PCM based Statistical Modeling using BPV approach

1
A cumulative distribution function-based method
for yield optimizationof CMOS ICs
M. Yakupov, D. Tomaszewski Division of Silicon
Microsystem and Nanostructure Technology,Institut
e of Electron Technology, Warsaw, Poland
Currently MunEDA GmbH, Munich, Germany
2
Outline

• Introduction
• Introductory example
• Cumulative-distribution function-based approach
• Application of the CDF method for inverter and
  opamp design
• Remarks on CDF-based method implementation
• Conclusions

3
Introduction

• Importance of the yield optimization task
• Time-to-market, and cost effectiveness,
• Robustness of design
• Corner (worst case WC) methods
• 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
• Time consuming, many simulations,
• Does not actively manipulate / improve a design
• 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

4
Introductory example
Given length L (design), sheet resistance Rs
(model) being a random variable of normal
distribution Task Design a resistor, which
fulfills the condition Rminlt R ltRmax Find W
(design)
5
Introductory example
Variant I Rmin 950 ? Rmax 1100 ? L 10010-6
m Rs,mean 10 ?/sq. ?Rs 0.3 ?/sq. Wopt?9.810-
6 m Variant II Rmin 900 ? Rmax 1100 ? L 10010
-6 m Rs,mean 13 ?/sq. ?Rs 0.3 ?/sq. Wide
acceptable range of W
6
Introductory example

• Variant III
• Rmin 1060 ?
• Rmax 1100 ?
• L 10010-6 m
• Rs,mean 10 ?/sq.
• ?Rs 0.6 ?/sq.
• Wopt?0.9310-6 m,
• but very low yield
• 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.

Question Can CDF be used for real more complex
tasks ?
7
CDF-based approach

• D - design parameter vector
• M - process parameter vector
• X - circuit performance vector
• S specification vector
• Yield optimization task

Xf(D, M)
Partial yields
8
CDF-based approach
Process par. variations
Nominal design i-th performance
Sensitivities

• Issues
• Determine Mnom
• influences performance of the nominal design
• influences performance sensitivities
• Determine ?M

• Remarks
• Relation to BPV method used for statistical
  modelling, i.e. for extraction of ?M
• ?Mj random variables

9
CDF-based approach
Yield optimization task may be reformulated Dopt
should maximize joint probability

• Issues
• Select reliable method for yield optimization,
  taking into account specific features of the task

• Remarks
• An optimization problem has been formulated

10
CDF-based approach
If Mj are uncorrelated normally-distributed
random variables, then is a random variable
I.
Sensitivities
Process par. variations

• Issue
• Selection of uncorrelated process parameters is
  very important

11
CDF-based approach
Due to the unavoidable correlation between
performances, direct yield and product of partial
yields are not equal.
II.

• Issues
• Take into account correlations between
  performances.
• or
• Assume, that

12
CDF-based approach
Based on assumptions I, II a joint probability
(parametric yield)
may be calculated as a product of CDFs Fi of
normal distributions
13
CDF-based approach
Interpretation If NX1 case is considred the
task may be illustrated "geometrically"
Maximize shaded area
14
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.
15
CDF-based approach vs BPV method
BPV
Functional block performance (PCM) sensitivities
_at_ nominal process parameters
Process parameter variances
Functional block performance variances determined
experimentally
Functional block performance sensitivities _at_
nominal process parameters
CDF of functional block performance variances
Process parameter variances
16
CDF method - Inverter
Inverter performances J.P.Uemura, "CMOS Logic
Circuit Design", Kluwer, 2002
Inverter threshold
Propagation delay
17
CDF method OpAmp
OpAmp performances 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,
• Sk - input-referred noise power spectral
  densities, consisting of thermal and 1/f
  components.

Equivalent input-referred noise power spectral
density
18
CDF method Inverter, OpAmp

• Monte-Carlo method 1000 samples
• simple MOSFET model
• 0.8 ?m CMOS technology
• tox20 nm,
• Vthn0.7 V, Vthp-0.9 V
• process parameters varied
• gate oxide thickness tox,
• substrate doping conc. Nsubn, Nsubp,
• carrier mobilities?on, ?op,
• fixed charge densities Nssn, Nssp

19
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)
A "valley" results from the specification of
tP,min constrain.
20
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)
21
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
22
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.

23
Conclusions (1)

• 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
23
24
Conclusions (2)

• 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.

24
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.