2.1.

3D Mask Effects

A characteristic of masks in EUV lithography is their thickness relative to the projection wavelength. EUV masks are composed of a mask absorber, which constitutes the mask’s features, placed on top of a multi-layer mirror of 40 bi-layers. With the standard EUV wavelength being 13.5 nm, EUV mask absorbers have a thickness in the range of 60 nm, making them more than four wavelengths thick. The material properties of the bi-layer and absorber, in addition to the 3D profile of the thick EUV mask ensue several aberration-like effects and deformations to the wavefront and phase of the light as it interacts with the edges of the mask’s features.⁶ Additionally, the multitude of reflections of the diffracted light from the multilayer stack leads to further deformations at the masks near field and consequently at the resulting image.⁶ The reflective nature of EUV masks necessitates an oblique incidence onto the mask, which causes an asymmetry across certain aspects of EUV imaging. The mask absorber being larger than the wavelength increases the severity of these effects. Another factor that aggregates the extent of these effects is the advancement of the feature sizes toward sizes approaching the wavelength or smaller than it. These effects are commonly referred to as the 3D mask effects, and they play a fundamental role in understanding and predicting the often non-linear behavior of an EUV lithography process.

One of the main 3D mask effects is an asymmetry in process windows. Process windows are indications of the operable ranges of defocus and dose (or threshold) variation that yield changes in the imaged feature sizes or CDs within a specified tolerance range. In other words, a process window is an indication of the dose and defocus latitudes of a given lithographic process. A symmetric process window for a given process means that shifts of the projector from the focal plane that are either toward the mask or away from the mask yield the same change in the printed feature size. However, certain cases in both DUV and EUV lithography lead to asymmetric process windows, namely cases with semi-dense or isolated features. The degree of the asymmetry of a process window is dependant on the feature type and pitch. A primary consequence of the process window asymmetry is a layout-dependent shift in the best focus position which yields the image with the sharpest contrast. Orientation dependency is a 3D mask effect that is a result of the oblique incidence on the EUV mask. EUV projection systems presently employ a chief ray angle of incidence (CRA) of 6 deg. The CRA is defined in the $yz$-plane, whereas the mask surface is the $xy$-plane. Consequently, features that are oriented perpendicular to the incidence plane (horizontal lines) will experience different shadowing effects from features that are parallel to the incidence plane (vertical lines). This results in a pronounced asymmetric shadowing that occurs only with imaging horizontal features and not vertical ones.^6,16

Non-telecentricity is a characteristic effect of EUV lithographic imaging that is attributed to the thick masks and the oblique incidence. Non-telecentricity is a focus-dependent shift of the feature position. The extent of the feature position variation through focus increases in magnitude as the angle of incidence (CRA) increases.⁶ This 3D mask effect is numerically described by the gradient of the feature position versus focus curve at the best focus position.

A bias in imaging for dense features versus more isolated features as a result of the discrepancy in the number of contributing diffraction orders is a standard feature of projection lithography. In other words, isolated and dense features of the same size are projected with varying fidelities due to the diffraction limitation of the system. The iso-dense feature bias is one of the optical proximity effects that can be, to a certain extent, remedied by OPC techniques. EUV masks exacerbate the extent of the iso-dense feature bias. Furthermore, isolated EUV mask features are more sensitive to focus variation, which in turn further exacerbates the iso-dense bias in EUV lithography.

Accurate representations of the aforementioned 3D mask effects necessitate a rigorous numerical solution of the EMFs as they reflect from the thick EUV mask. Although 2D representations of masks have been widely used, at smaller technology nodes they become increasingly inaccurate. Therefore, the formulation of a viable deep learning model to be utilized for EUV systems requires the consideration of such 3D mask effects. Recent deep learning approaches for EUV lithography have tackled specific 3D mask effects by involving additional modeling or learning steps, such as in LithoGAN where an additional CNN was implemented to predict the position shift of the feature that is predicted by a cGAN.¹⁵ In our approach, we aim to encompass all the mentioned 3D effects in the training of an efficient generative network.

2.2.

Generative Networks

3.1.

