Download Electron on a Spring Model

Electrical Properties of Matter
Fields and Waves
EE 6316
Spring 2008
1-30-2008
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
Q
+
+
l
No field
pi = Ql
External field
r
r r
r
r
r
D = ε 0 E + P = ε 0 (1 + χ e ) E = ε 0ε r E = εE
where
r
1 P
r
χe =
ε0 E
r
P = lim
pi
∑i ΔV
ΔV → 0
In terms of molecular polarizability,α
r
r
P = NαElocal
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
-
E field
+
l
Mechanical Equivalent model of CEO
Mathematical development of CEO Model
In the presence of an applied electric field: Abraham Lorentz Equation
r
r
d 2l
dl
jω t
m
d
sl
Q
E
t
Q
E
e
+
+
=
=
(
)
0
dt 2
dt
d: Damping coefficient
d 2l
dl
Q r
jω t
2
s: spring constant
l
E
e
α
ω
⇒
+
+
=
2
0
0
2
dt
dt
m
where
d
α =
2m
s
ω0 =
m :Resonant frequency
Q r
E 0 e jω t
m
l ss = l 0 e j ω t =
Steady state solution
d
⎛
⎞
ω 02 − ω 2 + j ω ⎜
⎟
m
⎝
⎠
(
)
CEO Model :Damping Conditions
Condition
Classification of Solution
1) α>ω0
overdamped
2) α=ω0
Critically damped
3) α<ω0
Underdamped
Frequency dependent dielectric response: Dispersion
⎛ Q2 ⎞ r
N ⎜ ⎟E
⎝m⎠
r
Polarization: P =
(ω
2
0
)
⎛d⎞
− ω 2 + jω ⎜ ⎟
⎝m⎠
r
P
Permittivity: ε = ε 0 + r = ε 0 +
E
(ω
⎛ Q2 ⎞
N⎜ ⎟
⎝m⎠
2
0
)
⎛d⎞
− ω 2 + jω ⎜ ⎟
⎝m⎠
= Re(ε ) − j Im(ε )
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
r
r
r
dv
m
= −eE − mv f
dt
r
r
eE
⇒v =−
For sinusoidal variation:
m ( f + jω )
r
2
r
r
ne E
Convection Current:
J = − nev =
m ( f + jω )
r
2
r
r r
r
ne E
∇XH = jωε 0 E + J = jωε 0 E +
Inserting in Curl Eqn:
m ( f + jω )
⎡⎛
⎤r
⎞
ne 2
ne 2 f
⎟− j
= jω ⎢⎜⎜ ε 0 −
E
2
2 ⎟
2
2 ⎥
m( f + ω ) ⎠
mω ( f + ω ) ⎦
⎣⎝
Real Part
Imaginary part=0 as f
0
Plasma Frequency
⎛ ω p2 ⎞
ne 2
ε = ε0 −
= ε 0 ⎜1 − 2 ⎟
2
⎜ ω ⎟
mω
⎝
⎠
where
Plasma frequency
ne 2
ωp =
ε 0m
Permittivity is negative for frequencies below plasma frequency. Physically
this means wave is reflected off of plasma and attenuated inside.
Anisotropic Media: Dielectric tensors
[]
[]
r
r
D = [ε ] E
⎡ Dx ⎤ ⎡ε xx
⎢ D ⎥ = ⎢ε
⎢ y ⎥ ⎢ yx
⎢⎣ Dz ⎥⎦ ⎢⎣ε zx
ε xy ε xz ⎤ ⎡ E x ⎤
⎥⎢ ⎥
ε yy ε yz ⎥ ⎢ E y ⎥
ε zy ε zz ⎥⎦ ⎢⎣ E z ⎥⎦
Dx = ε xx E x + ε xy E y + ε xz E z
D y = ε yx E x + ε yy E y + ε yz E z
Dz = ε zx E x + ε zy E y + ε zz E z
Electro-Optic Effect: Modulator
Modulation Index:
− ωlr n Em
ΔΦ m =
2c
3
33 e
Suggested Reading
• Chapter 2 :Advanced Engineering
Electromagnetics by Constantine A.
Balanis
• Chapter 13: Fields and Waves in
Communication Electronics by Simon
Ramo