Fitzgerald-1999 • Can increase strength with second phase particles • As long as distance between second phase< l, conductivity marginally effected Example: Conductivity Engineering
Trang 13.225 21
Example: Conductivity Engineering
• Objective: increase strength of Cu but keep conductivity high
τ
τ
µ τ
σ
v
m
e
m
ne
=
=
=
l
2
Scattering length connects scattering time
to microstructure
Dislocation (edge)
l decreases, τ decreases, σ decreases
e-© E.A Fitzgerald-1999
• Can increase strength with second phase particles
• As long as distance between second phase< l, conductivity marginally effected
Example: Conductivity Engineering
L
S
L+S
L
α+L β+L
α+β
S
microstructure
Material not strengthened, conductivity decreases
α
L
L>l Dislocation motion inhibited by second phase; material strengthened; conductivity about the same
Trang 2
• Scaling of Si CMOS includes conductivity engineering
• One example: as devices shrink…
– vertical field increases
– increased doping in channel need for electrostatic integrity: ionized impurity scattering
– τSiO2< τimpurity if scaling continues ‘properly’
Example: Conductivity Engineering
Evert
Ionized impurities (dopants)
G SiO2
© E.A Fitzgerald-1999
Determining n and µ : The Hall Effect
Vx, Ex
I, Jx
Bz
+ + + + + + + + + + +
B v q
E
q
F r = r + r × r
z D
y ev B
F = −
Ey
y
y eE
In steady state,
H Z D
Y v B E
E = = , the Hall Field Since vD=-Jx/en,
Z X H Z x
ne
ne
Trang 3Experimental Hall Results on Metals
• Valence=1 metals look like
free-electron Drude metals
• Valence=2 and 3, magnitude
and sign suggest problems
Trang 43.225 1
Response of Free e- to AC Electric Fields
• Microscopic picture
e
-t
i O
Z E e
E = −ω
B=0 in conductor, and F r ( E r ) F r ( B r )
>>
t i
e eE t
p
dt
t
τ
−
−
−
)
(
t i e p t
p ( ) = 0 −ω
0
0
p
−
τ
ω
try
τ
0
0
−
=
i
eE
p
ω >>1/ τ , p out of phase with E
ω <<1/ τ i ω , p in phase with E
eE
τ
0
p =
© E.A Fitzgerald-1999
Complex Representation of Waves
sin(kx- ω t), cos(kx- ω t), and e-i(kx-ωt) are all waves
real
imaginary
A
θ
e iθ=cos θ +isin θ
Trang 53.225 3
• Momentum represented in the complex plane
Response of e- to AC Electric Fields
real
imaginary
p
p (ω<<1/τ)
Instead of a complex momentum, we can go back to macroscopic
and create a complex J and σ ωτ
i
e J t
2
0
0
) 1 (
E i
m
ne
m
nep nev
J
ω
=
−
=
−
=
m
ne
i
τ
σ
ωτ
σ
−
=
© E.A Fitzgerald-1999
• Low frequency (ω<<1/τ)
– electron has many collisions before direction
change
• High frequency (ω>>1/τ)
– electron has nearly 1 collision or less when
direction is changed
– J imaginary and 90 degrees out of phase with
E, σ is imaginary
Response of e- to AC Electric Fields
Qualitatively:
ωτ<<1, electrons in phase, re-irradiate, Ei=Er+Et, reflection
ωτ>>1, electrons out of phase, electrons too slow, less interaction,transmission ε=εrε0 εr=1
Hz cm
x cm
x
8
10
10
5000 sec / 10
3 , sec,
τ E-fields with frequencies greater than visible light frequency expected to be
beyond influence of free electrons
Trang 63.225 5
• Need Maxwell’s equations
– from experiments: Gauss, Faraday, Ampere’s laws
– second term in Ampere’s is from the unification
– electromagnetic waves!