Training Data

The mask layout patterns used in the training data are horizontal and vertical lines, contact arrays, attacking, and butting line ends. Additionally, the pattern bar-in-U is utilized as a validation pattern to verify the model’s ability to generalize. The implemented layouts are industrially relevant ones specified by Synopsys Inc. The pattern layouts and the ranges of variations in their geometries are shown in Fig. 1. The network’s training data consist of 2D model aerial images provided as inputs, and rigorously computed 3D model aerial images provided as outputs. The 3D model aerial images are obtained by a rigorous solution of the diffracted light from the thick mask using the waveguide method, which solves the EMFs in the spatial frequency domain.¹ This method decomposes the thick mask into slices that are homogeneous in the $z$-direction. Similar to the analysis of optical waveguides and optical fibers, the waveguide method transfers Maxwell’s equations to the Helmholtz wave equation shown in Eq. (5).

Fig. 1

Mask pattern layouts utilized (top row) and samples thereof (middle row) and their ranges of geometry variation (bottom row). All units are nm on mask scale.

We perform the waveguide method’s simulations without the Hopkins approach. The Hopkins approach is an approximation that is commonly utilized for image simulations, which assumes that the diffraction orders from the mask at small angles of incidence relative to the normal are equal to the diffraction orders obtained at the normal incidence. Accurate EUV simulations require a definition of the source without the Hopkins approximation, in which the mask diffraction spectrum is computed from a number of representative source points. In our training data, we use four non-Hopkins orders (1 per pole) referring to the representative source points. The 2D model aerial images are obtained via the Kirchhoff approach, which is based on the Kirchhoff diffraction and boundary condition. This approach ignores the 3D extent of the mask in the $z$-direction and only considers the geometry of the mask’s features in the $xy$-plane. In other words, the mask is assumed to be infinitely thin and is simply defined as a 2D array of binary transmission values. The aerial image is then computed using a Fourier transformation of the mask layout.

The optical settings for the training data of both the 2D model and 3D model are shown in Table 1.

Table 1

Optical settings employed in simulations for the generation of training data.

3.2.

Network Architecture

Pix2Pix-cGAN is the variant of cGANs proposed by Isola et al.,¹³ which we implement to train a model to translate lithographic aerial image data from the 2D model to the 3D model. We propose a strategy for training this network to predict the 3D mask effects, which involves formulating sufficiently varied training data in a way that highlights the 3D mask effects, in addition to concatenating the defocus information to the inputs at the network’s bottleneck layer, similar to the concatenation of the one-hot encoding vector done in TEMPO.³ The generator and discriminator models for this cGAN architecture both involve enhancements over traditional cGAN generators and discriminators. Both the generator and discriminator employ modules of the form convolution-batch normalization-rectified linear unit (ReLU). As the name indicates, these modules are composed of a convolutional layer, followed by a batch normalization layer, followed by a ReLU activation function. This module formation has proven to be efficient for training deep neural networks.¹⁹

The generator model of the Pix2Pix-cGAN is based on a series of downsampling or decoding layers followed by an equal number of encoding or upsampling layers, where each encoding layer is connected to the reciprocal decoding layer that has the same dimensions. This encoder-decoder architecture with skip connections follows the structure of U-Nets.²⁰ Those skip connections serve the purpose of transferring low-level information that is shared between the input and output images.¹³ Considering that for our application, the structures of the input and output images are largely aligned, therefore, a significant amount of low-level information is shared between mask patterns and aerial images.

The discriminator model consists of six down-sampling blocks involving convolutional layers. The special consideration with this discriminator is the focus on local image patches. This means this discriminator works to classify if each $N\times N$ patch in the image corresponds to a real or fake mapping, and generating a decision probability thereof. This localized focus of the discriminator serves to improve the learning of high-frequency features.¹³ The corresponding patch size ($N\times N$) is determined by number of convolutional layers in the discriminator and their filter sizes, in addition to the size of the input images. In this implementation, the input to the discriminator is a concatenation of two $512\times 512$ images (2D and 3D model aerial image) and the patch size is $70\times 70\text{pixels}$.

