High Order Overlay Modeling and APC simulation with Zernike- Legendre Polynomials

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1 High Order Overlay Modeling and APC simulation with Zernike- Legendre Polynomials JawWuk Ju a, MinGyu Kim a, JuHan Lee a, Stuart Sherwin b, George Hoo c, DongSub Choi d, Dohwa Lee d, Sanghuck Jeon d, Kangsan Lee b, David Tien b, Bill Pierson c, John C. Robinson c, Ady Levy b, Mark D. Smith c a SK Hyni, 2091, Gyeongchung-daero, Bubal-eub, Icheonsi, Gyeonggi-do, KOREA b KLA-Tencor Corp., 1 Technology Drive, Milpitas, CA, c KLA-Tencor Corp., 8834 N. Capital of Teas Hwy, Austin, TX d KLA-Tencor Korea, Starplaza bldg., 53 Metapolis-ro, Hwasung city, Gyeonggi-do, Korea ABSTRACT Feedback control of overlay errors to the scanner is a well-established technique in semiconductor manufacturing [1]. Typically, overlay errors are ured, and then modeled by least-squares fitting to an overlay model. Overlay models are typically Cartesian polynomial functions of position within the wafer (X w, Y w), and of position within the field (X f, Y f). The coefficients from the data fit can then be fed back to the scanner to reduce overlay errors in future wafer eposures, usually via a historically weighted moving average. In this study, rather than using the standard Cartesian formulation, we eamine overlay models using Zernike polynomials to represent the wafer-level terms, and Legendre polynomials to represent the field-level terms. Zernike and Legendre polynomials can be selected to have the same fitting capability as standard polynomials (e.g., second order in X and Y, or third order in X and Y). However, Zernike polynomials have the additional property of being orthogonal over the unit disk, which makes them appropriate for the wafer-level model, and Legendre polynomials are orthogonal over the unit square, which makes them appropriate for the field-level model. We show several benefits of Zernike/Legendre-based models in this investigation in an Advanced Process Control (APC) simulation using highly-sampled fab data. First, the orthogonality property leads to less interaction between the terms, which makes the lot-to-lot variation in the fitted coefficients smaller than when standard polynomials are used. Second, the fitting process itself is less coupled fitting to a lower-order model, and then fitting the residuals to a higher order model gives very similar results as fitting all of the terms at once. This property makes fitting techniques such as dual pass or cascading [2] unnecessary, and greatly simplifies the options available for the model recipe. The Zernike/Legendre basis gives overlay performance (mean plus 3 sigma of the residuals) that is the same as standard Cartesian polynomials, but with stability similar to the dual-pass recipe. Finally, we show that these properties are intimately tied to the sample plan on the wafer, and that the model type and sampling must be considered at the same time to demonstrate the benefits of an orthogonal set of functions. Keywords: Overlay, Accuracy, Zernike, Legendre 1. INTRODUCTION Overlay is rapidly becoming the leading challenge for device yield for current and future technology nodes. One indispensable technique is process control, both in the form of statistical process control (SPC) and advanced process control (APC) [1]. For SPC, the key requirement is the ability to characterize the epected amount of noise in the process, so that ecursions can be detected quickly. If two (or more) coefficients in the overlay model are strongly correlated in the overlay model, then there may be relatively large changes in the coupled coefficients when the actual overlay on the wafer is relatively stable. It is highly desirable to eliminate correlation between overlay model parameters so that clean signals can be monitored in SPC charts. For APC, coupling between coefficients is also inefficient rapid fluctuations lead to similar signal-to-noise problems: there may be situations where coupling between multiple terms masks an underlying trend in the overlay that could be corrected, but instead it is interpreted as noise, and no action is Metrology, Inspection, and Process Control for Microlithography XXIX, edited by Jason P. Cain, Martha I. Sanchez, Proc. of SPIE Vol. 9424, 94241Y 2015 SPIE CCC code: X/15/$18 doi: / Proc. of SPIE Vol Y-1

