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Resolution improvement through signal processing techniques for integrated circuit imaging is becoming more crucial as the rapid decrease in integrated circuit dimensions continues. Although there is a significant effort to push the limits of optical resolution for backside fault analysis through the use of solid immersion lenses, higher order laser beams, and beam apodization, signal processing techniques are required for additional improvement. In this work, we propose a sparse image reconstruction framework which couples overcomplete dictionary-based representation with a physics-based forward model to improve resolution and localization accuracy in high numerical aperture confocal microscopy systems for backside optical integrated circuit analysis. The effectiveness of the framework is demonstrated on experimental data.
© 2015 Optical Society of America
1. Introduction
Integrated circuit (IC) manufacturers continue to introduce new process nodes which define new design rules with smaller dimensions for IC components. This continuous reduction in size introduces the need for higher and higher resolution optical defect localization techniques such as laser voltage imaging (LVI) [1], laser voltage probing (LVP) [2] and photon emission microscopy [3]. These techniques are limited to backside analysis due to the opaque metal interconnect layers and flip-chip bonding in ICs. The state of the art technique to increase resolution in non-invasive backside analysis is to use an aplanatic solid immersion lens (aSIL) due to its high numerical aperture (NA ≈ 3.5) capability [4]. In high NA IC imaging, spatial resolution improvement in selected directions has been achieved using linearly polarized light for illumination [5]. A further improvement of spatial resolution has been shown through the use of radially polarized light for illumination [6] and the use of apodization masks. Signal processing algorithms are essential to complement the hardware improvements in optical microscopy to increase resolution in order to meet the requirements of defect detection in new IC technologies.
In high NA systems, the properties of focused polarized light and the properties of the observed images cannot be explained using scalar optics. Therefore in order to explain the properties of the images, full vectorial analysis is needed. The full vectorial analysis has three components: analysis of the focusing fields [7,8], electromagnetic analysis of the interaction between the sample of interest and the focused fields, and vectorial propagation of the field resulting from this interaction are needed [8–11]. Altering the polarization direction of linearly polar-ized illumination in high NA systems provides polarization diversity enabling collection of multiple observations with varying spatial resolution in different directions [5]. We have previously proposed a novel image fusion framework that benefits from this polarization diversity and from prior knowledge about the structures in ICs in order to achieve resolution improvement [12, 13]. In this fusion framework, the prior knowledge about the structures in ICs, such as the fact that small features in ICs are composed of lines and rectangular structures resulting in piecewise-constant images, is incorporated into the reconstruction framework by using non-quadratic regularization functionals. These non-quadratic functionals preserve the sparsity and the edges of the underlying features. In [13], we extended the optical model for high NA subsurface imaging in order to efficiently represent the system for a wider range of materials and different size objects. Although resolution improvement can be achieved by these generic sparsifying priors, additional improvement is needed. In response, we have proposed another framework which is based on overcomplete dictionaries and which further exploits the highly structured nature of IC features [14]. In [14], we introduced the idea of using overcomplete dictionaries for image reconstruction in IC imaging and we demonstrated a proof of concept on simulated data.
In this work, we propose dictionary-based sparse image reconstruction techniques to improve resolution of static IC images where objects of interest have smaller separation than the diffraction limit. Increasing resolution and localization accuracy in these static IC images is necessary to better register optical measurements of device activity to circuit layout and ultimately to better localize defects. The optical measurements of device activity such as LVI, LVP, which are used for fault localization, are co-registered to static IC images by the design of optical system. Therefore, improving the resolution of these static IC images enables better localization of fault region in circuit layout. The domain of IC imaging is particularly suitable for the application of overcomplete dictionaries in an image reconstruction framework. Predefined overcomplete dictionaries can be easily built using a limited set of building blocks derivable from computer aided design (CAD) layouts. Specifically, ICs are mostly composed of horizontal or vertical lines of varying, constrained and known widths and lengths. Therefore, the scene in the field of view can be sparsely represented using this predetermined overcomplete dictionary and this sparse representation can provide increased robustness to model mismatches, noise and resolution limits, since it imposes strong priors for the structures in ICs. We present an image reconstruction framework which couples overcomplete dictionary based representation with an extended optical forward model based on vectorial analysis and electromagnetic modeling. The efficiency of the method is demonstrated on experimental data. Besides the demonstration on experimental data, this work also extends our previous work in [14] by proposing a different sparse formulation which enforces a higher degree of sparsity and affords an increased robustness to noise. This dictionary-based sparse representation approach can also be extended to provide localization accuracy improvement for optical faults analysis data on active chips [15].
