Kalman Filtering Methods for Semiconductor Manufacturing

Transcription

1 Kalman Filtering Methods for Semiconductor Manufacturing Kameshwar Poolla Mechanical Engineering Electrical Engineering University of California Berkeley

2 Outline Kalman Filtering Overview Ingredients Applications Case Study: Thin Film Metrology Conclusions

3 The Filtering Problem Measurement noises, Disturbances, etc with known statistics Known Inputs Process Measured outputs Unknown Process State Problem: Estimate the process state

4 Example Measurement noises, Disturbances, etc Pressures, powers, flow rate, bias Plasma Chamber Wafer Temperature Reflected power Thickness metrology Problem: Etch Rates, Selectivities, Ion energies, ion densities Estimate the process state

5 Ingredients Process Model Could be linear (Kalman Filtering) Or Nonlinear (Extended Kalman Filtering) Could have unknown parameters in it Parameters could vary slowly with time Known input signals Measured output signals Statistics of noises and disturbances

6 Noises and Disturbances Noises e Measurement noise Quantization noise Disturbances w Exogenous signals that affect the process which we don t measure ex: leaks in backside He cooling system Also used to capture inaccuracies in process model: ex: sidewall deposition changes etching process

7 The Basic Idea Noises e Disturbances w Known Inputs u Actual Process Measured outputs y Kalman Gain K + Model of Process Estimated Process State

8 The Basic Idea Two mechanisms to estimate process state Use the process model Use the measurements Kalman Gain Captures the tradeoff between these two mechanisms Very little measurement noise K is large Very good model K is small

9 The formulae (Linear case) Process Model x& y = = Ax Cx + + B u 1 D u B 2 w D e 2 Kalman Gain Ricatti Equation 0 K = A T = B P + T 2 P PA + B 2 B T 2 PD T 2 D P 2

10 Extensions Extended Kalman Filtering For Nonlinear Process Models Uses a Jacobian Linearization EM Filtering Methods For Models with unknown parameters Joint estimation and filtering Can treat slow variations in parameters EWMA is a very special case of these methods

11 Applications A. Model-based Metrology Improvement (details in case study) B. Removing artifacts from metrology C. Indirect metrology D. End-pointing (details in case study)

12 A. Model Based Metrology Improvement Sensor observes a dynamic system Material model (layers/stacks) may be changing with time Traditional method: Some averaging to reduce noise Not too much because of bias Does not use all available data Dynamic nature of object being measured can be exploited More later in Case Study

13 B. Removing Artifacts from Metrology Spatially resolved Thermal metrology for PEB Using Integrated Wafer Onboard battery, electronics etc in a module Module has thermal mass and affects the measurements Want to remove this loading effect

14 Data Filtering Module Loading Thermal mass of module distorts temperature map Transient effect (30 second time constant) Correction based Kalman Filtering using FEM heat conduction equation Module Filter off Module Filter on

15 Module Effect Filtering Ideal Case Actual Case = bake plate temp (not measured) b T ) ( 2 R T R c R b + = α & ) ( ) ( 2 T T T T T c T m b + + = β α & ) ( 2 T T E E c E m + = β α & R T E = temp (meaured) = module m T

16 C. Indirect Metrology Cannot directly measure what is of interest Must use model to infer variable of interest Noises and disturbances Inputs Process Available measurements Variable of interest This is just Kalman Filtering!

17 Case Study: Optical Thin Film Metrology

18 Optical Thickness Measurements Light of one or more wavelength and/or polarization is focused on the wafer Thin film interference modulates the amount of reflected light for each wavelength/polarization. Reflectometry V t) = A Rt ( ) Io + B 2 I o I = R() t I Ellipsometry ( t) p i ( t ) ( ρ( t) = = tan( Ψ( t)) e R R s ( t) R o

19 Reflectance varies with λ and t Reflectance Reflectance Thickness (nm) Wavelength (nm)

20 in-situ optical sensors: Applications Deposition Deposition rate Uniformity Film quality as determined by refractive index Endpoint Etch Etch rate Uniformity Endpoint Photoresist development Development rate/depth

21 Reflectance Model Standard (simple) reflectance model: parallel, homogeneous, isotropic stack of thin films. Each film parameterized by complex refractive index and thickness } Layers under top layer are combined into an effective (possibly non-physical) complex refractive index Film Parameters: 5 or 7 real numbers Measurement parameters: polarization, angle of incidence, wavelength

22 Inverse Problem Given: Measurement data, measurement parameters (wavelength, polarization, incidence angle) Find: Film parameters Basic problem: Data points at given λ: 1 or 2 per angle of incidence Unknowns: 3 complex refractive indices 1 thickness

23 Regularizing the Inverse Problem Multiple angles of incidence Assume underlying model for refractive index Measure during deposition and assume constant refractive index

24 Dynamic Regularization Ellipsometry - Aspnes (1993) - uses derivative Urban and Tabet (1993) - uses 5 ellipsometric measurements to obtain 10 equations and 10 unknowns Reflectometry - Breiland and Killeen (1995) Assume constant rate of change in thickness: d Find parameters which best fit the data in a least squares sense + + = ) ) (,,,, ( ) ) (,,,, ( ),,,, ( 2 1 d T n t k n k n r d T t k n k n r t k n k n r r r r eff eff eff eff eff eff n M M