2 taken by the APC system. One standard approach for removing coupling between different terms in an overlay model is cascading [2], which basically consists of fitting the model in two or more sequential steps instead of fitting all of the model parameters at once. For eample, the wafer-level terms might be fitted first, and the field-level terms fitted in a second step. This technique reduces the interactions between model terms, but it leads to a model fit that has slightly higher residuals. Thus, the trade-off is a small reduction in the fitting capability of the model in echange for less correlation between the model terms. In this paper, we investigate orthogonal polynomials as an alternative to cascading, with a goal of reducing correlation without a reduction in accuracy. The outline of the paper is as follows: First, we outline the properties of orthogonal polynomials, and eplain how they might help better model overlay. Second, we describe a simple APC algorithm, and apply this APC simulator to production wafers. Results are presented for several models, all of which are third-order polynomials for grid and third-order polynomials for field. The first model uses standard Cartesian polynomials with cascading, so the wafer level terms are fit separately from the field level terms. The second model is a single-pass fit to standard Cartesian polynomials, and the third model is a single-pass fit to Zernike/Legendre polynomials. We also investigate a hybrid single-pass model that uses a Zernike polynomial for the wafer, and standard Cartesian model for the field. Finally, in the last section, we present our summary and conclusions. 2. PROPERTIES OF ORTHOGONAL POLYNOMIALS Mathematically, orthogonal functions are usually defined in terms of an integral (or summation) over a specific region. The term orthogonal means that if you pick any two of these special functions, the integral is zero unless they are the same function. For eample, products of two Zernike polynomials are orthogonal when integrated over the unit disk, which makes them useful for describing the wafer-level portion of the overlay model. The Legendre polynomials are orthogonal on the interval from -1 to 1, so a product of a Legendre polynomial in X f with a second Legendre polynomial in Y f creates a set of polynomial functions orthogonal over a square-shaped region, which is useful for the field-level portion of the overlay model. It is sometimes helpful to demonstrate abstract mathematical properties with a concrete eample. The eample here is for fitting regular polynomials and Legendre polynomials over the interval -1 to 1, and the data we will fit is a shifted cosine function evaluated at equally spaced intervals, as shown in Figure 1. The functional form for the regular polynomial is 2 3 P = C + C + C + C +K (1) And the functional form for the Legendre polynomial is 2 1 ( 3 1) + L ( 5 ) + K 1 3 P = L + L1 + L (2) Where we will use least-squares fitting to determine the unknown coefficients C i and L i. We can write these polynomials to arbitrarily high order, but we will start with a fit to a quadratic polynomial, as shown in Figure 1. Two important results are demonstrated in this eample. First, the actual polynomial best-fit curves are identical, although the coefficients C i and L i are numerically different. The root-mean-square (RMS) error and fitted coefficients are shown in Table 1. The RMS error is identical because the fitting capability of the two models is the same (both are quadratic). Second, the regular polynomial has regions where different terms must cancel this is most apparent around the fitting at =1, where the constant and linear terms are approimately -0.7, while the quadratic term is approimately 1.5. By contrast, the individual terms in the Legendre polynomial cover a smaller range from approimately -0.5 to 1.0. The large amount of cancellation between terms in the regular polynomial will lead to stronger coupling between the fitting terms. If we fit the same shifted-cosine data to a cubic polynomial, as shown in Figure 2, then we again see that the final fitted functions are equivalent, but that there are again larger cancellations in the fit of the regular polynomial compared with the orthogonal polynomial. Another interesting property of orthogonal polynomials is shown in Table 1, where we show the fitted coefficients for the quadratic and cubic fits in Figures 1 and 2. Note that when the order of the polynomial is increased from quadratic to cubic, the linear coefficient changes for the regular polynomial the coefficient changes by more than 100% (from to ) when the cubic term is added to the fit. By contrast, the coefficient only Proc. of SPIE Vol Y-2

