We introduce a level-set strategy to find the geometry of confinement that will guide the self-assembly of block copolymers to a given target design in the context of lithography. The methodology is based on a shape optimization algorithm, where the level-set normal velocity is defined as the pressure field computed through a self-consistent field theory simulation. We present numerical simulations that demonstrate that this methodology is capable of finding guiding templates for a variety of target arrangements of cylinders and thus is an effective approach to the inverse directed self-assembly problem.

We derive functional level-set derivatives for the Hamiltonian arising in self-consistent field theory, which are required to solve free boundary problems in the self-assembly of polymeric systems such as block copolymer melts. In particular, we consider Dirichlet, Neumann and Robin boundary conditions. We provide numerical examples that illustrate how these shape derivatives can be used to find equilibrium and metastable structures of block copolymer melts with a free surface in both two and three spatial dimensions.

We introduce a framework for simulating the mesoscale self-assembly of block copolymers in arbitrary confined geometries subject to Neumann boundary conditions. We employ a hybrid finite difference/volume approach to discretize the mean-field equations on an irregular domain represented implicitly by a level-set function. The numerical treatment of the Neumann boundary conditions is sharp, i.e. it avoids an artificial smearing in the irregular domain boundary. This strategy enables the study of self-assembly in confined domains and enables the computation of physically meaningful quantities at the domain interface. In addition, we employ adaptive grids encoded with Quad-/Oc-trees in parallel to automatically refine the grid where the statistical fields vary rapidly as well as at the boundary of the confined domain. This approach results in a significant reduction in the number of degrees of freedom and makes the simulations in arbitrary domains using effective boundary conditions computationally efficient in terms of both speed and memory requirement. Finally, in the case of regular periodic domains, where pseudo-spectral approaches are superior to finite differences in terms of CPU time and accuracy, we use the adaptive strategy to store chain propagators, reducing the memory footprint without loss of accuracy in computed physical observables.

Directed self-assembly using block copolymers for positioning vertical interconnect access in integrated circuits relies on the proper shape of a confined domain in which polymers will self-assemble into the targeted design. Finding that shape, i.e., solving the inverse problem, is currently mainly based on trial and error approaches. We introduce a level-set based algorithm that makes use of a shape optimization strategy coupled with self-consistent field theory to solve the inverse problem in an automated way. It is shown that optimal shapes are found for different targeted topologies with accurate placement and distances between the different components.

A major challenge in the application of block copolymer directed self-assembly (DSA) to advanced lithography is the exploration of large design spaces, including the selection of confinement shape and size, surface chemistry to affect wetting conditions, copolymer chain length and block fraction. To sweep such large spaces, a computational model is ideally both fast and accurate. In this study, we investigate various incarnations of the density functional theory (DFT) approach and evaluate their suitability to DSA applications. We introduce a new optimization scheme to capitalize on the speed advantages of DFT, while minimizing loss of accuracy relative to the benchmark of self-consistent field theory (SCFT). Although current DFT models afford a 100-fold reduction in computational complexity over SCFT, even the best optimized models fail to match SCFT density profiles and make extremely poor predictions of commensurability windows and defect energetics. These limitations suggest that SCFT will remain the gold standard for DSA simulations in the near future.

Computational modeling is increasingly becoming part of the directed self-assembly (DSA) development. Modeling is used to optimize block copolymer formulations, estimate process windows, predict defect density, or just visualize the polymer morphology inside a specific feature. This chapter describes the use of mesoscale field-theoretic block copolymer models, specifically self-consistent field theory (SCFT) in the context of block copolymer DSA. In recent years, SCFT has been widely used as a tool to predict equilibrium block copolymer morphology both in bulk and in confined geometries, and its predictions are shown to agree well with experiments. Examples reviewed in this chapter include chemoepitaxy (lamellar PS–PMMA on brushed surfaces) and graphoepitaxy (contact hole shrink and line-space applications).

