1

Task Leader : Alex Orailoglu, UC San Diego

Students : Rasit Onur Topaloglu, UC San Diego, 2007

Industrial Liaisons : Hosam Haggag, National Semiconductor Corp.Patrick Drennan, Freescale Semiconductor, Inc.Mien Li, Advanced Micro Devices, Inc.

“Mismatch analysis for high speed, deep sub-micron blocks and simulation methodology”

Task ID:906.001

2

Technical Thrust : Circuit design

Anticipated Result : Mismatch simulation and testing methods, with possible implementation in an EDA environment

Task Description : Provide measurement, simulation, test and verification methods for mismatch for deep-submicron technologiesTask Deliverables : Report on developing a mismatch test methodology Report on developing level 1 sensitivity functions

3

Accomplishments During the Past Year : Devised a test generation methodology to target mismatch Devised a general methodology to derive sensitivity functions for mismatch Devised Forward Discrete Probability Propagation Method for estimation of high level parameter probability distributionsFuture Direction : Implementation of these techniques at behavioral levels : will enable ability to use along with HDL, ex.Verilog-AMS

Executive Summary

4

Technology Transfer & Industrial Interactions : Monthly telephone communications to National

Semiconductor on project progress

Internship at National Semiconductor

Publications : SRC Deliverable Reports : ( P007960 and P009498 )

On Mismatch in the Deep Sub-micron Era : From Physics to Circuits , ASPDAC 2004

Executive Summary

5

Task Leader : Alex Orailoglu, UC San Diego

Students : Ayse K. Coskun, UC San Diego, 2008

Chengmo Yang, UC San Diego, 2008

Industrial Liaisons : Hosam Haggag, National Semiconductor Corp.

“Mismatch for Next Generation”Task ID: 1184.001

6

Technical Thrust : Circuit design

Anticipated Result : A wafer-aware and design-to-avoid mismatch design flow for mixed-signal and RF circuits implemented in an EDA environment. Task Description : Provide mismatch-immune design and analysis methodologies including parasitics and passives Task Deliverables : Report on MINT modelsReport on mismatch verification and diagnosis, Nov04

7

Technology Transfer & Industrial Interactions : Monthly telephone communications to National

Semiconductor on project progress

Executive Summary

Future Direction : Discovery of mismatch integrated models and diagnosis Techniques to target mismatch

8

Outline

Mismatch Amplification

Test Generation

Test of Mismatch

Forward Discrete Probability PropagationProbability Discretization TheoryQ, F, B, R Operators and r-domainExperimental Results

Conclusions

Motivation

Excitation PlotsMismatch Factor

9

Test of Mismatch

10

Motivation for Testing

Find an analogous specialized test for mismatch

•Functional testing is not the only method for digital circuits

•While testing for stuck-at faults, other faults typically discovered also

GOALS

Low cost : measured in terms of speed and price of tester

Separate design from test : to earn test engineers time

Determinism : to provide pass and fail information

11

Mismatch AmplificationActivate the defect, then propagate

•Aim is to differentiate circuit response from the nominal

•Bias, voltage, temperature and input used to amplify mismatch

12

Excitation Plots Gain @100kHz vs. Widths of matched pair

•Dispersion from matched condition leads to appreciable reduction in observed parameter, ex. gain•Equal width variations in the pair => negligible reduction

no-m

ism

atch

dia

gona

l

max-mismatch diagonal

13

Deteriorating Effects of Mismatch Gain @100kHz vs. Widths of matched pair

•A wider range of equal variations on no-mismatch diagonal => still negligible reduction

no-m

ism

atch

dia

gona

l

14

Separation of Responses Frequency response DC response

•Fault-free responses are separated

•Fault-free responses sit on no-mismatch diagonal

•Vertical cuts are used in excitation plots for over-a-range plots

15

Mismatch Factor

•3-D response, when sampled, can be represented as a matrix

012314

123143

231432

314321

143210

stepsizeMF

|| 21

•Mismatch Factor (MF) gives a degree of mismatch effect in circuit for some parameter, ex. tox on an analysis, ex. sampled AC gain

∆1

∆2

stepsize

Matrix representation of response:

High MF => small mismatch causes appreciable impact

16

Other Observed Excitation Plots

•MF still effective due to symmetric nature

Sens. of AC gain to bias Sens. of AC gain to VDD

17

Test Algorithm

Mismatch (mm) pair,

physical parameter,

worst-case (V,T),

obtain MF’s;

select largest ones.

