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THEORETICAL LIMITS FOR SIGNAL
REFLECTIONS DUE TO INDUCTANCE
FOR ON-CHIP INTERCONNECTIONS
F. Huret, E. Paleczny, P. KennisF. Huret, E. Paleczny, P. Kennis
Institut d ’Electronique et de Microélectronique Institut d ’Electronique et de Microélectronique
du Nord, UMR CNRS 9929du Nord, UMR CNRS 9929
D. DeschachtD. Deschacht, G. Servel, G. Servel
Laboratoire d’Informatique, de RobotiqueLaboratoire d’Informatique, de Robotique
et de Microélectronique, UMR CNRS 5506.et de Microélectronique, UMR CNRS 5506.
SLIP ’2000, San Diego, April 8-9th.SLIP ’2000, San Diego, April 8-9th.
OUTLINE OF THE TALK
Introduction
Theoretical limits
Electromagnetic analysis :- Methodology- Application
Limits between RLC and RC models
Illustration of the theoretical limits :- in frequency-domain- in time domain
Comparison with previous work
Conclusion
INTRODUCTION
0.7µm, 2 metal layers Up to 100,000 devices on a chip Typical CPU frequency 50MHz
0.25µm, 6 metal Up to 10,000,000 devices on a chip Typical CPU frequency 400 MHz
1989 1999
IC
10 years of 10 years of evolutionevolution
10 years of 10 years of evolutionevolution
INTRODUCTION
With the continued scaling down of technology,increased die aera :
* cross-section decreases* interconnect length increases
interconnections : blocking point of performances improvement
Introduction of new materials such as Cu
inclusion of inductance ?
INTRODUCTION
Interconnect delay dominates gate delay
in current deep submicronic VLSI circuits.
More accurate interconnect models
and signal propagation characterization are required.
With faster on-chip rise times inductance
is becoming more important.
Electromagnetic analysis is needed.
THEORETICAL LIMITS
Long lines :
Static hypothesis
x
AeA l
30
12
30
gl
x
lln
Short lines :
Traveling wave leA
Range of lengths for inductance inclusion :
We have to determine
attenuation factor
phase factor
x : attenuation
coefficient
THEORETICAL LIMITS
x
lln
15
Propagation parameters of the waveguide
INTERCONNECTION = WAVEGUIDE
Phase factorrad/cm
Attenuation factordB/cm ou Np/cm
Zc Characteristic impedance
ELECTROMAGNETIC ANALYSIS Methodology
Full wave analysis
Finite Element Method
Zc
Wire length L
v1 v2
i1 i2
v1 v2
i1 i2
Z
v1 v2
i1 i2
Y
v
i
L Z LL
ZL
v
i
c
c
1
1
2
2
cosh( ) sinh( )sinh( )
cosh( )
v
i
Z v
i1
1
2
2
1
0 1
v
i Y
v
i1
1
2
2
1 0
1
j
a
b
c
ELECTROMAGNETIC ANALYSIS Methodology
Definitons of the voltage-current matrices used in this analysis
ELECTROMAGNETIC ANALYSISMethodology
Vin(t) Vout(t)
Vin(freq)
(freq) (freq) Zc(freq)
Matched Load Impedances
Chain Matrix Vout(freq)
F.F.T.-1
+
* F.F.T = Fast Fourier Transform
F.F.T.
