• If we can move charge temporarily without current flow, can store even more • Bound charge around ion cores in a material can lead to dielectric properties •Two kinds of charge can cre
Trang 1Artificially Modulated Structures
Quantum Wells
EC
n=2
EV
L
If we approximate well as having infinite potential boundaries:
k = nπ for standing waves in the potential well
L
E =
2m * =
© E Fitzgerald-1999
Trang 2Photodetectors/Solar Cells
E-h pairs generated by photons with energy
h γ ≥ Eg
are separated by the built-in potential gradient at the p-n junction
The current voltage characteristics are given by
I = Io [ exp (qV kT ) −1 ] − Ip
where Ip is the photo-induced “reverse current.”
Junctions/Functions
P/n
Metal/semiconductor
Injection/diffusion/collection Blocking (reverse bias)
p-n rectifier, switch p-n-p transistor Acceleration/breakdown
Tunneling
Avalanche and tunnel diodes
Quantum devices
Trang 3
The Capacitor
d
A
C
A
Qd V
C
Q
A
Qd Edx
V
A Q t dx
E
E
o
o
o
d
d
o
o
d
t
d o
o
ε
ε
ε
ε
ε
ρ
ε
ρ
ε
ρ
=
=
=
=
=
=
=
=
=
⋅
∇
∫
∫
−
−
−
−
2
2
2
2
+V
+ + + + +
+
-
-I=0 always in capacitor
ρ
E
V
t d/2 d/2
© E Fitzgerald-1999
The Capacitor
• The air-gap can store energy!
• If we can move charge temporarily without current flow, can store even more
• Bound charge around ion cores in a material can lead to dielectric properties
•Two kinds of charge can create plate charge:
•surface charge
•dipole polarization in the volume
•Gauss’ law can not tell the difference (only depends on charge per unit area)
Trang 43.225 3
Material Polarization
χ
ε
ε
ε
ε
ε
ε
ε
+
= +
=
=
= +
=
1
1
E
P
E P
E
D
o
r
o
r
o
P is the Polarization
D is the Electric flux density or the Dielectric
displacement
χ is the dielectric or electric susceptibility
+ + + + +
+ +
- + + +
-E
P
d
A
C ε r εo
=
All detail of material response is in εr and therefore P
© E Fitzgerald-1999
Origin of Polarization
• We are interested in the true dipoles creating polarization in materials (not
surface effect)
• As with the free electrons, what is the response of these various dipole
mechanisms to various E-field frequencies?
• When do we have to worry about controlling
– molecular polarization (molecule may have non-uniform electron density)
– ionic polarization (E-field may distort ion positions and temporarily create dipoles) – electronic polarization (bound electrons around ion cores could distort and lead to polarization)
• Except for the electronic polarization, we might expect the other mechanisms
to operate at lower frequencies, since the units are much more massive
• What are the applications that use waves in materials for frequencies below the visible?
Trang 53.225 5
Application for Different E-M Frequencies
Methods of detecting
these frequencies
Cell phones
λ =14-33cm DBS (TV) λ =2.5cm
Other satellite, 10-50GHz
λ =3cm-6mm (‘mm wave’)
Fiber optics
λ =1.3-1.55 µ m
‘MMIC’, pronounced ‘mimic’
mm wave ICs
In communications, many E-M waves travel in insulating materials:
What is the response of the material (εr) to these waves?
© E Fitzgerald-1999
Wave Eqn with Insulating Material and Polarization
(
)
t
E B
x
t P E J B
x
t
D J
H
x
t
B E
x
insulating o
nonmag
r r r
r
r
r r
r r
r
r r
r
r
r
ε
ε
ε
ε
= +
=
∂
∂
=
∇
∂ +
∂ +
=
∇
→
∂
∂ +
=
∇
∂
∂
−
=
∇
2
2 2
2
2 0
0
2
t
E c
t
E
r
∂
∂
=
∂
∂
=
k
n
c k
c
k
c
c r
E r
E
e r E e e E e
E
E
optical
r
r
r
t t
r ik t r
i
→
=
=
−
=
∇
=
=
ε
ω
ε
ω
ε
ω
ϖ ϖ
ϖ
2
2
2
2
2
2
0 ) (
0
) ( )
(
) (
So polarization slows down the velocity of the wave in the material
Trang 63.225 7
Compare Optical (index of refraction) and Electrical Measurements of ε
Material Optical, n2 Electrical, ε
Only electronic polarization
Electronic and ionic polarisation
Electronic, ionic, and molecular polarisation
