Buffer sizing
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This document discusses buffer sizing to reduce delay uncertainty caused by process variations. It shows that increasing buffer size reduces uncertainty from variations in both the buffer and interconnect. Variations in device parameters introduce delay uncertainty in CMOS inverters. Increasing buffer size reduces this uncertainty by increasing the current sourced by the buffer. Variations in interconnect parameters also affect delay, but buffer sizing can decrease this uncertainty as well.
![BUFFER SIZING FOR DELAY UNCERTAINTY INDUCED BY PROCESS VARIATIONS
Dimitrios Velenis and Ramyashree Sundaresha Eby G. Friedman
Department of Electrical Department of Electrical
and Computer Engineering and Computer Engineering
Illinois Institute of Technology University of Rochester
Chicago, Illinois 60616-3793 Rochester, New York 14627-0231
ABSTRACT higher circuit performance has pushed clock frequencies
deep into the gigahertz frequencies range, reducing the pe-
Controlling the delay of a signal in the presence of various
riod of the clock signal well below a nanosecond. Uncer-
noise sources, process parameter variations, and environ-
tainty in the propagation delay of the clock signal can cause
mental effects represents a fundamental problem in the de-
violations of the set-up and hold timing constraints at the
sign of high performance synchronous circuits. The effects
data path registers. In addition, delay uncertainty of the data
of device parameter variations on the signal propagation de-
signals can also cause violations of these constraints. To
lay of a CMOS buffer are described in this paper. It is shown
prevent these violations, either the tight timing constraints
that delay uncertainty is introduced due to variations in the
should be relaxed, or the uncertainty in the signal delay
current flow through a buffer. In addition, the variations in
should be reduced. Relaxing the timing constraints, espe-
the parasitic resistance and capacitance of an interconnect
cially at the most critical paths of a circuit, increases the
line also affect the buffer delay. A design methodology that
clock period and reduces the circuit performance. Alterna-
reduces the delay uncertainty of signals propagating along
tively, reducing delay uncertainty can prevent the violations
buffer-driven interconnect lines is presented. The proposed
of the timing constraints, thereby improving the robustness
methodology increases the current flow sourced by a buffer
of a circuit and enhancing the circuit performance. In order
to reduce the sensitivity of the delay on device and intercon-
to develop design methodologies that reduce delay uncer-
nect parameter variations.
tainty, the effects that cause uncertainty in the signal propa-
gation delay should be investigated.
1. INTRODUCTION Delay uncertainty is introduced by a number of factors
that affect the signal propagation, examples of which in-
The rapid scaling of on-chip feature size has produced phe- clude process and environmental parameter variations [3, 4]
nomenal advances in circuit density, chip functionality, and and interconnect noise [5]. Effects such as the non-uni-
operational clock frequency [1, 2]. This development has formity of the gate oxide thickness and imperfections in
also resulted in increased sensitivity of the circuit charac- polysilicon etching [6] can cause variations of the current
teristics on parameter variations. Imperfections in the man- flow within a transistor, thereby introducing delay uncer-
ufacturing process and environmental effects degrade the tainty. The geometrical characteristics of interconnect lines
quality of signals within a circuit and affect the signal prop- are also affected by imperfections in the photolithography,
agation delay. Deviations of a signal from the target delay planarization and metal etching processes [4]. Environmen-
can cause the loss of critical data, thereby causing a system tally induced parameter variations caused by changes in the
to malfunction. These deviations of the signal propagation ambient temperature [7] and external radiation [8] also in-
delay from a target value are described as delay uncertainty. troduce delay uncertainty.