Sparse signal representation based on overcomplete dictionaries is a well studied topic in the image reconstruction literature. However, the way the overcomplete dictionaries are built varies according to the application domain. One approach is to use a set of training images in order to learn the overcomplete dictionary [16–18]. In another approach, a predetermined overcomplete dictionary can be built to sparsely represent the scene being imaged, such as a wavelet-based dictionary [19, 20], a point and region-based dictionary or a shape-based dictionary [21]. The predetermined dictionary that we are using in this work is an adaptation of the shape and region based dictionary approach.
Sparse image reconstruction as a superresolution technique for optical imaging has been proposed in [22] and been used in high NA microscopy [23–25]. However, to the best of our knowledge, it has not been used in combination with multiple polarization data in high NA microscopy to take advantage of polarization diversity for resolution improvement, an idea we developed in our earlier work [12]. Additionally, there is no prior work that incorporates sparse representations and dictionaries for resolution enhancement in IC, demonstrated here and shown as a proof of concept on simulated data in [14]. This type of representation is especially useful for this application field because of the information coming from the CAD layouts, which contains all the underlying structures and building blocks in the IC. Unlike general dictionary-based problems, this constraint serves to effectively limit the corresponding problem size and allows the use of global, rather than local, patch-based dictionaries. This work demonstrates the efficiency of the framework on experimental data where objects of interest have smaller separation than the diffraction limit.
Our paper is organized as follows. In Section 2, we provide details of the proposed framework. The construction of the overcomplete dictionaries is given in Section 2.1. The observation model is explained in Section 2.2 and the sparse representation framework is presented in Section 2.3. The details of the physics-based PSF model are given in Section 2.4. We present results on experimental data in Section 3. In Section 4, we provide summary and conclusions.
2. Dictionary-based image reconstruction framework for IC imaging
2.1. Construction of dictionaries
We can construct global IC image dictionaries by using the information in the corresponding CAD layouts. The structures in ICs are composed of horizontal and vertical lines of varying width and height as can be seen in the CAD layout in Fig. 1. Additionally, these dimensions come from a limited set defined by IC design rules, so ideally it is possible to specify the minimum and maximum width and length of the rectangles. In order to construct a dictionary suitable for a CAD layout, rectangles of different sizes and all possible locations need to be considered. The columns of the dictionary Φ would consist of vectorized versions of all these different-size and different-location rectangles. For example, in order to sparsely represent the design shown in Fig. 2, the object is first divided into 6 building blocks each consisting of rectangles with fixed width and height as shown in Fig. 3. Then, each building block is repeated for every possible spatial location. Finally, these dictionary elements composed of horizontal and vertical lines of various widths, lengths and locations are vectorized and used as the columns of dictionary Φ. In this work, we assumed that the sample is aligned with the x − y plane and dictionary elements do not need to be aligned. However, when the sample is rotated, all possible rotations of dictionary elements should also be included in the dictionary.
Fig. 3 Examples of dictionary elements for the design from Fig. 2: Horizontal and vertical lines of various widths, lengths and locations.
