Systems Approaches to Estimation Problems in Thin Film Processing
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1 Systems Approaches to Estimation Problems in Thin Film Processing Tyrone Vincent October 20, 2008 T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
2 Acknowledgements Matt Hilt, Numerica Corporation Renee Spinhirne, Lockheed-Martin Bharat Joshi, University of North Carolina, Charlotte Lin Simpson, NREL Kameshwar Poolla, UCB Costas Spanos, UCB T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
3 The Problem of Interest State Estimation of Dynamic Systems x 3 u x 2 y x 1 goal: determine current internal state from observable measurements T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
4 The Problem of Interest State Estimation of Dynamic Systems x 3 u(t) y(t) u t x 2 t y x 1 goal: determine current internal state from observable measurements T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
5 The Problem of Interest State Estimation of Dynamic Systems x 3 u(t) y(t) u t x 2 t y x 1 goal: determine current internal state from observable measurements T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
6 The Problem of Interest State Estimation of Dynamic Systems x 3 u(t) y(t) u t x 2 t y x 1 goal: determine current internal state from observable measurements T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
7 The Problem of Interest State Estimation of Dynamic Systems x 3 u(t) y(t) u t x 2 t y x 1 goal: determine current internal state from observable measurements T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
8 The Problem of Interest State Estimation of Dynamic Systems x 3 u(t) y(t) u t x 2 t y x 1 goal: determine current internal state from observable measurements T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
9 The Problem of Interest Examples given input/output data, estimate processing conditions given in-situ reflectance measurement, estimate deposition rate given partial measurement of wafer parameter, estimate parameter value across the wafer Systems Questions Experiment Design Sensor Selection Measurement Sequencing T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
10 The Problem of Interest Examples given input/output data, estimate processing conditions given in-situ reflectance measurement, estimate deposition rate given partial measurement of wafer parameter, estimate parameter value across the wafer Systems Questions Experiment Design Sensor Selection Measurement Sequencing T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
11 Outline 1 Enhancement of Estimation of Relative Deposition Rate 2 Observability Enhancement for Optical Measurements 3 Measurement Sequencing in Semiconductor Manufacturing 4 Conclusion T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
12 Enhancement of Estimation of Relative Deposition Rate Outline 1 Enhancement of Estimation of Relative Deposition Rate 2 Observability Enhancement for Optical Measurements 3 Measurement Sequencing in Semiconductor Manufacturing 4 Conclusion T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
13 Enhancement of Estimation of Relative Deposition Rate Multi-Zone Co-Evaporation Deposition on moving substrate. Effusion sources create metal plume via resistive heating. Different materials can be deposited in each zone Composition measured at exit T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
14 Enhancement of Estimation of Relative Deposition Rate Deposition Model T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
15 Enhancement of Estimation of Relative Deposition Rate Multi-Zone Model with Transport Delay Compensation T set 1 L therm T 1 N T set 2 L therm T 2 N r 1 L plume + r 2 L plume + y With constant set-points, individual deposition rates are not observable. T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
16 Enhancement of Estimation of Relative Deposition Rate Perturbation Experiment idea: add small perturbations to source set-points output perturbations encode relative sensitivity of deposition rate to temperature perturbations. perturbations chosen to meet manufacturing tolerances T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
17 Enhancement of Estimation of Relative Deposition Rate Effusion Source Model effusion rate assuming supersaturation: r e = κ exp(a) ( exp b ) (τ melt ) c+0.5 τ melt deposition rate to good approximation ( r = δη exp b τ melt ) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
18 Enhancement of Estimation of Relative Deposition Rate Rate Sensitivity biased measurement τ of melt temperature ατ + β = τ melt deposition rate sensitivity to temperature dr dτ = αb τ 2 melt r T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