Delay uncertainty can be critical for the operation of In this paper, the delay uncertainty of a signal propa-
high speed synchronous circuits. The continuous quest for gating along a CMOS buffer due to process, environmen-
tal, and system parameter variations is investigated. It is
This research was supported in part by the National Science Founda-
tion under Grant No. MIP-9610108, the Semiconductor Research Corpora- shown that increasing the buffer size reduces the uncertainty
tion under Contract No. 99-TJ-687, the DARPA/ITO under AFRL Contract in the signal delay due to parameter variations in both the
F29601-00-K-0182, a grant from the New York State Science and Tech- buffer and the interconnect. The variations in the delay of a
nology Foundation to the Center for Advanced Technology - Electronic
Imaging Systems, and by grants from Xerox Corporation, IBM Corpora-
CMOS inverter due to device parameter variations are dis-
tion, Intel Corporation, Lucent Technologies Corporation, and the Eastman cussed in Section 2. The effect of buffer size on the delay
Kodak Company. uncertainty introduced by interconnect parameter variations
0-7803-8715-5/04/$20.00 ©2004 IEEE. 415](https://image.slidesharecdn.com/buffersizing-120109055349-phpapp02/85/Buffer-sizing-1-320.jpg)
![Y ` b d
is described in Section 3. Finally, some conclusions are pre-
sented in Section 4.
2. DELAY UNCERTAINTY DUE TO DEVICE f g i p r t
g u
i v v
PARAMETER VARIATIONS
The signal propagation delay of a logic gate depends upon
the gate size and the magnitude of the current flowing within
the gate. Variations of the device parameters can cause the y € ‚ „ † ‡ ˆ
g u
i v v
magnitude of the current flow to change, introducing de-
lay uncertainty. Therefore, reducing the effect of the device
Figure 1: Dependence of device parameters on process, en-
parameters on the current flow can reduce delay uncertainty.
vironmental, and system parameters.
In particular, the delay uncertainty of the active components
of the most critical data paths and interconnect lines within a
circuit should be reduced to prevent tight timing constraints In order to evaluate the effect of the variation of the de-
vice parameters on the inverter propagation delay, is
!
¡ #
from being violated.
In the case of a CMOS inverter, a first order relation- substituted from (3) into (1), yielding
ship between the signal propagation delay and the current
¡ ¡
(4)
§ ©
£
¡ ¥ ¥
through an inverter is presented in [9] as ¦
§
¦
’
” – –
5 B ˜
¥
@
9
¡ ¡
(1) Differentiating (4) with respect to ,
5 B
§ ©
¡ £ ¥ ¥
@
¦ § ¦ ! 9
¡ # %
¥
™
¡ ¡ ¡
(5)
£
where
™
9
@
5 B
§
¦
’
” – –
5 B
˜
9 e
@
(2)
'
£
©
%
¡ ¡
As shown in (5), the sensitivity of inverter delay on vari-
and is the signal propagation delay, is the velocity sat-
¡
ations of the device parameters is inversely propor-
” m
f g h
uration index described in [9], is the transition time of the
tional to the inverter size. The delay uncertainty due to vari- f j k
signal at the input of the inverter, is the supply voltage,
¡ ¡
ations in the drain current can therefore be reduced by in-
and is the capacitive load at the output of a CMOS in-
creasing the size of the inverter. In addition, as shown in
verter. is the threshold voltage of the active transistor
'
(4), the propagation delay of an inverter also decreases with
during a signal transition and is the drain current flow-
!
¡ #
) 1 1
increasing inverter size. The reduction in delay is shown
ing through that transistor (defined at ).
£ ¡ £ ¡ ¡
in Fig. 2 where the inverter size is increased by up to five
is often used as an index of the current drive of a MOS-
!
¡ #
times. The inverter delay is calculated analytically from (1) o
FET transistor and depends upon the transistor size, and simulated using Spectre R simulator1 [11]. The differ-
o
1 Spectre R is a registered trademark of Cadence Design Systems Incor-
(3)
5 B
!
£ 3
¡ #
4 5 6 6 9 @
%
porated.
where and represent the width and the effective
4 5 6 6
3
channel length of a transistor, respectively, and repre-
5 B
85
tPHL(NMOS on)
@
9
sents the effect of the device parameters on the current flow. 75 tPLH(PMOS on)
According to MOSFET transistor models first proposed
by Shockley in [10] and Sakurai and Newton in [9], the 65
Buffer Delay (ps)
primary factors that affect the drain current within a tran- 55
Simulation
sistor are the carrier mobility ( ), the gate oxide capac- E F
45
itance ( ), the threshold voltage ( F ), and the power
G
'
supply voltage ( ). The dependence of these factors
¡ ¡
35
on process, environmental, and system parameters is shown
25 Analytical
in Fig. 1. These parameters are temperature ( ), gate I P R T
oxide thickness ( ), substrate doping density ( ), and
F G
V W
15
1 2 3 4 5
power supply voltage ( ). A variation of these param-
¡ ¡
Buffer Size
eters causes a change in the magnitude of the current flow
within a transistor and, eventually, uncertainty in the propa-
gation delay. Figure 2: Reduction in delay with increasing inverter size.
