Post on 25-Mar-2022
1
Hagen-Rubens: from the solution of Maxwellβs equations ( = n) for small
frequencies
Drude: free damped electrons (classical electron theory), determine the color of
materials
Lorentz: strongly bound electrons (classical electron theory for dielectric materials)
Reflectivity of metals
100%
2
Hagen-Rubens RelationRelationship between the optical reflection and the electrical
conductivity
In the IR* range ( < 1013 s-1): /
2
0
22
2
0
22
22
1
22
1n
2
0
2
4
1
n
0
2
2222
22
22
222
412
112 small1
12
41
12
412
1
1
1
1
nnnn
knn
n
knn
nknn
n
n
n
n
R
R
0
0
41
R
Metals with good electrical
conductivity have a large
reflectivity in the IR range
(small π)
* Infrared
3
Hagen-Rubens: from the solution of Maxwellβs equations ( = n) for small
frequencies
Drude: free damped electrons (classical electron theory), determine the color of
materials
Lorentz: strongly bound electrons (classical electron theory for dielectric materials)
Reflectivity of metals
100%
4
Free Electrons (Classical Drude Theory
of Electrical Conductivity)Electron gas within the material
M
NN A
Number of atoms/electrons in alkali metals per m3
πA β¦ Avogadro constant
πΏ β¦ density
π β¦ molar mass
eEvdt
dvm
eEdt
dvmF
Free electrons β¦
Interaction with the crystal lattice β¦
π£ β¦ drift velocity
π β¦ electron mass
πΈ β¦ electric field
πΎ β¦ damping
5
Free Electrons (Classical Drude Theory
of Electrical Conductivity)
t
v
vF
m
eEv
eE
mv
tmv
eEvv
eEvv
eE
dt
dvm
eEvdt
dv
eEvdt
dvm
F
F
F
F
F
F
exp1
0
β¦ equation of motion
β¦ maximum drift velocity
β¦ solution of the equation of motion
β¦ time between two collisions
β¦ Fermi velocity
EevNj FF
m
eNF
2
6
Free Electrons without Damping (Classical Theory)
Excitation of electrons via
electromagnetic wave (light):
Equation of motion:
This equation can be solved by the substitution:
Dipole moment of an electron:
Total polarization:
N β¦ number of free electrons (number of electron at the Fermi surface)
πΈ = πΈ0 exp πππ‘
ππ2 π₯
ππ‘2= ππΈ = πΈ0 exp πππ‘
π₯ = π₯0 exp πππ‘
π₯0 = βππΈ0
ππ2= β
ππΈ0
4π2ππ2
π· = π π₯
π = ππ π₯
7
Free Electrons without Damping(Classical Theory)
Permittivity:E
P
0
1~
2
0
2
2
41~
m
Nen
2
2
0
2
2
~
411~ n
m
Ne
E
exN
Frequency
Frequency
Free electrons without damping
8
Free Electrons without Damping (Classical Theory)
11
11
nn
nnR
Reflection:
Reflective Transparent
22
2
41~
m
Nen
f
πf β¦ number of free
electrons per cmΒ³
For high frequencies
π becomes real.
Therefore imag(π) = 0
For low frequencies
π becomes imaginary.
Therefore real(π) = 0
11
1
)1(
)1(2
2
22
22
k
k
kn
knR
Free electrons without damping
Frequency
Refe
lctivity
(%)
9
The Plasma Frequency
m
Ne
m
Ne
m
Nen
ff
f
2
2
2
12
1
2
2
22
2
41
4
41
Good compliance with the
experiment for alkali metals
11
Free Electrons with Damping (Classical Drude Theory)
Excitation of electrons via an
electromagnetic wave (Light): tiEE exp0
tieEeEdt
dx
dt
xdm exp02
2
Equation of motion:
02
2
dt
xdConstant velocity of electrons:
Equation of motion with maximal drift velocity:
t
v
vF
eEF v
Drift velocity:
fF
eN
jv
Ohmβs law: Ej 0
Damping:
0
2
fNe
12
Free Electrons with Damping (Classical Drude Theory)
This equation can be solved by the substitution: tixx exp0
2
0
20
0
miNe
eEx
f
Dipole moment of an electron:
Total polarization:
tieEeEdt
dxNe
dt
xdm
f
exp0
0
2
2
2
Equation of motion:
20
0m
eEx
Complex amplitude of oscillations
π· = π π₯
π = ππ π₯
13
Free Electrons with Damping (Classical Drude Theory)
Total polarization:
e
mi
eN
EeNP
f
f
2
0
2
22
0
0
2
2
0
0
00 42
11
111
eN
mi
eN
mi
E
P
ff
Permittivity:
2
2
2
1
2
2
2
12
0
2
10
2
0
2
102
2
1
2
1
2
0
0
2
1
0
2
2
112
21
2
11
4
ii
iim
eN f
14
Free Electrons with Damping (Classical Drude Theory)
22
2
2
2
121
22
2
2
2
2
1
2
2
2
22
1
2
1
2
2
2
2
2
2
2
1222
2:Im;1:Re
112
nkkn
i
i
i
iinkknn
Permittivity (dielectric function, dielectric constant):
0
0
41
R
15
Free Electrons with Damping (Classical Drude Theory)
22
2
221
21
1
i
n
0
2
102
0
2
2
2
1
2
4
m
eN f
1 β¦ Plasma frequency
2 β¦ Damping
frequency
Free electrons with damping
Frequency
Frequency
17
Free Electrons with Damping (Classical Drude Theory)
11
11
nn
nnR
Reflectivity:
Reflective Transparent
Absorption
Free electrons with damping
Frequency
Reflectivity (
%)
18
Free Electrons with Damping (Classical Drude Theory)
Small spectrum for the
absorption of light
(absorption band),
investigated for metals and
nonmetals
100%
19
Strongly Bound Electrons(Electron Theory for Dielectric Materials)
Bonding between electron and atom is quasi-elastic harmonic oscillator
with natural frequency and damping
20
Strongly Bound Electrons
(Electron Theory for Dielectric Materials)
tieEeEkxdt
dx
dt
xdm exp02
2
Equation of motion:
m β¦ electron mass, Β΄β¦ damping, k β¦ spring constant (bond to the core)
This equation can be solved by the substitution:
:
tixx exp0
0
2
020
0
22
00
220
020
0
0
a
a
Ne
m
kmk
iNe
m
eE
im
eE
imk
eEx
Drude theory
21
Strongly Bound Electrons
(Electron Theory for Dielectric Materials)
Total polarization:
im
ENeP a
220
2
im
Nen
im
Ne
E
P
a
a
22
00
22
22
00
2
0
221
11Permittivity:
Index of refraction:
π0 β¦ Eigenfrequency of electrons
πΎβ¦ damping (electrical conductivity,
emission of photons)
22222
0
22
0
2
222222
0
22
0
22
0
2
1
42;
41
m
Ne
m
mNe aa
π = ππ π₯
22
Model of Strongly Bound ElectronsPermittivity
eigenfrequency
Bound electrons with damping and eigenfrequency
Frequency
Frequency
23
Model of Strongly Bound ElectronsIndex of Refraction
eigenfrequency
Bound electrons with damping and eigenfrequency
Frequency
Frequency
24
Model of Strongly Bound ElectronsReflectivity
eigenfrequency
Bound electrons with damping and eigenfrequency
Frequency
Reflectivity (
%)
25
Free Electrons with Damping and Bound Electrons
with Damping and Natural Frequency
eigenfrequency
2
bound
2
freetotal
boundfreetotal
nnn
Frequency
Frequency
Free and bound electrons with damping
26
Free Electrons with Damping and Bound Electrons
with Damping and Natural Frequency
IR absorption
(reflection)Absorption of
visible light
Frequency
Free and bound electrons with damping
Reflectivity (
%)
28
Dispersion CurvePolarizability (proportional to permittivity) as function of frequency (wavelength)
Slow permanent dipoles
Interaction between ions
Interaction between electrons and
atomic nuclei
Polarizability
Ion resonance
Electron resonance
Frequency
Microwave
radiation
Infrared
radiation
Ultraviolet radiation
and x-rays
Valid
ity o
f M
axw
ellβ
s e
quations
Dipole
relaxation
29
Optical AbsorptionConducting electrons
Especially in metals
Ionic crystals and
insulators are normally
transparent
Lattice vibrations
Absorption in IR range β small
natural frequencies of lattice
vibrations
IR and Raman spectroscopy β
investigation of lattice
dynamics
Core electrons
Interaction between electron
and atomic nucleus
High natural frequency
Absorption and emission of
radiation in the x-ray range
(selective filters,
fluorescence spectroscopy)
X-rays
hard soft Visible light
Core
electrons
Vibrations
Conducting
electrons
Absorp
tion c
oeff
icie
nt
Wavelength π
30
Overview of scattering processes
Raman process
Photon
, k
Phonon
, K
Photon
Β΄, kΒ΄
IR absorption with
two phonons
Photon
, k
Phonon
, K
Photon
Β΄, kΒ΄
Photon β light quantum
Phonon β quasiparticle to describe lattice vibrations
Electron
spectroscopy with
x-rays
XPS
X-ray
photon
Photoelectron
βπ = βπβ² Β± βΞ©
βπ = βπβ² Β± βπΎ
31
Overview of scattering processes
Thomson process
Photon
, k
Photon
Β΄, kΒ΄
Elastic scattering β
x-ray diffraction,