3 1.5 - Data -OrthoQonaIVoNnomlal 1.5 Figure 1: Fit of a regular quadratic polynomial (left) and an orthogonal quadratic polynomial (right) to a shifted-cosine function. The dashed curves show the constant, linear, and quadratic contributions from each term in the polynomial. Figure 2: Fit of a regular cubic polynomial (left) and an orthogonal cubic polynomial (right) to the same shifted-cosine function as in Figure 1. The dashed curves show the constant, linear, quadratic, and cubic contributions from each term in the polynomial fit. Polynomial ( 2 ) Polynomial ( 3 ) Legendre ( 2 ) Legendre ( 3 ) C 0/L C 1/L C 2/L C 3/L RMS Table 1: Fitted coefficients for polynomials shown in Figures 1 and 2. Proc. of SPIE Vol Y-3

4 slightly changes for the Legendre polynomial, only 18% (from to ). Note that if we were to add more data points to the fit, so that we more closely approimate an integral from -1 to 1, the interaction for the Legendre polynomial would go to zero. This simple eample shows that there are fewer interactions between terms in the orthogonal polynomial when compared to an equivalent regular polynomial, and the reduction in these interactions make the fitting process more stable when terms are added. Net, we apply these same ideas to overlay modeling, and investigate the impact of these properties on APC. 3. DESCRIPTION OF SIMPLE MODEL FOR APC Here we describe a simple APC simulator for overlay feedback control. First, we start with an overlay model and a corresponding sample plan. The overlay model serves two purposes to act as a filter to remove noise from the spatial overlay signature on each ured wafer, and to obtain correction parameters that can be fed back to the scanner in order to improve overlay on subsequent wafers. The sample plan locations are designated as r i, which represents the location on the wafer. This location is typically a test location within a field, which can be specified by local field coordinates (X f, Y f), along with a wafer coordinate that specifies the location of the middle of the field (X w, Y w). The symbol r i implicitly includes both field and wafer coordinates for the i th urement. We can then write the overlay model evaluated at each of the urement locations as A = (3) b model Where A is the design matri with rows that represent the model evaluated at each wafer location r i and with columns that represent the polynomial terms in the model. The variable is a vector containing the model parameters, and b model is the overlay value predicted by the model. We can define the error between the model and the ured data as The squared error can be written as e T e = b model (4) b T e = A b ) ( A b ) (5) ( And the error will be minimized when the following conditions are met: We can then solve for the parameters in the overlay model: A = T T A = A b (6) T 1 T ( A A) A b Second, we use the following simple feedback control scheme in our APC simulator. When a wafer is ured for overlay, we can solve for the set of model parameters that would minimize the RMS error this is identical to the parameters found by solving equation (6). If we were to strip the photoresist from the wafer and re-work the wafer with these parameters as corrections to the scanner, then we would see much reduced overlay errors. This is called the rework case: b rework = b = b b rework model A The rework case represents the best result possible with the specified overlay model and sample plan. However, rework is not practical because it requires processing each wafer twice. Instead, we could use urements from the current wafer to predict the scanner correction values for the net set of wafers. For eample, we could use urements from lot 0 to correct lot 1. This would (hopefully) lead to improved overlay urements on the wafer: With b rework 1 1 0, APC b A = (9) (7) (8) Proc. of SPIE Vol Y-4

5 0 = T 1 T 0 ( A A) A b Note that the subscript indicates urements without any APC correction, while, APC indicates the results with the APC correction. For the net set of wafers, we could use the urements for wafer 1 to calculate a new correction, but we must be careful to de-correct the ured overlay values by removing the APC correction applied with the coefficients 0. We achieve this by solving equation (9) for the overlay errors that would have occurred without APC: Which can be generalized as We can then solve for the a new set of corrections using the general relation b (10) = b, APC + A (11) n n n 1 b = b, APC + A (12) n = T 1 T n ( A A) A b While this simple APC scheme should remove systematic offsets and slow drifts, it will be very susceptible to noise. This noise will appear in the ured overlay. Common sources of noise would include wafer-to-wafer variations due to slightly different processing paths through the fab (e.g., variations between different etchers, or different