mos-ak/gsa workshop at essderc/esscirc, helsinki, sept 16, 2011 a cumulative distribution...
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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
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 2
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 3
Importance of the yield optimization tasko Time-to-market, and cost effectiveness,o Robustness of design;
Corner (worst case – WC) methodso 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) methodo Time consuming, many simulations, o Does not actively manipulate / improve a design;
Worst-case distance (WCD) methodo Operates in process parameter space, not obvious from design
perspective; Cumulative distribution function (CDF) – based method
o Operates in design parameter space;
Introduction
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 4
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):
Introductory example
)LW
RRLW
R(Pmax
)RWL
RR(Pmax
)RRR(Pmax
maxsminW
maxsminW
maxminW
Rsmean,ss ,RR N
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 5
Variant IRmin 950 Rmax 1100 L 100×10-6 mRs,mean 10 /sq.Rs 0.3 /sq.
Wopt9.8×10-6 m
Variant IIRmin 900 Rmax 1100 L 100×10-6 mRs,mean 13 /sq.Rs 0.3 /sq.
Wide acceptable range of W
Introductory example
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 6
Variant IIIRmin 1060 Rmax 1100 L 100×10-6 mRs,mean 10 /sq.Rs 0.6 /sq.
Wopt0.93×10-6 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.
Introductory example
Question: Can CDF be used for real more complex tasks ?
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 7
D - design parameter vector
M - process parameter vector
X - circuit performance vector
S – specification vector
Yield optimization task
CDF-based approach
SXDD
Pmaxargopt
XMDXX
XMDXX
XMDXX
SX
max,NNmin,N
max2,2min2,
max1,1min1,
XXX,
,
,
X=f(D, M)Partia
l yie
lds
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 8
CDF-based approach
N
1jj
j
i
,
i
ii
MM
MX,X
,X,X
MD
MD
MMDMD
nom
nom
nom
Issues:Determine Mnom
o influences performance of the nominal design
o influences performance sensitivities
Determine M
Nominal design
i-th performance
SensitivitiesProcess par. variations
Remarks:Relation to BPV method used for statistical modelling, i.e. for extraction of M
Mj – random variables
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 9
CDF-based approach
Yield optimization task may be reformulated:
Dopt should maximize joint probability:
MD
MD
MD
nom
nom
nom
,XX
NM
MX,XXP
imax,i
1i
N
1jj
j
i
,
imin,iX M
Issues:Select reliable method for yield optimization, taking into account specific features of the task
Remarks:An optimization problem has been formulated
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 10
CDF-based approach
If Mj are uncorrelated normally-distributed random variables, then
is a random variable
N
1jj
j
i
,
MM
MXY
MD nom
N
1j
2
j
i
,
2
2YY
M
jMMXwhere,,0Y
MD nom
N
I.
Sensitivities Process par. variations
Issue:Selection of uncorrelated process parameters is very important
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 11
NN
i
xx,,
1imaxi,imini,
1maxi,imini, XMDXXPXMDXXP
CDF-based approach
Due to the unavoidable correlation between performances, direct yield
and product of partial yields are not equal.
NN
i
xx,,
1imaxi,imini,
1maxi,imini, XMDXXPXMDXXP
Issues:Take into account correlations between performances.
orAssume, that
II.
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 12
CDF-based approach
Based on assumptions I, II a joint probability (parametric yield)
MD
MD
MD
nom
nom
nom
,XX
NM
MX,XXP
imax,i
1i
N
1jj
j
i
,
imin,iX M
N
1inomimin,iinomimax,ii
N
1i
,XX
,XXiii
X
X nomimax,i
nomimin,i
,,
d
XXFXXF
xxf
MM DD
MD
MD
may be calculated as a product of CDFs Fi of normal distributions
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 13
0
0.1
0.2
0.3
0.4
0.5
0.6
-3 -2 -1 0 1 2 3x [arb. units]
prob
abili
ty d
ensi
ty f
unct
ion Xmax - X(D,Mnom)Xmin - X(D,Mnom)
CDF-based approach
Interpretation
If NX=1 case is considredthe task may be illustrated "geometrically"
Maximize shaded area
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 14
Calculations of standard deviations of process parameters based on
variations of performances and on performance sensitivities.