Polarization that is active depends on material and frequency
© E Fitzgerald-1999
Microscopic Frequency Response of Materials
• Bound charge can create dipole through charge displacement
• Hydrodynamic equation (Newtonian representation) will now have a restoring force
• Review of dipole physics:
r
d
Dipole moment:
+q -q
p r Applied E-field rotates dipole to align with field:
E x
p r r
r = τ Torque
θ cos
E p E p
=
⋅
−
=
Potential Energy
Trang 73.225 9
• For a material with many dipoles:
Microscopic Frequency Response of Materials
) ( p E E N p N
α
α
=
=
(polarization=(#/vol)*dipole polarization)
α=polarizability
0
so ,
ε
α
χ
ε
E
P
o
=
r
E
p r α= r Actually works well only for low density of dipoles, i.e gases: little screening
For solids where there can be a high density: local field
Eext
For a spherical volume inside (theory of local field),
o ext loc
P E
E
ε
3
r r
r
+
=
© E Fitzgerald-1999
=
• We now need to derive a new relationship between the dielectric constant and the polarizability
Microscopic Frequency Response of Materials
+
=
−
=
+
=
=
3
ext loc
ext o ext o
r
ext o ext o
r
E
E
E E
P
P E E
D
ε
ε ε
ε
ε ε
ε
Plugging into P=NαEloc:
(
3
1
3
2 +
=
−
+
=
−
r o
r
ext
r ext
o ext o
r
N
E N
E
E
ε α ε
ε
ε α ε
ε
ε
Clausius-Mosotti Relation:
o o
r
νε
α
ε
α
ε
ε
3 3
2
1 = = +
−
Where v is the volume per dipole (1/N)
Macro Micro
)
Trang 83.225 11
Different Types of Polarizability
• Atomic or electronic, α e
• Displacement or ionic, α i
• Orientational or dipolar, α o
Highest natural frequency
Lowest natural frequency
Lightest mass
Heaviest mass
o i
e α α α
t
i
o e E
As with free e-, we want to look at the time dependence of the E-field:
Kx
eE
t x
m
t
x
∂
∂
=
∂
∂ τ
2
2
Response Drag Driving Force
Restoring Force
(
m
K
m
eE
m
K
m
eE
x
Kx eE x
m
e
x
x
Kx eE
x
m
o
o o
o
o
o o
o
t
i
o
=
−
=
−
=
−
−
−
=
−
−
=
−
ω
ω ω
ω
ω
ω
2
2
2
2 )
(
&&
So lighter mass will have a higher critical frequency
© E Fitzgerald-1999
+
)
Classical Model for Electronic Polarizability
• Electron shell around atom is attached to nucleus via springs
+
E r
+
E r
p r
r
t
i o loc
i
Z && = − − , assume = − ω
Zi electrons,
mass Zim
Trang 93.225 13
Electronic Polarizability
−
=
i
o
o
mZ
K
m
eE
r
2
ω
2
2 ,
oe
i
e
oe
m
e
Z
ω
α
ω
ω << =
0
, e
oe α
ω
ω >> =
( 2 2
2
oe
i
e
m
e
Z
ω
ω
α
−
=
( 2 2
2
o e
o
o
i
m
e
Z
ω
=
( 2 2 ;
i
oe
oe
o
o
mZ
K
m
eE
ω
=
; i t
o
i er p p e
Z
qd
2
2
2 1
m e NZ
N
oe
o
i
o
e
− +
= +
=
ω ω ε
ε
α
ε
εr
ω
ωoe
1
( 2
2
1
oe
o
i
m e
NZ
ω
ε +
© E Fitzgerald-1999
)
)
)
)
)
QM Electronic Polarizability
• At the atomic electron level, QM expected: electron waves
• QM gives same answer qualitatively
• QM exact answer very difficult: many-bodied problem
( )
h
0
1 10 2
2
10
10
2
f
m
e
e
−
=
−
ω
ω ω
α
E1
E0
f10 is the oscillator strength of the transition (ψ1 couples to ψo by E-field)
For an atom with multiple electrons in multiple levels:
( )
h
0 0
0 2 2
10
0
2
j
f
m
j
j
j
e
−
=
−
ω ω
α
Trang 103.225 15
Ionic Polarizability
• Problem reduces to one similar to the electronic polarizability
• Critical frequency will be less than electronic since ions are more massive
• The restoring force between ion positions is the interatomic potential
E(R)
R Nuclei repulsion
Electron bonding in between ions
Parabolic at bottom near Ro
) ( 2 )
o
o
R R k R
E F
R R k E
−
=
∂
∂
=
−
=
kl ijkl
ij C kx
© E Fitzgerald-1999
Ionic Polarizability
Eloc
+
-p
u+
u-•2 coupled differential eqn’s
•1 for + ions
•1 for - ions
(
2
2 2
2 , , 2
1 1 1 ,
ω ω α
α
ω ω ω
ω ω
−
=
=
=
=
−
=
=
=
+
−
= +
=
−
=
−
=
−
−
− +
− +
− +
oi i
o i o o
oi oi
o o
t o t o loc
loc
M e
E ew p
M
K M
eE w
e w w e E E
eE Kw w M
M M M
u u w u u w
&&
&&
&&
&&
Ionic materials always have ionic and
electronic polarization, so:
2
ω ω α α
α
α
α
− + +
=
+
oi e
i
e
© E Fitzgerald-1999
) ) )