deposition tools), scanner matching issues, or metrology noise. One approach to etract the best corrections from a noisy signal is to use an estimator for the underlying drift signal in the overlay urement. A very common estimator is an eponentially weighted moving average (EWMA), given by: ˆ n n 1 ( 1 λ) ˆ (13) n = λ + (14) Where the symbol with the carrot indicates the EWMA estimator given by the recursion relationship in the above equation. The parameter λ is used to adjust the amount of averaging: smaller λ values will give a larger amount of historical averaging, and larger λ values will give a smaller amount of historical averaging. The complete APC model is given applying equation (13), calculating the estimator in equation (14), and then using a generalization of equation (9) to obtain the simulated overlay errors on the wafer: b ˆ n + 1 n 1 n, APC = b + A (15) 4. EVALUATION OF ZERNIKE-LEGENDRE MODELS BY APC SIMULATION We evaluate Zernike and Legendre polynomials as overlay model equations by using APC simulation. This is done by taking overlay urements from production wafers, and de-correcting the APC used in production by using equation (12). (The correction parameters represented by that were applied to the wafers in each lot can be obtained from scanner log files.) Now that we have the raw overlay for the production wafers, we can apply equation (13) to calculate model corrections for a specific wafer, apply the EWMA filter given by equation (14), and then calculate the remaining overlay errors after application of APC by using equation (15). We present APC simulation results for four cases: POR: Cascade model where the linear terms are fit first and then the higher-order terms are fit to the residuals of the linear fit. This set of results is labelled POR because it represents the previously known method [2] for stabilizing interactions between high-order wafer and high-order field models. XY (Single Pass): regular XY polynomials where the wafer and field terms are the same order as the POR model. ZL (Single Pass): Zernike/Legendre polynomials where the wafer and field terms are the same order as the POR model. This model has the same fitting capability as the XY (Single Pass) model. Z+iHOPC (Single Pass): Zernike polynomials for the wafer terms, and regular XY polynomials for the field terms. Proc. of SPIE Vol Y-5

6 Figure 3: APC simulation results for the four overlay models (POR, XY, ZL, and Z+iHOPC) as well as the raw decorrected overlay (for no APC) and the ideal rework case. The EWMA filter was λ=0.3 for all APC results ecept for the Measured POR Overlay case, which is a more sophisticated APC scheme used in actual production. Note that the XY (single pass), the ZL (single pass), and the Z-iHOPC (single pass) results are identical for this APC simulation E 6!Ili!I'll] III ''' '''''''''''''''''''''''''' XY (POR), mean : XY (single pass), mean : IL (single pass), mean : ZeSHOPC (single pass), mean : Measured POR Overlay, mean Ideal (Rework Case), mean : Raw Deeorrecled, mean : cc i '',-''' N -1' '''''. ''''''' -, ol'ivii. j 1,;'1-irmioul'ilril.1-1 ''''''''''''''''''''''''''' Ighg Z.gth'.'61'41g.Fgltngtf.igT6Ifilfihri21%111 * g liii 1 i 1 i I I I I ii -XY (POR), mean XY (single pass), mean : (single pass), mean : i1-10PC (single pass), mean : Measured POR Overlay, mean : Ideal (Rework Case), mean : Raw Deeorrecled, mean : r +. ' '''. ' ' I I 1 I I I ii 1 HEMINMUMMEMPAHRIENI 1 I MIHNIIRMEM i I I I I I I I I I I I I I I I I I I Figure 4: APC simulation results for the four overlay models (POR, XY, ZL, and Z+iHOPC) as well as the raw decorrected overlay (for no APC) and the ideal rework case. Separate EWMA filter values were used for each model coefficient. For each model, the λ coefficients were optimized to minimize the average OPO (mean plus three sigma) over the sequence of 60 lots. Proc. of SPIE Vol Y-6

7 Note that the order of the polynomials used for the wafer and field level terms is the same for all four cases. The first set of results are shown in Figure 3, where the EWMA filter was set to λ=0.3 for all of the models. For this choice, all of the single pass models give basically the same result. This is because the fitting capability of the models is equivalent. All single pass models perform better than the cascade model (POR) when we average the mean plus three sigma value over all 60 wafers. Net, we allow the EWMA values to be different for each coefficient in the model, as shown in Figure 4. We optimized the λ values so that the predicted overlay (mean plus three sigma) would be minimized over the sequence of 60 lots. For this case, there are