2)N(NSUBln
2)P(DW
2)P(DL
2)P(UO
2)P(NSUBln
2TOX
2221
1211
2Prds
2Ngm
2NIdsat
2NVth
2Prds
2Pgm
2PIdsat
2PVth
SS
SS
Backward Propagation of Variance Method
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 (σe
2)
Backward propagation method used to calculate variances of model technology parameters (σm
2) Least squares algorithm Sequental quadratic programming
(SQP) method
j
i2
ij2m
2e m
eS;σSσ
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 15
CDF-based approach vs BPV method
Functional block performance variances determined experimentally
Functional block performance (PCM) sensitivities @ nominal process parameters
Process parameter variances
BPV
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 16
CDF method - Inverter
Inverter performances J.P.Uemura, "CMOS Logic Circuit Design", Kluwer, 2002
pn
n,Tpnp,TDDI 1
VVVV
tt5.0t
CR1V
VV4ln
VV
V2t
CR1V
VV4ln
VV
V2t
LH,PHL,PP
outpDD
p,TDD
p,TDD
p,TLH,P
outnDD
n,TDD
n,TDD
n,THL,P
VVII IinDDmax,DD
Inverter threshold
Propagation delay
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 17
CDF method – OpAmp
4
1j j
c
ptgarcPM
gg
g
gg
gA
7o6o
6m
4o2o
2mv
SSg
gSSS 2
423
1m
3m2
22
21
2in
OpAmp performancesM.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
Equivalent input-referred noise power spectral density
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 18
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 19
Contour plots of yield in design parameter space
tP y ie ld 0 .2 0 .9
ID D m a x y ie ld 0 .8 0 .4 0 .2
CDF method - Inverter
open - partial yieldsclosed - product of partial yields (CDF)
solid - product of partial yields (CDF)dashed - product of partial yields (MC)dotted - yield (MC)
A "valley" results from the specification of tP,min constrain.
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 20
Av
yiel
d 0.
7
0.9
P M y ie ld 0 .2 0 .5 0 .7 0 .9
S in yie ld 0 .9 0 .7 0 .5 0 .2
CDF method – OpAmp
Contour plots of yield in design parameter spaceopen - partial yieldsclosed - product of partial yields (CDF)
solid - product of partial yields (CDF)dashed - product of partial yields (MC)dotted - yield (MC)
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 21
CDF-based method implementation
Objective function maximizationClose 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 22
CDF-based method implementation
Objective function maximizationObjective 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 2323
1. The presented CDF-based method may be used for IC block design optimizing parametric yield,
2. The method may predict parametric yield of the design,
3. The CDF-based method gives results very close to the time consuming Monte Carlo method,
4. 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),
5. 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,
6. The method may be used for performances determined both analytically, as well as via Spice-like simulations (batch mode required),
Conclusions (1)
* Pierre Dautriche, "Analog Design Trends & Challenges in 28 and 20 nm CMOS Technology", ESSDERC'2011
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7. Basic requirements for automatization of design task based on CDF method have been formulated,
8. 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,
9. 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,
10. Thus the method may be very useful for evaluation if the process is efficient enough to achieve a given yield.
11. 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,
12. Statistical modelling, i.e. determination of process parameter distribution is a critical issue, for MC, CDF, WCD methods of design.
Conclusions (2)
MOS-AK/GSA Workshop at ESSDERC/ESSCIRC, Helsinki, Sept 16, 2011 25
Acknowledgement
Financial support of EC within project PIAP-GA-2008-218255 ("COMON")
and
partial financial support (for presenter) of EC within project ACP7-GA-2008-212859 ("TRIADE")
are acknowledged.