19 Enhancement of Estimation of Relative Deposition Rate The advantage of ratios ratio of sensitivities dr 1 dτ dr 1 dτ ( τ 2 ) 2 = melt α1 b 1 r 1 τmelt 1 α 2 b 2 r 2 same metal implies b 1 b 2 same type of thermocouple implies α 1 α 2 ratios cancel (most) unknowns dr 1 dτ dr 1 dτ ( τ 2 ) 2 = melt r1 τmelt 1 r 2 melt temperature ratio τ 2 melt τ 1 melt is fairly well known ( ). T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
20 Enhancement of Estimation of Relative Deposition Rate Manufacturing Tolerances During processing, perturbations are undesired in Total amount deposited Cu/(In+Ga) (controls microstructure) Ga/(In+Ga) (controls bandgap) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
21 Enhancement of Estimation of Relative Deposition Rate Example Experiment Data from production chamber courtesy Global Solar Energy, Tucson, AZ Independent perturbation applied to sources in two zones depositing the same material. Experiment increased output standard deviation from 20 to 40 Angstroms. Temperature Change (K) Relative Thickness (A) Perturbation Sequences Composition Variation Zone 1 Zone Time (s) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
22 Enhancement of Estimation of Relative Deposition Rate Experiment Analysis T set 1 L therm T 1 dr 1 dτ T set 2 L therm T 2 dr 2 dτ r 1 L plume + r 2 L plume + y Find gains to best match small scale input/output behavior T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
23 Enhancement of Estimation of Relative Deposition Rate Relative Deposition Rate Estimate filtered output model fit 100 Ratio estimated from shutter experiment: 3.5 Ratio estimated from system identification experiment: 2.4 Magnitude Time (s) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
24 Enhancement of Estimation of Relative Deposition Rate Block diagram in terms of input variance assumption: input has characteristics of i.i.d random sequence with covariance R. T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
25 Enhancement of Estimation of Relative Deposition Rate Estimate Quality Analysis s N( s, S 1 ) g = s 1 s s s 1 3 p g (g) g T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
26 Enhancement of Estimation of Relative Deposition Rate Estimate Quality Analysis s N( s, S 1 ) g = s 1 s s s 1 3 p g (g) g T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
27 Enhancement of Estimation of Relative Deposition Rate Estimate Quality Result T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
28 Enhancement of Estimation of Relative Deposition Rate Convex approximations for ratio distributions E(S) = { s (s s) S(s s) δ 2} s s 1 3 p g (g) g T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
29 Enhancement of Estimation of Relative Deposition Rate Convex approximations for ratio distributions 2 φ max (S) = { max subject to ratio(s) s E(S) 1.5 s min s s 1 3 p g (g) 2 1 φ min g T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
30 Enhancement of Estimation of Relative Deposition Rate Convex approximations for ratio distributions 2 φ min (S) = { min subject to ratio(s) s E(S) 1.5 s min s s max s 1 3 p g (g) 2 1 φ min φ max g T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
31 Enhancement of Estimation of Relative Deposition Rate Convex approximations for ratio distributions φ(s) = φ max (S) φ min (S) s min s s max s 1 3 p g (g) 2 1 φ min φ max g T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
32 Enhancement of Estimation of Relative Deposition Rate Convex approximations for ratio distributions ρ = s s min 0.5 s max s 1 3 p g (g) 2 1 φ min φ max g T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
33 Enhancement of Estimation of Relative Deposition Rate Properties of ratio metrics Input and Noise Covariances Estimate Covariance φ for ratio estimate is a convex function of the parameters of the input covariance Input Covariance and Disturbance Covariance Performance Variable Covariance φ for ratio of performance variable is a quasi-linear function of the parameters of the input covariance. optimal input covariance found by solving convex optimization problem. T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
34 Enhancement of Estimation of Relative Deposition Rate Example Two zone/two metal process goal: optimize ratio estimate for indium T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
35 Enhancement of Estimation of Relative Deposition Rate Example Case 1: Indium perturbations only, no correlation R = T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
36 Enhancement of Estimation of Relative Deposition Rate Example Case 2: Indium perturbations only R = T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
37 Enhancement of Estimation of Relative Deposition Rate Example Case 2: Indium and Gallium perturbations R = T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
