Efficient Bayesian inversion for shape reconstruction of lithography masks
Nando Farchmin, Martin Hammerschmidt, Philipp-Immanuel Schneider, Matthias Wurm, Bernd Bodermann, Markus Bär, Sebastian Heidenreich
EEfficient Bayesian inversion for shape reconstruction oflithography masks
Nando Farchmin a,b , Martin Hammerschmidt c,d , Philipp-Immanuel Schneider c,d , MatthiasWurm a , Bernd Bodermann a , Markus B¨ar a , and Sebastian Heidenreich aa Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin b Technische Universit¨at Berlin, Institute of Mathematics c JCMwave GmbH d Zuse Institute Berlin
ABSTRACT
Background:
Scatterometry is a fast, indirect and non-destructive optical method for quality control in theproduction of lithography masks. To solve the inverse problem in compliance with the upcoming need for im-proved accuracy, a computationally expensive forward model has to be defined which maps geometry parametersto diffracted light intensities.
Aim:
To quantify the uncertainties in the reconstruction of the geometry parameters, a fast to evaluate surro-gate for the forward model has to be introduced.
Approach:
We use a non-intrusive polynomial chaos based approximation of the forward model which increasesspeed and thus enables the exploration of the posterior through direct Bayesian inference. Additionally, thissurrogate allows for a global sensitivity analysis at no additional computational overhead.
Results:
This approach yields information about the complete distribution of the geometry parameters of asilicon line grating, which in return allows to quantify the reconstruction uncertainties in the form of means,variances and higher order moments of the parameters.
Conclusion:
The use of a polynomial chaos surrogate allows to quantify both parameter influences and re-construction uncertainties. This approach is easy to use since no adaptation of the expensive forward model isrequired.
Keywords: uncertainty quantification, polynomial chaos, inverse problem, parameter reconstruction, scatterom-etry
ACKNOWLEDGEMENTS
This project has received funding form the German Central Innovation Program (ZIM) No. ZF4014017RR7.
1. INTRODUCTION
Scatterometry is an optical scattering technique frequently used for the characterization of periodic nanostruc-tures on surfaces in semiconductor industry (determination of critical dimensions).
In contrast to other tech-niques like electron microscopy, optical microscopy or atomic force microscopy, scatterometry is a non-destructiveand indirect method. In the last decades both feature sizes and the required measurement uncertainty decreasedcontinuously, hence advanced scatterometry techniques are required. Recently, deep ultraviolet (DUV) scat-terometry, extreme ultraviolet (EUV) scatterometry,
1, 8 imaging scatterometry and combinations with whitelight interferometry are developed. In these approaches, geometrical parameters and associated uncertaintiescan be determined from diffraction patterns by solving a statistical inverse problem. For an overview on themetrology of surfaces in semiconductor industry see and references therein.We emphasize that scatterometry is an integral measurement method, which means that information ofvariances within the probe are lost due to an averaging over the spot size of the beam. These parameter variations E-mail: [email protected] a r X i v : . [ phy s i c s . d a t a - a n ] M a y ypically lead to a broadening of the diffracted beam. This stochastic effect was not taken into account by themodel of the line structure used in this work. Instead the diffraction efficiencies were calculated from the integralover the whole beam.The inverse problem of scatterometry is in general ill-possed and regularization techniques have to be applied.The geometry is typically parametrized and sought-after parameters are obtained by weighted least squaresminimization, with weights derived directly from uncertainties in the measurements. However, the quality ofthese weights depends highly on the measurements used and itself influences the reconstruction results of thegeometry parameters. An alternative approach is to apply a maximum likelihood estimate, which introducesa likelihood function based on an error model and optimizes weighting terms as hyper parameters instead ofusing predefined values. Based on the same principle but additionally including some prior knowledge is themaximum posterior approach, which is a state of the art method in parameter reconstructions. In the aboveframeworks, uncertainties are typically obtained from the Fischer information or covariance matrix , which relieson an assumed shape of the posterior. However, the shape of the posterior is generally unknown, hence this canlead to significant errors in the estimation of uncertainties if the actual posterior shape differs from the assumedone.The Bayesian approach allows to integrate prior knowledge and approximates the probability density func-tion of the geometry parameters independent of any shape assumptions. Uncertainties obtained by employingBayesian inference are thus much more robust. On the other hand it requires a large number of evaluations ofthe forward model which is not feasible for expensive computations as in the case of scatteromery. To obtaina surrogate model that mitigates the computation time, we employ a polynomial chaos expansion, that is anexpansion into an orthonormal polynomial basis in the parameter space to approximate the forward model witha global polynomial.