ELECTROMAGNETIC ANALYSISApplication
Interconnection geometry and environment
0.8 m
M5
0.8 m
2.4 m
M5
7.3 m
passivation
SiO2
Si bulk 7cm 500 m
2.4 m
M5
7.3 m
passivation
SiO2
2nd configuration1st configuration 3rd configuration
Metal 5 : W=1 m
T= 1 mAluminium or Copper
ELECTROMAGNETIC ANALYSISApplication
Frequency behavior of the attenuation factors
0
0.5
1
1.5
2
2.5
3
0 5 10 15 20 25 30Frequency (GHZ)
(
Np
/cm
)
1st configuration
2nd configuration
3rd configuration
Al
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0 5 10 15 20 25 30Frequency (GHZ)
(
Np
/cm
)
1st configuration
2nd configuration
3rd configuration
Cu
ELECTROMAGNETIC ANALYSISApplication
Frequency behavior of the phase factors
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25 30Frequency (GHZ)
(
rad
/cm
)
1st configuration
2nd configuration
3rd configuration
Al
0
2
4
6
8
10
12
14
16
18
20
0 5 10 15 20 25 30Frequency (GHZ)
(
rad
/cm
)
1st configuration
2nd configuration
3rd configuration
Cu
ELECTROMAGNETIC ANALYSISApplication
Attenuation determination
Traveling wavex
AeA l
0
5
10
15
20
25
30
35
40
0 5 10 15 20Length (mm)
Atte
nuat
ion
valu
e
1st configuration
2nd configuration
3rd configuration
Al
0
5
10
15
20
25
0 5 10 15 20Length (mm)
Atte
nuat
ion
valu
e
1st configuration
2nd configuration
3rd configuration
Cu
Attenuation value of the wave, for 10 GHz, versus interconnection length
Theoretical limits :
We have determined To determine x :
comparison output signal
between RC and RLCG models
x
lln
15
ELECTROMAGNETIC ANALYSISApplication
OUTLINE OF THE TALK
Introduction
Theoretical limits
Electromagnetic analysis :- Methodology- Application
Limits between RLC and RC models
Illustration of the theoretical limits :- in frequency-domain- in time domain
Conclusion
LIMIT BETWEEN RLC AND RC MODELS
The RLCG line model deduced from the electromagnetic analysis :
)..).(..(. CjGLjRj
..
..
CjG
LjRZc
240
260
280
300
320
340
360
380
400
0 5 10 15 20 25 30 35 40 45Frequency (GHZ)
R (
cm)
1st configuration
3rd configuration
2nd configuration
0
200
400
600
800
1000
1200
1400
1600
1800
0 5 10 15 20 25 30 35 40 45Frequency (GHZ)
C (f
F/cm
)
1st configuration
2nd configuration
3rd configuration
These calculated values are used to build the distributed RC model
LIMIT BETWEEN RLC AND RC MODELS
n
Rline
n
Cline
.2 n
Cline
n cells
COMPARISON BETWEEN :
HSPICE simulations : RC modelElectromagnetic analysis : RLC model
LIMIT BETWEEN RLC AND RC MODELS
Waveform of input and output signals in the range of lengths with inductance effect
2nd configuration - L=6 mm - Cu
0
0,5
1
1,5
2
2,5
3
3,5
0 50 100 150 200 250 300Time (ps)
Vo
lta
ge
(V
)
Vin
Vout1st reflection on the output
1st reflection on the input
2nd reflection on the output
LIMIT BETWEEN RLC AND RC MODELS
Waveform of input and output signals in the range of lengths with inductance effect
2nd configuration - L=10 mm - Cu
0
0,5
1
1,5
2
2,5
3
3,5
0 50 100 150 200 250 300 350 400Time (ps)
Vo
lta
ge
(V
)
Vin
Vout1st reflection on the output
1st reflection on the input
2nd reflection on the output
1st structure - L=10mm - Al
0
0.5
1
1.5
2
2.5
3
3.5
0 100 200 300 400 500 600
Time (ps)
Vol
tage
(V)
Input signal
Ouput RC model
Output RLC Model
1st structure - L=16 mm - Cu
0
0.5
1
1.5
2
2.5
3
3.5
0 100 200 300 400 500 600 700 800Time (ps)
Vo
ltag
e (V
)
Input Signal
Ouput RC model
Output RLC Model
LIMIT BETWEEN RLC AND RC MODELS
Attenuation determination :
Limit : the amplitude of the reflected wave is sufficiently low to give the reflection effect negligible
OUTLINE OF THE TALK
Illustration of the theoretical limits :- in frequency-domain- in time domain
Theoretical limits :
We have determined x
x
lln
15
Aluminium
0
0,5
1
1,5
2
2,5
3
0 5 10 15 20 25 30Frequency (GHz)
Lo
we
r li
mit
(m
m)
1st configuration
2nd configuration
3rd configuration
Copper
0
0,5
1
1,5
2
2,5