We use three-dimensional self-consistent field theory (SCFT) to study the directed self-assembly (DSA) of cylinder-forming block copolymers in a peanut-shaped (also called egg-box) prepattern. The design of the prepattern shape will target the pitch reduction of the contact holes. The idea is that the DSA of block copolymers will not only lead to reduced critical dimensions relative to the template but will also repair defects in the guiding prepatterns and produce defect-free contact holes. We also study blends of block copolymers and homopolymers with various lengths and volume fractions. Using SCFT simulations, we establish the effects of the added homopolymer on defectivity, the process window, and the properties of the formed cylinders. In an attempt to quantify the effect of thermal fluctuations on the placement of the cylinders, we resort to complex Langevin simulations and perform a stochastic sampling of the assembled morphologies. (C) 2015 Society of PhotoOptical Instrumentation Engineers (SPIE)

ABSTRACTWe investigate the directed self‐assembly (DSA) of cylinder‐forming block copolymers inside extended guiding templates of different sizes and shapes. We focus in particular on templates that promote the formation of linear arrays of two or three cylinders for contact hole multiplication. Using self‐consistent field theory (SCFT) simulations, we uncover the existence windows for two‐ and three‐cylinder arrays and assess their stability relative to other defective morphologies present in the templates. From SCFT simulations, templates in which three DSA cylinders are the lowest free energy configuration are more prone to defective morphologies in comparison to narrower templates with a two cylinder ground state. The size, placement and pitch of the defect‐free structures are estimated for different template sizes and shapes. Beyond the thermodynamic description of the self‐assembly process and the ensuing equilibrium properties, we also consider the equally important issue of the kinetics of melting of defects into perfect structures. Using the string method, we compute the most probable kinetic pathways that anneal elementary defects into a desired perfect array of cylinders. Similar to our findings on defectivity levels, defects in three‐cylinder‐friendly templates are associated with high kinetic barriers > 8 kT while their counterparts in narrower templates more readily dissipate into two‐cylinder states after crossing barriers < 2 kT. © 2014 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys. 2015, 53, 317–326

We investigate the directed self-assembly (DSA) of cylinder-forming block copolymers inside cylindrical guiding templates. To complement and corroborate our experimental investigations, we use field-theoretic simulations to examine the fluctuation-induced variations in the size and position of the cylindrical microdomain that forms in the middle of the guiding hole. Our study goes beyond the usual mean-field approximation and self-consistent field theory simulations (SCFT) and incorporates the effects of thermal fluctuations in the description of the self-assembly process using complex Langevin (CL) dynamics. In addition to CL simulations, we present an efficient SCFT-based approach that can inform about the positional error of the formed cylinders. In this new scheme, an external chemical-potential field is applied to displace the inner cylinder away from its centered, lowest energy configuration. In both our experimental and modeling efforts, we focus on two wall-wetting conditions: (1) minor-block-attractive sidewalls and bottom substrates and neutral top surfaces and (2) neutral sidewalls, substrates, and top surfaces. For both cases, we explore the properties of the formed cylinders, including fluctuations in the center position and the size of the domain, for various prepattern conditions. Our results indicate robust critical dimensions (CDs) of the DSA cylinders relative to the prepattern CD, with a standard deviation <0.9 nm. Likewise, we find that the DSA cylinders are accurately registered in the center of the guiding hole, with deviations in the hole-in-hole distance on the order of similar to 0.7 to 1.4 nm, translating to errors in the hole-to-hole distance of similar to 1 to 2 nm. (C) 2015 Society of Photo-Optical Instrumentation Engineers (SPIE)

We investigate energy barriers and minimum energy paths (MEPs) for transitions from dislocation-pair defects to perfect lamellae in self-assembly of AB-diblock copolymer plus A-or B-homopolymer blends using self-consistent field theory (SCFT) and the numerical string method. For neutral substrates, all minimum energy paths discovered by the string method show two successive energy barriers. The two-barrier qualitative nature of the MEPs appears not to depend on the presence or absence of small amounts of homopolymer. For the first energy barrier, the barrier height shows pronounced increase with addition of A-homopolymer due to localization of A-homopolymer on the T-junction core of the dislocation. For chemo-epitaxially patterned substrates (stripes of A-attractive substrate alternating with neutral substrate), the presence of A-attractive stripes helps draw the system towards a perfect lamellar configuration, and energy barriers along the MEP are reduced, in some cases disappearing entirely. Our findings provide guidance on how the presence of homopolymer and chemo-epitaxial prepatterns affect the stability of defective morphologies.