Response of Light to Interaction with Material
SI Units (MKS)
M H
B P E
D
t
D
c
J
c H
t
B
c E
x
B
D
r r
r r r
r
r r
r
r
r
r
r
π
π
π
πρ
4
4
1
4
1
0
4
+
= +
=
∂
∂ +
=
∇
∂
∂
−
∇
=
•
∇
=
•
∇
0
0 0
0
0
;
0
ε ε ε µ µ
µ
µ µ
µ
ε
ε
ρ
r
r
H M H
B
E P E
D
t
D J H
x
t
B E
x
B
D
=
=
= +
=
= +
=
∂
∂ +
=
∇
∂
∂
−
=
∇
=
•
∇
=
•
∇
r r r
r
r r r
r
r r
r
r
r
r
r
Gaussian Units (CGS)
© E.A Fitzgerald-1999
Waves in Materials
• Polarization non-existent or swamped by free electrons, P=0
t
E J
B
x
t
B
E
x
∂
∂ +
=
∇
∂
∂
=
∇
r r
r
r
r
0
0
0 µε
µ
t
B
x
E
x
x
∂
∂∇
−
∇
∇
r
r
)
(
2
2 0 0
0
2
0 0
0
t
E
t
E
E
t
E
J
t
E
∂
∂ +
∂
∂
=
∇
∂
∂ +
∂
∂
−
=
∇
−
ε
µ
σ
µ
ε µ
µ
For a typical wave,
) ( )
( )
(
) (
2 0 0
0
2
0 ) (
0
r E r
E i r
E
e r E e e E e
E
ω ε µ σ
ϖµ
ϖ ϖ
ϖ
−
−
=
∇
=
=
Wave Equation
ω
σ ω
ε
ω ε
ω
0
2
2
2
1 ) (
) ( ) ( ) (
i
r
E
c r
E
+
=
−
∇
) (
) (
) (
2
2
2
0
ω
ε
ω
ω ε
ω
c k
v
c
k
e E r
E ik r
=
=
=
Trang 73.225 7
Waves in Materials
) 1 ( 1 1 ) (
0
0
σ ω
σ ω
ε
i i
i
− +
= +
=
m
ne
i
i
p
p
0
2
2
2
2
1 ) (
ε
ω
τ ω
ω τ ω ω
ε
=
− +
= Plasma Frequency For ωτ >>>1, ε(ω) goes to 1 For an excellent conductor ( σ0 large), ignore 1, look at case for ωτ <<1
ω τ
ω τ ω
ω τ ω ω
ε
2
2
2
)
i
i
≈
−
≈
© E.A Fitzgerald-1999
Waves in Materials
For a wave E = E0 ei( kz− ωt) Let k=kreal+kimaginary=kr+iki
[ k r z te k i z
e E
E = −ω −
0 The skin depth can be defined by
+
=
+
=
=
=
2
0
0
2
0
0
0
0
0
0
2 2
2
1
) (
c
i
c
i
c
k
i c
c
k
ε ω
σ
ε ω
σ
ωε σ
ω
ωε σ ω ω ε
ω
δ
ω µ σ
σω
ε
δ
o o
o
o
i
c
k
2 2
=
=
=
]
Trang 83.225 9
Waves in Materials
For a material with any σ0, look at case for ωτ >>1
( ) 1 2 2
ω
ω ω
R
ω
ωp
© E.A Fitzgerald-1999
Success and Failure of Free e- Picture
• Success
– Metal conductivity
– Hall effect valence=1
– Skin Depth
– Wiedmann-Franz law
• Examples of Failure
– Insulators, Semiconductors
– Hall effect valence>1
– Thermoelectric effect
– Colors of metals
K/ σ =thermal conduct./electrical conduct.~CT
τ
2
3
1
therm
v v c
=
Κ
m T
k v
nk
T
E
therm
b
v
v
3
;
2
=
∂
∂
=
m T
nk
m T k
nk b b
b
τ
2 3
3
2
3
3
=
Κ
m
ne τ
σ = 2
T
e
k b 2
2
3
=
Κ σ
Therefore :
~C!
Luck: cvreal=cvclass/100;
vreal 2=vclass 2*10
0
Trang 93.225 11
Wiedmann-Franz ‘Success’
Exposed Failure when
cv and v2 are not both
in property
Thermoelectric Effect
T Q
E = ∇
e
nk
ne
nk
ne
c
2
3
2
3
−
=
Thermopower Q is
Thermopower is about 100 times too large!
© E.A Fitzgerald-1999
Waves in Vacuum
0
J
0
0 ; ε ε
µ
2
2 0
0
2
t
E
E
∂
∂
=
For typical wave:
t r ik
e
E
2 0
0
2 = µ ε ω
k
0
0
−
=
ω k
For constant phase: ω t)
→
=
vphase ω
⇒
•
•
0
0
−
= µ
c
Example:
Violet light (ν = 7.5 x 1014 Hz)
λ = c/ν = 400 nm k=2π/ λ = 1.57 x 107 m -1
ω = 2 π ν = 4.71 x 1015 s-1
ε
)
ε
(kx-)
ε
Trang 103.225 13
Waves in Materials; Skin Depth
δ
The skin depth is defined by
2
2
=
ωµσ
δ
ωµσ µε
k 2 = 2 +
Conductive materials
ωµσ
i
k ≈
(
δ
±
=
+
±
2
1 1 2
( kx ω t i( ωt xδ xδ
e
E
0
µ
µ ≅
t
E
t
E
E
∂
∂ +
∂
∂
=
After Livingston
)
)
µ
Plasma Frequency
Remember: k 2 = ω 2µε + i ωµσ
1
0
−
ωτ
σ
ωτ
σ
σ
i
i
then
−
=
−
2 2
0 2
ω
ω µε ω
τ
µσ µε
k
≅
2
2
ε
ω
m
ne
For ω > ω p; k is real number no attenuation!
ω < ω p; k contains imaginary component, wave reflected
⇒
Criteria for transparent electrode?
⇒
(Example: 8 x 1027/m3; ωp = 4.3 x 1015s-1)
)
ω
n=5