The filter sizes and layer sequence for both the generator and discriminator models are shown in Figures 2 and 3, respectively. An encoder indicates a 2D-convolution layer followed by a batch normalization layer followed by a leaky ReLU activation. A decoder indicates a 2D-deconvolution layer followed by a batch normalization layer followed by a concatenation with the skip connection of the reciprocal encoder, followed by a ReLU activation. All convolutional and deconvolutional layers have filter sizes of $4\times 4$ and stride sizes of $2\times 2$. The last four decoders in the generator include an additional 50% dropout layer following the batch normalization layers. The inclusion of the dropouts allows the generator model to achieve such depth with minimized over-fitting. The activation function following the final layer is a sigmoid for the discriminator and a tanh for the generator, as proposed by Isola et al.¹³

Fig. 2

Generator architecture. An encoder indicates a 2D-convolution layer followed by a batch normalization layer followed by a leaky ReLU activation. A decoder indicates a 2D-deconvolution layer followed by a batch normalization layer. The number of filters for each encoder/decoder is shown on the bottom of each block. The dimensions of the layers are shown on the bottom corner of the blocks. Plotted using PlotNeuralNet framework.²¹

Fig. 3

Discriminator architecture. An encoder indicates a 2D-convolution layer followed by a batch normalization layer followed by a leaky ReLU activation. The number of filters for the convolutional layers for each encoder is shown on the bottom of each block. The dimensions of the layers are shown on the bottom corner of the blocks. Plotted using PlotNeuralNet framework.²¹

3.3.

Cost Functions and Update Scheme

The loss of the cGAN described in Eq. (4) is addressed by updates of both the discriminator and generator models. The discriminator’s updates are based only on the accuracy of its estimate of whether the image translations are real or fake, and this is achieved using a binary cross-entropy loss. The updates of the generator loss are based on minimizing an L1 loss, in addition to an adversarial loss that aims to maximize the loss of the discriminator by minimizing its ability to distinguish real or fake translations.¹⁷ The generator loss is defined as a weighted sum of the adversarial loss and the L1 loss. This weighing is determined by the parameter $\lambda$ in Eq. (4). A $\lambda$ of 1/100 means a weighing of 100:1 in favor of the L1 loss to encourage image generations that are closer to the target is recommended by the authors of the model.¹³ In the following section, we discuss the impact of the $\lambda$ on the accuracy of the generator models in predicting the 3D mask effects. The metrics we use to evaluate the network’s training are the mean absolute error (MAE) of the generated images, in addition to the errors of the CDs in comparison to those of the 3D model images. Moreover, we further evaluate the model by assessing its predictions of 3D mask model process metrics such as telecentricity, best focus shifts, and iso-dense bias.

3.4.

U-Net Model and Generator-to-Discriminator Weighing

To identify optimum ratios of generator-to-discriminator loss weighing for lithographic aerial image translations, we compare the performance of different Pix2Pix cGAN models that have varying $\lambda$ values. The $\lambda$ value is the ratio of generator loss weight divided by the discriminator loss weight, and this dictates the contribution of each of their losses to the overall loss of the composite model. We benchmark the performance of the models based on the CD-fidelity of contact holes for a test dataset of 500 pairs of 2D model and 3D model aerial images. Figure 4 shows the average relative CD errors of generated images for models with $\lambda$ values of 100, 500, 1000, and infinity. A lambda value of infinity means that only the generator’s loss is taken into account, which effectively turns the cGAN into a pure generator U-Net. Relative CD errors indicate the deviation of the measured CD in the model predicted aerial image from the CD of the simulated 3D mask model aerial image. We utilize the mean of relative CD errors for predictions of a test dataset to represent the accuracy of a given model. Errors in Fig. 4 are computed from predictions of a test dataset of 500 contact array layouts.

Fig. 4

Effect of the generator-to-discriminator loss weighing ratios on the prediction accuracy of the network. Training data are 3600 pairs of contact array 3D model and 2D model aerial images. Contact array layout variation is shown in Fig. 1. The model architecture is shown in Figs. 2 and 3, optical settings shown in Table 1.

