Modeling Algorithm for Test Cost Reduction of Analog/RF Circuits - - PowerPoint PPT Presentation

Slide 1

A Fast Wafer-Level Spatial Variation Modeling Algorithm for Test Cost Reduction

• f Analog/RF Circuits

Hugo Gonçalves1,2, Xin Li1, Miguel Correia2 and Vitor Tavares2

1ECE Department, Carnegie Mellon University, USA 2Faculdade de Engenharia, Universidade do Porto, Portugal

09/07/2014

Slide 2

Outline

 Motivation and background

 Virtual probe (VP)

 Proposed approach

 Dual Augmented Lagrangian method (DALM)  Two-pass test flow

 Experimental results  Conclusions

Slide 3

Process Variation

Small size Large variation 45nm 32nm 22nm

L L L V V V

TH TH TH

     

ΔVTH ΔL Parametric variations Doping fluctuation Line edge roughness

Slide 4

Wafer Probe Test

 Multiple test items must be measured for each die  An industrial example of dual radio RF transceiver

 ~1 second testing time per die  ~6500 dies per wafer  ~ 2 hour testing time per wafer

 Measuring all test items is time-consuming

~2h per wafer

Slide 5

Test Cost Reduction by Spatial Variation Modeling

 Measure a small number of dies at selected spatial locations  Recover the full wafer map by statistical algorithm

 [Chang11], [Kupp12], [Huang13], [Hsu13], etc.

Measured delay values (normalized) from 282 industrial chips

5 10 15 5 10 15 20 X Axis Y Axis 80 100 120 140 160

Measured dies Recovered wafer map

???

Slide 6

Virtual Probe (VP)

 List a set of linear equations based on measurement data

Measured delay values (normalized) from 282 industrial chips

5 10 15 5 10 15 20 X Axis Y Axis 80 100 120 140 160

                                                            

DCT coefficients Performance measurement DCT basis function

Results in an under-determined linear equation, since we have less measurements than unknown DCT coefficients

   





 

N i i i

y x b y x f

1

, ) , ( 

Slide 7

Virtual Probe (VP)

 Additional information is required to uniquely solve under-

determined linear equation

Measured delay values (normalized) from 282 industrial chips

DCT DCT coefficients (sparse)

5 10 15 5 10 15 20 X Axis Y Axis 80 100 120 140 160 10 20 10 20 100 200 300 X Axis Y Axis DCT Coefficients (Magnitude)

If process variations are spatially correlated wafer maps show sparse patterns in frequency domain

Slide 8

Virtual Probe (VP)

 Solve sparse DCT coefficients by L1-norm regularization

 DCT coefficients can be uniquely determined from a small

number of measurements

                                                            

DCT coefficients α (sparse) Performance measurement f DCT basis function B Sum of absolute values of all elements Regularization parameter

2 2 1

1 min 2      

α

B α f α

Slide 9

Virtual Probe (VP)

 There is no closed-form solution  A standard interior-point solver is not computationally efficient  We aim to develop an application-specific solver to reduce

computational time and, hence, testing cost

Linear regression problem:

2 2 1

1 min 2      

α

B α f α

Slide 10

Outline

 Motivation and background

 Virtual probe (VP)

 Proposed approach

 Dual Augmented Lagrangian method (DALM)  Two-pass test flow

 Experimental results  Conclusions

Slide 11

Dual Problem

 Key idea: form a dual problem to reduce the number of unknowns  Primal problem

 # of unknowns = # of DCT coefficients  # of dies

 Dual problem

 # of unknowns = # of measurements

 Since we have substantially less measurements than unknowns,

solving the dual problem is significantly more efficient

2 2 1

1 min 2      

α

B α f α

Primal problem:

Slide 12

Strong Duality

) (



α P ) (



x D ) ( ) (

  

x α D P

2 2 1

1 ( ) 2 P     α Bα f α

α x

Dual variable of P Dual variable of D Primal function Dual function

2 2 2 2

1 1 ( ) 2 2 . .