416](https://image.slidesharecdn.com/buffersizing-120109055349-phpapp02/85/Buffer-sizing-2-320.jpg)
![delay caused by 10% variation in the line resistance and 7.0
Uncertainty in Buffer Delay (ps)
r
±10% width variation at entire length
capacitance is investigated. The following three scenarios 6.0 ±10% width variation at first half
are considered, depending on the part of the line affected by ±10% width variation at second half
5.0
parameter variations:
4.0
i. r 10% variation in line resistance and capacitance at 3.0
the first half of the line closer to driving buffer.
2.0
ii. r 10% variation in line resistance and capacitance at 1.0
the second half of the line away from the buffer.
0.0
1 2 3 4 5
iii. r 10% variation in line resistance and capacitance at Buffer Size
the entire line length.
Figure 7: Reduction in delay uncertainty due to interconnect
parameter variations with increased buffer size
First half Second falf
[2] S. I. A., “The National Technology Roadmap for Semicon-
Driver
ductors,” Technical Report, Semiconductor Industry Associ-
ation, 2001.
[3] S. Natarajan, M. A. Breuer, and S. K. Gupta, “Process Vari-
ations and Their Impact on Circuit Operation,” Proceedings
of the IEEE International Symposium on Defect and Fault
Figure 6: Circuit model to evaluate the effect of line param- Tolerance in VLSI Systems, pp. 73–81, November 1998.
eter variations on buffer delay [4] V. Mehrotra, S. L. Sam, D. Boning, A. Chandrakasan, R. Val-
lishayee, and S. Nassif, “A Methodology for Modeling
The uncertainty in the buffer delay for these three cases the Effects of Systematic Within-Die Interconnect and De-
is illustrated in Fig. 7, where the buffer size is increased by vice Variation on Circuit Performance,” Proceedings of the
up to five times. It is shown that for each case the delay ACM/IEEE Design Automation Conference, pp. 172–175,
uncertainty is reduced with increasing buffer size. Note in June 2000.
Fig. 7 that the uncertainty in buffer delay is less when the [5] K. T. Tang and E. G. Friedman, “Interconnect Coupling
line parameters are affected at the second half of a line. This Noise in CMOS VLSI Circuits,” Proceedings of the ACM In-
reduction in delay uncertainty is caused by the shielding ef- ternational Symposium on Physical Design, pp. 48–53, April
1999.
fect of the first half of the line that isolates the buffer output
from the parameter variations at the second half. [6] R. Sitte, S. Dimitrijev, and H. B. Harrison, “Device Parame-
ter Changes Caused by Manufacturing Fluctuations of Deep
Submicron MOSFET’s,” IEEE Transactions on Electron De-
4. CONCLUSIONS vices, Vol. 41, No. 11, pp. 2210–2215, November 1994.
[7] S. Sauter, D. Schmitt-Landsiedel, R. Thewes, and W. Weber,
The effects of device parameter variations on the delay un- “Effect of Parameter Variations at Chip and Wafer Level on
certainty of a signal propagating through a CMOS buffer are Clock Skews,” IEEE Transactions on Semiconductor Manu-
investigated. It is demonstrated that delay uncertainty is in- facturing, Vol. 13, No. 4, pp. 395–400, November 2000.
versely proportional to the transistor current. Increasing the [8] J. F. Chappel and S. G. Zaky, “EMI Effects and Timing
inverter size, therefore, reduces the delay uncertainty caused Design for Increased Reliability in Digital Systems,” IEEE
by device parameter variations. In addition, the delay uncer- Transactions on Circuits and Systems I: Fundamental The-
tainty due to interconnect parameter variations is examined. ory and Applications, Vol. 44, No. 2, pp. 130–142, February
1997.
