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Limits of Directed Self-Assembly in Block Copolymers Karim Raafat Gadelrab, Yi Ding, Ricardo Pablo-Pedro, Hsieh Chen, Kevin W Gotrik, David G Tempel, Caroline A Ross, and Alfredo Alexander-Katz Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b00997 • Publication Date (Web): 18 May 2018 Downloaded from http://pubs.acs.org on May 18, 2018
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Nano Letters
Limits of Directed Self-Assembly in Block Copolymers K. R. Gadelrab1, Yi Ding1, Ricardo Pablo-Pedro2, Hsieh Chen1, Kevin W. Gotrik1, David G. Tempel1, Caroline A. Ross1, and Alfredo Alexander-Katz*1 1
2
Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA.
Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
*E-mail:
[email protected] Abstract Understanding the conditions under which defects appear in self-assembling soft-matter systems is of great importance, for example, in the development of block-copolymer (BCP) nanolithography. Here, we explore the limits of directed self-assembly of BCPs by deliberately adding random imperfections in the template. Our results show that defects emerge due to local “shear-like” distortions of the polymer-template system, a new mechanism that is fundamentally different from canonical mechanisms of 2D melting. Furthermore, our results provide a general criterion for melting, obtaining the highest tolerance to random deviations from the perfect template at about 0.1L0, where L0 is the natural BCP periodicity. These findings establish the limits of directed self-assembly of BCPs and can be extended to other classes of materials with soft interactions. Introduction Self-assembly of soft materials is ubiquitous in nature. It was this autonomous organization of soft materials that enabled the existence of life in the first place1. Self-assembly is relevant to many different research areas and has been explored as a means to construct hierarchical functional materials1, 2. The directed self-assembly (DSA) of block-copolymers (BCPs) is one 1 ACS Paragon Plus Environment
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process belonging to this category3, in which the use of templates can guide the microphase separation of a BCP film and form complex patterns on a substrate4-7. In this sense, this process is analogous to epitaxy in having a substrate that guides the crystallization of a thin film, but takes place at the mesoscale5. Two types of templates have been used, those with modulated surface chemistry4 (so-called chemoepitaxy), or those with surface topography (so-called graphoepitaxy) that includes trenches5, 8, post arrays9, and ridged surfaces10. Templating has resulted in a variety of nanostructures with an exquisite control over the order and the geometry such as line patterns, jogs, T-junctions, periodic dot patterns, etc11. Due to this versatility and controllability, the DSA of BCP has been included in the International Technology Roadmap for Semiconductors as a promising method for fabricating devices with sub-10 nm feature size11. Despite the significant progress made in the past decades, it is still not clear what the fundamental relation between template imperfections and defects in the self-assembled array is. This is a critical issue for future lithographic technologies, particularly as feature sizes decrease. For example, semiconductor device manufacturing requires extremely low defectivity (0.01 defects cm2)11. Yet due to physical limits of lithography and subsequent development procedures, a certain degree of fluctuation in the placement accuracy of template features is inevitable12, which we refer to here as jitter or noise. This noise can arise from edge roughness 13, 14
in line patterns or random displacement in post arrays. The ability of the BCP system to
generate a defect-free structure, given a certain level of noise, depends on the coarseness of the template and the degree of coupling between the template and BCP5, 15. Hence, a 2D spherical BCP assembly guided by a post template can serve as a model system to understand how noise in the template affects directed self-assembly.
