PCM based Statistical Modeling using BPV approachof CMOS ICs
Institute of Electron Technology, Warsaw, Poland
* Currently MunEDA GmbH, Munich, Germany
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Outline
Introduction
Application of the CDF method for inverter and opamp design
Remarks on CDF-based method implementation
Conclusions
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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):
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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.
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Variant I
Rmin 950
Rmax 1100
Introductory example
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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.
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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
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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.
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D - design parameter vector
M - process parameter vector
X - circuit performance vector
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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.
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CDF-based approach
influences performance sensitivities
Remarks:
Relation to BPV method used for statistical modelling, i.e. for extraction of M
Mj – random variables
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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.
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CDF-based approach
Dopt should maximize joint probability:
Issues:
Select reliable method for yield optimization, taking into account specific features of the task
Remarks:
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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.
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CDF-based approach
is a random variable
N
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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.
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CDF-based approach
Due to the unavoidable correlation between performances, direct yield and product of partial yields are not equal.
Issues:
or
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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.
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CDF-based approach
Based on assumptions I, II a joint probability (parametric yield)
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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.
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CDF-based approach
Maximize shaded area
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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.
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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
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· Selection of PCM performances (e)
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(
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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,
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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;
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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.
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estimation of PCM data variances (
σ
e
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