It is shown that variations in line parameter in the proximity
of a driving buffer have a greater effect on buffer delay. In- [9] T. Sakurai and A. R. Newton, “Alpha-Power Law MOS-
creasing the buffer size alleviates these effects and reduces FET Model and its Applications to CMOS Inverter Delay
and Other Formulas,” IEEE Journal of Solid State Circuits,
delay uncertainty.
Vol. 25, No. 2, pp. 584–593, April 1990.
[10] W. Shockley, “A Unipolar “Field-Effect” Transistor,” Pro-
5. REFERENCES ceedings of the Institute of Radio Engineers, pp. 1365–1377,
November 1952.
[1] G. E. Moore, “Progress in Digital Integrated Circuit,” Pro-
[11] Affirma Spectre Circuit Simulator User Guide, Cadence De-
ceedings of the IEEE International Electron Devices Meet-
sign Systems, Inc., 2000.
ing, pp. 11–14, December 1975.
418](https://image.slidesharecdn.com/buffersizing-120109055349-phpapp02/85/Buffer-sizing-4-320.jpg)
Buffer sizing
- 1. BUFFER SIZING FOR DELAY UNCERTAINTY INDUCED BY PROCESS VARIATIONS Dimitrios Velenis and Ramyashree Sundaresha Eby G. Friedman Department of Electrical Department of Electrical and Computer Engineering and Computer Engineering Illinois Institute of Technology University of Rochester Chicago, Illinois 60616-3793 Rochester, New York 14627-0231 ABSTRACT higher circuit performance has pushed clock frequencies deep into the gigahertz frequencies range, reducing the pe- Controlling the delay of a signal in the presence of various riod of the clock signal well below a nanosecond. Uncer- noise sources, process parameter variations, and environ- tainty in the propagation delay of the clock signal can cause mental effects represents a fundamental problem in the de- violations of the set-up and hold timing constraints at the sign of high performance synchronous circuits. The effects data path registers. In addition, delay uncertainty of the data of device parameter variations on the signal propagation de- signals can also cause violations of these constraints. To lay of a CMOS buffer are described in this paper. It is shown prevent these violations, either the tight timing constraints that delay uncertainty is introduced due to variations in the should be relaxed, or the uncertainty in the signal delay current flow through a buffer. In addition, the variations in should be reduced. Relaxing the timing constraints, espe- the parasitic resistance and capacitance of an interconnect cially at the most critical paths of a circuit, increases the line also affect the buffer delay. A design methodology that clock period and reduces the circuit performance. Alterna- reduces the delay uncertainty of signals propagating along tively, reducing delay uncertainty can prevent the violations buffer-driven interconnect lines is presented. The proposed of the timing constraints, thereby improving the robustness methodology increases the current flow sourced by a buffer of a circuit and enhancing the circuit performance. In order to reduce the sensitivity of the delay on device and intercon- to develop design methodologies that reduce delay uncer- nect parameter variations. tainty, the effects that cause uncertainty in the signal propa- gation delay should be investigated. 1. INTRODUCTION Delay uncertainty is introduced by a number of factors that affect the signal propagation, examples of which in- The rapid scaling of on-chip feature size has produced phe- clude process and environmental parameter variations [3, 4] nomenal advances in circuit density, chip functionality, and and interconnect noise [5]. Effects such as the non-uni- operational clock frequency [1, 2]. This development has formity of the gate oxide thickness and imperfections in also resulted in increased sensitivity of the circuit charac- polysilicon etching [6] can cause variations of the current teristics on parameter variations. Imperfections in the man- flow within a transistor, thereby introducing delay uncer- ufacturing process and environmental effects degrade the tainty. The geometrical characteristics of interconnect lines quality of signals within a circuit and affect the signal prop- are also affected by imperfections in the photolithography, agation delay. Deviations of a signal from the target delay planarization and metal etching processes [4]. Environmen- can cause the