17, 18
We additionally show how this surrogate allows for a Bayesian approach to the inverseproblem.In a recent publication, it has been demonstrated that a surrogate of a forward model for scatterometry basedon a polynomial chaos expansion enables Bayesian inversion and the use of Markov Chain Monte Carlo (MCMC)sampling. In this approach, cubature rules on sparce grids of Smolyak type are used to determine the expansioncoefficients and to construct the surrogate model.
19, 20
However, cubature rules and sparce grids are adapted tothe stochastic distributions chosen and less accurate for correlated stochastic input parameters. In the presentwork we used optimal linear regression to obtain the coefficients of the polynomial chaos expansion. This novelapproach uses optimal sampling points, is much more flexible and well suited for extensions to adaptive systems.
Figure 1. Cross section of the photomask with description of the stochastic parameters. The dimensional parametervector is given by ξ = ( h, cd , swa , t, r top , r bot ). The pitch of the computational domain, i.e. the period is fixed to 50 nm. In this paper, we determine the geometry parameters of a photomask that consists of multilayered, periodic,straight absorber lines of two optically different materials. The period of the line structure (pitch) is 50 nm andthe geometry parameters of interest are the height of the line h , the width at the middle of the line (criticalimension) CD, the sidewall angle SWA, the silicon oxide layer thickness t and the radii of the rounding at thetop and bottom corners of the line r top and r bot , respectively. A cross section of the geometry for one period ofthe structure is depicted in Fig. 1. The photomask was illuminated by a light beam of wavelength λ = 266 nm(DUV) for different angles of incidence θ = 3 ◦ , ◦ , . . . , ◦ for perpendicular ( φ = 0 ◦ ) and parallel ( φ = 90 ◦ )orientation of the beam with respect to the grating structure as well as S and P polarization. The refractiveindices used are n si = 1 .
967 + 4 . i for silicon, n ox = 1 . . i for the top oxide layer and n air = 1 . µ r = 1 .
2. FORWARD MODEL
In principle, the propagation of electromagnetic waves is described by Maxwell’s equations, but for our simplegrating geometry and in the time-harmonic case, Maxwell’s equations reduce to a single second order partialdifferential equation,
15, 21 ∇× µ ( r ) − ∇× E ( r ) − ω ε ( r ) E ( r ) = 0 . (1)Here, ε and µ are the permittivity and permeability, r is the spatial coordinate and ω is the frequency of theincoming beam. We employ the finite element method (FEM) implemented in the JCMsuite software packageto discretize and solve the corresponding scattering problem formulation on a bounded computational unit cell inweak formulation as described in. This formulation yields a splitting of the complete R n into an interior domainhosting the total field (incident and scattered) and an exterior domain where only the purely outward radiatingscattered field is present. Appropriate boundary conditions are applied at the boundary of the computationaldomain as depicted in Fig. 1. As the geometry is periodic in lateral direction Bloch-periodic boundary conditionsare applied. In vertical direction the geometry is assumed to be unbounded and thus requires the satisfaction of atransparent boundary condition at the interface. We use an adaptive perfectly matched layer (PML) method
23, 24 to realize the transparent boundary condition and to satisfy the radiation condition for the scattered field. Theemployed vectorial FE method uses high-order polynomial ansatz functions defined on the spatial discretizationof the computational domain. The triangulation allows to geometrically resolve the material interfaces and thetangential continuity of the electromagnetic fields across these interfaces is automatically enforced by the method.The forward model is given by a map of geometry parameters onto S and P polarization of zeroth orderintensities of the scattered light. The parameters ξ used for modelling the grating geometry are depicted inFig. 