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In this paper, we examine the limits imposed by noise for DSA of BCPs through both experiments and simulation. In particular, we use a sphere-forming polystyrene-blockpolydimethylsiloxane (PS-b-PDMS) thin film that naturally assembles into a close-packed array of spherical PDMS microdomains, templated by a hexagonal array of posts with period greater than that of the BCP. By adding noise to the template we find the limits on the amount of disorder that the system can tolerate before spontaneous defects start to appear in the microdomain array, which set the fundamental constraints of DSA. It is also noteworthy that besides BCPs, other systems with noise (also known as quenched disorder) exist, such as vortices in type II superconductors16, 17, Ising spin chains18, 19, charged density waves20, and colloids21, 22, and they all have very similar underlying physics. Thus, our results may serve as a general guide for establishing the limits of directed assembly of these systems as well. Main Article
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Figure 1. Self-assembled monolayer of a sphere-forming BCP: a) fabrication procedure for directed self-assembly of BCP; from step 1 to step 4 are HSQ posts fabrication, PDMS brush layer deposition, spin-coating of BCP and annealing, and plasma etching; respectively. For experimental details, please refer to Methods in the Supplementary Materials; b) large scale SEM image of BCP film without template; c) schematics depicting the ‹11› and ‹20› phases; d) SEM image of a noise free template without the BCP film; e) BCP film self-assembled on template with no deliberately introduced noise. The posts are apparent as brighter spots in the image; f) BCP film self-assembled on template at noise level ζ/L0 = 0.15; g) Enlarged image (square outline in f) of a template with noise. The absolute value of the displacement of the posts from their original positions is denoted by r. Notice the appearance of a dislocation pair highlighted in dashed lines. All scale bars correspond to 100nm.
Our experimental system consists of a sphere-forming BCP film containing a monolayer of microdomains self-assembled on top of a topographical template consisting of posts that are functionalized with the minority block. Fig. 1a illustrates the major steps of the experimental method. First the templates were fabricated through electron beam lithography (EBL) of 4 ACS Paragon Plus Environment
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hydrogen silsesquioxane (HSQ). Then a brush layer of the minority block PDMS was grafted onto the substrate. Next, a film of the BCP was applied by spin-coating and then annealed. The samples were plasma etched and imaged using SEM to show the oxidized PDMS spheres (see Methods and Supplementary Materials for detailed fabrication and analysis procedure). We chose PS-b-PDMS, (minority volume fraction fPDMS = 16.5%, molecular weight 51.5 kg/mol, PDI = 1.04) for these experiments. Generating a monolayer of PDMS spherical microdomain requires tuning the film thickness to be ~ L0 of the BCP9, 23, 24. A film thickness of 38 nm resulted in a monolayer of spheres with periodicity of L0 = 37.2 nm (i.e. center-to-center spacing) for the annealing condition employed. We maintained this film thickness throughout experiments to demonstrate the effect of the posts on the BCP. The resulting L0 was large enough that 2D sparse close-packed post array templates could be fabricated with controlled jitter at inter-post distances Lp comparable to L0. Furthermore, this material is well studied in terms of annealing and etching procedures to obtain equilibrated microdomain arrays on post templates9. Fig. 1 demonstrates typical examples of BCP self-assembly without a template (Fig. 1b) or with a template (Fig. 1e), and Fig. 1d shows a noise-free template before spin coating the BCP. Qualitatively, Figs. 1b and 1e show that the untemplated monolayer is composed of multiple grains of close-packed spheres with different orientations with a typical grain size on the order of 10L0, whereas the post array template can produce excellent long range order. The focus of this study is the degree of imperfection of the template characterized by noise level ζ, which measures the average random displacement of a post from its original location in a noise-free close-packed lattice with ζ= 0. More specifically, we have added random displacements to each post, compared to the original perfect lattice site (group p6mm). The magnitude of the random displacements r follows a Gaussian distribution (see figure S1) with 5 ACS Paragon Plus Environment
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mean zero and variance ζ2L02, while the direction is selected randomly from a uniform distribution over [0, 2π]. We have utilized different levels of noise ranging from ζ = 0.0 to ζ = 0.25. The introduction of this noise disrupts the BCP domain order and can yield defects (Fig. 1f and g). Our study is conducted at different values of commensurate inter-post distances Lp in the noise-free post lattice. Without loss of generality, we chose ‹11› and ‹20› lattice types as examples (the definition for the lattice types is the same as in9), which means that the inter-post distances Lp were varied from √3L0 (forming the ‹11› phase) to 2L0 (forming the ‹20› phase) as shown in Fig. 1c.
