Electrical Properties of Matter

Fields and WavesEE 6316

Spring 20081-30-2008

Topics:• Review of Dipole, Polarization ,Susceptibility etc in isotropic medium

• Classical Harmonic Oscillator Model

•Abraham Lorentz Equation

•Damping

•Dispersion plots •Application of the model : some examples

•Plasma

• Dielectric behavior in anisotropic medium

•Permittivity tensor

• Example: Modulator

Review+

+

External fieldNo field

EP

whereEEEPED

e

re

r

r

rrrrrr

0

000

1

)1(

εχ

εεεχεε

=

==+=+=

∑ Δ=

→Δ i

i

V VpP

0lim

r

Qlpi =l

Q

In terms of molecular polarizability,α

localENPrr

α=

α has contribution due to electronic, ionic and permanent dipole polarization

Classical Electron Oscillator (CEO)Model

• electron cloud is modeled as a spring mass system, with attractive electric force between nucleus and electron cloud as the spring providing the restoring force

-

+-

l

E field

Mechanical Equivalent model of CEO

Mathematical development of CEO Model

( ) ⎟⎠⎞

⎜⎝⎛+−

==

=

=

=++⇒

==++

mdj

eEmQ

ell

ms

md

where

eEmQl

dtdl

dtld

eEQtEQsldtdld

dtldm

tj

tjss

tj

tj

ωωω

ω

α

ωα

ω

ω

ω

ω

220

0

0

0

0202

2

02

2

2

2

)(

r

r

rr

In the presence of an applied electric field: Abraham Lorentz Equation

d: Damping coefficient

s: spring constant

:Resonant frequency

Steady state solution

CEO Model :Damping Conditions

Underdamped3) α<ω0

Critically damped2) α=ω0

overdamped1) α>ω0

Classification of SolutionCondition

Frequency dependent dielectric response: Dispersion

( )

( ))Im()Re(

220

2

00

220

2

εεωωω

εεε

ωωω

j

mdj

mQN

EP

mdj

EmQN

P

−=⎟⎠⎞

⎜⎝⎛+−

⎟⎠⎞

⎜⎝⎛

+=+=

⎟⎠⎞

⎜⎝⎛+−

⎟⎠⎞

⎜⎝⎛

=

r

r

r

rPolarization:

Permittivity:

Dispersion Relation of Permittivity

Similar relationship for relative permittivity hence refractive index can be derived.The real and imaginary parts of refractive index are related to each other through Kramers-Kronig relationship.

Dispersion Plots

Application of CEO Model: Example

Ref: APPLIED OPTICS / Vol. 27, No. 12 /pp.2549-2553/1998

Further Application of CEO Model

• This model can be extended to model optical processes in semiconductors– Spontaneous and Stimulated emission– Rabi Oscillation– Collision Broadening – Radiative lifetimes etc

Plasma• Plasma is a sea of free electrons in a

background of positive ions of same density.

• Due to existence of free electrons, they are very conductive.

• Plasma response to electrical field is very strong too

Dynamics of Plasma• Motion of free electrons is governed by collision frequency f

Efm

fnejfmnej

jfmEneEjJEjHX

jfmEnevneJ

jfmEev

fvmEedtvdm

r

rrrrr

rrr

rr

rrr

⎥⎦

⎤⎢⎣

⎡

+−⎟⎟⎠

⎞⎜⎜⎝

⎛+

−=

++=+=∇

+=−=

+−=⇒

−−=

)()(

)(

)(

)(

22

2

22

2

0

2

00

2

ωωωεω

ωωεωε

ω

ωFor sinusoidal variation:

Convection Current:

Inserting in Curl Eqn:

Real Part Imaginary part=0 as f 0

Plasma Frequency

mne

where

mne

p

p

0

2

2

2

02

2

0 1

εω

ωω

εω

εε

=

⎟⎟⎠

⎞⎜⎜⎝

⎛−=−=

Plasma frequency

Permittivity is negative for frequencies below plasma frequency. Physically this means wave is reflected off of plasma and attenuated inside.

Anisotropic Media: Dielectric tensors

[ ] [ ][ ]EDrr

ε=

zzzyzyxzxz

zyzyyyxyxy

zxzyxyxxxx

z

y

x

zzzyzx

yzyyyx

xzxyxx

z

y

x

EEED

EEED

EEED

EEE

DDD

εεε

εεε

εεε

εεεεεεεεε

++=

++=

++=

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

Electro-Optic Effect: Modulator

cEnlr me

m 2

333ω−

=ΔΦModulation Index:

Suggested Reading

• Chapter 2 :Advanced Engineering Electromagnetics by Constantine A. Balanis

• Chapter 13: Fields and Waves in Communication Electronics by Simon Ramo