25 Parameter estimation Least squares - Maximum Likelihood estimate under white/gaussian noise assumptions Static etch rate: V s.t. ' (θ, d) t k + 1 r k = = t N k = 1 k = rˆ k n k + Td k 2 ( θ, t ) + n n k ~ N( 0, I ) k ˆ θ, dˆ = arg minv ' ( θ, d)

26 Dynamic Models Breiland and Killeen: t k + 1 r k = = rˆ( θ, t Extension to variable etch rate: t d k + 1 r k k ~ = = t t p k k d rˆ( θ, t + Td k ) ) ( d; k) + Td k k θ = [ n k n k] eff eff

27 Extended Kalman Filtering V ( θ ) min nk 1 + wk 1 w R Q 1 =, Lw N N k = Extended Kalman Filter is approximate, recursive solution for fixed θ. Computationally very efficient. Caveats: Can be sensitive to noise model Correct noise model does not necessarily give the best results

28 Estimation Applications Photoresist development Carroll and Ramirez (1990) observations are fringe counts Palmer, Spanos, Poolla (1996) parameter estimator Etch Vincent et al. (1995) reflectometry Galarza et al. (1997) ellipsometry Deposition Woo et al. (1996) reflectometry

29 EKF versus NLLSQ Simulated etch of amorphous silicon on known substrate, unknown a-si refractive index Single wavelength reflectometry Etch rate Reflectance Time Time

30 NLLSQ Least squares fit to data True n = 4.38, k = Estimated n = 4.65, k = etch rate=19.98 A/s Reflectance Reflectance Data Fit Time (s)

31 EKF Minimization of True n = 4.38 k = Estimated n = 4.37 k = V (θ ) Etch rate Etch rate Fit Time (s)

32 Extended Model for Reflectometry Disturbances Noise Process Etch/Dep Rate Thickness Optical Model Sensor Model Actuator Inputs Composition Variable sensor gain, offset - augmented state for parameter estimation Includes reactor model Gain and Offset

33 15 Gain Estimates Voltage (V) Simulation Simulation of SiO 2 /Si etch with initial errors and drifts in sensor gains and offsets Simulated Dual Wavelength Reflectometry Time (s) Etch Rate (Å/s) Depth Error (Å) Estimate Actual Time (s) Etch Rate from Simulated Etch Estimated Etch Rate Actual Etch Rate Offset Estimates Estimate Actual Depth Error Time (s)

34 Limited Region of Convergence SiO 2 /Si etch with varying initial thickness estimate True initial thickness: 4000 Å Correct process model Thickness Estimate (Å) Time (s)

35 Applications and Expt Results Etch rate/depth estimation Etch rate stabilization SiO 2 etching Process development/optimization Real time uniformity with distributed dual wavelength reflectometry Accurate Endpointing a-si(n+)/a-si(i) interface for TFT backchannel recess etch

36 Dual Wavelength Reflectometry Real-time, in-situ measurement of etch rate and depth for SiO 2 /Si etch Etch Rate (Å/s) Depth (Å) Etch Rate 900 W Etch Rate Power W Depth Time (s)

37 Active Etch Rate Stabilization Etch Rate (Å/s) Etch Rate Feedback Control Etch Rate Estimate Reference Forward Power Set-point Watts Time (s)

38 Spatially Distributed Reflectometry Multi-point laser reflectometry system installed on Plasmatherm 7000 RIE reactor Beam Splitters Optical Fiber Choppers Red Lock-In Green Lock-In Photo Diode Glass Plate Chamber Optical Fiber Green HeNe Red HeNe

39 Real Time Uniformity Optimization Distributed sensors allow for real time uniformity measurements Process Optimization Diagnostics Control if distributed actuation Etch Rate (Å/s) Distributed Etch Rate Measurement Op 1 Op 2 Op 3 Port 1 Port 2 Port Time (s)

40 Endpoint for a TFT Recess Etch Back channel recess process is simpler (lower cost, higher throughput) but demands accurate endpoint. Mask a-si:h(n+) a-si:h(i) SiN x Ta Gate Glass Substrate Requirement: Etch through a-si:h(n+) and endpoint at a-si:h(n+)/a-si:h(i) interface Manufacturing challenge

41 Accurate Endpoint Experiment 20 Etch Experiments 10 timed etches each of 100 seconds 10 with endpoint called using estimated etch depth from EKF-R Final film thickness measured by ex-situ spectral ellipsometer Expt Stack a-si:h (i) Ta Glass Substrate Goal: Etch 500 Å of 1000 Å a-si film Can not use OES to solve this endpoint problem.

42 Endpoint Result Result: 90% reduction in standard deviation of remaining film thickness by using EKF-R endpoint. Remaining a-si thickness (Å) Remaining a-si thickness (Å) σ=177Å Timed Etches σ=18Å EKF-R Endpointed Etches

43 Conclusions Extended Kalman Filtering is useful for real time optical metrology Enables self calibration of Reflectometry Estimates of etch/depth rates, film parameters Fast, recursive algorithm Demonstrated Experimental Validation Etch rate estimation/stabilization Process Optimization Endpointing Many other interesting and important applications of these ideas in Semiconductor Manufacturing

44 Acknowledgements Optical Thin Film Metrology examples from Tyrone Vincent, Colorado School of Mines Thermal metrology example Thanks to OnWafer Technologies for permission to use data