Error Analysis in Inverse Scatterometry I: Modeling

Transcription

1 Error Analysis in Inverse Scatterometry I: Modeling Rayan M. Alassaad and Dale M. Byrne i Erik Jonsson School of Engineering and Computer Science, University of Texas at Dallas, MS EC33, Richardson, TX ABSTRACT Scatterometry is an optical technique that has been studied and tested in recent years in semiconductor fabrication metrology for critical dimensions. Previous work presented an iterative linearized method to retrieve surface relief profile parameters from reflectance measurements upon diffraction. Using the iterative linear solution model in this work, rigorous models are developed to represent the random and deterministic or offset errors in scatterometric measurements. The propagation of different types of errors from the measurement data to the profile parameter estimates is then presented. The improvement in solution accuracies is then demonstrated with theoretical and experimental data by adjusting for the offset errors. In a companion paper a new improved optimization method is presented to account for unknown offset errors in the measurements based on the offset error model. 1-Introduction and Objectives 1.1 Overview of Scatterometry A continously increasing demand for ultra large-scale integration (ULSI) in the semiconductor industry imposes great challenges on all aspects of the fabrication process. According to the semiconductor roadmap 004 update 1, one of these challenges is the requirement of nanometer and sub-nanometer accuracy for critical dimension (CD) measurements. To meet this particular challenge, optical measurement techniques suitable for non-invasive, rapid and in-situ operations have been explored extensively. One optical technique under constant investigation is scatterometry,3 based on the diffraction of an incident wave by a periodic structure. Scatterometry is a term that is typically used in the semiconductor manufacturing process to refer to measurements of reflected (or transmitted) power upon the interaction of an incident beam with some periodic surface (grating). The characteristics of the diffractive surface affect directly the acquired measurements. Therefore these measurements contain information about the grating profile and the period. In view of the highly nonlinear relation between the grating profile geometry and the scatterometric data, library search and neural network approaches have been initially used to estimate CD and other features offering some degree of success but also revealing a number of limitations, mainly in the requirement of large databases. More recently nonlinear inverse solution techniques that do not require databases have also been investigated revealing other challenges such as speed and accuracy. In recent work, comparisons in accuracies of results for various computational methods to solve the inverse scatterometry problem have been presented 4. The study concluded that the most accurate results were achieved using a library search approach. However the remaining optimization techniques are i Correspondence: byrne@utdallas.edu; URL: Telephone: ; Fax:

2 attractive since they do not require the construction of large databases and offer more flexibility to sample diversity. Moreover, computational techniques such as the previously established linearized inverse method 5-8 are more suitable for thorough mathematical analysis of the errors in parameter values. Mathematical models of errors in the measurements and the profile parameters allow careful monitoring of the error propagation in the solution method. Consequently useful techniques for controlling and improving the accuracies in the results can be developed. In this paper, a brief review of the main configurations used in scatterometry is first presented, as well as a description of a common test structure and the profile parameters under investigation. Next, the formalism of the linearized method is revisited followed by a conceptual representation for the iterative procedure in the solution method. The modeling of the standard deviation and offset errors in the problem is then described and a complete view of the progression of the parameter estimates and the associated errors in the iterative model is illustrated. Enhanced accuracies in the parameter estimates are then presented by accounting for the offset errors in the measurements for simulated angular reflectance data as well as experimentally acquired data. - Measurement Configuration and Test Structure Profile.1 Measurement Types There are two common types of scatterometry instruments: the angular configuration and the spectral configuration. The first configuration is usually referred to as the -θ configuration which is shown in Fig. 1. The incident beam and the normal to the diffracting surface form the plane of incidence which is usually perpendicular to the directions of the grooves of the grating. The direction of propagation of all reflected diffracted orders (designate by angle θ m ) is given by the grating equation mλ sin θ inc + sinθ m =. (1) a The angles θ inc and θ m correspond respectively to the incident and the m th order reflected beams measured relative to the normal to the grating surface in the plane of incidence, λ is the wavelength of the incident beam, a is the period (pitch) of the grating, and m is an integer that designates the specific diffracted order ( m = 0, ± 1, ±, ). When the incident beam is linearly polarized such that the electric field strength oscillates perpendicular to the plane of incidence and parallel to the grooves of the diffracting structure the polarization state is referred to as TE or s. The polarization state orthogonal to TE is known as TM or p where the electric field strength oscillates parallel to the plane of incidence and perpendicular to the grooves. It is common, although not necessary, to measure only the specular order (m = 0). The resulting grating equation reduces to θ0 = θ inc. In the angular configuration, measurements are made by varying the angle of incidence over a chosen range and recording the reflected zero th (specular) order power at each corresponding angle of reflectance maintaining a constant wavelength. As for the spectral configuration, the wavelength is changed in a chosen range commonly the visible region of the spectrum for sub-micron profile dimensions while the angle of incidence of the beam is usually fixed at some value.

