Lecture 3: Optical Properties of Bulk and Nano

5 nm

The Previous LectureOrigin frequency dependence of χ in real materials

Today optical properties of materials• Insulators (Lattice absorption, color centers…)

• Semiconductors (Energy bands, Urbach tail, excitons …)

• Metals (Response due to bound and free electrons, plasma oscillations.. )

• Lorentz model (harmonic oscillator model)

( )n ω'n ''n ' 1n =

ω0ωNucleus

e-

+

0ω

Optical properties of molecules, nanoparticles, and microparticles

Classification Matter: Insulators, Semiconductors, MetalsBonds and bands

• One atom, e.g. H. Schrödinger equation:

• Two atoms: bond formation

EH+

H+ H+?

• Equilibrium distance d (after reaction)

Every electron contributes one state

Classification Matter

~ 1 eV

• Pauli principle: Only 2 electrons in the same electronic state (one spin & one spin )

Classification Matter

Empty

outer orbitals

Partly filled

valence orbitals

FilledInnershells

Distance between atoms

Ene

rgy

Outermost electrons interact

Atoms with many electrons

Electrons in inner shells do not interact

Form bands

Do not form bands

Classification MatterInsulators, semiconductors, and metals

• Classification based on bandstructure

Dispersion and Absorption in Insulators

Electronic transitions Atomic vibrations No transitions

2.0

3.0

1.0

3.4

0.1 1.0 10 λ (μm)

Ref

ract

ive

inde

x: n

’Refractive Index Various Materials

Color Centers

• Ion beam irradiation or x-ray exposure result in beautiful colors!

• Due to formation of color (absorption) centers….(Homework assignment)

• Insulators with a large EGAP should not show absorption…..or ?

Absorption Processes in Semiconductors

Phononω

EEC

EV

Photon

Phonon

Absorption spectrum of a typical semiconductor

Excitons: Electron and Hole Bound by CoulombAnalogy with H-atom

• Electron orbit around a hole is similar to the electron orbit around a H-core

• 1913 Niels Bohr: Electron restricted to well-defined orbits

+n = 1-13.6 eV

n = 2-3.4 eV

n = 3-1.51 eV

• Binding energy electron:( )

4

2 20

13.6 , 1, 2,3,...2 4

eB

m eE eV nnnπε

= − = − =h

Where: me = Electron mass, ε0 = permittivity of vacuum, = Planck’s constantn = energy quantum number/orbit identifier

h

Binding Energy of an Electron to Hole

*

2

1 13.6 , 1, 2,3,...Be

mE eV nm ε

= =

Electron orbit “around” a hole

• Electron orbit is expected to be qualitatively similar to a H-atom.

• Use reduced effective mass instead of me:

• Correct for the relative dielectric constant of Si, εr,Si (screening).

Binding energy electron:

*1/ 1/ 1/e hm m m= +

• Note: Exciton Bohr radius ~ 5 nm (many lattice constants)

10 100BE meV meV= −• Typical value for semiconductors:

e-

h-

εr,Si

Optical Properties of Metals (determine ε)Current induced by a time varying field

( ) ( ) ( ){ }Re expt i tω ω= −E E• Consider a time varying field:

• Equation of motion electron in a metal:2

2

d dm m edt dt τ

= = − −x v v E

relaxation time ~ 10-14 s

• Look for a steady state solution: ( ) ( ) ( ){ }Re expt i tω ω= −v v

• Substitution v into Eq. of motion: ( ) ( ) ( )mi m e

ωω ω ω

τ− = − −

vv E

( ) ( ) ( )1

em i

ω ωτ ω−

=−

v E• This can be manipulated into:

• The current density is defined as: ( ) neω = −J v

Electron density

• It thus follows: ( ) ( )( ) ( )

2

1ne m

iω ω

τ ω=

−J E

Optical Properties of Metals

• Definition conductivity:

( ) ( )( ) ( )

2

1ne m

iω ω

τ ω=

−J E

Determination conductivity

• From the last page:

( ) ( ) ( )ω σ ω ω=J E

( ) ( )( ) ( )

20

1 1ne m

i iσσ ω

τ ω ωτ= =

− −

2

0nemτσ =where:

( )B tε∂∂∇× = + = +

∂ ∂EDH J J

t t• From the curl Eq.:

( ) ( ) ( ){ }Re expt i tω ω= −E E• For a time varying field:

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )00

BB B

ti i

iε σ ω

ωε ω ω σ ω ω ωε ε ω ωε ω

⎛ ⎞∂∇× = + = − + = − −⎜ ⎟∂ ⎝ ⎠

EH J E E E

t

Both bound electrons and conduction electrons contribute to ε

( )EFFε ω

0 0EFF B B i

iσ σε ε εε ω ε ω

= − = +

Currents induced by ac-fields modeled by εEFF

• For a time varying field:

Bound electrons Conduction electrons

Optical Properties of MetalsDielectric constant at ω ≈ ωvisible

• Since ωvisτ >> 1: ( ) ( )( )

( )00 0 0

2 22 2

11 1

ii

iσ ωτσ σ σσ ω

ωτ ω τ ωτω τ+

= = ≈ +− +

0 03 2 2

0 0 0EFF B Bi i σ σσε ε ε

ε ω ε ω τ ε ω τ= + = + −• It follows that:

• Define: ( )2

2 0

0 0

10pne eV for metals

mσωε τ ε

= = ≈

2 2

2 3p p

EFF B iω ω

ε εω ω τ

= − +

• What does this look like for a real metal?

Bound electrons Free electrons

Optical Properties of Aluminum (simple case)

Aluminum

ε’’

ε’

n’ n’’

2 2

2 3p p

EFF B iω ω

ε εω ω τ

= − +

• Only conduction e’scontribute to:

1Bε ≈2 2

, 2 31 p pEFF Al i

ω ωε

ω ω τ≈ − +

0 5 10 15

Photon energy (eV)

0

2

4

6

8

10

-2

Die

lect

ric fu

nctio

ns

-4

-6

-8

0 5 10 15

-10

0

2

4

6

8

10

-2

-4

-6

-8

-10

ħωpEFFε

( )ε ω'ε ''ε ' 1ε =

ω0ω

• Agrees with:

Ag: effects of Interband Transitions

• Ag show interesting feature in reflection

• Feature caused by interband transitions

Ref

lect

ance

(%)

ε’

Photon energy (eV)

Excitation bound electrons

2 2

2 3p p

EFF ib iω ω

ε εω ω τ

= − +

• For Ag: 0B ibε ε= ≠

Photon energy (eV)

2

2' 1 pf

ωε

ω= −

ibε

EFFε

Ag

0 2 4 6 8 10 12

1020

4060

100

46

2

0

2

4

6

-2

-4

-60 2 4 6 8 10 12

• Both conduction and bound e’s contribute

EFFεto

interband

Summary

Microparticles

• Particles with dimensions on the order of λ

• Insulators…Rayleigh Scattering (blue sky)

• Semiconductors....Resonance absorption at ħω ≥ EGAP , size dependent fluorescence…)

• Metals…Resonance absorption at surface plasmon frequency, no light emission)

Nanoparticles

Light interaction with small objects (d < λ)• Light scattering due to harmonically driven dipole oscillator

• Microspheres with diameters much larger than λ

λ-independent (white clouds)

Enhanced forward scattering

Applications: resonators, lasers, etc…

Intuitive ray-picture useful

Rainbows due to dispersion H20