Efficient Sensitivity-Based Capacitance Modeling for Systematic and Random Geometric Variations

Transcription

1 Effcent Senstvty-Based Capactance Modelng for Systematc and Random Geometrc Varatons 16 th Asa and South Pacfc Desgn Automaton Conference Nck van der Mejs CAS, Delft Unversty of Technology, Netherlands January 6, 011 Yokohama, Japan Delft Unversty of Technology Challenge the future

2 Outlne Introducton Systematc & random varatons Modelng method (panel senstvty) Modelng for systematc varatons Modelng for random varatons Panel senstvty based statstcal modelng method Experments and results Case study: 8-bt bnary-scaled charge-redstrbuton DAC Modelng for both varatons Dagram Experment and result Concluson

3 Process Varablty Systematc Varaton Lthography, Etchng, CMP Layout dependent Random Varaton [Scheffer - 006] 3

4 Process Varablty Systematc Varaton Lthography, Etchng, CMP Layout dependent Random Varaton Lne-edge roughness Spatal correlaton [Asenov - 003] 4

5 Process Varablty Systematc Varaton Lthography, Etchng, CMP Layout dependent Random Varaton Lne-edge roughness Spatal correlaton Varabltes of R & C!? How to model BOTH varatons? [Asenov - 003] 5

6 Modelng Method BEM-based capactance extracton Partal short-crcut capactances C Nomnal capactance C 0 : sum of assocated partal short-crcut capactances C a C a, b b 6

7 Modelng Method Panel senstvty have been produced n calculaton of the nomnal C 0 usng BEM S C 1 j k, j = = Ck,aCk,b ρk εak a N b N j [B-CICC-009] ρ - : a small dsplacement of a panel k ε k - : materal permttvty around panel k - A k : the area of panel k C k, a C k, a a N - : an entry n the partal short-crcut capactance matrx - : capactance between a panel k and a node 7

8 Modelng Method Panel senstvty have been produced n calculaton of the nomnal C 0 usng BEM S C 1 j k, j = = Ck,aCk,b ρk εak a N b N j [B-CICC-009] no extra costly computaton partal capactances (data for standard capactance extracton) BEM FAST! 8

9 Modelng for Systematc Varaton Lnear / senstvty model C C = + j C C0 Δp = S p p k s p : the set of panels ncdent to the geometrc parameter p s p k,j 9

10 Modelng for Systematc Varaton Lnear / senstvty model C C = + j C C0 Δp = S p p k s p : the set of panels ncdent to the geometrc parameter p Varance of capactance due to the dmensonal varaton: s p k,j var(c) j sys = ( k s p S k,j ) σ p [B-CICC-009] σ p : standard devaton of parameter p 10

11 Modelng for Random Varatons Modelng the effects of Lne-Edge Roughness () on capactances 11

12 Statstcal Model of Capactances Capactance Modelng ΔC = L n l = 1 = 1 l S ρ - ΔC : capactance varaton nduced by the - ρ : a sequence of random varables (panel dsplacements) - S : panel senstvty assocated wth panel dsplacement - n l : the number of devaton panels for rough lne l - L : the number of rough lnes ρ 1

13 Statstcal Model of Capactances Capactance Modelng Varance of ΔC ΔC = L n l = 1 = 1 l S ρ σ ΔC = var( L n l = 1 = 1 l S ρ ) Some ρ s are NOT ndependent!! 13

14 Statstcal Model of Capactances Capactance Modelng Varance of ΔC ΔC = L n l = 1 = 1 l S ρ σ ΔC = var( L n l = 1 = 1 l S ρ ) σ ΔC = L nl s + var(ρ ) l= 1 = 1,j: < j s s j cov(ρ,ρ j ) 14

15 Random Varaton σ ΔC = var (ρ ) = σ L nl s + var(ρ ) l= 1 = 1,j: < j s s j cov(ρ x,ρ j ) σ y 15

16 Random Varaton σ ΔC = L nl s + var(ρ ) l= 1 = 1,j: < j s s j cov(ρ,ρ j ) var (ρ ) = σ x Gaussan correlaton functon: cov(ρ,ρ r, y η r r,y j,y j ) = σ exp( η Any correlaton functon! s the y-coordnate of the poston assocated wth ρ s the correlaton length n y-drecton ) σ η y 16

17 Random Varaton σ ΔC = L nl s + var(ρ ) l= 1 = 1,j: < j s s j cov(ρ,ρ j ) var (ρ ) = σ x Gaussan correlaton functon: cov(ρ,ρ r r,y j,y j ) = σ exp( η ) r, y η s the y-coordnate of the poston assocated wth ρ s the correlaton length n y-drecton σ η y Two characterzaton parameters 17