Figure 4 shows that increasing the ratio in favor of the generator provides favorable results for this application range. Predictably, the pure generator U-Net provides a training runtime speedup of more than $2\times$ compared to the full cGAN with a discriminator. For a test dataset of 2800 2D and 3D model image pairs, the U-Net generator trained for 50 epochs in under 50 min, while the full cGAN trained for 50 epochs in over 110 min. The hardware and setup utilized for the training are discussed in Sec. 4.1. The rest of the applications and investigations in this work are implemented using the pure generator U-Net architecture that is shown in Fig. 5.

Fig. 5

Input-output framework of the generator U-Net model implemented. The value of the defocus position is adapted (expanded) to match the dimensions of the network’s bottleneck layer and then concatenated to it. The decoding and encoding portions of the model are composed of seven blocks with 2D convolutional layers and seven blocks with 2D deconvolutional layers, respectively. Details of the generator’s layers are shown in Fig. 2.

3.5.

Training Strategy

The motivation for using the 2D model images as inputs instead of the mask layouts is their computational efficiency and their high structural similarity to the 3D model aerial images, which would allow for a higher prediction accuracy with minimal computational overhead. The defocus of the images is concatenated as an additional input at the bottleneck layer of the generator or U-Net, considering that this defocus information cannot be accurately represented with the input 2D images alone. The concept behind our training strategy is to incorporate sufficiently varied imaging scenarios that highlight the different 3D mask effects in the training data. This would allow the network to learn process variation trends from these scenarios which are represented in the output 3D model aerial images. This is achieved by including a variety of mask pattern layouts that involve different feature orientations to highlight the horizontal-vertical feature bias, different pitches to highlight the isolated-dense feature bias, in addition to different defocus ranges to learn the process variations and tendencies through focus. For each mask layout pattern, we simulate 2D and 3D model images at varying defocus positions. The network then trains on the 2D model images at the input, the defocus offset value concatenated at the bottleneck layer, and the 3D model images as the targets at the output layer of the generator.

To optimize the learning of the defocus-based effects, we first train the network on images generated at five fixed defocus positions with a relatively large separation, namely $-100$, $-50$, 0, $+50$, and $+100\mathrm{nm}$. Then we retrain this network on images that are generated at randomly selected non-fixed defocus values within the same range of defocus offsets ($-100$ to $+100\mathrm{nm}$). This would allow the network to learn the more pronounced structural patterns and trends from the large defocus steps in the first training iteration. The retraining serves as a fine-tuning stage to allow the network to learn the less pronounced trends from the intermediate and smaller defocus steps.

4.1.

Training Details

The training runs and tests are performed on an Nvidia Tesla V100 GPU and an Intel Xeon Gold 6134 CPU. For the first training, the generator U-Net model is trained on a set of 7500 pairs of aerial images obtained by the 2D model and the 3D model, generated at defocus positions of $-100$, $-50$, 0, 50, and 100 nm. The model is then retrained on another set of 7500 aerial image pairs generated at randomly selected defocus positions ranging from $-100$ to 100 nm for fine-tuning. It is worth noting that relevant accuracy levels can be achieved with training data that are less than half the amount of this dataset in cases where the range of data is less varied as shown in Fig. 4. The optimizer utilized is of the type Adam, with a batch size of 4 and a step size of 1. A set of an aerial image pair for each mask layout pattern is used as a test set to evaluate the performance of the training. The MAE between the 3D model aerial images and the network-generated images are utilized as the main metric to evaluate the training. Each training spans 200 epochs. The generator model is then extracted at the epoch where it generates images with the lowest MAE values. The optimum MAE point was reached in the first training at 124 epochs and in the retraining at 180 epochs.

4.1.1.

