DESCRIPTION
A cumulative distribution function-based method for yield optimization of CMOS ICs. M . Yakupov*, D . Tomaszewski Division of Silicon Microsystem and Nanostructure Technology, Institute of Electron Technology, Warsaw, Poland * Currently MunEDA GmbH, Munich, Germany. Outline. Introduction - PowerPoint PPT Presentation
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PCM based Statistical Modeling using BPV approachof CMOS ICs
Institute of Electron Technology, Warsaw, Poland
* Currently MunEDA GmbH, Munich, Germany
*
*
Outline
Introduction
Application of the CDF method for inverter and opamp design
Remarks on CDF-based method implementation
Conclusions
*
*
Importance of the yield optimization task
Time-to-market, and cost effectiveness,
Fast design space exploration in terms of PVT variations,
Increase of number of corners with increase of types of
devices,
Lack of correlations between different parameters sets;
Monte-Carlo (MC) method
Worst-case distance (WCD) method
Operates in process parameter space, not obvious from design
perspective;
Cumulative distribution function (CDF) – based method
Operates in design parameter space;
Introduction
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*
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
*
Given: length L (design), sheet resistance Rs (model) being a
random variable of normal distribution:
Task: Design a resistor, which fulfills the condition: Rmin< R
<Rmax;
Find W (design):
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
Variant I
Rmin 950
Rmax 1100
Introductory example
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
Variant III
Rmin 1060
Rmax 1100
Conclusions:
yield depends on relation between parameter (R) constraints and
process (Rs) quality;
cumulative distribution function (CDF) has been successfuly used to
determine optimum design.
Introductory example
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
D - design parameter vector
M - process parameter vector
X - circuit performance vector
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
CDF-based approach
influences performance sensitivities
Remarks:
Relation to BPV method used for statistical modelling, i.e. for
extraction of M
Mj – random variables
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
CDF-based approach
Dopt should maximize joint probability:
Issues:
Select reliable method for yield optimization, taking into account
specific features of the task
Remarks:
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
CDF-based approach
is a random variable
N
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
CDF-based approach
Due to the unavoidable correlation between performances, direct
yield and product of partial yields are not equal.
Issues:
or
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
CDF-based approach
Based on assumptions I, II a joint probability (parametric
yield)
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
CDF-based approach
Maximize shaded area
*
There are two main approaches to design of robust circuits, which
are widely used in Analog Design. First is the less time consuming
Corners Files Method, or another name of this method is Worst-Case
Files. This method combines the device parameters which maximize or
minimize a single digital block performances, as speed or power
consumption of simples digital block invertor. But due to the lack
of correlation between model parameters in specific corner, this
approach gives too pessimistic estimation of analog circuit
performances.
*
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
*
Calculations of standard deviations of process parameters based on
variations of performances and on performance sensitivities.
Backward Propagation of Variance Method
*
· Selection of PCM performances (e)
(
(
(
Backward propagation method used to calculate variances of model
technology parameters (σm2)
· Least squares algorithm
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
*
CDF-based approach vs BPV method
Functional block performance variances determined
experimentally
Functional block performance (PCM) sensitivities @ nominal process
parameters
Process parameter variances
Functional block performance sensitivities @ nominal process
parameters
Process parameter variances
CDF method - Inverter
Inverter threshold
Propagation delay
CDF method – OpAmp
M.Hershenson, et al.,"Optimal Design of a CMOS Op-Amp via Geometric
Programming", IEEE Trans. CAD ICAS, Vol.20, No.1
Low-frequency gain
Phase margin
gmi, goi - input and output conductances of i-th transistor,
c is a unity-gain bandwidth,
pj - j-th pole of the circuit,
*
CDF method – Inverter, OpAmp
Monte-Carlo method 1000 samples
process parameters varied:
carrier mobilities
on, op,
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CDF method - Inverter
open - partial yields
dotted - yield (MC)
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CDF method – OpAmp
open - partial yields
dotted - yield (MC)
CDF-based method implementation
Objective function maximization
Close to the maximum the objective function may exhibit a
plateau;
Optimization task based on gradient approach requires in this case
2nd order derivatives of yield function, but…
this makes optimization based on gradient methods useless;
*
CDF-based method implementation
Objective function maximization
Objective function may exhibit more than one plateau or more local
maxima;
thus
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The presented CDF-based method may be used for IC block design
optimizing parametric yield,
The method may predict parametric yield of the design,
The CDF-based method gives results very close to the time consuming
Monte Carlo method,
The results of yield optimization (Yopt) based on CDF method have
direct interpretation in design parameter space (problem of
selection of design parameters: explicit or combined),
The design rules of the given IP and also discrete set of allowed
solutions* may be directly used and shown in the yield plots in the
design parameter space,
The method may be used for performances determined both
analytically, as well as via Spice-like simulations (batch mode
required),
Conclusions (1)
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Basic requirements for automatization of design task based on CDF
method have been formulated,
If the performance constraints are mild with respect to process
variability, a continuous set of design parameters, for which yield
close to 100% is expected,
If the constraints are severe with respect to process variability,
the method leads to unique solution, for which the parametric yield
below 100% is expected,
Thus the method may be very useful for evaluation if the process is
efficient enough to achieve a given yield.
The methodology may be used for design types (not only of ICs)
taking into account statistical variability of a process and aimed
at yield optimization,
Statistical modelling, i.e. determination of process parameter
distribution is a critical issue, for MC, CDF, WCD methods of
design.
Conclusions (2)
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Acknowledgement
and
are acknowledged.
*
)
¯
estimation of PCM data variances (
σ
e
2