

ininninn

222

121

222

ˆ

2ˆˆ

nnn o 24 222

1

)()ˆ(czntiz

co

cznti

ox eeEeEE

damping plane wave

oncc 22EI From

Reflectivity22

222

)1()1(

1ˆ1ˆ

nn

nn

IIRo

r

In a metal with low frequencies and dielectric values less than 10:

13

17

1010

2 o

oo

iin2

ˆ 2 22

2

o

n

onn 422

oR 41

2/121

1224122

1212

1ˆ1ˆ

22

2

22

222

nnn

nnnnn

nnnn

nn

IIRo

r

Hagan-Rubens relation

Theoretical Classical Drude model Electrons are free within a band and can be accelerated by external fields and can loose energy by scattering. They transition between energy states within a band. Typically phonons are involved. Transitions are caused by phonon-electron scattering or by impurity-electron scattering (in insulators).

The phenomenological model by Drude includes these contributions

results in

And from there to the complex current density

The real and imaginary parts are

Eqvmvt

m ˆˆˆ

)ˆRe()ˆRe( titi evveEE

Eim

qvEqvmivmvmi ˆ11ˆˆˆ1ˆˆ

Eim

nqvnqj ˆ11ˆˆ

2

io

1

)(

221)(Re

o 221

)(Im

o

complex

Estimate of scale

At =1THz=1012/s and

and for the non-complex versions:

s

CmSkg

nqm o 14

219328

731

2 104.2106.1105.8108.5101.9

1024.0

EEmnqE

imnqvnqj o

ˆˆˆ11ˆˆ

22

221)(Re

o 221

)(Im

o

Conductivity Damping

Au

Ag

Al

5um 2um 1um 500nm 200nm

60 150 300 600 1500 THz

oo

pp m

Ne

2

Theoretical backgroundLorentz theory of bound electrons

electrons are bound to nuclei by “springs”, which determine the natural frequency. Recall the harmonic oscillator

An external electric field displaces charges and creates a dipole. This is assumed to be the oscillator. Vibrations are forced by an external AC field.

At the resonance frequency the maximum amount of energy is absorbed.

The combination explains free electrons with high absorption (R near zero) for low frequencies ( ) for the IR region of the light spectrum. The bound electrons, oscillator explain the absorption bands.Insulators and semiconductors are explained by the harmonic oscillator of bound electrons.

FailuresWhy should electrons be free at low frequencies and bound at higher frequencies?

Quantum MechanicsSolves the dilemma and explains the absorption (or not) of light with intra-band and inter-band transitions as well as direct and indirect energy gaps.

mkkxF o

0o

This is mostly observed in metals, where small energy transitions are possible due to partially filled bands. In semiconductors and insulators this does not occur. Exceptions may be highly doped semiconductors where the Fermi Energy is right at the conduction (or valence) band.At low frequencies this effect dominates (not quantized).

Window coatings with materials such as ITO are used to transmit visible light but reflect IR light. Loss of heat is minimized in the winter or the room temperature remains cooler in the summer.

Conclusions

Inter-band transitions correspond to the bound-state version of the Lorentz model, whileIntra-band transitions correspond to free electron effects of the Drude model.

The sharp absorption lines from atoms (i.e. single resonance frequencies) give way to broad bands and hence absorption bands and not absorption frequencies. The plasma frequency corresponds to the edge where the reflectance of a metal turns up.

Materials Work function (eV)

threshold frequency (1014/s

Ceasium 1.91 4.62

Rubidium 2.17 5.25

Potassium 2.24 5.42

Lithium 2.28 5.51

Sodium 2.46 5.95

Zinc 3.57 8.63

Copper 4.16 10.06

Tungsten 4.54 10.98

Silver 4.74 11.46

Platinum 6.30 15.23

0 1 2 3 4 5 6 70

2

4

6

8

10

12

14

CaesiumRubidiumPotassiumLithiumSodium

Zinc

CopperTungsten

Silver

Platinum

Thre

shol

d fre

quen

cy (1

014H

z)

Work function (eV)

Inter-band transitionsElectrons transition from one band to another (usually from the valence to the conduction band)

Direct gap GaN vs Indirect gap Si

Direct: Electrons transition “vertically” without the “assistance” of phonons. The momentum vector k remains constant (the momentum of the photon is very much smaller and insignificant here). There is a vast number of near continuous transitions possible. The band gap merely represents the lower minimum.

Indirect: Phonons are created or absorbed to accommodate the required change on momentum vector k. In metals these transitions play a miniscule role (100 to 1000 times smaller in intensity). However, in semiconductors they play a big role. Keep in mind that jumps to higher bands (not shown) are also possible.