loss of critical data, thereby causing a system tally induced parameter variations caused by changes in the to malfunction. These deviations of the signal propagation ambient temperature [7] and external radiation [8] also in- delay from a target value are described as delay uncertainty. troduce delay uncertainty. Delay uncertainty can be critical for the operation of In this paper, the delay uncertainty of a signal propa- high speed synchronous circuits. The continuous quest for gating along a CMOS buffer due to process, environmen- tal, and system parameter variations is investigated. It is This research was supported in part by the National Science Founda- tion under Grant No. MIP-9610108, the Semiconductor Research Corpora- shown that increasing the buffer size reduces the uncertainty tion under Contract No. 99-TJ-687, the DARPA/ITO under AFRL Contract in the signal delay due to parameter variations in both the F29601-00-K-0182, a grant from the New York State Science and Tech- buffer and the interconnect. The variations in the delay of a nology Foundation to the Center for Advanced Technology - Electronic Imaging Systems, and by grants from Xerox Corporation, IBM Corpora- CMOS inverter due to device parameter variations are dis- tion, Intel Corporation, Lucent Technologies Corporation, and the Eastman cussed in Section 2. The effect of buffer size on the delay Kodak Company. uncertainty introduced by interconnect parameter variations 0-7803-8715-5/04/$20.00 ©2004 IEEE. 415
- 2. Y ` b d is described in Section 3. Finally, some conclusions are pre- sented in Section 4. 2. DELAY UNCERTAINTY DUE TO DEVICE f g i p r t g u i v v PARAMETER VARIATIONS The signal propagation delay of a logic gate depends upon the gate size and the magnitude of the current flowing within the gate. Variations of the device parameters can cause the y € ‚ „ † ‡ ˆ g u i v v magnitude of the current flow to change, introducing de- lay uncertainty. Therefore, reducing the effect of the device Figure 1: Dependence of device parameters on process, en- parameters on the current flow can reduce delay uncertainty. vironmental, and system parameters. In particular, the delay uncertainty of the active components of the most critical data paths and interconnect lines within a circuit should be reduced to prevent tight timing constraints In order to evaluate the effect of the variation of the de- vice parameters on the inverter propagation delay, is ! ¡ # from being violated. In the case of a CMOS inverter, a first order relation- substituted from (3) into (1), yielding ship between the signal propagation delay and the current ¡ ¡ (4) § © £ ¡ ¥ ¥ through an inverter is presented in [9] as ¦ § ¦ ’ ” – – 5 B ˜ ¥ @ 9 ¡ ¡ (1) Differentiating (4) with respect to , 5 B § © ¡ £ ¥ ¥ @ ¦ § ¦ ! 9 ¡ # % ¥ ™ ¡ ¡ ¡ (5) £ where ™ 9 @ 5 B § ¦ ’ ” – – 5 B ˜ 9 e @ (2) ' £ © % ¡ ¡ As shown in (5), the sensitivity of inverter delay on vari- and is the signal propagation delay, is the velocity sat- ¡ ations of the device parameters is inversely propor- ” m f g h uration index described in [9], is the transition time of the tional to the inverter size. The delay uncertainty due to vari- f j k signal at the input of the inverter, is the supply voltage, ¡ ¡ ations in the drain current can therefore be reduced by in- and is the capacitive load at the output of a CMOS in- creasing the size of the inverter. In addition, as shown in verter. is the threshold voltage of the active transistor ' (4), the propagation delay of an inverter also decreases with during a signal transition and is the drain current flow- ! ¡ # ) 1 1 increasing inverter size. The reduction in delay is shown ing through that transistor (defined at ). £ ¡ £ ¡ ¡ in Fig. 2 where the inverter size is increased by up to five is often used as an index of the current drive of a MOS- ! ¡ # times. The inverter delay is calculated analytically from (1) o FET transistor and depends upon the transistor size, and simulated using Spectre R simulator1 [11]. The differ- o 1 Spectre R is a registered trademark of Cadence Design Systems Incor- (3) 5 B ! £ 3 ¡ # 4 5 6 6 9 @ % porated. where and represent the width and the effective 4 5 6 6 3 channel length of a transistor, respectively, and repre- 5 B 85 tPHL(NMOS on) @ 9 sents the effect of