Image predictions and image fidelities

The network is then capable of generating images that are demonstrably similar to the 3D model simulated aerial images, with median MAE values consistently below 0.005. Using an Intel Xeon CPU with $2\times 8$ cores @3.20 GHz and an Nvidia Tesla V100 GPU with 32 GB of VRAM, the average runtime for an aerial image prediction by the model is 0.017 s, compared to the 3D model EMF simulations which can last from 9 s to more than 1 min, depending on the mask layout. Therefore, the speedup achieved by the U-Net model ranges from $500\times$ to more than $3000\times$ compared to the rigorous simulations. On the same hardware, the computation time of the 2D model aerial images ranges from 0.15 to 0.65 s. Therefore, the total end-to-end speedup achieved by the model is $100\times$ on average, while it ranges from a minimum of $3\times$ up to $400\times$. The accuracy of the network’s prediction of the 3D model is verifiable via the predicted CDs and the EDEs. The EDE is a metric originally proposed to guide mask synthesis in inverse lithography applications.²² The similarity between the model-predicted images and the target 3D model images is visualized via the aerial images contors in Fig. 6. Furthermore, the model’s ability to generalize is demonstrated by its ability to predict images from a pattern that was not included in the training, as shown in the rightmost contor plot in Fig. 6. The ability to generalize indicates a minimal presence of overfitting and can be attributed to the utilization of 50% dropout layers in the last four decoders of the generator.

Fig. 6

Comparison of the 3D and 2D model and predicted aerial images using contors extracted at the same intensity threshold. The filled gray line represents the 3D model image, the hollow gray line represents the 2D model image and the hollow white line represents the predicted aerial image. The rightmost contor plot is from the validation layout pattern bar-in-U, which was not included in the model’s training data.

To extrapolate a more global representation of the image fidelity from the CDs, we define the CD error as the average relative CD error of multiple features for each image. In contact arrays, the error is an average of the $x$- and $y$-direction CD values of 5 features, the contact at the center, top, bottom, left, and right. In the attacking and butting line end patterns, the error is defined as the average of the CD error at the center gap, top, and bottom gaps. Considering that the bar-in-U pattern is not associated with a particular CD of interest, we do not include it in the CD error evaluations, rather in the global evaluations. Figure 7 demonstrates the mean values of relative CD errors obtained by the model for a test dataset of 1200 image pairs in the case of training with the proposed retraining framework.

Fig. 7

Average CD errors for the models prediction of 3D model aerial images for different mask pattern layouts using a test dataset of 1200 3D and 2D model aerial images of the patterns shown in Fig. 1. Line ends include both attacking line ends and butting line ends.

The model’s prediction accuracy is also evaluated using the EDE as a fully global metric based on the image contors, given the shortcomings of the CD when it comes to describing more complex mask layouts. The EDE is computed as per Eq. (7) and is measured by units of distance. While the EDE is not sufficient as a standalone metric since it may fail to represent feature shifts, it provides valuable information on the reciprocity of the imaged feature sizes in a global manner. Therefore, the EDE can be a viable metric to be used alongside others such as MSE, CD errors, telecentricity errors, etc. Average EDE values for a test dataset of 600 images are shown in Fig. 8.

Eq. (7)

$$\mathrm{EDE}=\frac{\text{Generated area}-\text{Target area}}{\text{Target perimeter}}.$$

Fig. 8

Average EDE values for the models prediction of 3D model aerial images for different mask pattern layouts using a test dataset of 600 3D and 2D model aerial images of the patterns shown in Fig. 1. Line ends include both attacking line ends and butting line ends.

The generator U-Net is able to predict 3D model images accurately with relative CD errors averaging consistently sub-1% across various mask layout patterns, in addition to EDE values averaging below 0.4 nm for all the mask patterns that are implemented, including the pattern bar-in-U, which was not included in the training data. The relative CD errors from Fig. 7 are shown excluding two extreme outlier values from each selection, while the EDE values from Fig. 8 involve no outlier exclusion.

4.1.2.

Prediction of 3D model process variations

The trained model is capable of accurately predicting the process variation tendencies as dictated per the 3D mask model simulations with sufficient accuracy for a variety of mask layouts and settings. As detailed in Sec. 2.1, EUV process windows of semi-dense and isolated features involve a significant degree of asymmetry and shift of the best focus position. The 2D mask model is incapable of capturing such asymmetric process windows. Figure 9 shows process windows simulated with the 2D and 3D mask model in addition to the process windows from the network’s predicted images. The process windows are computed by calculating the threshold-to-size that provides a CD value on target, $+10\%$ and $-10\%$ from the target CD for 31 aerial images simulated with 2D and 3D model and predicted by the network for defocus values from $-150$ to 150 nm.