some small differences, but it is difficult to tell if these differences are statistically significant. It is somewhat surprising that the overlay models with orthogonal polynomials did not show different performance compared with standard XY polynomials. The eplanation for this result is that all of the single pass models have the same fitting capability. For each ured wafer, the root-mean-square (RMS) error between the ured overlay and the model is the same for the XY single pass, XL single pass, and the Z+iHOPC single pass models. The only difference was observed for the XY (POR) model which uses a cascading scheme. It is interesting to investigate the differences between the cascading and orthogonal polynomial approach. The investigation by Ullah et al. [2] showed lower correlation coefficients with cascading. The correlation coefficient is given by r ij = N n n ( i i )( j j ) N N n 2 n ( i i ) ( j j ) n= 1 n= 1 n= 1 Note the subscripts in this equation represent different terms in the fitted model, such as wafer-level -mag and fieldlevel rotation. The summation is over the wafer sequence in the APC simulation. However, cascading leads to slightly larger RMS errors between the model and ured overlay, so it is desirable to find models that lead to lower RMS errors and low correlation at the same time. Shown in Figure 5 are some properties of the term correlations for the four models in this study. As shown in the figure, the four models have similar degree of correlation between the different terms in the model. The reason that we see similar results for all four models is that the correlation coefficient ures correlation from wafer-to-wafer, while the benefits of orthogonal polynomials is to reduce interactions within a single wafer. An eample that displays the difference between orthogonality of the overlay model and correlation coefficient is shown in Figure 6. In this eample, we show a sequence of two model coefficients that could easily be corrected by an APC scheme (left part of figure). Perhaps these model parameters were determined by one of the four models in our study. However, if we were to scramble the order of the wafers (shown in the right part of the figure), the correlation coefficients calculated by equation (16) would be the same, but now the signals appear to be almost entirely noise, and the APC correction would be ineffective. We now propose another method to evaluate the benefits of orthogonal polynomials, and this new metric ures the interactions between the terms used to fit a single wafer. This metric is inspired by the eample results shown earlier where we fit to the shifted cosine term in Figures 1 and 2, where the regular polynomials showed regions where individual terms had large values, and cancellations between terms were required to fit the data. One way to quantify these large values is to take the maimum (absolute value) of the contribution for each model term over the region of the fit. We call this the impact of each term. As an eample, the impact for the cubic fitting to the shifted cosine is shown in Figure 7, and we see that the orthogonal polynomial has a smaller impact for the constant and quadratic terms than for the regular polynomial. We see similar results if we eamine the fitting coefficients from our APC study. The overlay impact metrics for the third-order wafer terms for each wafer lot in the APC simulation are shown in Figure 8 for the POR (XY polynomial with cascade), for the standard XY single pass, and for the Zernike-Legendre single pass models. As shown in the figure, the range of the overlay impact parameter is much smaller for POR and the ZL models compared with the standard XY polynomial. In addition, the fluctuations in the overlay impact parameter are smaller for the Zernike- Legendre model. We can take the standard deviation of each overlay impact parameter in the model as a way to quantify the size of these fluctuations in the parameters. Shown in Figure 9 is the average standard deviation for each group of terms for all of the models. Here we see that the fluctuations in overlay impact are the largest for the standard XY 2 (16) Proc. of SPIE Vol Y-7