3
0 5 10 15 20 25 30Frequency (GHz)
Lo
we
r li
mit
(m
m)
1st configuration
2nd configuration
3rd configuration
Aluminium
0
10
20
30
40
50
60
0 5 10 15 20 25 30Frequency (GHz)
Up
pe
r li
mit
(m
m)
1st configuration
2nd configuration
3rd configuration
Copper
0
10
20
30
40
50
60
0 5 10 15 20 25 30Frequency (GHz)
Up
pe
r li
mit
(m
m)
1st configuration
2nd configuration
3rd configuration
ILLUSTRATION OF THEORETICAL LIMITSin the frequency-domain
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 5 10 15 20 25 30 35 40 45 50Frequency (GHz)
Re
lati
ve
mo
du
le
0
0,5
1
1,5
2
2,5
3
0 100 200 300 400 500Time (ps)
Vo
lta
ge
(V
)
1
2
3
4
rtf
1Frequency Time domain
ILLUSTRATION OF THEORETICAL LIMITSin the frequency-domain
Aluminium
0
5
10
15
20
25
30
0 20 40 60 80 100 120 140 160 180 200tr (ps)
Up
pe
r lim
it (
mm
)
1st configuration
2nd configuration
3rd configuration
Copper
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100 120 140 160 180 200tr (ps)
Up
pe
r lim
it (
mm
)1st configuration
2nd configuration
3rd configuration
Aluminium
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 50 100 150 200 250 300 350tr (ps)
Lo
we
r lim
it (
mm
)
1st configuration
2nd configuration
3rd configuration
Copper
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 50 100 150 200 250 300 350tr (ps)
Lo
we
r lim
it (
mm
)
1st configuration
2nd configuration
3rd configuration
ILLUSTRATION OF THEORETICAL LIMITSin the time-domain
OUTLINE OF THE TALK
Introduction
Theoretical limits
Electromagnetic analysis :- Methodology- Application
Limits between RLC and RC models
Illustration of the theoretical limits :- in frequency-domain- in time domain
Comparison with previous work
Conclusion
COMPARISON WITH PREVIOUS WORK
The two figures of merit can be combined into a two sided inequality that determines the range of the length of interconnect in which inductance effects are significant :
C
L
Rl
CL
tr
2
.2
« Figures of Merit to characterize the Importance of On-chip Inductance »DAC 98, June 1998
1st configuration :
R = 17300 /mC = 170 pF/mL = 490 nH/mG # 0
2nd configuration :
R = 17300 /mC = 63.6 pF/mL = 655 nH/mG # 0
1st configuration - L=2mm - Cu
0
0,5
1
1,5
2
2,5
3
3,5
4
0 50 100 150 200 250Time (ps)
Vo
ltag
e (
V)
Vin
Vout
1st configuration - L=1mm - Cu
0
0,5
1
1,5
2
2,5
3
3,5
4
0 50 100 150 200 250 300Time (ps)
Vo
ltag
e (
V)
Vin
Vout
1st configuration - L=10mm - Cu
0
0,5
1
1,5
2
2,5
3
3,5
4
0 100 200 300 400 500Time (ps)
Vo
ltag
e (
V)
Vin
Vout
COMPARISON
WITH PREVIOUS WORK
1st configuration - Copper
0
5
10
15
20
25
0 20 40 60 80 100 120 140 160 180 200tr (ps)
Lim
it (
mm
)
Upper limit
Lower limit
DAC Limit
2nd configuration - L=15mm - Cu
0
0,5
1
1,5
2
2,5
3
3,5
4
0 50 100 150 200 250 300 350 400Time (ps)
Vo
ltag
e (
V)
Vin
Vout
2nd configuration - L=25mm - Cu
0
0,5
1
1,5
2
2,5
3
3,5
4
0 100 200 300 400 500 600 700 800Time (ps)
Vo
ltag
e (
V)
Vin
Vout
2nd configuration - L=2mm - Cu
0
0,5
1
1,5
2
2,5
3
3,5
4
0 50 100 150 200 250Time (ps)
Vo
lta
ge
(V
)
Vin
Vout
COMPARISON
WITH PREVIOUS WORK
2nd configuration - Copper
0
5
10
15
20
25
30
35
40
0 20 40 60 80 100 120 140 160 180 200tr (ps)
Lim
it (
mm
)
Upper limit
Lower limit
DAC Limit
CONCLUSION
A full-wave electromagnetic analysis have been presented
to build accurate interconnect models,
including inductance effects.
New limits for signal reflections due to inductance
for on-chip interconnections have been proposed.
CONCLUSION
These limits have been illustrated
with typical interconnection geometries,
for Al and Cu wires.
This study shows evidence demonstrating that a range
exists for which inductance effects cannot be neglected
and requires a transmission line model.
CONCLUSION
FUTURE WORK :
Interconnect coupling : taking into account
not only the coupling capacitance, but also
the impact of inductance and mutual inductance.