Fig. 9

Model-predicted process windows compared to 3D and 2D model process windows for horizontal lines and spaces patterns (top row) and contact array patterns in the $y$-direction (bottom row). The feature sizes are 32 nm for lines and $32\times 32{\mathrm{nm}}^2$ for contacts. The left column shows process windows for dense feature settings (pitch = 50 nm), and the right column shows process windows for isolated features (lines pitch = 110 nm, contacts pitch = 150 nm). Other optical settings are shown in Table 1. Legend is in the bottom right plot. 2D model process windows are rescaled with factors from 55% to 70%, to take into account the transmission losses from the 3D mask.

As shown in Fig. 9, the network’s ability to more accurately predict the 3D mask model process windows is highlighted in the cases of isolated features. With a simple re-scaling, the 2D model may appear to accurately represent the 3D model process window for certain feature pitches. However, upon further investigation, the shapes of the 2D model process windows shift further away from those of the 3D model at larger pitches. The noisy behavior of the predicted curves is attributed to the fact that the curves are formulated from network predictions of numerous images across a defocus range, while the 3D model curves are smooth because they are formulated from images following the physical model. Although a complete match between the physical and predictive models across a range of focus variations is practically impossible, the network approaches the physical models’ curves with a certain level of noise. However, the noise in the curves corresponds to relative errors fall within the achievable error ranges of the network shown in Fig. 7. Moreover, the noisy curves can be efficiently smoothed using a spline or curve-fitting functionality in practical scenarios.

The optical proximity effect (OPE) curves refer to the variations of the printed feature size across different pitches. While the 2D mask model is capable of representing some trends of the optical proximity effects, the OPE curves differ significantly between the 2D and 3D mask models. Our investigations of the predicted aerial images across a range of pitches have shown that the model is capable of representing the OPE curves of the 3D mask model with demonstrable accuracy, as shown in Fig. 10. Additionally, the network-predicted OPE curves capture the discrepancy between the behavior of the horizontal and vertical features as described by the 3D model’s curves. On the other hand, OPE curves of the horizontal and vertical features as per the 2D model are identical. It can also be observed from the bottom row of Fig. 10 that the model may exhibit better accuracy predicting an orientation for certain mask patterns better than others. For example, CDs of vertical lines are predicted with smaller errors compared to vertical CDs of contact arrays. This prediction discrepancy is likely a result of the unequal representation of the different patterns in the dataset, as the layouts in the dataset were distributed in a random manner. The oscillatory behavior in the curves can be attributed to the sub-optimal sampling of the illumination source.

Fig. 10.

Model-predicted OPE curves compared to 3D and 2D model process windows for lines and spaces and contact array masks. The top row shows OPE curves of (a) 32-nm horizontal lines and (b) 32-nm vertical lines. The bottom rows show the OPE curves for $32\times 32{\mathrm{nm}}^2$ contact arrays in (c) $y$-direction and (d) $x$-direction. Optical settings are shown in Table 1. Legend is in the top-left plot. 2D model process windows are rescaled with factors from 55% to 70%, to take into account the reflectivity losses of the 3D model.

Telecentricity, or focus-dependent position shifts, is another mask topography effect that the 2D mask model fails to represent. The U-Net generator predicts the position shifts through focus accurately for mask layouts that involve a significant degree of telecentricity such as horizontal lines. Figure 11 shows that the network’s predictions of the position shifts are more accurate for the more isolated case, in which the telecentricity error is larger in magnitude. Although the position shifts are predicted slightly less accurately in the dense case, the trend through focus is adequately captured and the relative error of the position shift remains within the previously demonstrated range of relative CD errors of $\sim 1\%$.

Fig. 11

Feature position behavior through focus using the 3D mask model and the network’s predictions for 32-nm horizontal lines at (a) pitch of 70 nm and (b) pitch of 110 nm. Optical settings are shown in Table 1. Legend is on the right. 2D model curves are not demonstrated, as they are constant.