2.2. Observation model
In high NA optical systems, a full vectorial analysis of fields is required in order to assure an accurate observation model. In [13], we proposed using such vectorial analysis in order to model the PSF of the high NA subsurface imaging system. Given the PSF, the assumed linear convolution model is expressed as:
where j indicates the polarization direction of the linearly polarized input light source, gj is the vectorized discrete observation data, f is the discrete vectorized underlying object image, Hj is the Toeplitz matrix that implements convolution as a matrix operation based on the PSF in polarization direction j.2.3. Sparse representation framework for resolution-enhanced IC imaging
Our goal is to obtain high resolution images through sparse image reconstruction techniques. For this reason, we propose a sparse representation created by using the information in CAD layouts. Additionally, we combine the information coming from high-resolution orientation information in each observation by incorporating the multiple observations from Section 2.2 into the sparse representation framework. The unknown underlying scene f can be represented as:
where Φ is the appropriate overcomplete dictionary composed of building blocks of the structures in the IC and ω is the vector of representation coefficients. The dictionary Φ can be predetermined by using the CAD layouts. We know the dimensions of structures in the ICs under consideration a priori. These structures are composed of lines of specified width and varying length. Combining this sparse representation with the observation model in Eq. (1), the overall model can be rewritten in the presence of noise wj as:When reconstruction problems use overcomplete dictionary-based representations, the inversion problem suffers from the non-uniqueness of the solution. We have both a selection problem to choose which dictionary elements need to be used and an estimation problem to choose the weights of the dictionary elements that are used. In order to overcome the non-uniquness issue, constraints on the solution must be set. In other domains, sparsity constraints have been demonstrated to produce robust and accurate dictionary estimates [18,19,21]. There are a variety of methods that have been shown to produce sparse solutions [27–30]. One of these, which we use here, is to pose the problem as a regularized inversion based on lp regularization with p ≤ 1. Therefore, we can create an estimate of the underlying IC scene by posing this as an lp regularization problem; that is, a sparse reconstruction problem with respect to the given circuit dictionary Φ:
where p ≤ 1, n is the total number of observed images at various polarizations and λ is a regularization parameter that determines the overall level of problem sparsity. It has been shown in spectral analysis that higher resolution spectral estimates can be obtained using lp-norm, where p < 2 rather than the l2-norm [31]. Previous studies have shown that lp-regularization with p < 1 can produce sparser solutions than l1-regularization [32, 33]. However, in the case of p < 1 the objective function of the minimization problem in Eq. (4) is non-convex, whereas it is convex for the case of p ≥ 1. The minimization problem with p = 1 is solved with the l1-ls solver provided by the authors of the interior-point method for l1-regularized least squares in [34]. For the case of p < 1, the p-Shrinkage algorithm in [35] has been used.2.4. PSF model
A full vectorial analysis of fields is required in order to accurately model the PSF of a high NA imaging system [13]. The electromagnetic field at the detector plane of the high NA microscope is given by the following equation:
where , given in [11], is the Green’s function taking into account the interface effects, is the scattered field calculated using the Finite Difference Time Domain (FDTD) method [26], is the reflected field from the interface calculated using the Angular Spectrum Representation [7], r,θaSIL and ϕ are the radial coordinate, the polar angle and the azimuthal angle with respect to the aSIL coordinate center. We use this equation in order to model the effective PSF of the optical system as follows: where α is a coefficient which accounts for the increase in the scattered field as the size of the objects of interest is increased. In order to calculate the PSF of the system, a simulation is performed: a spherical scatterer with a radius of 25 nm is placed near the interface where the objects of interest are located. Then, the PSF of the system is calculated using the scattered field of this spherical object obtained with a FDTD solver and the formulation given in Eq. (6).3. Experimental results
3.1. PSF generation
In our previous work in [12], we have shown that the optical model needs to be improved in order to account for interference effects in metal structures and intensity changes for different size particles. For this reason, in this work we present a more accurate representation for the PSF of the system given in Eq. (6). The intensity of the scattered light from an object scales with the size of the object, which is modeled by parameter α in Eq. (6). We simulated different PSFs varying α and compared cross sections from these PSFs with cross sections from experimental images of lines with different sizes, thus determining the range of α values appropriate for this range of objects. Figure 4 shows the PSFs as a function of α by comparing cross sections from PSFs with different values of α for both aluminum and polysilicon structures. Figure 5 compares a cross section from the simulated PSF and a cross section from a reflection experiment of a thin structure. The structure, shaped as number 2, is made of aluminum and placed at the interface of silicon and air. The cross section is along a line which is thin compared to the PSF and can be considered as a line response. Therefore, the PSF is consistent with this experimental data cross section. Both have similar shapes and show similar dips caused by interference effects.
Fig. 4 Comparison of PSF cross section for different values of α for aluminum objects (a) horizontal cross section, (b) vertical cross section, and for polysilicon objects (c) horizontal cross section, (d) vertical cross section.