18 Experment - I σ = 3. 5nm η = 16nm - Measurement data from IMEC [Stucch - 007] 00nm Relatve std. devaton: σ C C Monte-Carlo smulaton 1000 samples 18

19 Experment - I σ = 3. 5nm η = 16nm - Measurement data from IMEC [Stucch - 007] 00nm Relatve std. devaton: σ C C Monte-Carlo smulaton 1000 samples σ / C C Error CPU Tme MC smulaton 0.681% Proposed model 0.603% 11.5% 50 19

20 Experment - II Usng the proposed model, one can easly study: 1. The relatonshp between and the conductor length;. The mpact of parameters σ and η on. σ C C σ C C Same structure as Experment-I 5 examples of msmatches Combnaton of varous σ and η Applcaton: Hgh accuracy vs. low power consumpton 0

21 Experment - II Usng the proposed model, one can easly study: 1. The relatonshp between and the conductor length;. The mpact of parameters σ and η on. σ C C σ C C Two sweepng parameters Proposed method: an hour MC approach: 43 days 1

22 Experment - II Usng the proposed model, one can easly study: 1. The relatonshp between and the conductor length;. The mpact of parameters σ and η on. σ C C σ C C The proposed modelng method provdes a fast and practcal tool for crcut desgners to estmate msmatches and optmze dmensons of crtcal structures accordngly.

23 A Case Study Novel passve devces wth hgh-precson structures 8-bt charge-redstrbuton DAC 55 dentcal unt capactors Mn. value of a unt capactor for hgh power effcency (0.5fF) σ C 0 Man consderaton: msmatch of capactors ( ) C 0 Unt cap. [Harpe-ESSCIRC-010] 3

24 A Case Study Novel passve devces wth hgh-precson structures 8-bt charge-redstrbuton DAC 55 dentcal unt capactors Mn. value of a unt capactor for hgh power effcency (0.5fF) σ C 0 Man consderaton: msmatch of capactors ( ) C 0 Desgn requrement: msmatch of the unt capactor < 1% 4

25 A Case Study Desgnrequres: msmatch of the unt capactor < 1% Smulatonshows: The msmatch of the unt capactor caused by the s around 0.5%; Measurementndcates: A random msmatch of the unt capactor beng better than 0.6%; Smulatonand measurement together conclude: Smulaton results are very reasonable; The structure can be used for more accurate desgns: e.g. 10-bt DAC (desgner s plan!); Desgn tool enables a new desgn, based on proper modelng but NOT guessng 5

26 Senstvty Based Modelng for Both Systematc and Random Varatons Desgn For Manufacturng Layout Tech. fle Dmensonal Parameters σ sys Rough lnes σ, η Senstvty-based modelng Systematc varablty Panel senstvty Random varablty Nomnal capactances Systematc msmatch Random msmatch BEM-based LPE Tool 6

27 Experment - III Two parallel conductors wdth/space = μm/μm; thckness = μm; length = 8μm on four edges of two conductors: σ = 0.03μm, η =. 00 μm σ = 0.04μm, η =. 88 μm Systematc varaton of the two conductors: σ = 0.03μm, σ = μm sys sys Parameters are chosen based on pure assumpton 7

28 Experment - III Two parallel conductors wdth/space = μm/μm; thckness = μm; length = 8μm on four edges of two conductors: σ = 0.03μm, η =. 00 μm σ = 0.04μm, η =. 88 μm Systematc varaton of the two conductors: σ = 0.03μm, σ = μm sys sys Parameters are chosen based on pure assumpton 3 Monte Carlo smulatons wth 1000 samples each Systematc varaton Random varaton Superposton of the above two 8

29 Experment - III σ C / C sys σ C / C σ C / C snr MC smulaton Proposed method.%.05% (7.7% error) 0.1% 0.4% (14.3% error).%.06% (7.18% error) CPU Tme 38h5 58 The systematc varaton s the domnant one Some desgns are senstve to both varatons and some (e.g. 8-bt DAC) are only vulnerable to random varatons Beng able to apply the approprate modelng technques s essental Extremely hgh effcency + good enough accuracy = a fast and convenent tool for DFM! 9

30 Concluson Senstvty based method for statstcal property of capactances due to both systematc and random varatons Modelng method for the effect of on capactance Smulatons & measurement on chps Good enough accuracy & hgh effcency Useful and convenent tool for msmatch estmaton & crcut optmzaton Overall pcture of the senstvty based method for both varatons Extenson of BEM LPE tools Servng DFM! 30