T

D S T 



     x x f f B x

Size: # of coefficients Size: # of measurements

Slide 13

Dual Augmented Lagrangian

 Define an auxiliary variable z to form an equality constraint  Solve the augmented Lagrangian of the dual problem

 

 

 

2 2 2 2 2 2 ,

1 1 max , , 2 2 2

T T T A

L



              

x z

x z α x f f α z B x z B x z

Primal variable size = # of DCT coefficients Dual problem

2 2 2 2

1 1 max ( ) 2 2 . .

T

D S T 



    

x

x x f f B x

           

  

z z z , , ) (

2 2 2 2 ,

1 1 max ( ) 2 2 . .

T

D S T 



     

x z

x x f f z B x z

Dual problem w/ equality constraint

Slide 14

Alternating Direction Method

 Solve optimization with alternating direction method [Yang10]

Augmented Lagrangian

 

 

 

2 2 2 2 2 2 ,

1 1 max , , 2 2 2

T T T A

L



              

x z

x z α x f f α z B x z B x z

   

 

, ,

k k A

L    x z α z

   

 

1

, ,

k k A

L



   x z α x

       

 

1 1 1 k k k k T



  

    α α B x z

Optimality conditions Variable update

z ) (z P



  

 

) ( ) ( ) 1 (

/

k T k k

P x B α z  

 





AL step

   

) 1 ( ) ( 1 ) 1 (   

     

k k T k

Bz Bα f BB I x  

Slide 15

Fast Matrix Inverse

 Since DCT basis functions are used, we have  Hence, we do not need to explicitly calculate matrix inverse

T 

BB I

   

) 1 ( ) ( 1 ) 1 (   

     

k k T k

Bz Bα f BB I x  

 

) 1 ( ) ( ) 1 (

1 1

 

     

k k k

Bz Bα f x  

Slide 16

Fast Matrix-Vector Multiplication

 Since DCT basis functions are used, we can calculate these

matrix-vector multiplications by fast DCT or IDCT transform

 

   

 

1 k

k k T

P 





  z α B x

       

 

1 1 1 k k k k T



  

    α α B x z

 

) 1 ( ) ( ) 1 (

1 1

 

     

k k k

Bz Bα f x  

Slide 17

Two-Pass Test Flow

 Measure all dies on one wafer if its spatial pattern cannot be

predicted by a number of pre-selected dies

Pre-test analysis Test cost reduction

Predictable? N Extract spatial pattern by VP Measure all dies for all following wafers Y Measure few dies on a following wafer Predict spatial pattern by VP Measure all dies on first wafer Error is small? N Y Determine pass/fail by VP Measure all dies

Slide 18

Error Estimation

 Modeling error by VP must be sufficiently small to ensure small

escape rate and yield loss

Escape Rate Yield Loss lb ub

f f ~ f ~ f f measured values from current wafer

expected values from training lb ub lb ub

 



    

  

lb f ub f ub f lb

f d df f f ER

~

~ ~ , pdf

 



    

  

lb f ub f ub f lb

f d df f f YL

~ ~

~ ~ , pdf

 

f f ~ , pdf

Slide 19

Outline

 Motivation and background

 Virtual probe (VP)

 Proposed approach

 Dual Augmented Lagrangian method (DALM)  Two-pass test flow

 Experimental results  Conclusions

Slide 20

Experimental Setup

 Production test data of an industrial dual radio RF transceiver

 9 lots and 176 wafers in total  6766 dies per wafer and 51 test items per die – test items were

selected by [Chang11]

 1,089,120 good dies and 101,696 bad dies

[Chang11]: H. Chang, K. Cheng, W. Zhang, X. Li and K. Butler, “Test cost reduction through performance prediction using virtual probe,” ITC, 2011