We use three-dimensional self-consistent field theory (SCFT) to study the self-assembly of cylinder forming diblock copolymers confined in elongated templates. This situation arises in contact holes where the goal is to produce a contact hole with reduced dimensions as well as with narrowed pitch of the center-to-center distance of cylinders. In this study, we focus on systems where two minor-block cylinders form in inside of the elongated templates. A defective bridge structure is extensively studied in this work and we evaluate the defect in various wall affinities such as "all PMMA-attractive templates", "all neutral templates" and "PMMA-attractive sidewall with the neutral substrate". According to our SCFT simulations, the defect formation energy of the bridge is typically above 20 kT, or fewer than 2 defects per billion in the "all neutral" template and "PMMA-attractive sidewall with the neutral substrate", while the defect preferably forms in the "all PMMA attractive" template.

Directed self-assembly (DSA) of block copolymers has attracted attention for its use as a simple, cost- effective patterning tool for creating vertical interconnect access (VIA) channels in nanoelectronic devices.1, 2 This technique supplements existing lithographic technologies to allow for the creation of high-resolution cylindrical holes whose diameter and placement can be precisely controlled. In this study, we use self-consistent field theory (SCFT) simulations to investigate the equilibrium configurations of under-filled DSA systems with air-polymer interactions. We report on a series of SCFT simulations of our three species (PMMA-b-PS diblock and air) model in cylindrical confinement to explore the role of template diameter, under-fill fraction (i.e. volume fraction of air), air-polymer surface interaction and polymer-side wall/substrate interactions on equilibrium morphologies in an under-filled template with a free top surface. We identify parameters and system configurations where a meniscus appears and explore cases with PMMA-attractive, PS-attractive, and all-neutral walls to understand the effects of wall properties on meniscus geometry and DSA morphology. An important outcome is an understanding of the parameters that control the contact angle of the meniscus with the wall, as it is one of the simplest quantitative measures of the meniscus shape. Ultimately, we seek to identify DSA formulations, templates, and surface treatments with predictable central cylinder diameter and a shallow contact angle, as these factors would facilitate broad process windows and ease of manufacturing.

We use self-consistent field theory to investigate the directed self-assembly of cylinder-forming block copolymers for the purpose of a graphoepitaxial pitch multiplication. We focus in particular on conditions where two minor-block cylinders form inside confining rounded templates with neutral sidewalls and substrates. In addition to perfect cylinders, various defective morphologies can also self-assemble in the guiding templates. According to SCFT simulations, the formation energy of defects is typically above 20kT, or fewer than 2 defects per billion structures. We also use calculations based on the string method to explore the most probable transition pathways during the melting of defects into perfect cylinders. From our string calculations, the kinetic barrier associated with the melting process is often less 1.5kT, suggesting relatively short annealing times for the complete removal of defects.

We use self-consistent field theory (SCFT) to study shape rectification in overlapped cylindrical and non-cylindrical prepatterns. Specifically, we examine the potential of directed self-assembly (DSA) of block copolymers to not only reduce critical dimensions relative to the template, but also repair defects in the guiding prepatterns and produce defect-free contact holes. In our study over a wide range of prepattern dimensions, we found that defects in the central minor-block domain arise with decreasing center-to-center distance of the prepattern. Increasing the minor-block fraction in the block copolymer was observed to remove some of the defects. We also studied the effect of adding homopolymer to the block copolymer melt and show how blends can successfully eliminate defects and increase the range of the process window relative to the neat diblock case without influencing domain properties such as the critical dimension and the hole-to-hole distance.

We have studied the self-assembly of PS-PMMA block copolymers by means of SCFT simulations in elongated templates and established commensurability windows for the formation of single rows of two and three cylindrical VIAs. Our results indicate that VIAS with increasing CD form inside templates of increasing dimensions. The VIAs are symmetrically arranged inside the template and the resulting hole-to-hole distances range from 24 nm to 34 nm for template lengths between 80 nm and 140 nm. We emphasize that the present work assumed perfect templates; future studies will examine VIA positioning within templates with line edge roughness. While encouraging for both CD and placement aspects, our results nonetheless indicate low defect formation energies leading to defect densities well above the targets of the lithography community. The development of novel strategies, including alternative polymer architectures, to reduce defectivity is thus critical for the success of DSA in VIA lithography and contact multiplication.

To conclude, linear block copolymers are promising materials for DSA in cylindrical templates. However, for contact multiplication in extended and rounded templates, low defect formation energies and correspondingly high defect levels are anticipated for AB diblocks. In contrast, A2B-type miktoarm copolymers show potential for achieving the required defectivity threshold. Future work will include computational studies of miktoarm copolymers with n > 2 such as A3B and A4B, and confinement in rows of many cylinders beyond two.

