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5/1/2011
1
Session 2: Fundamentals
1
by definition is the ratio of potential difference of the wire ends to the total current flowing through it.
2
12.
.
L
A
dVR
I dsσ
−=∫
∫
E l
E
≜
W
T
H
1R
L WTσ=
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At high frequencies, current tends to distribute near the surface of a conductor
3
1
fδ
π µσ≜
Scale of GSI interconnections is continually shrinking toward dimensions
comparable with the mean free path of the electrons.
At the same time, interconnects operate at higher frequencies such that skin
depth becomes in the same order of mfp of electrons.
4
2002 2004 2006 2008 2010 2012 2014 2016 201810
100
1000
Technology node
Dim
en
sio
n
s
12 pitch
skin depth
3 Fclkδ
×
nm
Cuλ
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3
5
Nucleus
+
Core electrons
Valance electrons
In D-L-S model the metal is divided into 2 different subsystems
The kinetic theory of gas is applied to the metal gas
t
v
2τ
v
4τ2
*
neJ nev E E
m
τσ= = =
Fvλ τ=
v eE mτ=
v
6
D
p = specularity parameter(the fraction of electrons that
have elastic collisions at the
wire surfaces) (0<p<1)
diffuse
scattering
specular
scattering
DJ
⇒J
⇒
Dλ << Dλ ≈
bρ ρ≈ (1 (1 ) )
bp
D
λρ ρ≈ + −
0p =
1p =
λ λ
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7
Skin effectAnomalous skin effect
D
D δ λ> >>
λ λ
δ δ
D
bfδ ρ πµ=
D δ λ> ≈
fδ ρ πµ= (1 )b
λρ ρ
δ= +
12R f∝
23R f∝
w λ
δ λSize Effects
Skin Effects
Anomalous
Skin Effects
LowTemp
GSI
@ room temperature:
39Cu
nmλ ≈ nmAl 21≈λ
nmGHzCu 600
10≈δ
1≈
1≈
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o A capacitor is a passive
electronic component that stores
energy in the form of an
electrostatic field.
9
In its simplest form, a capacitor consists of two conducting
plates separated by an insulating material called the
dielectric. The capacitance is directly proportional to the
surface areas of the plates, and is inversely proportional
to the separation between the plates.
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S
L
dsQC
V d
⋅=
−
∫
∫
D
E l
≜i
�
10
1 10 12 13 12 13 1
2 12 20 12 23 23 2
3 13 23 30 13 23 3
Q C C C C C V
Q C C C C C V
Q C C C C C V
+ + − − = − + + − − − + +
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11
B.C.
0v =1.
0dv dn =2.
mC
C C
, ; ,m mc c c c⇒ ⇒1. 2.ր ց ց ր
Volume-based method: (finite-element, finite-difference)
+ : accuracy, any complex structure
- : time consuming
Software: Maxwell, HFSS, Raphael
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1( , )
4G r r
r rπ′ =
′−
Surface-based method: (integral equation)
Green’s function
Software: Boundary Elements Method, FASTHENRY
( , ) ( )surface
V G r r r daσ′ ′ ′= ∫Integral equation(panel method, method of moments)
( )surface
Q r daσ ′ ′= ∫
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13
1
1( )
N
center square i
i
V V xN =
= ∑
Random-walk method: (stochastic)
Software: Random Logic Corp, QuicCap
Random walk: best for self cap for complicated netSurface based: best for small coupling capacitanceVolume based: best for dealing with multiple dielectrics
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W
S
T
H
mC
C C
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15
m
W TC C
H Sε ε= =
16
0.222
1.08 0.32 1.38
1.15 2.8
1.82 2.8 0.43m
C W T
H H
C T W S
H T H
ε
ε
−
= +
= + +
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9
o An inductor is a passive
electronic component that
stores energy in the form of a
magnetic field.
17
In its simplest form, an inductor consists of a wire
loop or coil. The inductance is directly proportional
to the number of turns in the coil. Inductance also
depends on the radius of the coil and on the type of
material around which the coil is wound
18
Using parallel plate approximation
( ) (1 )t RC
c ddv t V e−= −
0.5 0.693t RC=
0.9 2.3t RC=
2
0.5 0.693L
tTH
ρε=
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19
( ) ( )( )
( , )sinh 1 coshsL
in
RC
s Lc r
V sV L s
src sRC R C s sRC=
+ + +
20
( )1 2
1 2( , ) 1t t
ddV t L V K e K eδ δ= + + +⋯
1
4
11 2
11.01
1.04
(2 )
T T
T T
T T T T
R CK
R C
RC R C R C
π
δσ
π
+ += −
+ +
−= =
+ + +
2
0.9
0.5
1( (2 ) ) ln( ) 0.1
1
2.3( )
0.69( ) 0.38
T T T T
s L s L
s L s L
t RC R C R C RC
t R C R C RC RC
t R C R C RC RC
υ πυ
= + + + +−
= + + +
= + + +
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21
Two coupled lumped RC lines
Two coupled distributed RC lines
22
Combining these equations
Assuming and simplifies to:
2
1 1 1 22
1
2
2 2 2 12
2
1( , ) ( ) ( , ) ( , )
1( , ) ( ) ( , ) ( , )
m m
m m
V x t c c V x t c V x tr x t t
V x t c c V x t c V x tr x t t
