Reflectivity of metals - tu-freiberg.de

1

Hagen-Rubens: from the solution of Maxwell‘s 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 𝜈)

* Infrared

3

Hagen-Rubens: from the solution of Maxwell‘s 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

𝑁A … Avogadro constant

𝛿 … density

𝑀 … molar mass

eEvdt

dvm

eEdt

dvmF

Free electrons …

Interaction with the crystal lattice …

𝑣 … drift velocity

𝑚 … electron mass

𝐸 … electric field

𝛾 … 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

… equation of motion

… maximum drift velocity

… solution of the equation of motion

… time between two collisions

… 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 … number of free electrons (number of electron at the Fermi surface)

𝐸 = 𝐸0 exp 𝑖𝜔𝑡

𝑚𝑑2 𝑥

𝑑𝑡2= 𝑒𝐸 = 𝐸0 exp 𝑖𝜔𝑡

𝑥 = 𝑥0 exp 𝑖𝜔𝑡

𝑥0 = −𝑒𝐸0

𝑚𝜔2= −

𝑒𝐸0

4𝜋2𝑚𝑓2

𝐷 = 𝑒 𝑥

𝑃 = 𝑒𝑁 𝑥

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

𝑁f … number of free

electrons per cm³

For high frequencies

𝑛 becomes real.

Therefore imag(𝑛) = 0

For low frequencies

𝑛 becomes imaginary.

Therefore real(𝑛) = 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

10

Free Electrons without Damping

100%

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

Ohm’s law: Ej 0

Damping:

0

2

fNe

12


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

𝐷 = 𝑒 𝑥


13


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


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


22

2

221

21

1

i

n

0

2

102

0

2

2

2

1

2

4

m

eN f

1 … Plasma frequency

2 … Damping

frequency

Free electrons with damping

Frequency

Frequency

16


1

22

2

212

222

2

2122

1 2;1

nkkn

17


11

11

nn

nnR

Reflectivity:

Reflective Transparent

Absorption

Free electrons with damping

Frequency

Reflectivity (

%)

18


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 … electron mass, ´… damping, k … 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:

𝜔0 … Eigenfrequency of electrons

𝛾… 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


22

Model of Strongly Bound ElectronsPermittivity

eigenfrequency

Bound electrons with damping and eigenfrequency

Frequency

Frequency

23

Model of Strongly Bound ElectronsIndex of Refraction

eigenfrequency


Frequency

Frequency

24

Model of Strongly Bound ElectronsReflectivity

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


with Damping and Natural Frequency

IR absorption

(reflection)Absorption of

visible light

Frequency


Reflectivity (

%)

27


with Damping and Eigenfrequency

100%

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

ell’

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 – small

natural frequencies of lattice

vibrations

IR and Raman spectroscopy –

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 𝜆

30

Overview of scattering processes

Raman process

Photon

, k

Phonon

, K

Photon

´, k´

IR absorption with

two phonons

Photon

, k

Phonon

, K

Photon

´, k´

Photon – light quantum

Phonon – quasiparticle to describe lattice vibrations

Electron

spectroscopy with

x-rays

XPS

X-ray

photon

Photoelectron

ℏ𝜔 = ℏ𝜔′ ± ℏΩ

ℏ𝑘 = ℏ𝑘′ ± ℏ𝐾

31

Overview of scattering processes

Thomson process

Photon

, k

Photon

´, k´

Elastic scattering –

x-ray diffraction,

neutron diffraction,

electron diffraction

Compton process

Photon

, k

Photon

´, k´

Inelastic scattering –

x-rays, neutrons

Phonon (for

neutrons)

Electron (for

x-rays)

Emission of

characteristic x-rays +

absorption

X-ray

photon

Increase of

electron energy

…

X-ray

photon

ℏ𝜔 = ℏ𝜔′

ℏ𝑘 = ℏ𝑘′ ± ℏ𝐾ℏ𝜔 = ℏ𝜔′ + ℏΩ

ℏ𝑘 = ℏ𝑘′ + ℏ𝐾

ℏ𝜔 ≠ ℏ𝜔′

ℏ𝑘 ≠ ℏ𝑘′

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



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



2

bound

2

freetotal

boundfreetotal

nnn

Frequency


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:

„slow“ permanent dipoles can’t change

their polarization easily – decrease of

permittivity

𝜒e … electric susceptibility

𝜖r … relative permittivity

𝜖0 … vacuum permittivity

𝑁m … number density of molecules

𝛼 … polarizability

𝑘B … Boltzmann constant

𝑇 … 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

𝑄 … generated surface charge

𝑘 … material constant

… length of crystal

𝑑 … thickness of crystal

𝐹 … force

Crystal with external voltage



Temperature change of crystal


(thermal expansion)


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

40

PiezoelectricityMechanical stress

Mechanical stress

Mechanical stress

Mechanical stress

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 (½,½,½)

O: 3c (0,½,½)

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 – the microstructure plays an

important role

44

Ferroelectric Domains in a BaTiO3

Single Crystal

Total polarization of crystal increases with external voltage

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Reflectivity of metals - tu-freiberg.de