neutron diffraction,
electron diffraction
Compton process
Photon
, k
Photon
Β΄, kΒ΄
Inelastic scattering β
x-rays, neutrons
Phonon (for
neutrons)
Electron (for
x-rays)
Emission of
characteristic x-rays +
absorption
X-ray
photon
Increase of
electron energy
β¦
X-ray
photon
βπ = βπβ²
βπ = βπβ² Β± βπΎβπ = βπβ² + βΞ©
βπ = βπβ² + βπΎ
βπ β βπβ²
βπ β βπβ²
32
Special Cases
High Frequencies
Real (n) < 1, Real (n) 1, Imag (n) 0
Low reflectivity, high absorption
0.0 0.5 1.0 1.5 2.0 2.5 3.0
10-3
10-2
10-1
100
Refle
ctiv
ity
0.0 0.5 1.0 1.5 2.0 2.5 3.010
-4
10-3
10-2
10-1
TER
Pe
ne
tra
tion
de
pth
(mm
)
Glancing angle (o2Q)
Example: gold (CuKa)
= 1.5418 10-10 m
= 4.2558 10-5
b = 4.5875 10-6
11
21
1
21
211
0
2
2
b
in
fiffr
n
rn
ee
e
X-ray radiation
33
Special Cases
Weak Damping
22222
0
22
0
2
222222
0
22
0
22
0
2
1
42;
41
m
Ne
m
mNe aa
22
00
2
14
10
m
Ne a
aNe
2
202
2;00
34
Multiple Oscillators
Multiple electrons per atom with damping and natural frequency
0 0i, i
i ii
iia
i ii
iia
m
fNenk
m
fmNekn
22222
0
220
2
2
22222
0
22
22
0
0
222
1
422
41
i i
iia
i i
ia
f
m
Nenk
f
m
Nenkn
222
0
2
0
3
2
2
22
00
2
2222
1
82
41
Weak damping
35
Free Electrons with Damping and Bound Electrons
with Damping and Eigenfrequency
i ii
iia
i ii
iia
m
fNe
m
fmNe
22222
0
220
2
2
2
2
2
122
22222
0
22
22
0
0
2
2
2
2
2
11
42
41
36
Free Electrons with Damping and Bound Electrons
with Damping and Eigenfrequency
2
bound
2
freetotal
boundfreetotal
nnn
Frequency
Free and bound electrons with damping
Frequency
37
Quantum Mechanical Description of
Optical Properties
Quantum jump (band transition)
Direct IndirecthE
phononphoton
phononphoton
phonon
phonon
photon
photon
2
22
kk
k
p
h
pk
Phonon = lattice vibration
38
Polarizability
Tk
pN
B
pm
31
2
0
re a
Polarizability of molecules:
Simplified dispersion curve:
βslowβ permanent dipoles canβt change
their polarization easily β decrease of
permittivity
πe β¦ electric susceptibility
πr β¦ relative permittivity
π0 β¦ vacuum permittivity
πm β¦ number density of molecules
πΌ β¦ polarizability
πB β¦ Boltzmann constant
π β¦ temperature
Fig. 6.32. Dielectric orientation polarization: Top:
orientation of molecular dipoles for three different field
frequencies. Center: orientation distribution of dipoles
(length of arrows is equal to the probability of a
polarization in direction of the same arrow). Bottom:
resulting relaxation curve of the permittivity
39
Piezoelectricity and Pyroelectricity
Polarization without an external electric field
Change in length of the crystal
Polarization of dipole moments
Generation of a surface charge
FdkQ
π β¦ generated surface charge
π β¦ material constant
β¦ length of crystal
π β¦ thickness of crystal
πΉ β¦ force
Crystal with external voltage
Polarization of dipole moments
Change in length of the crystal
Temperature change of crystal
Change in length of the crystal
(thermal expansion)
Polarization of dipole moments
Generation of a surface charge
Fig. 6.38. Evidence of the
transversal piezoelectric effect
Fig. 6.37. Orientation of a piezoelectric quartz plate
to the parent crystal
41
FerroelectricitySpontaneous polarization (arrangement) of dipole moments
without an external electric field
Dielectric material
EEP
E
P
00
0
)1(
1
Ferroelectric material
sPEEP 00)1(
Spontaneous
polarization
42
Ferroelectric Crystals
Wyckoff positions:
Ca: 1a (0,0,0)
Ti: 1b (Β½,Β½,Β½)
O: 3c (0,Β½,Β½)
Perovskite structure
o a
b
c
Ferroelectric materials with perovskite structure:
SrTiO3, BaTiO3, PbTiO3, KNbO3, LiTaO3, LiNbO3
Ferroelectricity is connected to the crystal structure
43
Ferroelectric Domains
The whole polarization of a crystal with ferroelectric domains is smaller than
the polarization of a crystal without domains β the microstructure plays an
important role