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

A Fast Wafer-Level Spatial Variation Modeling Algorithm for Test Cost Reduction of Analog/RF Circuits Hugo Gonçalves 1,2 , Xin Li 1 , Miguel Correia 2 and Vitor Tavares 2 1 ECE Department, Carnegie Mellon University, USA 2 Faculdade de Engenharia, Universidade do Porto, Portugal 09/07/2014 Slide 1

Outline  Motivation and background  Virtual probe (VP)  Proposed approach  Dual Augmented Lagrangian method (DALM)  Two-pass test flow  Experimental results  Conclusions Slide 2

Process Variation 45nm 32nm 22nm Small size Large variation    V V V TH TH TH    L L L Doping fluctuation Δ V TH Parametric variations Δ L Line edge roughness Slide 3

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 4

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 Measured dies 20 160 140 15 Y Axis ??? 120 10 100 5 80 5 10 15 X Axis Measured delay values (normalized) Recovered wafer from 282 industrial chips map  [Chang11], [Kupp12], [Huang13], [Hsu13], etc. Slide 5

Virtual Probe (VP)  List a set of linear equations based on measurement data 20 160 DCT basis    function              140 15                  Y Axis     120 10                    100   5  Performance   measurement 80 DCT coefficients 5 10 15 X Axis N Measured delay values (normalized)         f ( x , y ) b x , y from 282 industrial chips i i  i 1 Results in an under-determined linear equation, since we have less measurements than unknown DCT coefficients Slide 6

Virtual Probe (VP)  Additional information is required to uniquely solve under- determined linear equation 20 160 DCT Coefficients (Magnitude) 300 140 15 200 Y Axis DCT 120 10 100 100 0 5 20 20 10 10 80 5 10 15 0 0 Y Axis X Axis X Axis Measured delay values (normalized) DCT coefficients (sparse) from 282 industrial chips If process variations are spatially correlated wafer maps show sparse patterns in frequency domain Slide 7

Virtual Probe (VP)  Solve sparse DCT coefficients by L1-norm regularization  DCT coefficients can be uniquely determined from a small number of measurements DCT basis function B      Sum of absolute values of all elements                             1        2    B α f α   min 2           2 1  α          Performance   Regularization parameter measurement DCT coefficients f α (sparse) Slide 8

Virtual Probe (VP) 1 Linear regression problem:    2    B α f α min 2 2 1 α  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 Slide 9

Outline  Motivation and background  Virtual probe (VP)  Proposed approach  Dual Augmented Lagrangian method (DALM)  Two-pass test flow  Experimental results  Conclusions Slide 10

Dual Problem 1    2    B α f α Primal problem: min 2 2 1 α  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 Slide 11

Strong Duality Primal function Dual function 1 1 1   2   α Bα f α    2  2 P ( ) ( ) x x f f D 2 1 2 2 2 2 2   T . . B x S T   D ( x )    α P ( ) D ( x )  α P ( ) Dual variable of P α x Size: Size: # of coefficients # of measurements Dual variable of D Slide 12

Dual Augmented Lagrangian  Define an auxiliary variable z to form an equality constraint Dual problem Dual problem w/ equality constraint 1 1    2  2 1 1 max D ( ) x x f f    2  2 max D ( ) x x f f 2 2 2 2 x z , 2 2 2 2 x  T S T . . z B x   T S T . . B x    z   Solve the augmented Lagrangian of the dual problem    1 1     2     2   2           x z α x f f α z B x z B x z T T T max L , , A 2 2 2 2 2 2 x z ,    , z 0      ( z ) Primal variable       , z  size = # of DCT coefficients Slide 13

Alternating Direction Method Augmented Lagrangian    1 1     2     2   2           x z α x f f α z B x z B x z T T T max L , , A 2 2 2 2 2 2 x z ,  Solve optimization with alternating direction method [Yang10] Variable update Optimality conditions P  ( z )           k z α k x , , L      α ( k 1 ) ( k ) T ( k )  A z P / B x   0  z z           k 1 α k     L x z , ,  1           A B α  ( k 1 ) T ( k ) ( k 1 ) x I BB f Bz 0  x                   α k 1 α k B x k 1 z k 1 T AL step Slide 14

Fast Matrix Inverse       1          B α ( k 1 ) T ( k ) ( k 1 ) x I BB f Bz  Since DCT basis functions are used, we have T  BB I  Hence, we do not need to explicitly calculate matrix inverse   1         B α ( k 1 ) ( k ) ( k 1 ) x f Bz   1 Slide 15

Fast Matrix-Vector Multiplication          k 1  P    k k α B x T z    1         B α ( k 1 ) ( k ) ( k 1 ) x f Bz   1                   α k 1 α k B x k 1 z k 1 T  Since DCT basis functions are used, we can calculate these matrix-vector multiplications by fast DCT or IDCT transform Slide 16

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 Measure all dies on first Measure few dies on a wafer following wafer Extract spatial pattern Predict spatial pattern by by VP VP Y N Predictable? Error is small? Y N Determine pass/fail by Measure all dies for all Measure all dies VP following wafers Slide 17

Error Estimation  Modeling error by VP must be sufficiently small to ensure small escape rate and yield loss   Yield Loss f ~ Escape Rate pdf , f ub ~ f lb lb ub f lb ub f     ~ ~ ~ ~         YL pdf f , f df d f ER pdf f , f df d f ~     lb f ub lb f ub ~ ~       f ub f lb f ub f lb ~ expected values from training f measured values from current wafer f Slide 18

Outline  Motivation and background  Virtual probe (VP)  Proposed approach  Dual Augmented Lagrangian method (DALM)  Two-pass test flow  Experimental results  Conclusions Slide 19

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 Lot ID 1 2 3 4 5 6 7 8 9 Wafer # 25 9 23 25 25 25 17 25 2 [Chang11]: H. Chang, K. Cheng, W. Zhang, X. Li and K. Butler, “Test cost reduction through performance prediction using virtual probe ,” ITC, 2011 Slide 20

Spatial Pattern Examples  Spatial pattern is observed for a subset of test items, but not all test items 1 1 100 100 80 80 Y Axis Y Axis 60 60 0.5 0.5 40 40 20 20 0 0 20 40 60 80 20 40 60 80 X Axis X Axis 1 1 100 100 80 80 Y Axis Y Axis 60 60 0.5 0.5 40 40 20 20 0 0 20 40 60 80 20 40 60 80 X Axis X Axis Test item #1 Test item #48 Slide 21