1. The forward model is represented by the function f ∗ : Ω → R d such that the parameters ξ ∈ Ω ⊂ R M are mapped to diffracted efficiencies for a set of azimuthal angles, incidence angles and polarizations. Each ofthe d components of f ( ξ ) represents a different combination of azimuth, incidence angle and polarization. Allother experimental conditions such as e.g. the wavelength are fixed for this forward model. In our approach, theexperimental data y ∈ R d are modelled with the error y j = f j ( ξ ) + ε j , j = 1 , . . . , d where ε j ∼ N (0 , σ j ) describesa normal distributed noise with zero mean, standard deviation and error parameter b , σ j ( b ) = b y j , for b > . (2)Choosing σ j to depend on a stochastic parameter itself instead of setting it to specific values allows for anestimation of the measurement error based on the measurement data in the parameter reconstruction and thusincorporates less prior knowledge. The inverse problem is defined as the determination of geometry parametervalues ξ and the error parameter (hyper parameter) b from measured efficiencies y .To obtain a fast evaluation of the surrogate, the function f ∗ is expanded into an orthonormal polynomialbasis { Φ α } α ∈ Λ ⊂ L (Ω; (cid:37) ) ∗ ( ξ ) ≈ f ( ξ ) = (cid:88) α ∈ Λ f α Φ α ( ξ ) with f α = (cid:90) Ω f ( ξ ) Φ α ( ξ ) d (cid:37) ( ξ ) . (3)The finite set Λ ⊂ N M of cardinality P ∈ N is a set of multiindices and (cid:37) denotes the multivariate parameterdensity for the parameters ξ . With this surrogate the evaluation of the model in different parameter realizationsis equivalent to the evaluation of polynomials.We want to emphasize, that this approach allows for a global sensitivity analysis of the parameters at almostno additional cost. ? , 17, 28–33
3. OPTIMAL LINEAR REGRESSION
A simple and non-intrusive approach to compute the expansion coefficients in (3) is linear regression. With thereasonable assumption that, for an arbitrary enumeration of Λ, the residuum R ( ξ ) = f ∗ ( ξ ) − (cid:80) P(cid:96) =1 f (cid:96) Φ (cid:96) ( ξ ) is azero mean random variable, we want to find coefficients that minimize the variance of the residuum R . In otherwords, we obtain the least-squares minimization problem:Find coefficients f (cid:96) , (cid:96) = 1 , . . . , P such that (cid:90) Ω R ( ξ ) d (cid:37) ( ξ ) = min . (4)To avoid the high dimensional numerical integration in (4), we approximate the integral in a Monte Carlosense by (cid:90) Ω R ( ξ ) d (cid:37) ( ξ ) ≈ N N (cid:88) i =1 R ( ξ ( i ) ) , (5)where ξ (1) , . . . , ξ ( N ) are N realizations of possible geometry parameter values (see the domain column inTab. 1). Since (4) is a quadratic minimization problem, the critical point of the first variation yields the wantedminimum. This critical point can be obtained by solving the linear system F = (Ψ T Ψ) − Ψ T F ∗ , (6)where the matrices Ψ and F ∗ are given by Ψ i(cid:96) = Φ (cid:96) ( ξ ( i ) ) and F ∗ i = f ∗ ( ξ ( i ) ). To guarantee that theempirical Gramian Ψ T Ψ is not ill-conditioned, the number of realizations N has to be sufficiently large. Thechoice of sampling points is in principle arbitrary, but Cohen and Migliorati showed, that sampling from aspecific weighted least-squares distribution leads to an optimal (minimal) number of samples for a guaranteedwell-conditioned Gramian matrix. Hence, we setd µ = w − d (cid:37) for w − ( ξ ) = 1 P P (cid:88) (cid:96) =1 | Φ (cid:96) ( ξ ) | (7)and draw samples ξ ( i ) ∼ µ . Note that µ is still a probability measure since the polynomials { Φ (cid:96) } areorthogonal and normalized. With this, the number of samples required to guarantee a low condition of theGramian reads N/ log( N ) ≥ c P for some c >
0. Here, we choose c = 4, motivated by the empirical results in. Applying this optimal sampling strategy allows us to compute a surrogate for the forward model using aminimal number of function evaluations. This surrogate will be employed to reconstruct geometry parametersand quantify the reconstruction uncertainties using Bayesian inference. . BAYESIAN APPROACH
The Bayesian approach provides a statistical method to solve the inverse problem. Following Bayes’ theorem,the posterior density is given by π ( ˆ ξ ; y ) = L ( ˆ ξ ; y ) π ( ˆ ξ ) (cid:82) L ( ˆ ξ ; y ) π ( ˆ ξ ) d ˆ ξ , (8)where the prior density π describes prior knowledge and the likelihood function L contains the informationobtained from the measurement under the assumption of a specific measurement error model. Since the priordensity allows for expert knowledge to influence the model, it has to be chosen appropriately not to introduce abias on the system. We choose a uniform prior to induce as less information as possible on the compact domainsof the geometry parameters. The computational and implementation efforts of the Bayesian approach are higherthan for the maximum likelihood or least squares methods. However, the posterior yields information about thecomplete probability density function of the geometry parameters and is thus more reliable for the determinationof uncertainties than merely using quantities such as mean and covariance. In addition, the combination of theresults from different measurement modalities within the Bayesian framework assures a consistent propagationof uncertainties through all measurement contributions
35, 36 (hybrid metrology) in a way that the posterior forone measurement can be used as the prior for the next.
