Institute of Solid State PhysicsTechnische Universität Graz

19. Optical Properties of Insulators and Metals

Dec. 9, 2019

Dielectric function of silicon

absorption

extinction K

refractive index n

( ) ( ) ( )n iK

4 fKc

Inter- and intraband transitions

( ) ( )c vE k E k

When the bands are parallel, there is a peak in the absorption (")

Optical spectroscopy has developed into the most important experimental tool for band structure determination. - Kittel

Ashcroft and Mermin

Dielectric function of BaTiO3

Dielectrics

Dielectrics used as electrical insulators should not conduct.

Large breakdown field.

Low AC losses.

Sometimes a low dielectric constant is desired (CMOS interconnects)

Sometimes a high dielectric constant is desired (supercapacitors).

Breakdown field

Typically 105-106 V/cm

AC losses - loss tangent

In an ideal capacitor, current leads voltage by 90o.

21 0 tan2V

Because the dielectric constant is complex, in real materials current leads voltage by 90o - .

Power loss =

Becomes more of an issue at high frequencies (microwaves)

0 0

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Loss tangent

Polarizability

• Orientation polarizability• Space charge polarizability

• Ionic polarizability• Electronic polarizability

Overdamped modes

Underdamped modes

Polarizability

Kittel

Orientation (dipolar) Polarizability

For materials (gases, liquids, solids) with a permanent dipole moment.

The theory is very similar to paramagnetism.

1T

Curie law

Orientation Polarizability

Ion jumps.

doubly ionized

Orientation (dipolar) Polarizability

For low frequencies the dipoles can reorient with the field but at high frequencies they can't respond fast enough.

Overdamped modeImpulse response function

Susceptibility

Space charge polarizability

Multiple phases are present where one phase has a much higher resistivity than the other. Charge accumulates at the interfaces of the phases.

Like a network of resistors and capacitors.

This results in an overdamped mode.

Ionic Polarizability

Displacement of ions of opposite sign. Only in ionic substances.

This is an underdamped mode in the infrared.

Electronic polarizability (all materials)

P Np N E

density

dipole moments

polarizability

Schematic dielectric function of water from Wikipedia

Water

Source: Classical Electrodynamics,J.D. Jackson

AC Conductivity

For constant voltage, conductors conduct and insulators don't.

For low ac voltages in a conductor, electric field and the electron velocity are in-phase, electric field and electron position are out-of-phase.

For low ac voltages in an insulator, electric field and the electron position are in-phase, electric field and electron velocity are out-of-phase.

At high (optical) frequencies the in-phase and out-of-phase component of the response is described by the dielectric function.

Conductivity / Dielectric function

0 0

P nexE E

0

1 1i

j nev i nexE E E

0i

v i x

Below about 100 GHz the frequency dependent conductivity is normally used.Above about 100 GHz the dielectric function is used (optical experiments).

, , i t i t i tv v e x x e E E e Harmonic dependence

Diffusive transport (low frequencies)

sc

dvF ma eE m

scd

e E vm

e dE v

scme

2nenem

Diffusive metal

n nm mj E

The current is related to the electric field

( ) ( ) ( )dv t ev tm eE tdt

The differential equation that describes how the velocity changes in time is:

n nm mv E

The impulse response function :

1( ) exp 0etg t tm m

Inertial term

Steady state solution

Diffusive metal

( ) ( ) ( )dv t ev tm eE tdt

Assume a harmonic solution E()eit, v()eit

The differential equation is:

1 ( ) ( )i m v Ee

1

12 2

11 11

v ii m iE e

2 2

11

j v ine neE E

low ne 2

high inem

me Scattering time

Diffusive metal

2 2

0 0

11

ne ii i

2 20

1 11

ne i

2

23 21 pi

Take the limit as goes to infinity

2

21 p

0 for 0 for 0

22

0p

nem

low frequency metal / high frequency insulator

Ibach & Lueth

Aluminum

Conducting transparent contacts for LEDs and Solar cells

Windows that reflect infrared

Reflection of radio waves from ionosphere

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ITO

http://lamp.tu-graz.ac.at/~hadley/ss2/linearresponse/dmetal.php

Ellipsometry

1 1 2 2sin sinn n

Ellipsometry

Ellipsometry measures the change of polarization upon reflection. The measured signal depends on the thickness and the dielectric constant.

http://en.wikipedia.org/wiki/Ellipsometry

Ellipsometry

The ratio of the two reflected polarizations is insensitive to instabilities of light source or atmospheric absorption.

Intraband transitions

( ) ( )c vE k E k

When the bands are parallel, there is a peak in the absorption (")

Ibac

h &

Luet

h

Intraband (d-band) absorption

1 R

y =

13.6

ev

Reflection Electron Energy Loss Spectroscopy

Fast electrons moving through the solid generate and a time dependent electric field. If the polarization moves out of phase with this field, energy will be lost. This is detected in the reflected electrons.

e-

Microwave engineering

Microwave frequencies are a few GHz

The wavelength is smaller than the circuit

Losses in metals increase with increasing frequency

Losses in dielectrics increase with increasing frequency

There is a characteristic length scale called the skin depth which tells us how far into a metal fields penetrate before they are reflected out.