3 Incident plane wave ( λ ) Plane of incidence Specular reflected order (m=0) θ inc θ 0 Potential higher order Fig. 1 A general -θ configuration.. Sample and Profile Description The general profile of the diffracting structures investigated in this study is shown in Fig.. Each structure consists of an array of parallel lines with trapezoidal profiles, characterized initially by three geometric parameters w, d and s that correspond respectively to the top width - commonly referred to as CD - the depth (or thickness) and the side-slope (or sidewall) angle of the structure profile. The basic trapezoid geometry can account for the deviation of the structure profile from a typically rectangular design due to imperfections in the lithographic process. The period of the array is a fixed value and is assumed to be accurately determined prior to scatterometry measurements. The grating surface is assumed to be composed of PMMA developed on a silicon substrate. Other sets of parameters have also been investigated to model the same geometry. These models involve using more than one width parameter at different depths in the structure (such as top, middle or bottom width) instead or in addition to the slope angle parameter. top width depth slope angle Structure Profile Fig. Cross section of a grating on a substrate. 3- Description of the Solution Method 3.1 Formalism of the Linearized Inversion Among the different methods of obtaining profile information from scatterometric data, the linearized approach described by Drege et al. 5-8 has been chosen. This method is simple and convenient in estimating the uncertainties in the retrieved profile parameter values. First the rigorous coupled wave 3

4 theory (RCWT) 9,10 is used to approximate the zero th order reflectance measurements using as input geometrical structure information and material constants in addition to measurement conditions such as incident angle and wavelength. The relationship between the geometrical structure information and the zero th order reflectance can be expressed with high fidelity using the RCWT model R( u, p) R ( u, p). () i RCWT i The subscript i indicates the i th measurement, R(, p) is the measurement of the zero th order reflectance at u i the physical variable u i (such as the wavelength or the angle of incidence), R RCWT ( u i, p) is the modeled value of the reflectance at u i, assuming a set of geometrical parameters p that describe the diffracting structure. In a highly controlled lithographic process an actual fabricated grating differs only slightly in its geometrical properties from the design structure. Hence it becomes possible to approximate the actual measured reflectance by the first two terms of a Taylor series expansion of the RCWT model about the design parameter value. The resulting mathematical expression for this process is L ( a) (0) Ru ( i, p) RRCWT ( ui, p ) RRCWT ( ui, p ) + pj. (3) p j= 1 j (0) The p j 's are the departures of the actual parameter values (as fabricated) from the design values, p is a column vector containing the design parameter values, p j is a column vector containing the (unknown) actual parameter values and L represents the number of profile parameters. When Eq. (3) is substituted into Eq. (), we arrive at an expression relating the departures of parameters from their design values and the differences between the measured and expected reflectance values. This expression becomes N a RRCWT u ( ) (0) ( i, p) R( ui, p ) RRCWT ( ui, p ) = R( ui ) p j. (4) p j = 1 j (0) Upon considering the entire measurement set, Eq. (4) can be written in matrix form as R = M p, (5) where the elements of the matrix M are then defined as RRCWT ( ui, p) M ij =. (6) p j (0 ) p j The departures of the parameters can be easily retrieved by inverting Eq. (5) if the matrix M is a square matrix (the number of measurements equals to the number of parameters). However, in most experimental situations, the number of measurements is greater than the number of unknown parameters, and hence M is not a square matrix, creating an over-determined system. In this case the departures of the 11 parameters from their design values is obtained by a Moore-Penrose pseudo-inverse T -1 T p = ( M M) M R. (7) The proposed linearized inversion technique expressed by Eq. (7) can also be described as a least squares solution; that is a solution obtained by minimizing the sum square of the differences between the actual reflectance measurements and the calculated reflectance values obtained from the linearized RCWT model. This is shown by a residual value in measurement i as the difference between the acquired reflectance value and the modeled reflectance value such as i expi ij j j= 1 l r = R M p, (8) where l is the number of parameters. The sum square error (sse) is defined as the sum of the squares of the residual values in all the measurements such that N N l r = i ij j, i R M p (9) i= 1 i= 1 j= 1 (a) p p j (0) 4