8 APC Simulation Coefficient Correlations XY (POR), mean Icc1= XY (single pass), mean Iccl = ZL (single pass), mean Iccl = Z +ihopc (single pass), mean lccl = Iccl < < Iccl < < 'cc' < < Iccl Figure 5: The correlation coefficient calculated over the sequence of 60 lots for all 4 models investigated in this study ' 13 Uncorrected Overlay Errors Scramble Lot Order Uncorrected Overlay Errors Mr11..!i'lNiiv y-r0r» 03 o O C O CI 6 N 1 1 R t Y R f* o! R C 1 N Y J Figure 6: Eample sequence of overlay model coefficients. For the plot on the left, the two model parameters are correlated, and they occur in a sequence where the APC system could compensate for the systematic drift present in the two overlay coefficients. For the plot on the right, the same coefficient values are presented, ecept that the lot sequence has been randomly scrambled. For this case, the correlation coefficient is eactly the same, but there is no underlying drift signal that can be corrected by APC. model, and smallest for the POR and ZL models. Specifically, we can see that the average fluctuations in the overlay impact are smaller for the high-order wafer terms and the high-order field terms for the POR cascade model compared with the standard XY single pass model this demonstrates that cascading is successful in stabilizing these terms. The ZL model is also successful in suppressing the fluctuations in the high-order terms, so it gives the combined benefit of lower RMS errors and stabilization. Finally, if we eamine the Zernike-iHOPC model, we see that the high-order wafer terms are stabilized, but the field terms have similar performance to the standard XY polynomial model, as epected because orthogonal polynomials are not used at the field level. Proc. of SPIE Vol Y-8

9 Í I I 1 1 Regular Polynomial IS OS o.. 41,4Ip1ron o, :,.1 C 80`.p5 1 S. -LS Orthogonal Polynomial.D.. -db,o.. S ' ; '' a =-, IS Figure 7: The impact of the different fitting terms to the shifted-cosine eample. On the left, the regular polynomial has an impact of 0.69 for the constant term, 0.63 for the linear term, and 1.45 for the quadratic term. By contrast, the orthogonal polynomial has impact values of 0.21, 0.63, and 0.97, as shown on the right I r r r r E 0 A. g 10 Ê É _I it I I I I XY POR. APC Simulated Coefficients. Third Order Wafer. Stability Metric: ZL (single pass), APC Simulated Coefficients, Third Order Wafer, Stability Metric: _ Í f I I I 1 I I 1 I 1 I 1 I I 1 rn n 1og mi mrv I I I 1 1 l l f XY (single pass). APC Simulated Coefficients. Third Order Wafer. Stability Metric: I I 1 1 I I 1 I 1 I 1 I I 1 I 1 I I T30, = 0.3 -T21, =0.3 T12, X=0.3 T03, X= 0.3 Ty30, = 0.3 Ty21, =0.3 -T19 1=03 - Ty03, =0.3.rom190- -TT301, 0.3, =03 -T12, =0.3 T03, = 0.3 Ty30, X= 0.3 Ty21,=0.3 TO? 1=113 - Ty03, = I I MO_ nt3jjt3djd9j3dnjjnjnsd órv or J,i Z3-3, = 0.3 Z3.1,= ,=0.3 Z33,= ,=0.3 Zy31,=0.3 Zy31,=0.3 mmmmmmmgd00mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmd70dgd00modgd00moa0dgommmmmd70dodo7 Figure 8: The overlay impact for the third-order wafer terms for the POR model (top), XY (middle), and Zernike- Legendre (bottom). Results are plotted for the sequence of 60 wafers used in the APC simulation. Proc. of SPIE Vol Y-9

10 35 3 XY Model (POP), Stability Metric 303 _XY Model (single pass), Stability Metric 37 MIL Model (single pass). Stability Metric 19 IMZ.rHOPC Model (single pass). Stability Metric 2 12 apc Simulation Coefficient Stabddy by Order O III 16 L IllKati Ori Order Vre5er I" Order Wafer `.n Order Wafer 3e Orde Wafer 1" Order Field 2. Order Field 3. Order Field Figure 9: Standard deviation of the overlay impact for different model terms in the APC simulation. The bars represent the average of standard deviation for all of the terms in the model. 5. SUMMARY We have eamined orthogonal polynomials as the basis for overlay models, using Zernike polynomials for wafer-level terms and Legendre polynomials for field-level terms. The primary benefit of orthogonal polynomials is that there are smaller interactions between fitting terms. This reduces the coupling between terms when fitting to a single wafer, and gives a benefit similar to cascading. We defined the overlay impact metric that quantifies the interactions between terms on a single wafer fit. For small values of the overlay impact, the model has smaller cancellations between terms when fitting ured overlay data. The Zernike and Legendre polynomial models gave smaller overlay impact metrics compared with standard XY polynomial models. This property can be useful in SPC charts, where smaller baseline fluctuations would make it easier to detect ecursions. Continuing investigations using Zernike polynomials include the impact of sampling, wafer edge effects, and process noise, and planned as the subject of future reports. REFERENCES 1. Levinson, H.J, Lithography Process Control, SPIE Press (1999). 2. Ullah. M.Z., et al. An Investigation of High-Order Process Correction Models and Techniques to Improve Overlay Control by Using Multiple-Pass Cascading Analysis at an Advanced Technology Node, Proc. SPIE, 8681 (2013). Proc. of SPIE Vol Y-10