38 Outline Observability Enhancement for Optical Measurements 1 Enhancement of Estimation of Relative Deposition Rate 2 Observability Enhancement for Optical Measurements 3 Measurement Sequencing in Semiconductor Manufacturing 4 Conclusion T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
39 n the next section, we describe how the optical data is currently used to obtain lm Observability Enhancement for Optical Measurements ness. in-situ Reflectometry Current Practice any optical measurements of lm thickness used in the semiconductor industry depend in lm interference eects. This behavior is depicted in Figure 3.4. In this example, ht wave traveling through air strikes a wafer made up of a stack of parallel thin lms. incident light wave is partially reected and partially transmitted at the boundary monitor of wafer surface during deposition via reflectance een two media with dierent refractive indices ni. Reections from lower in the stack ine and interfere with the primary reection o the top of the stack. The result is a ted light wave whose intensity varies as a function of the incident light wavelength, the of incidence, media refractive indices, the polarization state, and most importantly, hicknesses of the lms in the stack Maximum singular value Time (s) Output Time (s) (a) Single Wavelength Observability Max and Min Singular Values Minimum singular value Maximum singular value Time (s) Output Figure 3.4: Thin lm interference he following is an abstraction of the thickness measurement problem which contains the Time (s) r of many of the measurement techniques used today. A static nonlinear model for the T. Vincent (IMPACT) Systems Approaches to Estimation (b) Dual Wavelength October 20, / 45
40 Observability Enhancement for Optical Measurements System block diagram Disturbance Power Flow Throttle PECVD Thickness Composition r(t, C) + Voltage Gain Noise ẋ = f(x, u) + v y = g(x, λ) + n T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
41 Observability Enhancement for Optical Measurements Parameter extraction techniques Least Squares Moving Horizon Filter Extended Kalman Filter 12 Output Time (s) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
42 Observability Enhancement for Optical Measurements Parameter extraction techniques Least Squares Moving Horizon Filter Extended Kalman Filter 12 Output Time (s) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
43 Observability Enhancement for Optical Measurements Parameter extraction techniques Least Squares Moving Horizon Filter Extended Kalman Filter 12 Output Time (s) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
44 Observability Enhancement for Optical Measurements Observability 12 thickness, deposition rate, gains Output Time (s) observability rank of output mapping T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
45 Observability Enhancement for Optical Measurements Wavelength Selection Time (s) (a) Single Wavelength Observability Max and Min Singular Values 10 8 Minimum singular value Observability Max and Min Singular Values Minimum singular value Maximum singular value Time (s) Output 10 1 Maximum singular value Time (s) Output Time (s) (a) Single Wavelength Time (s) (b) Dual Wavelength Observability Max and Min Singular Values T. Vincent (IMPACT) Systems Approaches 3 to Estimation October 20, / 45 Figure 3.12: Maximum and minimum singular values of obse
46 Measurement Sequencing in Semiconductor Manufacturing Outline 1 Enhancement of Estimation of Relative Deposition Rate 2 Observability Enhancement for Optical Measurements 3 Measurement Sequencing in Semiconductor Manufacturing 4 Conclusion T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
47 Measurement Sequencing in Semiconductor Manufacturing Starting point: Hierarchical Spatial Variability Model Jason P. Cain and Costas J. Spanos, Electrical linewidth metrology for systematic CD variation characterization and causal analysis Kun Qian and Costas J. Spanos, A Comprehensive Model of Process Variability for Statistical Timing Optimization L gate (f, d, k) = L 0 + L a (k) + L b (f, d, k) + L c (f, k) +L d (d, k) + L e (d) + L f (f, d, k) L gate (f) (e) (d) (c) (b) (a) pos T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
48 Measurement Sequencing in Semiconductor Manufacturing Terms in the Model L 0 - nominal value L a (k) - random variable N(0, σ a ) L b (f, d, k) = φ 1 (f, d)θ 1 (k) - second order polynomial L c (f, k) - i.i.d random sequence N(0, σ c ) L d (f, k) = φ 2 (f)θ 2 (k) - second order polynomial L e (d) - known layout dependent function L f (f, d, k) - i.i.d random sequence N(0, σ f ) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