5 where N is the number of measurements. In order to find the solutions for sum square residual, the derivative of the right term in Eq. (9) with respect to p j pq s that would minimize the is set to zero N N l r = Ri Mij pj 0 p i q p i= 1 q =, (10) i= 1 j= 1 Performing the differentiation included in Eq. (10) results in N l R i Mij p j Miq = 0, (11) i= 1 j= 1 which can be further simplified to obtain N l N M R = p M M. (1) iq i j ij iq i= 1 j= 1 i= 1 If the differentiation (given by Eq. (10)) is applied to all l parameters, the group of resulting equations (identical to Eq. (1)) can be rewritten in matrix form ( ) T T T -1 T M R = M M p p = ( M M) M R. (13) It is clearly evident that the final result in Eq. (13) is equivalent to that obtained by the pseudo-inverse solution in Eq. (7) in the linearized model. Hence we conclude that our linearized method is totally equivalent to minimizing the sum squared residuals as defined by Eq. (8). 3. The Iterative Solution With the pseudo-inverse solution presented above, the differences between the actual reflectance measurements and those expected based on the grating design parameters are used to compute profile departures from their respective design values. If the parameter values of the actually built structure are close enough to the design values when the fabrication process is highly controlled one linear inversion will suffice to achieve a correct solution. However in realistic situations the departures are somewhat larger and an iterative algorithm is implemented until convergence to the actual parameter values is reached. At each step the increment in parameter departures ( p ) is calculated by the linear model the expected reflectance differences ( R ) are updated based on the new p resulting in a full non-linear regression solution. This iterative linear solution is commonly referred to in literature as a stepwise linear regression solution. Starting from some expected parameter values, usually based on design, the iterative algorithm is expected to update these values and to converge closer with every consecutive iteration towards the actual parameter values at the global minimum of the sum square errors. The progression of solutions at every iteration can be monitored by generating and plotting the sse function vs. the parameter values in a range that covers both the starting design point and the actual solution point. For example, the sse curve in Fig. 3 is generated by simulating first the angular reflectance measurements with no for a periodic structure with an actual depth of 510 nm, a top width value of 510 nm, a slope angle of 0 o and a period of 1 µm. A set of angular reflectance curves is then generated by changing only the depth value along a range from 509 nm to nm. The sums of squares of the differences between the simulated reflectance measurements of the actual structure and each generated reflectance curve in the set are calculated in order to obtain the sse values vs. the respective depth values. All reflectance curves were modeled with 90 data points from 0 o to 89 o (1 o increments) with 63.8 nm wavelength and TE polarization. The curve in Fig. 3 reflects the general behavior of the sse curve in the case of one unknown parameter. The propagation of the solution in this case from some starting point for the parameter value towards the global minimum where the actual solution is located is demonstrated by two arrows that point at two solution points in two consecutive iterations. Here it is said that the solution converges at the second point when the iterative linear technique reaches a minimum sse value and no change in the solution can occur for successive iterations. Note that in general the calculated departures become smaller for successive 5

6 iterations. This is due to the fact that as the algorithm progresses the differences between the experimental reflectance values and the updated calculated values and thus the obtained parameter departure become smaller. In the case of free measurements, it is expected that the minimum value encountered at convergence is indeed the global minimum value and not some local minimum solution. This would be the case when the iterative solution starts at some guessing point usually selected based on design that is close enough to the actual value. This sets a limit on the solution method where large departures between design and actual structure parameter values may render a correct solution unsolvable. In the case of solving for the depth and the top width parameters simultaneously, the sse values can be plotted as a surface in 3-dimensinal space. The sse surface in Fig. 4 is generated by simulating the angular measurements with no for the same periodic structure described above along with a set of angular reflectance curves by changing the depth as well as the width values from 500 nm to 515 nm. The actual period and slope angle values were set fixed for all data sets at 1 µm and 0 o respectively. The sums of squares of the differences between the simulated measurements of the actual structure and each generated reflectance curve in the set are calculated and the sse values vs. the respective depth and width values are shown in the figure. All angular reflectance curves were modeled with 90 data points from 0 o to 89 o with 1 o increments with 63.8 nm wavelength and TE polarization. The progression of the solution is shown on the sse surface where the starting values were set at 500 nm for both the depth and the top width. The solution converges at the second iteration where the global minimum is reached. In the case of solving for three parameters simultaneously: depth, top width and slope angle, the sse function is no longer a surface that can be plotted in 3-dimensional space vs. some range of parameter values. The sse function rather becomes a hyper-surface which resides in a 4-dimensional space. A graphical representation in such situation is difficult, yet a sse surface analysis considering separately all possible pairs from the set of three parameters is still useful to study the propagation of the solution. Residual vs. Depth Paramter residual depth in micrometers Fig. 3 The propagation of solution by the iterative linear inverse method along the sse curve for the depth parameter from a starting or design value based on design (500 nm) to the actual value (510 nm) in the absence of in measurements. The angular reflectance data sets used to generate the sse curve were modeled with 90 data points from 0 o to 89 o with 63.8 nm wavelength and TE polarization. The actual top width, period and slope angle values were set fixed for all data sets at 510 nm, 1 µm and 0 o respectively. 6

7 Fig. 4 The propagation of solution by the iterative linear inverse method along the sse surface for the depth and the top width parameters from starting values for both based on design (500 nm) to the actual values (510 nm) in the absence of in measurements. The angular reflectance data sets used to generate the sse surface were modeled with 90 data points from 0 o to 89 o with 63.8 nm wavelength and TE polarization. The actual period and slope angle values were set fixed for all data sets at 1 µm and 0 o respectively. 4- Derivation of Errors in Parameters 4.1 Types of Errors in Measurements The parameter values derived (or resolved) by the proposed inverse linear method are in reality estimates of the actual parameters and not exact solutions. This is mainly due to the presence of errors in measurements and to inherent limits in the solution method. Generally two types of errors can be distinguished in the solution for parameter values: random errors and deterministic errors such as offset errors. The first represent the amount of random uncertainty in reported parameter values (caused by random errors in the measurements) while the second represent deterministic differences (caused by differences between the true measurements and measurements in the presence of bias or measurement offset). The deterministic and random error quantities in measurements propagate to the parameter solutions and are manifested as offsets and random uncertainties in the inferred parameter values. Mathematical derivation of the parameter error quantities based on measurement errors is possible and imperative in order to control measurement error related aspects and subsequently improve accuracies in results. 4. Standard Deviations and Offsets Errors in Parameter Estimates Second order statistics are sufficient to characterize the random error as long as the process is considered wide sense stationary 1. This is true if the distribution is assumed to have a zero mean value for all measurement variables and if the second central moment is only dependent on the difference between any two measurement variables. Hence the covariance matrix, which reflects the second joint 7

