MOS-AK/GSA Workshop at ESSDERC/ESSCIRC,
Helsinki, Sept 16, 2011
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
1. Introduction
2. Introductory example
3. Cumulative-distribution function-based approach
4. Application of the CDF method for inverter and opamp
design
5. Remarks on CDF-based method implementation
6. Conclusions
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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Introduction





Importance of the yield optimization task
o Time-to-market, and cost effectiveness,
o Robustness of design;
Corner (worst case – WC) methods
o Fast design space exploration in terms of PVT variations,
o Increase of number of corners with increase of types of devices,
o Lack of correlations between different parameters sets;
Monte-Carlo (MC) method
o Time consuming, many simulations,
o Does not actively manipulate / improve a design;
Worst-case distance (WCD) method
o Operates in process parameter space, not obvious from design
perspective;
Cumulative distribution function (CDF) – based method
o Operates in design parameter space;
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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Introductory example
Given: length L (design), sheet resistance Rs (model) being a random variable of
normal distribution:
Rs N Rs,mean , Rs
Task:
Design a resistor, which fulfills the condition: Rmin< R <Rmax;
Find W (design):
max P (Rmin  R  Rmax ) 
W
L
 Rmax ) 
W
W
W
W
max P (Rmin 
 Rs  Rmax  )
L
L
W
max P (Rmin  Rs 
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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Introductory example
Variant I
Rmin
950
Rmax
1100
L
100×10-6
Rs,mean 10
Rs
0.3


m
/sq.
/sq.
Wopt9.8×10-6 m
Variant II
Rmin
900
Rmax
1100
L
100×10-6
Rs,mean 13
Rs
0.3


m
/sq.
/sq.
Wide acceptable range of W
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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Introductory example
Variant III
Rmin
1060
Rmax
1100
L
100×10-6
Rs,mean 10
Rs
0.6


m
/sq.
/sq.
Wopt0.93×10-6 m,
but very low yield
Conclusions:
1. yield depends on relation between parameter (R) constraints and process (Rs)
quality;
2. cumulative distribution function (CDF) has been successfuly used to
determine optimum design.
Question:
Can CDF be used for real more complex tasks ?
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF-based approach

D - design parameter vector

M - process parameter vector

X - circuit performance vector

S – specification vector
Yield optimization task
Dopt  arg max PX  S 
X=f(D, M)
D
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
Partial yields
X1,min  X1D,M  X1,max

X 2,min  X 2 D,M  X 2,max
X S  

X ,min  X D,M  X ,max
NX
NX
 NX
7
CDF-based approach
Xi D,M  Xi D,Mnom  M 
NM
 Xi D,Mnom  
j1
Nominal design
i-th performance
 Xi
 Mj
D,Mnom
  Mj
Process par.
variations
Sensitivities
Remarks:

Relation to BPV method used for
statistical modelling, i.e. for
extraction of M

Mj – random variables
Issues:
Determine Mnom
o influences performance of the
nominal design
o influences performance
sensitivities
 Determine M

MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF-based approach
Yield optimization task may be reformulated:
Dopt should maximize joint probability:
 NX 
NM 


P    Xi,min  Xi D,Mnom   Xi
  Mj 
 i1 
j1  M j D,
Mnom 


 
 
 Xi,max  Xi D,Mnom  
 
 
Remarks:

An optimization problem has been
formulated
Issues:

Select reliable method for yield
optimization, taking into account
specific features of the task
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF-based approach
I.
If Mj are uncorrelated normally-distributed random variables, then
Y
 Xi
  Mj
j1  M j D,
Mnom 
NM

is a random variable
2
Y  N 0, Y ,
where

NM 
  Xi

2
2




Y


Mj

j1   Mj D,
Mnom 

Sensitivities
Process par.
variations
Issue:
 Selection of uncorrelated process
parameters is very important
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF-based approach
II. Due to the unavoidable correlation between performances, direct yield
and product of partial yields are not equal.
 Nx
 Nx
P  X i,min  X i D,M  X i,max    PX i,min  X i D,M  X i,max 


 i1
 i 1
Issues:

Take into account correlations between performances.
or

Assume, that
 Nx
 Nx

P  X i,min  X i D,M  X i,max    PX i,min  X i D,M  X i,max 


 i1
 i 1
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF-based approach
Based on assumptions I, II a joint probability (parametric yield)
 NX 
NM 


P    Xi,min  Xi D,Mnom   Xi
  Mj 
 i1 
j1  M j D,
Mnom 


 
 