18

Input and Analysis Choices

•Bias, voltage, temperature and input signal Input Choices

•AC magnitude response : powerful for wide-band circuits

Analysis Choices

•DC response : to be used for digital circuits

•Sensitivities of these : wrt. circuit biases and inputs

•IDDQ : identified as being succesful for analog mismatch

Use circuit specs to constraint ranges: ex. AC or VDD range

19

Test Generation Ex. : high coverageAC100kHz AC2GHZ

DCVin=1.4V DCVin=1.5V

IDDQ

SAC100kHz

Vbias1SAC100kHz

Vbias2SAC2GHz

Vbias1SAC2GHz

Vbias2

SAC100kHz

Vbias1SAC100kHz

Vbias2SAC2GHz

Vbias1SAC2GHz

Vbias2

SIDDQ

Vbias1SIDDQ

Vbias2{VDD1, VDD2, T1, T2}W, mm1

W, mm2 ..VFB, mmN

..

..

Each entry excitation plot MF analysis type

Ana

lysi

s T

ypes

physical param. and mm pair, select highest MF in each row

20

Test Generation Example : low costAC100kHz AC2GHZ

DCVin=1.4V DCVin=1.5V

IDDQ

SAC100kHz

Vbias1SAC100kHz

Vbias2SAC2GHz

Vbias1SAC2GHz

Vbias2

SAC100kHz

Vbias1SAC100kHz

Vbias2SAC2GHz

Vbias1SAC2GHz

Vbias2

SIDDQ

Vbias1SIDDQ

Vbias2{VDD1, VDD2, T1, T2, mm1, mm2,..,mmN}W

tox ..VFB

..

..

Each entry excitation plot MF analysis type

Ana

lysi

s T

ypes

physical parameter, select highest MF in each row

21

Test Set for Low Cost ExampleAC2GHZ :Apply 1mV input AC at 3.3V, 300K, find AC gain

DCVin=1.4V : Apply 1.4V input DC 2.7V, 200K, find DC gain

IDDQ :At 3.3V, 300K, find power supply current

SAC2GHz

Vbias1

SAC100kHz

Vbias2

SIDDQ

Vbias2

W

•This test set targets the Width mismatch in the circuit

: Apply 1mV input AC at 2.7V, 200K; then change Vbias1 by 10% and repeat

: Apply 1mV input AC at 2.7V, 200K; then change Vbias2 by 10% and repeat

: At 2.7V, 200K, find power supply current;then change Vbias2 by 10% and repeat

If mismatch in Width parameter present, results differ appreciably

22

Test Set Size and Verification

•Reduction in number of test vectors intrinsic

•As simulation based, verification also intrinsic

•Apply this test set before any functional test, as this test catches most hard faults

•Test number can be reduced to analysis types*physical parameters

•Test number is analysis types*physical parameters*mismatch pairs for increased fault coverage

23

Outline

Mismatch Amplification

Test Generation

Test of Mismatch

Forward Discrete Probability PropagationProbability Discretization TheoryQ, F, B, R Operators and r-domainExperimental Results

Conclusions

Motivation

Excitation PlotsMismatch Factor

24

Forward Discrete Probability Propagation

25

Motivation for Probability Propagation

Find a novel propagation method

•Estimation of circuit parameters needed to examine effects of process variations•Gaussian assumption attributed to device parameters no longer

accurate

GOALS

Determinism : a stochastic output using known formulas

Algebraic tractability : enabling manual applicability

Speed & Accuracy : be comparable or outperform Monte Carlo

26

Shortcomings of Monte Carlo

•Non-determinism : Not manually applicable

•Limited for certain distributions : Random number generators only provide certain distributions

•Accuracy : May miss points that are less likely to occur due to random sampling; limited by the performance of random number generator