8 central moment, forms the basis of the error analysis. The covariance matrix for a set of N measurements x, is comprised of elements C defined by the expression The operator i ( )( ij C = E[ x x x x )]. (14) ij i i j j E[ ] represents the expected value operator and produces the mean value of the term within the brackets. Each of the x i ' s (or x j ' s ) represents one of the measurements and x (or x ) represents the mean or average of values for measurement i (or j). The covariance matrix describes the correlations of random variations (about the respective mean values) in pairs of variables. For small levels and small variations in reflectance values, the relationship between the random errors in the parameter departures and the random errors in the reflectance differences can be assumed linear (narrow probability density function) and Eq. (7) can be used to rewrite the parameter errors covariance matrix in terms of R such as T C = E[( p p)( p p) ] = p T T T T T T T T T E ( M M) M R ( M M) M R ( M M) M R ( M M) M R This relationship can be further simplified and reduced to the following form p ( T 1 ) T ( T R ) C = M M M C M M M 1. (16) The square roots of the diagonal elements of the covariance matrix represent the standard deviations in parameter values, and hence indicate the uncertainties in these values. In the case of uncorrelated measurements with equal variances and zero mean the relationship above between the covariance matrix of parameter departures and the covariance matrix of measurement differences simplifies to C p T ( M ) 1 = σ M, (17) where, σ represents the variance for each measurement. Hence, variances in the parameter departures are given by T ( M M) 1 p = σ j jj σ. (18) which makes it possible to define a magnification factor by which the measurement can be scaled to obtain the uncertainty in retrieved parameters. The magnification factor is useful to predict the maximum tolerated amount of uncertainty in measurements in order to achieve parameter values within a desired standard deviation range and is defined as, where p j m p j σ 1 m p j = = ( M T M). (19) σ jj The magnification factor derived using the linearized inversion model shows that, for small levels, the relationship between the uncertainties in retrieved parameter values and the measurement is linear. With the knowledge of the magnification factor in parameter error derived in Eq. (19), the maximum tolerated amount of uncertainty in measurements can be predicted in order to achieve parameter values within a desired standard deviation range. This is a useful criterion for controlling errors in measurements. Moreover, the parameter error magnification factor is a quantity independent of the amount of in the acquired set of measurements. The magnification values reflect the amplification of errors in the solution independent of the level of, which makes it most suitable for quantifying and comparing errors in results for different measurement conditions 8. The generic term measurement error may consist of a truly random quantity with zero mean as described above or a random quantity with non-zero mean, as well as a deterministic departure from a true value as a result of some bias in the instrument or other systematic errors. In this work the random errors are modeled as zero mean random variables plus some offsets (departure from true values). For small i j (15) 8

9 biases, the offsets in the measurement differences µ p that can be derived from Eq. (7) as T -1 T µ R will result in offsets in the parameter departures µ p = ( M M ) M µ R. (0) 4.3 Propagation of Errors in the Iterative Solution Even in the absence of and bias in measurements the retrieved parameter departures p' s in the first iterations may be subject to errors due to a poor parameter estimation model. This is the case when the difference between an actual and a design parameter value exceeds a certain limit so that the first two terms of a Taylor series expansion are no longer a sufficiently good approximation in Eq. (3). Such type of solution errors are referred to as estimation errors δ p ' s. The estimation errors constitute essentially the differences between the estimated and the actual parameters at a given iteration k in the absence of random and deterministic errors in measurements and are given by 1 ( ) k k k act est act est k p p + p = p p = δ p. (1) However if the chosen starting parameter values are close to the actual parameter values in the fabricated structure, the differences between the calculated and the acquired measurements and consequently the differences between the estimated and the actual parameter values are expected to decrease as the number of iterations increases. Hence, we expect that the estimation errors converge eventually to zero values at the global minimum of the sse hyper-surface and can be represented mathematically as ( ( 1 )) ( ) k k k k p act est act est act est k k k lim δ = 0 lim p p + p = lim p p = p p = 0. () For actual experimental situations however, errors exist in measurements and consequently the estimated parameter values may include random and deterministic errors. In this case, the minimum sse value is no longer equal to zero and a range of possible solutions rather than a single global minimum value exists. This range of possible solutions is represented by the uncertainties in the parameter estimates (as given by the standard deviations in the parameter values, which are a direct result of the uncertainties in the reflectance values due to ). The relationship between the actual parameter values and the estimated values (including the error quantities) can be represented as where k ε p 1 ( ) k k k k k k act est + = act est = p p + p p p p p ε µ δ p, (3) k p are the random parameter errors due to, µ are the parameter offset errors and are the estimation parameter errors. As the iterative solution converges, the estimation errors diminish to zero and the relationship in Eq. (3) converges to k k lim p= 0 lim ( ) δ pact pest = pact pest = ε p p k k k δ p µ. (4) Furthermore, the random parameter errors are assumed to be confined with a certain level of confidence between σ and σ, where σ are the standard deviations in the parameter values and are obtained K p + K p p from the standard deviation σ as explained earlier, and K 0 accounts for the level of confidence; e.g., K = 1, and 3 represents 68.3%, 95.5% and 99.7% confidence respectively. The relationship between the random parameter errors and the standard deviations in the parameter values is thus expressed as follows σ ε Kσ. (5) K p p p The uncertainty range for the parameter random errors expressed in Eq. (5), is then used to set a range of uncertainty on the differences in Eq. (4) between the actual parameter values and the estimated parameter values in the presence of offset errors such as σ p p + µ σ. (6) K p act est p K p In Eq. (6) the parameter estimate values are random in nature as a consequence of the random errors. 9