We have investigated the directed self-assembly (DSA) of cylinder-forming block copolymers inside cylindrical guiding templates. To complement and corroborate our experimental study, we use field-theoretic simulations to examine the fluctuations-induced variations in the size and position of the cylindrical microdomain that forms in the middle of the guiding hole. Our study goes beyond the usual mean-field approximation and self-consistent field theory simulations (SCFT) and incorporates the effects of thermal fluctuations in the description of the self-assembly process using complex Langevin (CL) dynamics. In both our experimental and modeling efforts, we focus on minor-block-attractive sidewalls and bottom substrates and neutral top surfaces and explore the properties of the formed cylinders, including fluctuations in the center position and the size of the domain, for various prepattern conditions. Our results indicate robust critical dimensions (CD) of the DSA cylinders relative to the incoming CD, with a sigma CD < 0.9nm. Likewise, we find that the DSA cylinders are accurately registered in the center of the guiding hole, with deviations in the hole-in-hole distance on the order of approximate to 0.7-1nm, translating to errors in the hole-to-hole distance of approximate to 1-1.5nm.

ABSTRACTWe use self‐consistent field theory to calculate the energy of defect formation in cylinder‐forming diblock copolymers in laterally confining thin channels. We focus on two isolated defects, dislocations and disclinations and explore a wide range of polymer and channel characteristics. Our findings suggest appropriate conditions for the design of optimized graphoepitaxial processes, where defects are limited to very low concentrations. Using the string method, we also investigate the energy barriers and kinetic pathways of the transition from defective to perfect states. © 2014 Wiley Periodicals, Inc. J. Appl. Polym. Sci. 2014, 131, 40790.

The directed self-assembly (DSA) of block copolymers offers an attractive route to produce one-dimensional rows of cylindrical similar to 10 nm scale contact holes for lithography. The success of DSA as a viable lithography technique hinges however on the ability to produce cylinders at target locations, with a minimal placement error. Attempts at fundamental understanding of placement error and center-to-center distance variations in the self-assembly process must therefore go beyond a mean-field-based description and incorporate thermal fluctutations. In this study, we use a LandauPeierls-type approach to build a simple model to investigate the effects of thermal fluctuations on the registration of one-dimensional (1D) arrays of cylinders. We also conduct experimental measurements of the placement of contact holes and compare our results to the model. Similar to reported LandauPeierls instabilities in 1D crystals, collective vibrational modes, or phonons, induce large deviations of the cylinders from their equilibrium distribution and can dramatically disrupt the periodicity of the structure as longer arrays are formed.

The directed self-assembly (DSA) of diblock copolymers in laterally confining channels is a promising avenue to produce line-and-space patterns with a sub-25 nm pitch. In this study, we use self-consistent field theory (SCFT) to investigate the DSA of both cylinder-and lamella-forming diblock copolymers in narrow trenches with corrugated sidewalls. Specifically, we focus on systems that form lying-down cylinder monolayers or standing-up lamellae parallel to the sidewalls of the channel. While previous experimental and computational studies highlighted well-ordered cylinders and lamellae in smooth channels, undesirable defective structures are also observed. In the present study, the wetting sidewalls of the channels are no longer planar surfaces. Rather, we consider undulating sidewalls and investigate the effect of the rough surfaces on defectivity and line edge roughness (LER) in the self-assembled morphologies. We use SCFT to investigate the formation free energy of isolated, meta-stable defects of both cylindrical and lamellar block copolymers inside channels with sinusoidal corrugations along the sidewalls. Parametric studies include the effects of the amplitude and the frequency of the sinusoidal wall shape function, the placement of the defect core, as well as the number of cylinders and lamellae in channels of varying widths. Our simulations indicate that the relative decreases in defect formation energy in rough channels compared to smooth channels are strikingly similar in both cylinder-and lamella-forming melts. Furthermore, using a suitable order parameter and the center-to-center displacement of the self-assembled lines, our complex Langevin (CL) simulations (beyond SCFT) show that the propagation of the LER is sensitive to the amplitude and the wavelength of the sidewall shape function, with an even stronger dependence in the lamellar case compared to the cylindrical case. More broadly, our study reveals the dependence of line edge roughness propagation on a wide range of parameters that must be carefully controlled in order to successfully implement a directed self-assembly process with block copolymers.