Lowering the Cross Correlation between Different Shape Parameters of the Inverse Grating Problem in Coherent Fourier Scatterometry S. Roy, N. Kumar, S.F. Pereira, and H.P. Urbach Optics Research Group, Dept. of Imaging Science and Technology Faculty of Applied Sciences, Delft University of Technology Lorentzweg 1, 2628 CJ Delft The Netherlands
[email protected]
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Introduction
Angular Fourier Scatterometry techniques to solve inverse-grating problems have been widely used in semiconductor metrology since the initial work by den Boef et al. [1] because it is simple, fast and robust and yet yield accurate measurements. These systems use incoherent light as a source and are commonly referred to as Fourier Scatterometry (FS). Coherent Fourier Scatterometry (CFS) was introduced a few years ago [2] as an alternative technique to the existing tools. This new method differs from Incoherent Fourier Scatterometry by the aspect of using coherent light as a source to illuminate the grating to be characterized. It is shown [3] that the use of the information about the phase difference of successive orders generates significant gain in sensitivity. This phase difference is implicitly obtained by laterally scanning the grating with respect to the optical axis of the setup within one grating period and obtaining a few intensity measurements. This lateral scan affects phases of only orders higher orders than zeroth. Using this property, besides common parameters of grating reconstruction without making any additional measurement, the parameter bias [3] (lateral misalignment of the grating in experiment and its numerical model) could also available in CFS enabling one to align the target with respect to optical axis of the system to an accuracy of few nanometers. However, this works only when higher orders are available. In this article, first we will discuss scanning, interfering and sectioning in CFS and then we will show their effects in the cross-correlation of shape parameters with numerical examples.
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Working Principles of CFS and Its Extensions
At this point, it is worthwhile to mention a basic description of CFS. Figure 1 shows the schematic of the setup. This schematic includes the original CFS W. Osten (ed.), Fringe 2013, DOI: 10.1007/978-3-642-36359-7_5, © Springer-Verlag Berlin Heidelberg 2014
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prototype [4] in the right side branch together with an optional interferometric extension, which has been discussed recently [5]. Considering the main branch in the right side we may note that a collimated beam from the intensity stabilized laser is focused by a high numerical aperture objective onto a target. This creates the incident wave. The target is an one-dimensional grating.
ߦ
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Fig. 1 The schematic diagram of CFS with the Interferometric modification. The main branch on the right is the conventional CSF. The sample is generally an one-dimensional grating which is scanned in the direction of its grating vector by translation of the precision stage. The branch on the left side (consisting of lens 2 and polarizer 2) is used for an phaseshifting Interferometric version of CFS. A fiber phase shifter (not shown) can be used to produce the required phase shift. In the inset a schematic pupil is shown with the orthogonal axis (ξ,η) and ‘planar’ sectioning, in which pixels on ξ axis are chosen only.