10 In the presence of random and offset errors in general, the range of possible values for the j th parameter estimate,, is found from Eq. (6) to be where p est j p + µ Kσ p p + µ + Kσ p act j act j p j p j est j act j p j p j p j, (7) is the actual j th parameter value, µ is the offset error in the j th parameter, and is the standard deviation in the j th parameter which can be obtained from Eq. (18). The inequalities in Eq. (7) reveals that the range of possible values determined by ± Kσ p j, is displaced from the actual parameter value by an amount equal to the offset value, estimate, can be related to mean value in the j th parameter estimate, p - p est est ε j j p j σ p j µ p j. Furthermore, The random error in the j th parameter p est j =. (8) As a result, the mean value in the j th parameter estimate, p = E[ p ] = p + µ est j est j act j p j When the j th parameter offset error, p = p µ adj j est j p j µ p j The mean value in the j th adjusted parameter estimate, parameter value such as p = E[ p ] = p adj est µ j j p j act j p est j, as, is found from Eq. (7) to be. (9), is known, a new adjusted parameter estimate can be defined as. (30) p adj j, is then found to be equal to the actual. (31) From Eqs. (9) & (31), we observe that the adjusted parameter estimate is expected to be closer to the actual parameter than the estimated parameter value and therefore improves the accuracy in the parameter solution. Note that in the case of zero offset error ( µ = 0) or unbiased measurements, Eq. (9) also reveals that the mean estimated value in every parameter j in the solution, actual parameter value,, at the global minimum of the sse curve, i.e. p j p est j pact j pest = p j act j, is found equal to the. The similarity between the two results (resulting from Eq. (9) and from Eq. (31)) indicates that adjusting the parameter estimates in the solution is equivalent to eliminating biases in the measurements. To illustrate, three sse curves are shown in Fig. 5. The sse curves were generated for a chosen periodic structure in the same manner as described previously for the example shown in Fig. 3. The structure profile is a trapezoid with an actual depth and top width of 00 nm, a slope angle of 5 o and a period of 300 nm. The sse values were obtained by changing the slope angle value along a range from 0 o to 10 o while keeping all remaining parameter values fixed. The reflectance values were modeled with 90 data points from 0 o to 89 o (1 o increments) with 63.8 nm wavelength and TE polarization. The bottom curve with smaller sse values is a result of error free simulated angular measurements. The curve in the middle was obtained with the same measurements with 1% level and no offset errors. The top curve was similarly generated with 1% level but with a constant offset error set (by choice) equal to σ for all angular measurements. For the measurements with 1% level ( K = 1) and a nonzero offset, the iterative solution converges to the minimum value in the corresponding sse curve (top) and not to the actual slope angle value located at the global minimum of the bottom sse curve generated with error free measurements. In the top curve, the minimum sse value is not equal to zero and represent the parameter estimate value, p est. The absolute difference value between the mean parameter estimate in the solution and the actual parameter value is found larger than σ p j. In order to improve the accuracy in the solution, the offset error in the 10