the device parameters on the current flow. 75 tPLH(PMOS on) According to MOSFET transistor models first proposed by Shockley in [10] and Sakurai and Newton in [9], the 65 Buffer Delay (ps) primary factors that affect the drain current within a tran- 55 Simulation sistor are the carrier mobility ( ), the gate oxide capac- E F 45 itance ( ), the threshold voltage ( F ), and the power G ' supply voltage ( ). The dependence of these factors ¡ ¡ 35 on process, environmental, and system parameters is shown 25 Analytical in Fig. 1. These parameters are temperature ( ), gate I P R T oxide thickness ( ), substrate doping density ( ), and F G V W 15 1 2 3 4 5 power supply voltage ( ). A variation of these param- ¡ ¡ Buffer Size eters causes a change in the magnitude of the current flow within a transistor and, eventually, uncertainty in the propa- gation delay. Figure 2: Reduction in delay with increasing inverter size. 416
- 3. ence between the calculated and simulated delay values is caused by considering in (5) only the of the active tran- ! ¡ # sistor during a signal transition. The short-circuit current flowing through the inactive transistor is ignored and there- fore the calculated delay is smaller. The delay reduction, however, is similar in both the calculated and the simulated propagation delay, as shown in Fig. 2. The effect of device parameter variations on the inverter propagation delay is shown in Fig. 3. Note that the vari- ation of different device parameters has a different effect on the delay uncertainty. Increasing the buffer size reduces the delay sensitivity on the variation of the device parame- ters, as described by (5). As shown in Fig. 3, the variation of the power supply voltage ( ) is the dominant effect @ @ that introduces delay uncertainty, followed by variations in the gate oxide thickness ( ) and temperature ( ). In F G P R T addition to these parameters, the variation of the channel Figure 4: Delay uncertainty due to 10% @ @ variation for doping concentration ( ) by has also been investi- V W r s u different input transition time. gated, however the introduced delay uncertainty is shown to be negligible. It is shown in Fig. 3 that the delay uncer- tainty caused by different parameter variations is inversely proportional to inverter size. 14 Simulation Analytical Buffer Delay Uncertainty (ps) 12 10 dVDD 10% (a) Delay uncertainty due to (b) Delay uncertainty due ~ ‚ variation for dif- to variation in tem- x y z | ~ x € 8 ferent input transition time. perature for different input 6 transition time. dtOX ±5% 4 Figure 5: Effect of input signal transition time on delay 2 dtemp ±10o C uncertainty. 0 1 2 3 4 5 Buffer Size 3. DELAY UNCERTAINTY DUE TO INTERCONNECT PARAMETER VARIATIONS Figure 3: Reduction in delay uncertainty due to device pa- rameter variations with increasing buffer size. In addition to the parameters of active devices, manufac- turing process variations affect also the geometrical charac- As shown in (1), the propagation delay of a signal tran- teristics of on-chip interconnect lines. Fluctuations in the sition also depends upon the transition time of the input metal etching process can cause variations in the wire width signal. The effect of a variation in on the delay uncer- and spacing among adjacent metal lines. Furthermore, im- tainty is illustrated in Fig. 4, where the delay uncertainty perfections in the chemical-mechanical planarization pro- cess introduce variations in the metal thickness of an inter- due to a variation of of 10% is shown for a variation @ @ of between 30 ps and 80 ps. Increasing the input signal connect line and in the dielectric thickness between metal transition time increases the delay uncertainty; however, the layers. These effects create variations in the parasitic resis- effect of increasing the inverter size is dominant, and the tance and capacitance of a metal line, therefore affecting the overall delay uncertainty is decreased. Similar delay uncer- effective load of a buffer. tainty trends are shown in Figs. 5(a) and 5(b) for a r s u In order to evaluate the effect of load variations on buffer variation in and a variation in temperature, re- delay, the interconnect line structure shown in Fig. 6 is sim- F F G r ¥ v o spectively. ulated using Spectre R . In particular, the effect on buffer 417