Fig. 5 Comparison of PSF (α = 2.753) cross section and data cross section: (a) experimental aSIL data, (b) data cross section (along the green line shown in panel (a)), (c) PSF cross section (blue for horizontal and red for vertical).
There are two types of experimental samples that we use. The first one has aluminum resolution structures fabricated on silicon, therefore located on a silicon-air interface. The second one has polysilicon structures located on a silicon-to-silicon-dioxide interface. Therefore, we generated two different PSFs one for each case following Eq. (6). Figure 6(a) shows the PSF for aluminum objects and Fig. 6(b) shows the PSF for polysilicon objects.
Fig. 6 Simulated PSFs with linearly polarized input light in the y direction with α = 2.53 (a) for aluminum objects (b) for polysilicon objects.
3.2. Comparison of sparse reconstruction approaches through simulated data
Our goal in this section is to compare different sparse reconstruction techniques on simulated data with different levels of noise. Although the signal-to-noise ratio (SNR) is not very low for experimental reflectivity images, there are image modalities such as LVI that have very low SNR levels. LVI [1] is a common failure analysis technique that maps the laser reflection at specific frequencies of operation to physical transistor locations. Therefore, it is useful to compare the performance of different sparse reconstruction approaches for different levels of noise. Three different image reconstruction approaches are compared: non-quadratic regularization studied in [12], dictionary-based l1-regularization, and dictionary-based lp-regularization with p = 1/2. The non-quadratic regularization can be formulated as the following optimization problem:
where n is the total number of acquired images, j indicates polarization direction, Hj is the Toeplitz matrix that implements convolution as a matrix operation based on the PSF in polarization direction j, gj is the vectorized discrete observation data, f is the discrete vectorized underlying object image, D is the discrete approximation to the gradient operator that computes first-order image differences in the horizontal and vertical directions and λ1 and λ2 are regularization parameters.The phantoms for the resolution targets used in the simulated experiments are shown in Fig. 7. These phantoms are chosen to represent the experimental samples used to evaluate the resolution of the optical system. In Fig. 8, simulated observation images of these two phantoms are shown. For each phantom 4 observations are created using linearly-polarized incident light with x– and y–polarization and 2 levels of additive Gaussian noise, 10 dB and 20 dB. The simulated observation images are created using convolution with the PSF for aluminum structures and with α = 2.53 (Fig. 6(a)). Figure 9 compares different image reconstruction approaches; non-quadratic regularization (Eq. (7)) that we studied in [12], dictionary-based l1-regularization, and dictionary-based lp-regularization with p = 1/2 all for the observations from Fig. 8. The regularization parameter is chosen empirically to minimize the MSE between the ground truth and the reconstructed object image and to yield images where structures were well localized. A structure is well localized in the reconstructed image if we can observe a dip between peaks of object lines which were blurred and merged in the observation data. For choice of p = 1, λ is ranged from 0.0045 to 0.08 for phantom 1 and from 0.02 to 0.25 for phantom 2 depending on the noise level. For choice of p = 1/2, a parameter τ replaces the parameter λ in the p-shrinkage algorithm and it is ranged from 4 · 10−4 to 4 · 10−4 for phantom 1 and from 0.003 to 0.0045 for phantom 2 depending on the noise level. According to our experience, the same regularization parameter gives the best results for a fixed noise level and for a fixed blur level. The reconstruction results show that when there are different sized structures, dictionary-based lp-regularization achieves better localization accuracy than non-quadratic regularization. Even the smallest structures are recovered in Figs. 9(b) and 9(c), whereas they cannot be lo-calized in Fig. 9(a). The plots in Fig. 10 compare the mean square error (MSE) values for different reconstruction approaches at various noise levels. The MSE values and the error bars are calculated using 10 different realizations. The MSE plots show that reconstruction performance for dictionary-based l1 regularization is higher than for non-quadratic regularization, and dictionary-based l1/2 regularization has the best reconstruction performance and robustness to noise.
Fig. 8 Simulated observation images obtained from the two phantoms from Fig. 7 with either x– or y–polarized light and two levels of noise, 10 dB and 20 dB.