11 parameter, µ p, is calculated (from the given constant offset error in the measurements, µ R ). The parameter estimate in the solution is then adjusted according to Eq. (30) and is found within ± uncertainty. The improvement in accuracy is due to the fact that the mean value in the random σ p parameter estimate is effectively shifted to coincide with the actual parameter value using Eq. (30). The adjusted parameter estimate is also shown in Fig. 5 to lie at the minimum sse value of the middle curve that represents the resolved estimated value with noisy measurements and zero offset errors. This indicates that adjusting the parameter estimates by accounting for the parameter offset errors is found equivalent to eliminating the bias or offset in the set of noisy measurements, which is consistent with the observation established above. 3.5 x 10-3 Sse vs. Slope Angle Paramter 3-3 x p est 5 noisy meas. ( µ = p σ ) 4σ p 4 3 sse = p p p µ adj est propagation of solution along the sse curve noisy meas. ( µ 0 ) = p p = 5 o act error free meas. slope angle in degrees Fig. 5 The propagation of the slope angle solution by the iterative linear inverse method along the sse curve for the slope angle parameter from a guessing value 0 o to the actual value 5 o at 1% level in measurements (top curve). The difference between the actual and the adjusted parameter estimate value is shown to be within ± σ uncertainty p range. 4.4 Implementation for Random and Deterministic Errors in Simulated Studies In the following studies angular measurements of reflectance data were simulated using the RCWT. In order to study the effects of random errors in the measurements on the uncertainties in results, random selected from a Gaussian distribution model was added to each simulated measurement. The standard deviation of the Gaussian model was set as a percentage of the mean of the simulated measurements. The probability density function used to represent (with zero mean) in measurements is ε R i 1 σ p( ε R i ) = e. (3) πσ The variable ε R i is the magnitude of the added to the i th measurement, and σ is the standard deviation in the distribution (population). In the case of angular measurements, the value of σ is 11

12 used to define the level by computing first the root mean square (rms) in the reflectance values over the angular range of interest with a fixed wavelength value R rms ( θ θ ) θ 1 = R. (33) 1 θ1 RCWT ( θ, p) dθ Then the value for σ is set to represent a fraction f of R rms, thus designating the level in measurements shown in Fig. 5 as σ = f R rms. (34) Typically for low levels f is set to 1% of the average reflectance. It should be noted that the fractional level varies from measurement to measurement in the same set due to the variation of the actual reflectance value and the random nature of the. For this reason, the level is defined in the manner indicated by Eq. (17), implying an additive model where the reflectance measurements and the values are uncorrelated. The error values are randomly generated using a zero mean Gaussian distribution model with the calculated σ value. Although the mean of the population is set equal to zero, the actual mean in the finite generated set of random values may be small but not equal to zero. For this reason the small nonzero mean value in the set is calculated and is subtracted from all the random values resulting in an actual zero mean random set. The resulting random values, ε R, are then added to the simulated reflectance measurements, R sim, obtained from the RCWT model. The resulting sum of the two quantities (the simulated reflectance measurements free of and the zero mean random values) models noisy reflectance measurements,, as expected in a real experimental situation with no bias. Modeling R deterministic errors was done by adding offset values, µ R, to the sum of the simulated measurements and the respective random error values, resulting in R = R + ε R + µ R, (35) sim σ R i = p( ε ) ε 1 σ e πσ R i θ 1 RCWT 1 θ1 ( θ θ ) ( θ, ) Rrms = R dθ p σ = fr rms Fig. 6 Representing random level in measurements and modeling its distribution. 1

13 5- Simulated Studies and Results In order to demonstrate how the parameter estimates in the solution can be adjusted using Eq. (30) to improve the accuracies in the solutions, angular measurements were simulated for a periodic structure of developed PMMA on a silicon substrate. Specular order reflectance values were modeled for 90 angles (from 0 o to 89 o and 1 o increments) with TE polarization, µm wavelength, and with varied levels. The sample grating has an actual trapezoid profile geometry with 1 µm period, 1 o slope angle, 510 nm depth and 510 nm top width. The starting values for the iterative pseudo-inverse model were based on design at a 1 µm period rectangular profile at 0 o slope angle, 500 nm depth and 500 nm width values. The resulting estimated parameter values and errors were then examined and compared for the various levels. Two cases were studied with two different sets of generated measurement random errors (). The first set was generated with 1% level of according to Eq. (34) and a constant offset in all measurements chosen equal to + 1σ ( µ R =+ 1 σ, for i = 1,,...,90). The second set was generated i with % level and an offset set equal to σ ( µ R = σ, for i = 1,,...,90). The two simulated measurement sets represent biased noisy measurements, R (simulating larger levels in both random and offset errors). The two noisy and biased data sets, and the reflectance values for the two corresponding solutions are shown in Fig. 7. The resolved parameter estimates from the two biased data sets, the associated standard deviations and offset errors, 1. i p est σ p and µ p respectively, are shown in Table Angular Reflectance for Two Noise Levels simulated meas. with 1% & +1sigma mean reflectance values for the parameter estimates. with 1% & +1sigma mean simulated meas. with % & -sigma mean reflectance values for the parameter estimates. with % & -sigma mean normalized reflectance angle of incidence (degrees) 70 Fig. 7 Simulated angular reflectance measurements with two different distributions and the reflectance curves for the obtained parameter estimates. 13