- 4. delay caused by 10% variation in the line resistance and 7.0 Uncertainty in Buffer Delay (ps) r ±10% width variation at entire length capacitance is investigated. The following three scenarios 6.0 ±10% width variation at first half are considered, depending on the part of the line affected by ±10% width variation at second half 5.0 parameter variations: 4.0 i. r 10% variation in line resistance and capacitance at 3.0 the first half of the line closer to driving buffer. 2.0 ii. r 10% variation in line resistance and capacitance at 1.0 the second half of the line away from the buffer. 0.0 1 2 3 4 5 iii. r 10% variation in line resistance and capacitance at Buffer Size the entire line length. Figure 7: Reduction in delay uncertainty due to interconnect parameter variations with increased buffer size First half Second falf [2] S. I. A., “The National Technology Roadmap for Semicon- Driver ductors,” Technical Report, Semiconductor Industry Associ- ation, 2001. [3] S. Natarajan, M. A. Breuer, and S. K. Gupta, “Process Vari- ations and Their Impact on Circuit Operation,” Proceedings of the IEEE International Symposium on Defect and Fault Figure 6: Circuit model to evaluate the effect of line param- Tolerance in VLSI Systems, pp. 73–81, November 1998. eter variations on buffer delay [4] V. Mehrotra, S. L. Sam, D. Boning, A. Chandrakasan, R. Val- lishayee, and S. Nassif, “A Methodology for Modeling The uncertainty in the buffer delay for these three cases the Effects of Systematic Within-Die Interconnect and De- is illustrated in Fig. 7, where the buffer size is increased by vice Variation on Circuit Performance,” Proceedings of the up to five times. It is shown that for each case the delay ACM/IEEE Design Automation Conference, pp. 172–175, uncertainty is reduced with increasing buffer size. Note in June 2000. Fig. 7 that the uncertainty in buffer delay is less when the [5] K. T. Tang and E. G. Friedman, “Interconnect Coupling line parameters are affected at the second half of a line. This Noise in CMOS VLSI Circuits,” Proceedings of the ACM In- reduction in delay uncertainty is caused by the shielding ef- ternational Symposium on Physical Design, pp. 48–53, April 1999. fect of the first half of the line that isolates the buffer output from the parameter variations at the second half. [6] R. Sitte, S. Dimitrijev, and H. B. Harrison, “Device Parame- ter Changes Caused by Manufacturing Fluctuations of Deep Submicron MOSFET’s,” IEEE Transactions on Electron De- 4. CONCLUSIONS vices, Vol. 41, No. 11, pp. 2210–2215, November 1994. [7] S. Sauter, D. Schmitt-Landsiedel, R. Thewes, and W. Weber, The effects of device parameter variations on the delay un- “Effect of Parameter Variations at Chip and Wafer Level on certainty of a signal propagating through a CMOS buffer are Clock Skews,” IEEE Transactions on Semiconductor Manu- investigated. It is demonstrated that delay uncertainty is in- facturing, Vol. 13, No. 4, pp. 395–400, November 2000. versely proportional to the transistor current. Increasing the [8] J. F. Chappel and S. G. Zaky, “EMI Effects and Timing inverter size, therefore, reduces the delay uncertainty caused Design for Increased Reliability in Digital Systems,” IEEE by device parameter variations. In addition, the delay uncer- Transactions on Circuits and Systems I: Fundamental The- tainty due to interconnect parameter variations is examined. ory and Applications, Vol. 44, No. 2, pp. 130–142, February 1997. It is shown that variations in line parameter in the proximity of a driving buffer have a greater effect on buffer delay. In- [9] T. Sakurai and A. R. Newton, “Alpha-Power Law MOS- creasing the buffer size alleviates these effects and reduces FET Model and its Applications to CMOS Inverter Delay and Other Formulas,” IEEE Journal of Solid State Circuits, delay uncertainty. Vol. 25, No. 2, pp. 584–593, April 1990. [10] W. Shockley, “A Unipolar “Field-Effect” Transistor,” Pro- 5. REFERENCES ceedings of the Institute of Radio Engineers, pp. 1365–1377, November 1952. [1] G. E. Moore, “Progress in Digital Integrated Circuit,” Pro- [11] Affirma Spectre Circuit Simulator User Guide, Cadence De- ceedings of the IEEE International Electron Devices Meet- sign Systems, Inc., 2000. ing, pp. 11–14, December 1975. 418