The reflected wave is captured in reflection by the same objective. We will refer to it as the scattered wave. This scattered intensity is directly measured. Polarizer 1 and 2 can be used to define the polarization of the incident and scattered wave, respectively, with respect to laboratory frame ξ-η, an orthogonal Cartesian basis located at the objective pupil parallel to x-y. The laboratory co-ordinate system xyz is defined so that the z-axis is the instrument optical axis and x-y plane is at the top of the grating, i.e., the grating-air (incident media) interface. The aforementioned scanning is performed by translating the grating in the direction of the grating vector (here it is X-axis) through small steps until the length of the pitch is covered. The number of scanning positions depends on the number of overlapping orders [3], and is expressed more conveniently through the overlap parameter, F defined as, λ F= (1) NA × Λ
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where, λ is the wavelength, Λ is the pitch of the grating and NA is the numerical aperture of the objective. When F ≥ 2 there is no other order than zeroth, and thus scanning is irrelevant. When 2 > F ≥ 1, the first order superposes on the zeroth order and scanning is necessary, requiring at least 3 scanning positions [3]. Intensity frames can be captured for each scanning position with all possible polarization combinations. Only when there is overlap, CFS is shown to have greater sensitivity than FS. This restricts CFS to be useful only for Λ that are larger than λ 2. To tackle this problem, the scattered wave can be made to interfere with a reference wave. This is done in the branch left of main CFS branch, creating a phaseshifting-interferometry extension. With this modification, CFS should be able to take advantage of the phase information in each order together with the information about the phase difference between them. Again, the number of phase steps and scanning positions depends on F1. Sufficient scanning and/or phase steps to obtain maximum sensitivity is implicitly assumed for all results in this article ([3,5]). One may note another important feature of CFS, which was first mentioned in [2], is the fact that in CFS essentially a number of plane waves of different angles of incidence is simultaneously incident on the sample. But it is always possible to section a specific fan of rays from the pupil, which, depending on the specific situation, may be more useful than analyzing of the entire pupil. In any case, sectioning will always increase the speed of optimization by a large margin. In the present context, we will restrict ourselves to ‘planar’ sectioning, i.e., the plane waves with planar incidences are considered only. This reduces the number of computational points by 2 π p , where p is the number of samples along the radius of the pupil. Moreover, planar sectioning also allows us to avoid computationally expensive conical incidence solutions. We assume an one dimensional grating with periodicity along x, which implies the ‘planar’ sectioning is equivalent to considering the pixels only along ξ axis (figure 2 inset). In the remaining part of the article we consider the effect on cross-correlation of the grating parameters for scanning, interfering and sectioning in CFS. We show numerical simulations for an illumination of λ = 633 nm and NA = 0.9. We vary the parameter F by varying the grating pitch to cover a wide range of F = 1 to 2.9. This range includes propagation and evanescence of first order, and is of most interest. The grating parameters are shown in figure 2. From the concept of scanning, it is clear that for 2 > F ≥ 1, CFS is sensitive to the actual position of the center of the pitch of the grating (as modeled numerically, see Fig. 2) and the optical axis. This lateral displacement is defined as bias [3], and acts as one additional parameter for the given range of F. However, this parameter is not of main interest as compared to shape parameters and has a smaller cross-correlation with other parameters. Nonetheless, we consider two cases separately, where F = [1.01,1.9] and F = [2.01,2.9], with and without considering bias, respectively. For each intervals, we consider 20 values of F. 1
Note that number of scanning positions is different from conventional CFS and interferometric CFS for same F [5].
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Results and Discussion
Our numerical simulations use Rigorous Couple Wave Equations RCWA method. The covariance matrix C of the parameters is first computed [3,5] from which the normalized cross-correlation Ccor follows as C cor = ( I C ) −1 C( I C ) −1
(2)
where, I is the identity matrix and implies Hadamard product. As an example, we consider a binary symmetric grating (height = 150 nm, swa1=swa2=90o, midcd = Λ/2 and bias=0) as nominal parameters.
Fig. 2 In left it is shown how the shape parameters are defined in numerical model. In the right side the normalized cross-correlation for SWA-HEIGHT per pixel in the pupil is plotted against overlap parameter F for four cases, CFS, CFS with scan, CFS with scan and interference and CFS with scan-interference-sectioning. The 2-norm of normalized crosscorrelations for these four cases are respectively 2.6879, 2.4262, 1.9954 and 1.7871, indicating steady decrease. The ‘zero-cross-correlation’ is available for F~1.6 and F~1.69, only for CFS with scan in this case.
The plot for normalized cross-correlation of swa-height is shown in figure 2. All the methods indicated in figure 2 (right) lowers the cross-correlation for this case as the 2-norm of the cross-correlation for the whole range steadily decrease as we move from CFS to CFS-scanning-interfering-sectioning. One more interesting point to note is when the cross-correlations change sign, or, the ‘zero-crosscorrelation’ points. These points are specific for a given situation and a general prediction of their appearance, if possible, is still not understood. However, if a specific problem is given, it can be analyzed numerically and the nominal parameters can be set near these ‘zero-cross-correlation’ points (if any) to obtain faster optimization. To obtain a general idea, Table 1 summarizes the results obtained from our analysis for a resist - silicon grating. They show the 2-norm for the four cases discussed before, for 20 samples in the specified range of F.