27

spdf(X) or (X)pdf(X)

p-domain r-domain

Probability Discretization Theory : QN Operator; p and r domains

•QN band-pass filter pdf(X) and divide into bins

))(()( XpdfQX N

N in QN indicates number or bins

Certain operators easy to apply in r-domain

28

spdf(X) or (X)

r-domain

Characterizing an spdf

Ni

ii wxpX..1

)()(

•can write spdf(X) as :

im

im

i dxXpdfp)1(

)(

2)1(

imwi

where :

pi : probability for i’th impulse

wi : value of i’th impulse

29

F Operator •F operator implements a function over spdf’s

spdf(X) or (X)

))(),..,(()( 1 rXXFY Xi, Y : random variables

r

r

rss

Xs

Xs

Xs

Xs wwfyppY

,..,1

1

1

1

1

1

1)),..,((..)(

pXs : Set of all samples s belonging to X

•Function applied to individual impulses•Individual probabilities multiplied

30

))(()(' XBX e

Band-pass, Be, Operator

))((]),[(:

)()(Xwnmwi

ii

ii

wxpX

•Eliminate samples having values out of rangeMargin-based Definition:

))(()

)(max(:

)()(Xp

e

ppi

ii

iii

i

wxpX

Error-based Definition:•Eliminate samples having probabilities least likely to occur

31

Re-bin, RN, Operator ))(()(' XRX N

•Samples falling into the same bin congregated in one

i

ii wxpX )()( ijs

ji bwstppj

.where :

Impulses after F Unite into one bin Resulting spdf(X)

32

The Necessity of Re-binning

•Non-linear nature of functions cause accumulation in certain ranges

Band-pass and re-bin operations needed after F operation

Impulses after F, before B and R

33

Error Analysis

12

2

jbjiji

ji pwmdi

)(:,

),(Total distortion:

dqQpdfqQEQ )(][2/

2/

222

Variance of quantization error:

•If quantizer uniform and small, quantization error random variable Q is uniformly distributed

2)(),( jiji wmwmd Distortion caused by representing samples in a bin by a single

sample:

mi : center or i’th bin

34

Connectivity Graph Used in Experiments

•Connectivity Graphs can tie physical parameters to circuit parameters

35

Algorithm Implementing the F Operator

While each random variable has its spdf computed

For each rv. which has all ancestor spdf’s computed

For each sample in X1

For each sample in Xr

Place an impulse with height p1,..,pr at x=f(v1,..,vr)

Apply B and R algorithms to this rv.

36

Algorithm for the B and R Operators

Divide this range into M bins

For each binPlace a quantizing impulse at the center of the bin with a height pi equal to the sum of all impulses within bin

Find maximum probability, pi-max, of quantized impulses within bins

Find new maximum and minimum values wi within impulses

Divide this range into N bins

Find maximum and minimum values wi within impulses

Eliminate impulses within bins which have a quantized impulse with smaller probability than error-rate*pi-max

For each binPlace an impulse at the center of the bin with height equal to sum of all impulses within bin

37

T NSUB

PHIf

Q, F, B, R on a Connectivity Graph

Q Q

F

B,R

•Repeated until we get the high level distributionUseful for device characterization also

38

Experimental Results

•Impulse representation for threshold voltage and transconductance are obtained through FDPP on the graph

(X) for gm(X) for Vth

39•A close match is observed after interpolation

Monte Carlo – FDPP Comparison

solid : FDPP dotted : Monte Carlo

Pdf of VthPdf of ID

40

Monte Carlo – FDPP Comparison with a Low Sample Number

•Monte Carlo inaccurate for moderate number of samples•Indicates FDPP can be manually applied without major accuracy degradation

solid : FDPP,100 samples

Pdf of FPdf of F

noisy : Monte Carlo, 1000 and 100000 samples respectively

41

P1

P2

Monte Carlo – FDPP Comparison one-to-many relationships and custom pdf’s

P3

P4

•Custom pdf’s not possible without a custom random number generator

•Monte Carlo overestimates for one-to-many relationships as same sample is used

42

Conclusions

•A specialized test selection mechanism for mismatch is introduced

•Forward Discrete Probability Propagation is introduced as an alternative to Monte Carlo based methods

•FDPP should be preferred when low probability samples are important, algebraic intuition needed, custom pdf’s are present or one-to-many relationships are present

•Test of Mismatch is a deterministic, general and low-cost methodology