Metals
Lecture 3: Optical Properties of Insulators,
Semiconductors, and Metals
5 nm
Course Info
Next Week (Sept. 5 and 7)
no classes
First H/W is due Sept. 12
The Previous Lecture
Origin frequency dependence of χ in real materials
• Lorentz model (harmonic oscillator model)
e-
n (! )
n'
n '' n ' = 1
!0
+
Nucleus
!0
!
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.. )
Optical properties of molecules, nanoparticles, and microparticles
Classification Matter: Insulators, Semiconductors, Metals
Bonds and bands
• One atom, e.g. H. Schrödinger equation:
E
H+
• Two atoms: bond formation
H+
?
+H
Every electron contributes one state
• Equilibrium distance d (after reaction)
Classification Matter
~ 1 eV
• Pauli principle: Only 2 electrons in the same electronic state (one spin & one spin )
Classification Matter
Atoms with many electrons
Empty
outer orbitals
Partly filled
Outermost electrons interact
Form bands
Energy
valence orbitals
Filled
Inner
shells
Distance between atoms
Electrons in inner shells do not interact
Do not form bands
Classification Matter
Insulators, semiconductors, and metals
• Classification based on bandstructure
Dispersion and Absorption in Insulators
Electronic transitions
No transitions Atomic vibrations
Refractive Index Various Materials
3.4
Refractive index: n’
3.0
2.0
1.0
0.1
1.0
10
λ (µm)
Color Centers
• Insulators with a large EGAP should not show absorption…..or ?
• Ion beam irradiation or x-ray exposure result in beautiful colors!
• Due to formation of color (absorption) centers….(Homework assignment)
Absorption Processes in Semiconductors
Absorption spectrum of a typical semiconductor
E
EC
Phonon
Photon
EV
!Phonon
Excitons: Electron and Hole Bound by Coulomb
Analogy 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
• Binding energy electron:
n=3
-1.51 eV
EB = #
me e 4
2
2 (4!" 0 hn )
13.6
= # 2 eV , n = 1, 2,3,...
n
Where: me = Electron mass, ε0 = permittivity of vacuum,
n = energy quantum number/orbit identifier
h = Planck’s constant
Binding Energy of an Electron to Hole
Electron orbit “around” a hole
• Electron orbit is expected to be qualitatively similar to a H-atom.
• Use reduced effective mass instead of me:
1/ m* = 1/ me + 1/ mh
• Correct for the relative dielectric constant of Si, εr,Si (screening).
e-
εr,Si
h-
m* 1
Binding energy electron: EB =
13.6eV , n = 1, 2,3,...
2
me !
• Typical value for semiconductors:
EB = 10meV ! 100meV
• Note: Exciton Bohr radius ~ 5 nm (many lattice constants)
Optical Properties of Metals (determine ε)
Current induced by a time varying field
• Consider a time varying field:
E (t ) = Re {E (! )exp ("i!t )}
• Equation of motion electron in a metal:
d 2x
dv
v
m 2 =m
= "m " eE
dt
dt
!
relaxation time ~ 10-14 s
• Look for a steady state solution:
• Substitution v into Eq. of motion:
• This can be manipulated into:
• The current density is defined as:
v (t ) = Re {v (! )exp ("i!t )}
mv (! )
#i! mv (! ) = #
# eE (! )
"
#e
v (! ) =
E (! )
m (1 " # i! )
J (! ) = "nev
Electron density
• It thus follows:
J (! ) =
(ne
2
m)
(1 " # i! )
E (! )
Optical Properties of Metals
Determination conductivity
• From the last page:
J (! ) =
(ne
2
m)
(1 " # i! )
E (! )
! (" ) =
• Definition conductivity: J (! ) = " (! )E (! )
(ne
2
m)
(1 # $ i" )
=
!0
(1 $ i"# )
ne 2!
where: " 0 =
m
Both bound electrons and conduction electrons contribute to ε
• From the curl Eq.:
#$ H =
"! E (t )
"D
+J = B
+J
"t
"t
{
}
• For a time varying field: E (t ) = Re E (! )exp ("i!t )
&! B E (t )
$
" (# ) %
'( H =
+ J = )i#! B (# ) E (# ) + " (# ) E (# ) = )i#! 0 * ! B (# ) )
+ E (# )
&t
i
!
#
0
,
! EFF (" )
Currents induced by ac-fields modeled by εEFF
• For a time varying field: " EFF = " B $
Bound electrons
!
!
= "B + i
i" 0#
" 0#
Conduction electrons
Optical Properties of Metals
Dielectric constant at ω ≈ ωvisible
• Since ωvisτ >> 1: ! (" ) =
• It follows that:" EFF
! 0 (1 + i"# ) ! 0
!0
!0
=
$ 2 2 +i
2 2
1
%
i
"#
"#
(
) (1 + " # ) " #
!0
!0
!
= "B + i
= "B + i
%
3 2
" 0#
" 0# $
" 0# 2$
! 0 ne 2
• Define: " =
=
(% 10eV for metals )
# 0$ # 0 m
2
p
" EFF = " B $
! p2
!
2
+i
! p2
! 3#
Bound electrons
• What does this look like for a real metal?
Free electrons
Optical Properties of Aluminum (simple case)
0
10
5
10
15
8
8
6
" EFF = " B $
6
4
Dielectric functions
10
4
ε’’
2
ħωp
0
-2
Aluminum
-4
0
-2
-4
ε’
-6
2
-6
-8
-8
-10
n’
0
-10
n’’
5
10
Photon energy (eV)
! p2
+i
2
!
! p2
! 3#
• Only conduction e’s
contribute to: ! EFF
!B "1
" EFF , Al $ 1 %
! p2
!
2
+i
! p2
! 3#
• Agrees with:
! (" )
!'
! '' ! ' = 1
15
!0
!
Ag: effects of Interband Transitions
Reflectance (%)
100
60
• Ag show interesting feature in reflection
Ag
40
• Both conduction and bound e’s contribute
20
10
6
4
2
0
to ! EFF
2
4
6
8
10
12
Photon energy (eV)
6
• Feature caused by interband transitions
! ib
4
Excitation bound electrons
interband
ε’
2
0
-2
• For Ag: ! B = ! ib " 0
! EFF
" f ' = 1#
-4
-6
0
2
4
6
8
! p2
!2
10 12
Photon energy (eV)
" EFF = " ib $
! p2
!
2
+i
! p2
! 3#
Summary
Light interaction with small objects (d < λ)
• Light scattering due to harmonically driven dipole oscillator
Nanoparticles
• Insulators…Rayleigh Scattering (blue sky)
• Semiconductors....Resonance absorption at ħω ≥ EGAP , size dependent fluorescence…)
• Metals…Resonance absorption at surface plasmon frequency, no light emission)
Microparticles
• Particles with dimensions on the order of λ
Enhanced forward scattering
λ-independent (white clouds)
• Microspheres with diameters much larger than λ
Intuitive ray-picture useful
Rainbows due to dispersion H20
Applications: resonators, lasers, etc…