Lot ID 1 2 3 4 5 6 7 8 9 Wafer # 25 9 23 25 25 25 17 25 2

Slide 21

Spatial Pattern Examples

 Spatial pattern is observed for a subset of test items, but not

all test items

20 40 60 80 20 40 60 80 100 X Axis Y Axis 0.5 1 20 40 60 80 20 40 60 80 100 X Axis Y Axis 0.5 1 20 40 60 80 20 40 60 80 100 X Axis Y Axis 0.5 1 20 40 60 80 20 40 60 80 100 X Axis Y Axis 0.5 1

Test item #1 Test item #48

Slide 22

Wafer Map Prediction

 Two different solvers are implemented for comparison purpose

 IPM: interior-point method  DALM: dual augmented Lagrangian method

 DALM achieves up to 37 runtime speedup in this example

Number of Dies IPM DALM Runtime (Sec.) Runtime (Sec.) Iteration # Speed-up 100 48.3 12.2 7027 3.96 250 62.7 10.3 5664 6.07 500 84.7 8.9 5083 9.52 1000 119.9 8.1 4504 14.88 2000 171.2 7.3 3922 23.56 4000 255.2 6.7 3580 37.86

Slide 23

Wafer Map Prediction

 IPM and DALM result in identical modeling errors

20 40 60 0.5 1 Test Item Index Normalized Error IPM DALM 10

2

10

3

10

4

0.5 1 Number of Measured Dies Normalized Error IPM DALM

Modeling error with 2000 measured dies per wafer Modeling error for test item #48

Slide 24

Wafer Map Prediction

 IPM and DALM predict identical wafer maps

Actual wafer map Predicted wafer map by IPM Predicted wafer map by DALM

20 40 60 80 20 40 60 80 100 X Axis Y Axis 0.5 1 20 40 60 80 20 40 60 80 100 X Axis Y Axis 0.5 1 20 40 60 80 20 40 60 80 100 X Axis Y Axis 0.5 1

Slide 25

Test Cost Reduction

 Total number of measured dies for each test item

 IPM and DALM yield identical results

Full IPM DALM Overall test cost 60M 32M 32M Test cost reduction  1.9 1.9 Escape rate  1.2103 1.2103 Yield loss  2.0103 2.0103

0.0E+0 2.0E+5 4.0E+5 6.0E+5 8.0E+5 1.0E+6 1.2E+6 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 Number of Measured Dies Test Item Index Full IPM DALM

Slide 26

Conclusions

 Reducing test cost is a critical task for nanoscale integrated

circuit design and manufacturing

 Virtual Probe (VP) is an efficient method for test cost reduction

based on wafer-level spatial variation modeling

 Propose an efficient Dual Augmented Lagrangian method

(DALM) to reduce the computational cost of VP

 Achieve up to 37 runtime reduction over the conventional

interior-point solver

 The proposed DALM solver can be further applied to a number

• f other analog CAD problems related to sparse approximation

 E.g., analog performance modeling, analog self-healing, etc.

Slide 27

References

 [Chang11]: H. Chang, K. Cheng, W. Zhang, X. Li and K. Butler, “Test cost

reduction through performance prediction using virtual probe,” ITC, 2011

 [Kupp12]: N. Kupp, K. Huang, J. Carulli and Y. Makris, “Spatial estimation

• f wafer measurement parameters using Gaussian process models,”

ITC, 2012

 [Huang13]: K. Huang, N. Kupp, J. Carulli and Y. Makris, “Handling

discontinuous effects in modeling spatial correlation of wafer-level analog/RF tests,” DATE, 2013

 [Hsu13]: C. Hsu, F. Lin, K. Cheng, W. Zhang, X. Li, J. Carulli and K. Butler,

“Test data analytics - exploring spatial and test-item correlations in production test data,” ITC, 2013

 [Yang10]: J. Yang and Y. Zhang, “Alternating direction algorithms for l1-

problems in compressive sensing,” Technical Report, TR09-37, Rice University, 2010