Table 1. Estimations of parameters and uncertainties obtained from the mean value (mean), the double standard deviation(2 σ ), relative double standard deviation (11) (rel-2 σ ), skewness (skew) and kurtosis of the posterior distribution. Thedomain indicates the support of the prior distribution chosen. parameter domain mean 2 σ rel-2 σ skew kurtosis h / nm [43 . , .
0] 48 .
35 3 .
11 0 . − . . / nm [22 . , .
0] 25 .
48 0 .
59 0 . . . / ◦ [84 . , .
0] 86 .
87 2 .
65 0 . − . . t / nm [4 . , .
0] 4 .
96 0 .
35 0 . . . r top / nm [8 . , .
0] 10 .
65 2 .
30 0 . − . . r bot / nm [3 . , .
0] 4 .
89 1 .
80 0 . − . . b [0 . , .
1] 0 .
01 0 . . . . ξ consists of geometry parameters ξ and the noise parameter, i.e. ˆ ξ = ( ξ, b ). Assuming normaldistributed zero-mean measurement errors, we choose the likelihood function L ( ˆ ξ ; y ) = d (cid:89) j =1 √ πσ j ( b ) exp (cid:32) − ( f ( j ) ( ξ ) − y j ) σ j ( b ) (cid:33) , (9)where f ( j ) is the j -th component of (the vector valued function) f . Note that the form of the measurementerror has to be chosen appropriately not to introduce a bias. A more general approach in our case would benot to impose a zero mean but introduce another random hyper parameter. However, the residuum in Fig. 4suggests that the noise is distributed around zero. In the Bayesian framework, the distributions of parametersare in general determined by Markov-Chain Monte Carlo (MCMC) sampling where for every sampling step,the forward model has to be evaluated. Normally, this means that equation (1) has to be solved which makesMCMC sampling impractical due to the large number of required sampling steps. Since the surrogate onlyrequires evaluations of polynomials, the Bayesian approach becomes practical for scatterometry measurementevaluations. igure 2. Marginal 1D and 2D densities for the posterior of the stochastic parameters. For the 1D densities, the mean(solid line) and the standard deviation (dashed line) are depicted as well. For Bayesian inversion, we have to choose a prior distribution for the parameters, calculate the likelihoodfunction and determine the corresponding posterior distribution. The posterior distribution contains the desiredparameter values and their associated uncertainties. When two or more measurement results from differentmeasurement sets y (1) , . . . , y ( K ) are combined, the posterior distribution of the first measurement can be used asthe prior distribution for the evaluation of the second measurement, i.e. π ( ˆ ξ ; y ( K ) , y ( K − , . . . , y (1) ) = π ( ˆ ξ ) (cid:81) Kk =1 L ( ˆ ξ ; y ( k ) ) (cid:82) π ( ˆ ξ ) (cid:81) Kk =1 L ( ˆ ξ ; y ( k ) ) d ˆ ξ . (10)Note that the model function f in the likelihood function is in general different for different measurementetups.
5. RESULTS
First we want to emphasize the efficiency of our approach. For the scattering problem at hand, it is sufficientto use a chaos expansion with 217 terms to achieve a relative empirical L -error of less than 1%. Therefore,in the sense of section 3, we generate approximately 10 samples for the FEM forward model to evaluate. Incomparison, the computation of the function mean, variance and Sobol indices or the generation of posteriorsamples, if done empirically, require more than 10 function evaluations each due to the slow convergence rateof Monte Carlo integration.We apply Bayesian inversion to the scatterometry measurements to estimate geometry parameters of the linegrating. More details of the measurement setup are described in previous works.
5, 15
A global sensitivity analysisfor the geometry parameters indicates that the reconstruction of all parameters is possible, i.e. the forwardmodel is sensitive to all of them. In particular, the oxide layer thickness and critical dimension should be possibleto determine precisely due to their high sensitivity.For Bayesian inversion it is necessary to chose prior distributions. In our investigations we have chosenuniform priors on the domains given in Table 1. To obtain the posterior distribution, we sampled with anMCMC random walk Metropolis-Hastings algorithm using the surrogate. We have chosen a sampling size of 10 samples and a burn in phase of 10 samples. For diagnostics, we applied the Gelman-Rubin criterion, to assurethat the chains have fully explored the posterior. Figure 3. Deviation of posterior marginals from Gaussian distribution with the same mean and variance. The distributionsare depicted in their respective reconstruction domains (see Table 1).