Fig. 9 Sparse image reconstruction results for Phantom 1 (a) non-quadratic regularization for SNR=10dB, (b) dictionary-based l1–regularization and SNR=10dB, (c) dictionary-based l1/2–regularization and SNR=10dB, (d) non-quadratic regularization result for SNR=20dB case, (e) dictionary-based l1–regularization for SNR=20dB case, (f) dictionary-based l1/2–regularization for SNR=20dB, and for Phantom 2 (g) non-quadratic regularization result for SNR=10dB, (h) dictionary-based l1–regularization and SNR=10dB, (i) dictionary-based l1/2–regularization and SNR=10dB, (j) non-quadratic regularization result for SNR=20dB case, (k) dictionary-based l1–regularization for SNR=20dB case, (l) dictionary-based l1/2–regularization for SNR=20dB
3.3. Reconstruction results for aSIL microscopy data
We have 2 sets of experimental aSIL data. The first set contains linear-polarization observations of a resolution structure made of 0.35μm polysilicon lines fabricated on a silicon-silicon-dioxide interface. The second set contains images of horizontal and vertical resolution lines with pitches of 282nm, 252nm, and 224nm. These are typical resolution targets we use experimentally to evaluate the resolution of the optical system. They are made of metal aluminum lines deposited on a double-side polished silicon wafer, hence they lie on a silicon-air interface. The structure design for the first resolution target and the Scanning Electron Microscope (SEM) images of the aluminum lines are shown in Fig. 11. The CNN-shaped resolution target is large enough that all lines can be localized in aSIL microscopy data. However, for the second resolution target with aluminum lines, the lines are not localized, since the separation between the lines is smaller than the spot size of the laser beam. The reconstruction of the first resolution target in Fig. 12 shows an overall image enhancement and a better edge resolution than the observation images. Moreover, the reconstruction of the aluminum lines resolution data from Fig. 14 shows image enhancement, improvement of edge resolution and improvement in localization accuracy. The aSIL microscopy images with linearly x–polarized and y–polarized light are shown in Figs. 12(a) and 12(b) for CNN-shaped polysilicon resolution target. The results of various sparse reconstruction techniques for these observations are given in Figs. 12(c), 12(d), and 12(e) for non-quadratic regularization, dictionary-based l1–regularization and dictionary-based l1/2–regularization, respectively. The PSF for polysilicon structures with α = 2.53 was used (Fig. 6). The regularization parameter was chosen empirically to yield reconstructions with sufficient quality and where structures were resolvable. For choice of p = 1, λ was chosen as 65. For choice of p = 1/2, p-shrinkage algorithm replaces λ with parameter τ and it was chosen as 0.009. Horizontal and vertical cross sections from the middle section are plotted in Fig. 13 for comparison and to show resolution improvement with respect to the observation. All reconstruction techniques provided enhancement in terms of edge resolution; the cross sections show that the edges became sharper. However, there is a smoothness loss around the corners in the dictionary-based technique. The reason for this is the rounding around the corners in the fabrication process. The highest contrast is provided by the dictionary-based l1/2–regularization.
Fig. 11 (a) CNN structure design and SEM images for lines resolution target with (b) 282nm (c) 252nm (d) 224nm separation
Fig. 12 CNN-shaped polysilicon resolution target observation data with (a) x– polarized input light (b) y–polarized input light, and reconstruction results with (c) non-quadratic regularization (d) dictionary-based l1–regularization (e) dictionary-based l1/2–regularization.
Fig. 13 Cross sections from the observation and reconstructions for CNN-shaped polysilicon resolution target (a) horizontal (b) vertical
Fig. 14 Observation data of resolution target of aluminum lines with 282nm pitch (a) x–polarized input light (b) y–polarized input light, and reconstruction results with (c) non-quadratic regularization (d) dictionary-based l1–regularization (e) dictionary-based l1/2–regularization, observation data of resolution target of aluminum lines with 252nm pitch (f) x–polarized input light (g) y–polarized input light, and reconstruction results with (h) non-quadratic regularization (i) dictionary-based l1–regularization (j) dictionary-based l1/2–regularization, observation data of resolution target of aluminum lines with 224nm pitch (k) x–polarized input light (l) y–polarized input light, and reconstruction results with (m) non-quadratic regularization (n) dictionary-based l1–regularization (o) dictionary-based l1/2–regularization.