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Table 1 2-norm of cross-correlations for Resist-Si grating For both tables, CFS+SC = CFSScanning, CFS+SC+I = CFS- Scanning-Interfering, CFS+SC+I+SE = CFS- ScanningInterfering-Sectioning Binary Symmetric Grating : Resist on Silicon
3.3910
2.5733
2.7139
2.4922
F = [2.01,2.9] CFS+I CFS+I+S E 4.2461 3.6449 4.0370
2.2513
0.9016
3.0311
2.8708
1.7179
2.8377
3.6466
2.0741
1.2352
0.7257
1.3403
1.7962
0.6146
2.3715
CFS swaheight heightmidcd swamidcd
F = [1.01,1.9] CFS+SC CFS+SC+I
CFS+SC+I+SE
CFS
The general conclusions can be summarized as 1.
2.
3.
For swa-height scanning, interfering and sectioning lowers the crosscorrelation for F = [1.01,1.9], whereas, for F = [2.01,2.9], it has different effects for different processes. For height-midcd any further processing than only CFS mostly increases the cross correlation. This may indicate that these parameters are physically quite correlated and any processing leads to larger cross-correlation values. For swa-midcd, further processing is helpful in most cases.
We should keep in mind that firstly, cross-correlations alone do not bring out the complete situation of an optimization as it also depends on uncertainty values of each parameter. Processes such as scanning and interfering, in general, always leads to decrease in uncertainty [5] and this may be taken into account first. Secondly, sectioning will lead to considerably faster optimization which will outweigh a small increase in cross-correlation in most practical cases. In figure 4 we show the computation times (64 bit intel i5 @ 3.1 GHz, 4 GB RAM) with p for CGS and CGS with sectioning. Table 2 gives the average uncertainty values of the parameters for the whole range of F. It shows considerable decrease in computation time is possible by sectioning without losing significant accuracies. However, less data points is expected to hinder noise reduction post-processing. Finally, to author’s experience, height and midcd have generally significantly smaller uncertainty than swa, making any parameter’s cross-correlation with swa more important than others. Keeping all these in mind, it not uncommon that one allows a small penalty in cross-correlation values to obtain an overall better and/or faster convergence, depending on the specific situation at hand.
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Table 2 Uncertainties CFS+ SC+I swa
2.6462
×10 midcd
2.4068
×10 height
3
-6
8.6477
×10
-7
CFS+SC+ I+SE 2.1807
×10
3
1.9850 -6 ×10 7.6251 -7 ×10
Table 2 and Fig 3: In table 2 the uncertainty values are shown for noise with unit standard deviation. Figure 3 shows the time taken for computation in CFS and CFS with sectioning. Acknowledgments. Authors acknowledge Mark van Kraaij for the RCWA routine.
References 1. den Boef, A.J.M., et al.: European Patent. EP1628164 (2006) 2. Gawhary, O.E., et al.: Performance Analysis of Coherent Optical Scatterometry. Appl. Phys. B. 105(4), 775–781 (2011) 3. Roy, S., et al.: Scanning Effects in Coherent Fourier Scatterometry. J. Europ. Opt. Soc. Rap. Public 7, 12031 (2012) 4. Kumar, N., et al.: Coherent Fourier Scatterometry: Tool for Improved sensitivity in Semiconductor Metrology. In: Proc. SPIE, vol. 8324, p. 83240Q (2011) 5. Roy, S., Kumar, N., Pereira, S.F., Urbach, H.P.: Interferometric Coherent Fourier Scatterometry: A Method for Obtaining High Sensitivity in Optical Inverse-grating Problem. J. Opt. 15, 075707 (2013)