In Fig. 2 the posterior (marginal) densities for all 6 stochastic parameters are shown. All posterior densitiesare characterized by sharp peaks with mean and standard deviation similar to the previous publication. Themean and double standard deviation for each parameter including the hyperparameter (error parameter) b areshown in Table 1. Since the domain sizes of the parameters vary due to their geometrical meaning, we introducethe relative double standard derivation (rel-2 σ ). The rel-2 σ of a stochastic variable η is the double standarddeviation divided by half the width of the parameter domain:el-2 σ = 4 σβ − α where η ∈ [ α, β ] . (11)The rel-2 σ shows how the posterior distribution is spread within the domain. For example, if the domain forthe critical dimension is [22 ,
28] nm, and the 2 σ is 0 . σ is 0 . σ in Table 1 shows that the smallestreconstruction uncertainties are obtained for the critical dimension with rel-2 σ about 20%, followed by the oxidelayer thickness with 35% rel-2 σ . The height has a rel-2 σ of about 62%. The posterior densities of the sidewallangle and the corner rounding are slightly wider distributed at about 90% rel-2 σ . This goes in line with theglobal sensitivity analysis. The results for the error parameter b depicted in Table 1 show that the relativemeasurement uncertainty is approximately 1%.One major advantage of the Bayesian inference is information about the complete posterior distributioninstead of just parameter values obtained from the global minimizer. Looking at the marginals in Fig. 2, it iseasy to verify that the posterior is not Gaussian. The densities of the rounding radii are not symmetric, themarginal distribution of the sidewall angle exhibits a plateau around the mean and the height even suggestsmulti-modalities. Another validation of these observations can be found in the skewness (third moment) andkurtosis (forth moment) of the posterior. These differ (except for the oxide layer thickness) quite significantlyfrom the skewness and kurtosis of a Gaussian, see Table 1. The deviation of the marginals from a Gaussian withthe same mean and standard deviation is shown in Fig. 3 for all parameters .In our case the marginal distributions of the posterior are similar enough to a Gaussian distribution thatthe 2 σ confidence interval contains roughly 95% of the mass, as displayed in Table 2. However, in general it ismore reasonable to directly compute intervals of mass concentration (confidence intervals) rather than relyingon the standard deviation to characterize the uncertainties of a distribution, because this can be misleading fornon-Gaussian distribution shapes occurring for example in.
20, 39
Table 2. Double standard deviation and 95% mass confidence intervals of all geometry parameters and the error hyper-parameter. parameter 2 σ interval 95% confidence interval h / nm (45 . , .
47) (45 . , . / nm (24 . , .
07) (24 . , . / ◦ (84 . , .
52) (84 . , . t / nm (4 . , .
31) (4 . , . r top / nm (8 . , .
96) (8 . , . r bot / nm (3 . , .
69) (3 . , . b (0 . , . . , . b , which describesthe mismatch between the surrogate of the forward model f and the measurements. In a previous work a Maximum Posterior Approach (MPA) incorporating the same measurement data was used to determine thegeometry parameters. The MPA searches for the global maximum of the posterior. Uncertainties were determinedlocally with an approximation of the covariance matrix around the maximum posterior point. The differencehere is that we calculated the whole posterior distribution. This has the advantage that even for multiple peakedand non-Gausian posterior distributions this scheme gives reliable uncertainty estimations. The results obtainedin are consistent to our findings since the posterior is relatively close to the assumed Gaussian shape. There arenly slight differences. For example, the marginal distribution for the height h is broad (non-Gaussian) yieldinglarger uncertainties. Similarly, the mean values for r top and r bot are slightly shifted due to the asymmetry of themarginal posterior (non-Gaussian). The deviation between the forward model values and the measurement dataof 2% is comparable with that found in. Figure 4. Scattered intensities for the two polarizations and different azimuthal angles. Compared are the measurements ofthe scatterometry experiment and the simulation of the PC surrogate for the mean values of the parameter reconstruction.The bottom graph shows the pointwise deviation.
6. CONCLUSION
In this paper we applied a polynomial chaos expansion as a surrogate for the forward model in scatterometry.Since the surrogate only requires the evaluation of polynomials instead of solving Maxwell’s equation, it wasfeasible to use a full Bayesian approach to determine the posterior distribution for all geometry parameters.To generate samples from the posterior distribution, we employed a MCMC Metropolis random walk samplingmethod and checked the overall independence of the samples obtained by the Gelman-Rubin criterion. Thereconstruction results obtained by the surrogate model compared to those obtained by a Maximum Posteriorestimate with a Gauss-Newton like method are consistent and are in line with the predictions from a globalsensitivity analysis. We conclude that a Bayesian approach based on the polynomial chaos surrogate givesaccurate and reliable estimations for silicon line grating parameters and uncertainties. REFERENCES [1] Hsu, S. and Terry, F., “Spectroscopic ellipsometry and reflectometry from gratings (scatterometry) forcritical dimension measurement and in situ, real-time process monitoring,”
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