∂ ∂ ∂= + −
∂ ∂ ∂
∂ ∂ ∂= + −
∂ ∂ ∂
2
1 2 1 22
2
1 2 1 22
( ) ( )
( ) ( 2 ) ( )m
V V rc V Vx t
V V r c c V Vx t
∂ ∂+ = +
∂ ∂
∂ ∂− = + −
∂ ∂
1 2r r r= = 1 2c c c= =
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23
Boundary conditions
Transformation
1 1 2 2
1 1 2 2
( ) (0, ) (0, ) ; (0, ) (0, )
( , ) ( , ) ; ( , ) ( , )
dd s s
L L
V u t I t R V t I t R V t
I L s C V L s I L s C V L st t
− = − =
∂ ∂= =
∂ ∂
2
2
( ) (0, ) (0, )2
( , ) ( , )
dds
L
V rc Vx t
Vu t I t R V t
I L t C V L tt
+ +
+ +
+ +
∂ ∂=
∂ ∂
− =
∂=
∂
1 2 1 2( ) 2 ; ( ) 2V V V V V V− += − = +
Solution:
2
2( 2 )
( ) (0, ) (0, )2
( , ) ( , )
m
dds
L
V r c c Vx t
Vu t I t R V t
I L t C V L tt
− −
− −
− −
∂ ∂= +
∂ ∂
− =
∂=
∂
24
Sakurai single line solution
Plus solution
1
4
1 2
11.01
1.04
(2 )
T T
T T
T T T T
R CK
R C
R C R C
π
σπ
+ += −
+ +
=+ + +
1
1( , ) (1 exp( ))t
dd RCV t x l V K
σ−= ≈ +
2
1.04 1
2 (2 )4
1( , ) 1 1.01 exp( )dd
T T T T
V T T tRC R C R C
T T
R CV t x L
R C ππ + +
+
−+ + + + +
+ += ≈ −
+ +
Minus solution
2
1.04 1( 2 )2 (2 )
4
1( , ) 1 1.01 exp( )dd
m T T T T
V T T tR C C R C R C
T T
R CV t x L
R C ππ − −
−
−− +− + + +
+ += ≈ −
+ +
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25
Active line1
2
V VV + −+
= Quiet line 22
V VV + −−
=
Pt
PV
26
Solving for t :
Assuming simplifies to:
1 1 1 1
1 1
ln2 ( 2 )
P
m m
K Ct
K C C R C C RC
σ σ σ
σ
− − − +
+ +
= −
+ +
2
4
21 11.01
2 2 2 2
mC C
dd m mTP
T m m m
V C CR CV
R C C C C C Cπ
+= ≈
+ + + +
LC C<<
1
2
T S
W H T S+
1
2
mP
dd m
CV
V C C≈
+
parallel plate
approximation
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Finite rise time?!
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Solution:
Transient voltage
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parameters
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Time delay expressions:
29
As Trise�0 converges to Sakurai
Generalized delay formula for RC>Trise
30
Coupled line solutions:
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Hspice comparison
31
Peak crosstalk expression
32
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Hspice comparison
33
Length dependence
34
1
2
T S
W H T S+
1
2
mP
dd m
CV
V C C≈
+parallel plate
approximation
Scaling independent:
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Scaling dependence
35
Material dependence
36
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Driver resistance dependence
37
o An inductor is a passive
electronic component that
stores energy in the form of a
magnetic field.
38
In its simplest form, an inductor consists of a wire
loop or coil. The inductance is directly proportional
to the number of turns in the coil. Inductance also
depends on the radius of the coil and on the type of
material around which the coil is wound
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39
Inductance of rectangular wires with return path at infinity
40
Inductance of rectangular wires with return path in
perfect ground plane
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41
Loop inductance for coplanar ground lines
42
In its simplest
SV S
R→ ∞
x← →
inf( , )V x t
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43
2 20inf 0
0
02
2 2 1 2
1
( , ) [ ( ( ) )
( )1
( ( ) ) 4 (1 ) ]1
t
S
S
k
k
k
k
ZV x t V e I t x lc
Z Ru t x lc
t x lcI t x lc
t x lc
σ σ
σ
−
∞−
=
= − +
+ −
− − − Γ + Γ − Γ +
∑
0
0
2 ; S
S
R Zr l
R Zσ
−= Γ =
+where
020inf
0
( , )rx Z
S
S
ZV x x lc V e
Z R
− =
+ Note that:
44
SV SR
l← →( , )finV l t
Delay model:
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45
Capacitive load
Delay model:
+
- CL
Rtrx=0 x=L
( )
2
0
max ,0.37 0.69
0.69 0.65 0.36
d F S
L S
t t rcL R cL
C rL R Z
≈ +
+ + +
46
• Transmission line effects should be considered when the rise or fall time of
the input signal (tr ,tf) is smaller than the time-of-flight of the transmission line
(tflight).
tr (tf) << 2.5 tflight
• Transmission line effects should only be considered when the total
resistance of the wire is limited:
R < 5 Z0
• The transmission line is considered lossless when the total resistance is
substantially smaller than the characteristic impedance
R < Z0 /2
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47
n=1 n=10
n=50 n=500
48
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49
50
2 coupled distributed RLC
interconnects
A (active) and Q (quiet).
3 coupled distributed RLC
interconnects
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51
n=1 n=10
n=50 n=500
52
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53
1
2
mP
dd m
CV
V C C≈
+
2 line RC:
4
mP
dd m
CV
V C C
π≈
+
2 line RLC:
43
3
mP
dd m
CV
V C C
π≈
+
3 line RLC:
