Computational Scaling Research Articles

Multiple-field-point pupil wavefront optimization in computational lithography.

Some pupil wavefront optimization (PWO) approaches were studied to compensate the thick mask effects considering only a field point, and these PWO methods neglect the inherent wave aberration in a realistic lithography system. Particularly, the wave aberration of lithography projection optics is exposure field dependent, and the wave aberrations at different fields of view (FOVs) would seriously and unevenly impact the results and effects of PWO. The current PWO method for single FOV cannot match full FOV. In this paper, we propose a multiple-field-point PWO (MPWO) method to improve lithography imaging quality for full FOV. A multiple-field-point cost function is built including the uneven impact of multiple aberrations on lithography imaging at full FOV. The comprehensive simulations demonstrate that the proposed MPWO method can effectively improve consistency of lithography imaging and enlarge the overlapped process window for full FOV. The most important point is that the optimized wavefront attained by MPWO can be realized via pupil wavefront manipulatorFlexWave in lithography equipment, which is significant in holistic lithography for the next technology node.

Apodization of Silicon Integrated Bragg Gratings Through Periodic Phase Modulation

The sinusoidal phase modulation apodization technique, owing to its high precision and resolution and very low phase noise, shows great promise for spectral tailoring of silicon integrated Bragg grating (IBG) devices for optical telecommunications and signal processing. Here, we extend this promising sinusoidal phase modulation technique by showing that phase modulation apodization of a silicon IBG can actually be accomplished based on any periodic function. This paper also shows a dependence of the apodization characteristic, physical grating structure, and actual apodized grating performance on the periodic function used. Then, we propose a general implementation process of the periodic phase modulation apodization to achieve a desired response on a silicon IBG, and study the limiting factors of the apodization performance, design tradeoffs and optimization, and grating robustness against fabrication constraints for different periodic phase functions, using a computational lithography model together with a structure-aware grating emulator. Finally, the extended periodic phase modulation apodization technique is validated by demonstrating a series of differently designed phase-modulated silicon IBGs, including Gaussian-apodized gratings, single- and multi-channel flat-top filters, and flat-top dispersion-compensating filters, using different periodic phase functions. The work offers an additional degree of freedom for the design and optimization of phase-modulated gratings, and has significant implications for practical implementation of the phase modulation apodization for spectral engineering of silicon Bragg grating devices.

Built-in Lens Mask Technology for Generating Three Dimensional Image based on Computational Lithography

Built-in Lens Mask Technology for Generating Three Dimensional Image based on Computational Lithography

The design of thermal management system for hypersonic launch vehicles based on active cooling networks

In this paper, a design method of thermal management system (TMS) for hypersonic vehicles is developed. The system consists of a passive thermal protection system (TPS) and an active cooling network (ACN) with coolant of kerosene. In most previous studies, the passive TPS and ACN are always designed separately and thus leads to over-conservative results that deviating from real engineering conditions. A coupled design method is developed in present work, and the process includes calculation of aerodynamic heat, determination of passive TPS concept distribution, computation of TPS and ACN scales, and iterative design. The coupled design is realized based on two keys, the active cooling is coupled in aerodynamic heating and heat transfer in TPS by an equivalent thermal equilibrium model, and the overall capacity of active cooling is indicated by an equivalent heat transfer coefficient. The model and the coefficient act as the rationale and the equivalent parameter of the whole process, respectively. The TMS of a reusable launch vehicle is established under a typical trajectory. The influences of equivalent heat transfer coefficient on aerodynamic heating, passive TPS and ACN are studied. The results show that the weight of passive TPS decreases, while the coolant mass flow rate increases with the enhancement of active cooling.

Mitigating the Impact of Mask Absorber Error on Lithographic Performance by Lithography System Holistic Optimization

The non-ideal mask absorber can cause an increase in critical dimension error (CDE) and decrease in process window (PW). However, the random mask absorber errors induced during mask fabricating and measuring are not considered in computational lithography. The problem cannot be neglected as the continuous scaling of lithography technology node. In this work, for the first time to our knowledge, a source, numerical aperture (NA), and process parameters co-optimization (SNPCO) method is developed to reduce the CDE induced by absorber errors and improve the PW. First, the source is represented by Zernike polynomials to balance computational burden and flexibility of source. Then a weighted cost function containing CDE and PW that incorporates the influences of absorber errors is created. Finally, a statistical optimization method is used to optimize the lithographic system parameters. Simulations of 1D mask pattern show that for the system with extreme absorber errors, the pattern errors of the proposed method are reduced by 62.1% and 58.9%, and the PWs are increased by 40.3% and 36.4%, respectively. The results illustrate that this method is effective in mitigating the CDE caused absorber errors and improving process robustness.

Optimization of THz diffractive optical elements thickness

Diffractive optical elements (DOEs) are strictly related to the design wavelength due to the fact that they must introduce particular phase delay of the wavefront propagating through the structure. Mostly the attenuation of the material is not taken into account. In this article we propose to optimize thickness of the DOE by reducing introduced phase retardation but also attenuation. The efficiency of DOEs is determined by the method of coding phase distribution and can be easily measured by using diffraction orders of corresponding diffraction grating. Here, we analyze binary phase diffraction gratings with assumed attenuation. Full Text: PDF ReferencesJ.-L. Coutaz, Optoélectronique térahertz (Les Ulis CEDEX A, France, EDP Sciences 2012). DirectLink D. Headland, Y. Monnai, D. Abbott, C. Fumeaux,and W. Withayachumnankul, "Tutorial: Terahertz beamforming, from concepts to realizations", APL Photonics 3, 5 (2018). CrossRef S. F. Busch, M. Weidenbach, M. Frey, F. Schäfer, T. Probst, nd M. Koch, "A 3D-Printable Polymer-Metal Soft-Magnetic Functional Composite—Development and Characterization", Journal of Infrared, Millimeter, and Terahertz Waves 35, 12 (2014) CrossRef A. Siemion, P. Kostrowiecki-Lopata, A. Pindur, P. Zagrajek, M. Sypek, "Paper on Designing Costless THz Paper Optics", Advances in Materials Science and Engineering 2016, 9615698 (2016). CrossRef A. Siemion, A. Siemion, M. Makowski, J. Suszek, J. Bomba, A. Czerwinski, F. Garet, J.-L. Coutaz, and M. Sypek, "Diffractive paper lens for terahertz optics", Opt. Lett. 37, 4320–4322 (2012). CrossRef J.-L. Coutaz, F. Garet, E. Bonnet, A. V. Tishchenko, O. Parriaux, and M. Nazarov, "Grating Diffraction Effects in the THz Domain", Acta Phys. Pol. A 107, 26-37 (2005). CrossRef M. S. Heimbeck, P. J. Reardon, J. Callahan, and H. O. Everitt, "Transmissive quasi-optical Ronchi phase grating for terahertz frequencies", Opt. Lett. 35, 21 (2010). CrossRef D. Li, S. Shu, F. Li, G. Ma, Y. Dai, and H. Ma, "Anomalous transmission of terahertz wave through one-dimensional lamellar metallic grating", Opt. Commun. 284, 10-11 (2011). CrossRef X. Li, and S. F. Yu, "Diffraction Characteristics of Concentric Circular Metal Grating Operating at Terahertz Regime", IEEE Journal of Quantum Electronics 46, 6 (2010). CrossRef B. Nöhammer, C. David, J. Gobrecht, and H. P. Herzig, "Optimized staircase profiles for diffractive optical devices made from absorbing materials", Opt. Lett. 28(13), 1087-1089 (2003). CrossRef V. Deuter, M. Grochowicz, S. Brose, J. Biller, S. Danylyuk, T. Taubner, D. Grutzmacher, and L. Juschkin, "Holographic masks for computational proximity lithography with EUV radiation", International Conference on Extreme Ultraviolet Lithography 2018 10809, 108091A (2018). CrossRef J. W. Goodman, Introduction to Fourier optics (Greenwood Village, USA, Roberts & Company Publishers 2005). DirectLink W. B. Veldkamp, "Optimized staircase profiles for diffractive optical devices made from absorbing materials", Appl. Opt. 21(17), 3209-3212W (1982). CrossRef W. B. Veldkamp, and C. J. Kastner, "Beam profile shaping for laser radars that use detector arrays", Appl. Opt. 21(2), 345-356 (1982). CrossRef https://www.mcortechnologies.com/de/3d-drucker/mcor-iris/ DirectLinkM. Sypek, M. Makowski, E. Hérault, A. Siemion, A. Siemion, J. Suszek, F. Garet, and J.-L. Coutaz, "Highly efficient broadband double-sided Fresnel lens for THz range", Opt. Lett. 37, 12 (2012). CrossRef

Lagrangian mixing simulation and quantification of scales

This paper introduces Lagrangian Mixing Simulation (LMS), which is a new method that uses particles to track the flow front between two fluids in a mixer. The line formed by an ensemble of particles enables the calculation of the instantaneous rate of interfacial area generation between the fluids and the computation of segregation scales. LMS uses the Eulerian velocity field from CFD simulation, which has a spatial resolution determined from the flow dynamics, for the simulation of mixing at scales that are several orders of magnitude below the CFD gridscale. Since only the interface between the two fluids is simulated, the LMS is orders of magnitude faster than the flow field simulation. In this work, LMS is applied to 2D CFD simulations of the flow in Confined Impinging Jets (CIJ) reactors for Re = 100, 300 and 500. LMS shows that the interfacial area generation is exponential for chaotic flow regimes (Re = 300 and 500) and linear for steady flows (Re = 100). Moreover, LMS is able to simulate segregation scales smaller than 10−8 m, i.e., LMS result spans and overcomes the entire range of spatial scales that have physical meaning.

Information theoretical approaches in computational lithography.

Computational lithography is nowadays playing an indispensible role in improving the imaging performance of optical lithography systems. This paper develops a new and powerful approach to computational lithography by introducing an information theoretical channel modeling in partially coherent lithography systems. A statistical model is built up based on the lithography imaging model to characterize the information transfer between the mask and print images. Then, this paper calculates the optimal information transfer (OIT) in partially coherent lithography systems, and derives the theoretical limit of image fidelity for optical proximity correction (OPC), which is used extensively in computational lithography. Finally, the proposed information theoretical approaches are applied to improve the OPC solutions obtained by the gradient-based algorithm. A set of simulations are provided to verify the proposed information theoretical model and approaches.

Multiple stages of regression to improve accuracy in calibrated lithography process models

A two-stage approach is introduced to improve the accuracy of compact patterning models used in large-scale computational lithography. Each of the two stages uses a separate empirically calibrated regression model whose accuracy at predicting printed feature dimensions has been proven in the usual standalone (single stage) mode. For the first model stage, we choose an established regularized regression model of the kind that accounts for resist non-idealities by suitably modifying the pre-thresholded exposing dose pattern, with the model basis functions taking the form of modified convolutions of adjustable kernels with the optical image. A different class of regression model is used in the second stage, namely a model that accounts for resist non-idealities by making a pattern-dependent local adjustment in the develop threshold, with the model basis functions being characteristic traits of the image trace along feature cutlines. However, rather than applying this second model in the usual mode where it adjusts the develop threshold applied to the exposing optical image, we use it to adjust a threshold that is applied to the improved effective dose distribution provided by the first-stage model. The effectiveness of the proposed method is verified by modeling pattern transfer of critical layers in 14- and 22-nm complementary metal–oxide–semiconductor (CMOS) technology. In our experience, little accuracy improvement is gained by expanding the complexity of standard single-stage models beyond the level of empirically proven model forms. However, even in a basic implementation, inclusion of a second stage of modeling will in itself reduce RMS error by ∼45 % in our 14-nm example. Moreover, accuracy improvement is further boosted to ∼55 % by adopting a minimax strategy in which the model is conservatively regularized according to the worst-case outcome in cross-validation tests but is calibrated according to the best-case outcome.

Tip-to-tip variation mitigation in extreme ultraviolet lithography for 7 nm and beyond metallization layers and design rule analysis

The authors develop and present computational lithography solutions to mitigate the tip-to-tip variations in 7 nm and beyond metallization layers. An array of patterns that represent lithography challenges are generated from basic design rules. The lithography process is optimized by source mask co-optimization to achieve high image qualities and maximum overlapping process windows for all patterns. By analyzing the role of diffraction orders in forming images with tight tip-to-tip, the authors identify the diffraction orders that should be admitted or rejected in the projection imaging system. This leads to analytically derived source profiles that match and explain numerical results. The authors have also found optimal minimum pitches that can achieve robust lithography process as well as design flexibility without losing throughput. Our work provides design for manufacturability guidance to metallization layers in advanced technology nodes so that patterning failures can be avoided before time-consuming optical rule check and expensive wafer validation.

Effects of stochastic exposure on critical dimension in electron-beam lithography

The computational lithography is a common approach to various optimizations of the electron-beam lithographic process. An essential step in most of the optimization problems is to estimate the exposure distribution in the resist, which is typically done through the convolution of the dose distribution of a circuit pattern with a point spread function (PSF). Although most of the computational lithographic methods employ a deterministic PSF, a PSF is stochastic in reality due to the shot noise and electron scattering. The feature size estimated from the stochastic exposure can be substantially different from that of the corresponding deterministic exposure. This difference stems from the fact that the rough development-front caused by the stochastic exposure makes the developing rate effectively larger. In this study, the effects of stochastic exposure on the critical dimension (linewidth) and their dependency on the lithographic parameters are analyzed in detail. Results obtained through an extensive simulation are presented and thoroughly discussed in this paper.

Aspherical Lens Design Using Computational Lithography

Optical lithography is often applied to fabricate three-dimensional (3D) patterns because of its capability on the scale of optical wavelengths. An aligner, a laser writer, and a stepper with an optical projection system are used as the exposure tool. The focusing capability must enable exposure of step heights of ≥50μm, which is normally achieved using an optical projection system with a large focal depth. Recently, a projection lens was developed that can produce a variable step height using an adjustable numerical aperture, which has led to the fabrication of micro-lens arrays (MLAs) with step heights as high as 100μm. These MLAs are used in various fields but mostly only as condensing lenses. However, wider applications await the realization of optical aberration reduction. The introduction of aspherical lenses would be a major step in this direction; however, aspherical lenses are difficult to produce because of the limitations in processing and polishing the aspherical surfaces of optical elements made of glass. Alternatively, the fabrication of 3D patterns using optical lithography involves a photomask whose optical density distribution corresponds to the 3D profile after exposure and development. A density distribution corresponding to the desired 3D shape can be achieved using a computer to control the dot density of a microdot mask pattern. In this study, we present a design method for low-aberration aspherical MLAs that satisfies the Rayleigh standard using computational lithography.

Design technology co-optimization assessment for directed self-assembly-based lithography: design for directed self-assembly or directed self-assembly for design?

We report a systematic study of the feasibility of using directed self-assembly (DSA) in real product design for 7-nm fin field effect transistor (FinFET) technology. We illustrate a design technology co-optimization (DTCO) methodology and two test cases applying both line/space type and via/cut type DSA processes. We cover the parts of DSA process flow and critical design constructs as well as a full chip capable computational lithography framework for DSA. By co-optimizing all process flow and product design constructs in a holistic way using a computational DTCO flow, we point out the feasibility of manufacturing using DSA in an advanced FinFET technology node and highlight the issues in the whole DSA ecosystem before we insert DSA into manufacturing.

Cellular Architecture Regulates Collective Calcium Signaling and Cell Contractility.

A key feature of multicellular systems is the ability of cells to function collectively in response to external stimuli. However, the mechanisms of intercellular cell signaling and their functional implications in diverse vascular structures are poorly understood. Using a combination of computational modeling and plasma lithography micropatterning, we investigate the roles of structural arrangement of endothelial cells in collective calcium signaling and cell contractility. Under histamine stimulation, endothelial cells in self-assembled and microengineered networks, but not individual cells and monolayers, exhibit calcium oscillations. Micropatterning, pharmacological inhibition, and computational modeling reveal that the calcium oscillation depends on the number of neighboring cells coupled via gap junctional intercellular communication, providing a mechanistic basis of the architecture-dependent calcium signaling. Furthermore, the calcium oscillation attenuates the histamine-induced cytoskeletal reorganization and cell contraction, resulting in differential cell responses in an architecture-dependent manner. Taken together, our results suggest that endothelial cells can sense and respond to chemical stimuli according to the vascular architecture via collective calcium signaling.

Extreme Ultraviolet Multilayer Defect Compensation in Computational Lithography.

For the extreme ultraviolet (EUV) lithography, multilayer (ML) defects such as bump and pit defects can disrupt the phase of reflected field and degrade aerial images on wafer. In this paper, a defect printability and repair simulator (DPRS) is introduced to predict and repair the effect of ML defects in EUV aerial images. DPRS is composed of multilayer growth by using Gaussian function and Stearns's method, mask simulation by using a scattering matrix (S-matrix) analysis method, and projection simulation by using Köhler's illumination. For bump and pit ML defects, the combining the modified absorber and the layer-by-layer ML peeling is better than other methods. This study can be helpful in understanding EUV defect and also give insight into the EUV defect compensation for the device volume production.

Two-layer critical dimensions and overlay process window characterization and improvement in full-chip computational lithography

Edge placement error (EPE) was a term initially introduced to describe the difference between predicted pattern contour edge and the design target for a single design layer. Strictly speaking, this quantity is not directly measurable in the fab. What is of vital importance is the relative edge placement errors between different design layers, and in the era of multipatterning, the different constituent mask sublayers for a single design layer. The critical dimensions (CD) and overlay between two layers can be measured in the fab, and there has always been a strong emphasis on control of overlay between design layers. The progress in this realm has been remarkable, accelerated in part at least by the proliferation of multipatterning, which reduces the available overlay budget by introducing a coupling of overlay and CD errors for the target layer. Computational lithography makes possible the full-chip assessment of two-layer edge to edge distances and two-layer contact overlap area. We will investigate examples of via-metal model-based analysis of CD and overlay errors. We will investigate both single patterning and double patterning. For single patterning, we show the advantage of contour-to-contour simulation over contour to target simulation, and how the addition of aberrations in the optical models can provide a more realistic CD-overlay process window (PW) for edge placement errors. For double patterning, the interaction of 4-layer CD and overlay errors is very complex, but we illustrate that not only can full-chip verification identify potential two-layer hotspots, the optical proximity correction engine can act to mitigate such hotspots and enlarge the joint CD-overlay PW.

Three-dimensional imaging approach using built-in lens mask lithography

Three-dimensional imaging using a complex transparency mask called a built-in lens mask is proposed in this paper. The mask is designed to have multiple focal planes at arbitrary depths for the seed patterns that form partial components of the three-dimensional structure. Three-dimensional imaging is demonstrated by the superposition of these seed patterns at various depths. Using computational lithography, the optical intensity profiles in space are demonstrated for several characters, and their three-dimensional imaging is confirmed. The separation characteristics of the optical images in the depth direction and the discretization of the complex transmittance of the built-in lens mask is discussed.

Efficient source mask optimization method for reduction of computational lithography cycles and enhancement of process-window predictability

Source-mask optimization (SMO) is used in advanced computational lithography to further enlarge the process margin. SMO provides the source for subsequent optical proximity correction (OPC) to generate the mask with reasonable manufacturability and functionality. Little attention is paid to the mask optimization procedure of SMO. The procedure may potentially cause significant mismatch between a source-mask optimized mask (SMOed mask) and an optical proximity-corrected mask (OPCed mask), which affects the efficiency of the optimization. We investigate and report a specific example of an efficient method to align the SMOed mask to the OPCed mask so as to reduce the cycles of computational lithography and improve the predictability of SMO. This method incorporates techniques of retargeting and manipulating the cost function (CF) into SMO to modify the CF and eventually change the mask shapes. Various defects can also be corrected to minimize the needed number of hotspots, which also improves the effectiveness of SMO and decreases the cycles of computational lithography. Our sample simulations performed on a metal layer with both diffractive optical element (DOE) and freeform illumination demonstrate that the proposed SMO further enhances the process window (PW) by more than 30% compared with conventional SMO. The optimized mask shape is also more similar to OPCed mask. Experimental verification is also performed to validate the proposed method.

The Trapping Index: How to integrate the Eulerian and the Lagrangian approach for the computation of the transport time scales of semi-enclosed basins

In this work, we investigated if the Eulerian and the Lagrangian approaches for the computation of the Transport Time Scales (TTS) of semi-enclosed water bodies can be used univocally to define the spatial variability of basin flushing features. The Eulerian and Lagrangian TTS were computed for both simplified test cases and a realistic domain: the Venice Lagoon. The results confirmed the two approaches cannot be adopted univocally and that the spatial variability of the water renewal capacity can be investigated only through the computation of both the TTS. A specific analysis, based on the computation of a so-called Trapping Index, was then suggested to integrate the information provided by the two different approaches. The obtained results proved the Trapping Index to be useful to avoid any misleading interpretation due to the evaluation of the basin renewal features just from an Eulerian only or from a Lagrangian only perspective.

Radio Channel Allocations With Global Optimality and Bounded Computational Scale

The radio channel assignment (RCA) in wireless networks is an optimization problem that is often found NP-complete. For networks of practical sizes, various heuristic algorithms are used to solve it. However, there are two major issues: finding a globally optimized solution without relying on specific interference models and estimating the computational complexity of general heuristic algorithms. In this paper, we propose a new simulated annealing (SA)-based RCA (SRCA) algorithm to find the globally optimized channel assignment in a distributed way but with bounded computational complexity. We propose using effective channel utilization (ECU) as the evaluation vector, whereas the objective function is to maximize the total ECU in a neighborhood. The ECU can be easily calculated by an access point (AP). The impact of interference is included in the ECU. We propose a hybrid method for estimating the algorithm's computational scale (CS), i.e., the number of channel reallocations until the network reaches a convergence state, by combining analytical and experimental methods. The resulting algorithm is a dynamic and distributed algorithm. Our extensive simulation results have demonstrated that it quickly achieves 99% of the global maximum with a chance over 95%, whereas its complexity is linear with the number of routers in the network.