14 parameter estimates p est from biased measurements R adjusted parameter estimates p = p µ adj est p parameter estimates p est from unbiased measurements R µ R offset parameter errors µ p standard deviations ± σ p 1% µ =+1σ R d = nm w = nm s = 1.49 o d = nm w = nm s = 1.1 o d = nm w = nm s = 1.10 o µ d = 0.77 nm µ w = nm µ s = o σ d = ± 0.0 nm σ w = ± nm o σ s = ± 0.15 % µ = σ R d = nm w = nm s = 0.79 o d = nm w = nm s = 1.0 o d = nm w = nm s = 1.10 o µ d = nm µ w = nm µ s = σ d = ± nm σ w = ± nm σ s = ± 0.35 o Table 1 Estimated parameter values and errors for two different levels. The values in the parameter estimates were then adjusted according to Eq. (30) using the parameter offset values µ p, which are in turn calculated from the given µ R values using Eq. (0). For a small bias in measurement values the linear relation expressed in Eq. (0) is expected to be a good approximation of the parameter offset errors. As was established earlier, the adjusted parameter estimates, shown in Table 1, are expected to be very close to the solution obtained from unbiased p adj measurements, R µ R, with zero mean or zero offset. The parameter estimates, p est, obtained from the noisy but unbiased sets of measurements are also shown in Table 1. The estimated values from the unbiased measurements, p est, shown in Table 1 are retrieved within the reported standard deviation ranges ( σ ) from the actual values in both measurement sets and ± p for all parameters. This is also found true for the adjusted parameter estimates (obtained from the biased measurements). The resolved parameter estimates from the biased measurements,, however were found with much larger errors. In the first set generated with a smaller bias ( + ) the p adj 1σ estimated values were found within ± 5σ d, ± 4σ w and ± 4σ s for the depth, top width and slope angle parameters respectively. In the second set generated with a larger bias ( σ ) the estimated values were found within ± 7σ d, ± 5σ w and ± 6σ s for the depth, top width and slope angle parameters respectively. The solution accuracies were hence improved considerably (up to 7 times in some cases) by adjusting the estimated values using Eq. (30), that was derived based on the error models proposed in this work. p est 6- Experimental Results It is shown from the simulated studies above that in the case of noisy and biased measurements, the estimated parameter values in the solution can be adjusted by accounting for the offset errors to improve the accuracies in results. In a real experimental situation however where levels might be larger and the offset values may not be constant but vary across measurements, the validity of the proposed error models need to be investigated. For this purpose measurements were acquired using the -θ configuration for two gratings which were fabricated lithographically. 14

15 The -θ configuration was set up using a red HeNe laser, a linear polarizer, a chopper, and a lens to focus the beam onto the center of the grating sample. The angular measurements are acquired by rotating the sample stage with a motor and a control box connected to a standard PC and controlled by a software (Labview). The reflectance measurements are collected by positioning a photodiode in front of the zero th reflected order. The two fabricated gratings were obtained by initially spinning 000 A of PMMA on a silicon wafer. The PMMA was exposed using an electron-beam pattern generator whose spot size was estimated at 0.5 µm. Each line was exposed using two passes. The spacing of the exposed lines was 1 µm. The multiple pass exposures created the trenches after development of the PMMA. The total area of the pattern was 3 mm. A profile image of one of the fabricated grating samples was captured using an SEM and is shown in Fig. 8. The grating sample was sputtered with a thin layer of gold to enhance the SEM image. Estimated parameter values, from the SEM picture, were chosen as starting values in the iterative solution. The depth was estimated at 00 nm, the width at 600 nm and the slope angle was estimated at 0 o. The period of the gratings was accurately determined prior to scatterometry measurements and found to be equal to the design value at 1 µm. o Fig. 8 SEM profile image and the resolved profile from scatterometry for a 1 µm grating composed of developed PMMA on a silicon wafer. Angular reflectance measurements were made with a TE and TM polarized HeNe source, in the case of the first and second grating sample respectively, at 63.8 nm and varying the angle of incidence from 7 o to 85 o with 1 o increments. The experimental reflectance data are shown in Fig. 9(a) for the two grating samples. The measurements were assumed noisy and biased with some mean offset value. The standard deviation error and the offset error in the measurements, mainly characteristics of the instrument, were estimated beforehand from additional angular measurements acquired for a bare silicon wafer sample as well as an unpatterned PMMA on a silicon substrate with known thickness. The standard deviation and the measurement mean offset value were estimated and found comparable at approximately 5.5% of the rms value in the measurements ( σ µ R Rrms ). These values represent a high level and biases in the measurements, probably exceeding most instrumental situations. Following the same steps in the simulated studies, the parameter estimates, p est, were first obtained from the noisy and biased measurement sets for the two grating samples using the iterative model. The reflectance values obtained from the solutions for the two samples are shown along with the biased experimental data in Fig. 9(a). The adjusted parameter estimates,, were obtained for the two samples using Eq. (30) and the estimated mean offset value, µ R, in the instrument. Third, in order to approximate unbiased measurements, the estimated mean offset value was subtracted form the acquired biased data sets and the corresponding new parameter estimates, p est, were obtained. The unbiased sets of measurements for the two samples and the reflectance values generated for the new parameter estimates are shown in Fig. 9(b). All three solutions: p, p and, p are shown in Table, as well as the associated parameter standard deviations and offset errors, σ p and µ p. est adj est p adj 15