The aSIL microscopy images with x–polarized and y–polarized input light are shown in Figs. 14(a) and 14(b) for the 282nm pitch lines, in Figs. 14(f) and 14(g) for the 252nm pitch lines, and in Figs. 14(k) and 14(l) for the 224nm pitch lines. The 252nm and 224nm pitch lines are not localized in the observation data. The non-quadratic regularization results are shown in Figs. 14(c), 14(h) and 14(m). There are some oscillations in the background, but the number of lines and the thickness of the lines for 282nm and 252nm pitch lines match the SEM data. The non-quadratic regularization for the 224nm pitch vertical lines is at the limit of localization, and some of the horizontal lines cannot be localized. The reason for this is a defect in the fabrication of horizontal lines. This defect can be seen in the SEM image in Fig. 11(d). The localization accuracy and resolution are improved by the dictionary-based reconstruction techniques compared to non-quadratic regularization. The results for dictionary-based l1–regularization are shown in Figs. 14(d), 14(i) and 14(n) and the ones for dictionary-based l1/2–regularization are shown in Figs. 14(e), 14(j) and 14(o). For all reconstruction techniques the PSF with α = 2.53 was used for 282nm and 252nm pitch lines and PSF with α = 2.43 was used for 224nm pitch lines. For choice of p = 1, λ is ranged from 20 to 100 depending on blur level, in other words depending on the spacing between lines. For choice of p = 1/2, p-shrinkage algorithm replaces λ with parameter τ and it is ranged from 0.01 to 0.03 depending on the spacing between lines. The imperfections in the defect regions started to appear in the dictionary-based l1/2–regularization for horizontal lines of 224nm pitch. The comparison of horizontal and vertical cross sections from the observation data and all the reconstructions techniques are shown in plots in Fig. 15. The cross sections show that the best resolution is achieved by the dictionary-based l1/2–regularization for all the lines data. One of the resolution metrics used for IC imaging is the Sparrow resolution criterion. The Sparrow criterion defines the resolution as the distance between the peaks of the two PSFs when the midpoint just becomes visible. Similarly, we can evaluate if a resolution structure is resolved or not, in other words, if a structure is localized or not, depending on whether we observe a peak. The resolution of the optical system was 282nm according to the Sparrow criterion, since we can localize lines with 224nm separation in the reconstructed image; the equivalent resolution was improved to 224nm by the reconstruction framework.
Fig. 15 Cross sections from observation data and reconstructions for aluminum lines resolution target for 282nm pitch (a) horizontal (b) vertical, for 252nm pitch (c) horizontal (d) vertical and for for 224nm pitch (e) horizontal (f) vertical
4. Conclusions
In this work, we proposed a dictionary-based reconstruction framework for image enhancement and resolution improvement of backside aSIL imaging of integrated circuits. Dictionary-based image reconstruction techniques are particularly suitable for IC imaging because predetermined dictionaries can be built using the information that is stored in CAD layouts. Additionally, since the building blocks of structures in ICs come from a limited set, mostly line segments of varying width and length, dictionary blocks pose strong priors for the reconstructed scene. Hence, enforcing sparsity on the dictionary coefficients, the resolution can be significantly improved. The framework incorporates polarization properties of high NA optical systems using vectorial optics and electromagnetic analysis for PSF modeling and enables fusion of multiple polarization observations to benefit from improved resolution in each set of observation data. The proposed framework was studied on simulated data and different sparse reconstruction approaches have been compared for different levels of noise. This study is crucial to evaluate the robustness of the framework to noise level since modalities in IC imaging have higher levels of noise, such as LVI Imaging. We have also shown resolution improvement in experimental aSIL data.
Acknowledgments
The authors would like to thank Abdulkadir Yurt for collecting the experimental data and for useful discussions. This project was supported by the Intelligence Advanced Research Projects Activity (IARPA) via Air Force Research Laboratory (AFRL) contract numbers FA8650-11- C-7105 and FA8650-11-C-7102. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright annotation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of IARPA, AFRL, or the U.S. Government.
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