 Xi,max  Xi D,Mnom  
 
 
may be calculated as a product of CDFs Fi of normal distributions
NX Xi,max XiD,Mnom 

 f i xi d xi 
i1 Xi,min XiD,Mnom 
NX

i1
 Fi Xi,max  Xi D,Mnom  Fi Xi,min  Xi D,Mnom 
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF-based approach
Interpretation
If NX=1 case is considred
the task may be illustrated
"geometrically"
probability density function
0.6
Xmin - X(D,Mnom)
0.5
Xmax - X(D,Mnom)
0.4
0.3
0.2
0.1
0
-3
-2
-1
0
x [arb. units]
1
2
3
Maximize shaded
area
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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Backward Propagation of Variance Method
Calculations of standard deviations of process parameters based on
variations of performances and on performance sensitivities.
C. C. McAndrew, “Statistical Circuit Modeling,” SISPAD 1998, pp. 288-295.
 Selection of non-correlated process
parameters (m)
 Selection of PCM performances (e)

Measurement of the PCM data

Statistical analysis of the PCM data;
estimation of PCM data variances (σe2)

Backward propagation method used to
calculate variances of model
technology parameters (σm2)
 Least squares algorithm
 Sequental quadratic programming
(SQP) method
 2



Vth
P


2




Idsat
P
 2



TOX
 2

 2

gmP 
ln NSUB(P ) 


2


 rds
P   S11 S12   2


UO(P ) 



  S21 S22  
2

2


DL(P ) 



Vth N
   2

  
DW (P ) 
2


Idsat N 
2


 2

ln NSUB(N)


 gmN 



 2

2
 rdsP  
 ei 
2
2

σ e  S σ m ; Sij  
  




 m j 
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF-based approach vs BPV method
Functional block performance
(PCM) sensitivities @ nominal
process parameters
BPV
Process parameter variances
Functional block performance
variances determined
experimentally
CDF of functional block
performance variances
Functional block performance
sensitivities @ nominal
process parameters
Process parameter variances
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF method - Inverter
Inverter performances J.P.Uemura, "CMOS Logic Circuit Design", Kluwer, 2002
Inverter threshold
VI 
VDD  V T,p  n p  V T,n
1  n p
IDD,max  IDD Vin  VI
Propagation delay
 2 V T,n
 VDD  V T,n  
 ln  4
 1   Rn  Cout
tP,HL  
VDD


 VDD  V T,n
 2 V T,p
 VDD  V T,p


 ln 4
 1   Rp  Cout
tP,LH  


VDD
 VDD  V T,p


tP  0.5  tP,HL  tP,LH
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF method – OpAmp
OpAmp performances
M.Hershenson, et al.,"Optimal Design of a CMOS OpAmp via Geometric Programming", IEEE Trans. CAD
ICAS,Vol.20, No.1
Low-frequency gain
 gm2   gm6 


A v  





g
g
g
g
 o 2 o 4   o 6 o7 
Phase margin
 
PM     arc tg  c 
 pj 
j1
 
4
Equivalent input-referred noise
power spectral density
2
2  2  2   gm3   2  2
Sin S1 S2 
 S3 S4
g
 m1 


 gmi, goi - input and output
conductances of i-th transistor,
 c is a unity-gain bandwidth,
 pj - j-th pole of the circuit,
 Sk - input-referred noise power
spectral densities, consisting of
thermal and 1/f components.
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF method – Inverter, OpAmp
 Monte-Carlo method 1000
samples
 simple MOSFET model
 0.8 m CMOS technology
o tox=20 nm,
o Vthn=0.7 V, Vthp=-0.9 V
 process parameters
varied:
o gate oxide thickness
tox,
o substrate doping conc.
Nsubn, Nsubp,
o carrier mobilities
on, op,
o fixed charge densities
Nssn, Nssp
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF method - Inverter
Contour plots of yield in design parameter space
open - partial yields
closed - product of partial yields (CDF)
solid - product of partial yields (CDF)
dashed - product of partial yields (MC)
dotted - yield (MC)
IDDmax yield
0.8
0.4
0.2
tP yield
0.2
0.9
A "valley" results from the specification of tP,min constrain.
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF method – OpAmp
Contour plots of yield in design parameter space
open - partial yields
closed - product of partial yields (CDF)
solid - product of partial yields (CDF)
dashed - product of partial yields (MC)
dotted - yield (MC)
Sin yield
0.9
0.7
0.5
0.2
0.9
Av yield
0.7
PM yield
0.2
0.5
0.7
0.9
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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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;
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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CDF-based method implementation
Objective function maximization
 Objective function may exhibit more than one plateau or more local
maxima;
thus
 a non-gradient global optimization method is required.
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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Conclusions (1)
1.
2.
3.
4.
5.
6.
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),
* Pierre Dautriche, "Analog Design Trends & Challenges in 28 and 20 nm CMOS Technology",
ESSDERC'2011
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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Conclusions (2)
7.
8.
9.
10.
11.
12.
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.
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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Acknowledgement
Financial support of EC within project PIAP-GA-2008-218255 ("COMON")
and
partial financial support (for presenter) of EC within project ACP7-GA-2008212859 ("TRIADE")
are acknowledged.
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011
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