16 Acquired Biased Masurements and Calculated Reflectance Values from Solution acquired (biased) measurements - TE polarized, for sample 1 calculated reflectance values (from solution), sse= acquired (biased) measurements - TM polarized, for sample calculated reflectance values (from solution), sse= Adjusted Unbiased Masurements and Calculated Reflectance Values from Solution, mean offset value = 0.01 acquired (biased) measurements - TE polarized, for sample 1 calculated reflectance values (from solution), sse= acquired (biased) measurements - TM polarized, for sample calculated reflectance values (from solution), sse= normalized reflectance normalized reflectance angle of incidence (degrees) angle of incidence (degrees) (a) (b) Fig. 9 Experimental data and generated reflectance values from solution vs. incident angle for sample 1 and sample gratings: (a) biased noisy measurements, (b) unbiased noisy measurements. parameter estimates p est from biased measurements R adjusted parameter estimates p = p µ adj est p parameter estimates p est from unbiased measurements R µ R offset parameter errors µ p standard deviations ± σ p sample 1 d = nm w = 7.8 nm s = o sse = d = nm w = nm s =.06 o sse =.0117 d = nm w = nm s =.1 o sse = µ d = 0.1 nm µ w = +6.9 nm µ s = 5.90 o σ d = ± 0.65 nm σ w = ± nm o σ s = ± 6.34 sample d = 06.1 nm w = 8.5 nm s = 3.36 o sse = d = 03.4 nm w = 14. nm s = o sse = d = nm w =.6 nm s = o sse = µ d = +.7 nm µ w = nm µ s = σ d = ± nm σ w = ± 4.45 nm o σ s = ± 5.95 Table Estimated parameter values and errors from biased and unbiased measurement sets for two fabricated grating samples. It can be observed from Table that the smallest sse values correspond to the solutions obtained from the unbiased measurements. The corresponding profile shows a good fit to the SEM image for sample 1 as shown in Fig. 7. This was also found true for sample (figure not shown). The adjusted parameter estimates, padj, are found to be close and within ± σ p accuracy with respect to the values derived from the unbiased measurements, p est, for both samples. This implies that the offset error model in Eq. (0) is sufficiently accurate even for a high level and biases in the measurements such as those considered here. 16

17 On the other hand, when the offset errors in the biased measurements are unaccounted for, the solutions are retrieved with larger sse values for both samples as can be seen in Table. In addition, the resolved parameter estimates, pest, were found with larger errors with respect to p est which were chosen as the best approximations for the actual values. For the first sample, the estimated values were found within ± 1σ d, ± 3σ w and ± 1σ s for the depth, top width and slope angle parameters respectively. For the second sample, the estimated values were found within ± 1 d, ± w and ± 3 s for the depth, top width and slope angle parameters respectively. The reported adjusted parameter estimates, σ σ σ, show clearly improved accuracies over the corresponding parameter estimates, p est. The improvements in the accuracies reported for the experimental studies were smaller than those reported for the simulated studies. This is due to the fact that the uncertainties in the experimental studies were reported with respect to the unbiased measurement results,, while the uncertainties in the simulated studies were reported with respect to the p est actual parameter values. Under the ± σ p uncertainties in results, adjusting the parameter estimates, p adj, is found in fact equivalent to eliminating a mean offset value in the measurements, as was similarly observed in the simulated studies. In any given experiment, it is necessary to obtain good approximations for the measurement offset errors in order to adjust properly the parameter estimates in the solution. A new optimization method based on the offset error model in Eq. (0) is presented in a companion paper to adjust the parameter estimates with no prior knowledge of offset errors in the measurements. p adj 7- Summary and Conclusions Mathematical models for random and deterministic errors in results were presented based on the iterative linearized solution to the inverse scatterometry problem. For small levels of errors, the standard deviations and offset errors in parameter departures were first related linearly to the standard deviations and offset errors in the measurement differences. The propagation of the errors in the iterative model was then presented and the resulting errors in the parameter estimates were defined in terms of the errors in the measurements. In the case of biased measurements, improved solution accuracies were predicted according to Eq. (30) by adjusting the parameter estimates when the offset errors in the measurements are known. Studies based on simulated angular measurements were then analyzed using the proposed error models. Noisy measurements with a constant bias as well as noisy unbiased measurements were considered. It was found that in the case of biased measurements, the accuracies in results can be significantly improved by adjusting the values of the parameter estimates values. It was also found that adjusting the parameter estimates is equivalent to eliminating the constant bias in the measurements as observed earlier in the error analysis section. In addition, two grating structures composed of patterned PMMA on silicon substrates were fabricated by means of electron-beam lithography. Two angular measurement sets were then acquired for the two samples. The acquired measurements were assumed biased with offset errors that vary across the angular range. A mean offset was therefore estimated empirically as a characteristic of the instrument. The estimated mean measurement offset was then subtracted from the biased measurements to obtain new unbiased measurement sets. Both biased and unbiased measurements were used to solve for the parameter estimates for the two samples. The solutions corresponding to the unbiased measurement were found with smaller sse values and constitute better fits to the sample profiles obtained by the SEM. The parameter estimates, from the biased measurements, were also adjusted using Eq. (30) to obtain closer values to the unbiased solutions, within σ uncertainties for 68.3% confidence. ± p 17