49 Measurement Sequencing in Semiconductor Manufacturing Are Lengths Correlated? Want to know: L gate (f, d, k) for all f, d, k. Does measurement of L gate (f 1, d 1, k) help me predict L gate (f 2, d 2, k)? Definition The covariance of random variables x 1 and x 2 is Result cov (x 1, x 2 ) = E [(x 1 E(x 1 ))(x 2 E(x 2 ))] If x 1, x 2 are Gaussian, and cov (x 1, x 2 ) = 0, p x1 (x 1 x 2 ) = p x1 (x 1 ). Measurement of x 1 does not help prediction of x 2 when cov (x 1, x 2 ) = 0. T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
50 Measurement Sequencing in Semiconductor Manufacturing Are Lengths Correlated? Want to know: L gate (f, d, k) for all f, d, k. Does measurement of L gate (f 1, d 1, k) help me predict L gate (f 2, d 2, k)? Definition The covariance of random variables x 1 and x 2 is Result cov (x 1, x 2 ) = E [(x 1 E(x 1 ))(x 2 E(x 2 ))] If x 1, x 2 are Gaussian, and cov (x 1, x 2 ) = 0, p x1 (x 1 x 2 ) = p x1 (x 1 ). Measurement of x 1 does not help prediction of x 2 when cov (x 1, x 2 ) = 0. T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
51 Measurement Sequencing in Semiconductor Manufacturing Are Lengths Correlated? Want to know: L gate (f, d, k) for all f, d, k. Does measurement of L gate (f 1, d 1, k) help me predict L gate (f 2, d 2, k)? Definition The covariance of random variables x 1 and x 2 is Result cov (x 1, x 2 ) = E [(x 1 E(x 1 ))(x 2 E(x 2 ))] If x 1, x 2 are Gaussian, and cov (x 1, x 2 ) = 0, p x1 (x 1 x 2 ) = p x1 (x 1 ). Measurement of x 1 does not help prediction of x 2 when cov (x 1, x 2 ) = 0. T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
52 Measurement Sequencing in Semiconductor Manufacturing Covariance Let L 1 = L gate (f 1, d 1, k), L 2 = L gate (f 2, d 2, k) cov (L 1, L 2 ) = var (L a ) (+ var (L c )) + φ 1 (f 1, d 1 )φ 1 (f 2, d 2 ) var (θ 1 ) + φ 2 (f 1 )φ 2 (f 2 ) var (θ 2 ) Prediction possible if L a, θ 1, and θ 2 vary with k. T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
53 Measurement Sequencing in Semiconductor Manufacturing Illustration of Reduction in Uncertainty L gate (f) (e) (d) (c) (b) (a) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45 pos
54 Measurement Sequencing in Semiconductor Manufacturing Illustration of Reduction in Uncertainty L gate (f) (e) (d) (c) (b) (a) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45 pos
55 Measurement Sequencing in Semiconductor Manufacturing Illustration of Reduction in Uncertainty L gate (f) (e) (d) (c) (b) (a) T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45 pos
56 Measurement Sequencing in Semiconductor Manufacturing Dynamic Model What is the variation from wafer to wafer? L gate (b) (a) pos T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
57 Measurement Sequencing in Semiconductor Manufacturing Dynamic Model What is the variation from wafer to wafer? L gate (b) (a) pos T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
58 Measurement Sequencing in Semiconductor Manufacturing Dynamic Model What is the variation from wafer to wafer? L gate (b) (a) pos T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
59 Measurement Sequencing in Semiconductor Manufacturing Dynamic Model What is the variation from wafer to wafer? L gate (b) (a) pos T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
60 Measurement Sequencing in Semiconductor Manufacturing Time Correlation R theta (τ) θ k+1 = Aθ k + w k τ T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
61 Measurement Sequencing in Semiconductor Manufacturing Choosing a Sampling Sequence Dynamic Model, x = [ θ1 θ 2 ]. x k+1 = Ax k + w k y k = C(f, d) k x k + e k Uncertainty Prediction - Kalman Filter T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
62 Measurement Sequencing in Semiconductor Manufacturing Sampling Sequence Selection Uncertainty for wafer k after measurement L gate (f) (e) (d) (c) (b) (a) pos T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
63 Measurement Sequencing in Semiconductor Manufacturing Sampling Sequence Selection Uncertainty for wafer k + 1 before measurement L gate (f) (e) (d) (c) (b) (a) pos T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
64 Outline Conclusion 1 Enhancement of Estimation of Relative Deposition Rate 2 Observability Enhancement for Optical Measurements 3 Measurement Sequencing in Semiconductor Manufacturing 4 Conclusion T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
65 Conclusion Conclusion Three illustrations of analysis of measurement accuracy Tools: uncertainty modeling and propagation optimization linking measurements over time using dynamic models Dynamic modeling of wafer processing may lead to more efficient measurement sequencing Supported in part by NSF grant ECS and DOE Programs DE-ZDO and DE-FG36-08GO T. Vincent (IMPACT) Systems Approaches to Estimation October 20, / 45
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