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Figure 2. Experimental data for BCP self-assembly in close packed templates with different inter-post distances Lp and levels of noise ζ from 0 to 0.15. The inter-post distance Lp that is commensurate with the ‹11› phase is √3L0, whereas for the ‹20› phase is 2L0. The incommensurate template shown has Lp = 1.82L0. Colored diagrams above each SEM image are obtained from a Delaunay triangulation. The blue dots represent a point with 6 neighbors, red with 5 neighbors, green with 7 neighbors and yellow with 8 neighbors. SEM images for additional noise levels between 0.0 and 0.25 and additional values of Lp are given in the Supplementary Materials.
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Fig. 2 provides a summary of our experimental results for varying inter-post distance Lp and noise ζ showing SEM images (bottom) together with their Delaunay triangulation images (top). The use of a Delaunay network helps in visually assessing the extent of order in the grid, the orientation of local domains, and BCP domain coordination among many other quantities. The BCP domains are sixfold (6-fold) coordinated in the crystalline phase. Domains with 5- and 7fold coordination are disclinations that can be paired to form an isolated 5-7 dislocation. In the noise-free commensurate cases, the system achieves a perfectly ordered dense array of BCP domains guided by the template. Straight lattice planes are seen in the Delaunay grid. There is a rotation of the unit vector of the BCP microdomain array with respect to the post array as it transitions from the ‹11› (30o) to ‹20› (0o) phase. As noise increased, the self-assembly on the commensurate post pattern was resilient against forming defects in response to jitter. Defects started to appear only above ζ ~ 0.10, and increasing ζ beyond this value caused a continuous reduction in the fraction of 6-fold coordinated domains in the systems (see Fig. 3) marking the creation of an increased number of dislocations (5-7 pairs). The ζ ~ 0.10 seems to be an emergent criterion observed in our experiments that mimics a Lindemann criterion for melting in solids25, 26. While a limiting ζ was not reported in previous work on a chemical pattern, 15 it certainly appears that the polymer film in chemoepitaxy is more resilient to jitter when observed from the top free surface. This is due to pattern rectification as polymer domains extend away from the substrate. Thus, it is film thickness dependent, and both scenarios should be compared near the substrate surface. In the case where the inter-post distance is Lp = 1.82L0, the template is incommensurate with the BCP. This post spacing is very close to the critical spacing where the ‹11› and the ‹20› phases
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have the same free energy9. In this incommensurate case, the system breaks into grains of the ‹11› and ‹20› phases with size ξG ~ 10L0. This happens even at the lowest levels of noise. The experimental results show an increase in disorder with increasing template noise and the spontaneous formation of dislocations. While isolated dislocations are sometimes observed in the commensurate templates, clusters of dislocations are more frequently formed. This is expected to reduce the energy penalty of dislocation creation as it reduces the overall strain energy of the system while maintaining the dislocation core energy8. Similar behavior has been observed for other hexagonally ordered systems such as defects in graphene27, 28. On the other hand, in the incommensurate template the dislocations are concentrated at the grain boundaries between the two phases present. Close inspection reveals 5-8-5 defect formation that corresponds to a vacancy. This type of defect group is characterized by an extended structure that appears to be a fusion of two nearby BCP microdomains. The rare incidence of such groups in experiments hints that it might be a metastable state that will split into two BCP microdomains upon further annealing. The transition of the ordered BCP array to a disordered arrangement can be considered as a melting transition. Unfortunately, the low dimensionality of the system (2D) adds significant complexity to the nature of the problem. Crystal lattices with dimension D < 3 are thermally unstable due to long-wavelength phonon modes29. Lattices in 2D will possess only quasi longrange translational order. On the other hand, local crystalline orientation is maintained and longrange orientational order can be achieved. One of the most popular theoretical frameworks that describe 2D crystal melting was given by Kosterlitz, Thouless, Halperin, Nelson, and Young30-33. The KTHNY theory predicts that melting of 2D crystal occurs in a two stage continuous transition mediated by defect generation. An intermediate phase, named hexatic, appears due to 9 ACS Paragon Plus Environment
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the fact that the system is expected to lose its translational symmetry prior to losing its 6-fold orientational symmetry during melting. In addition, the theory predicts that the melting scenario persists for systems with “quenched disorder” (i.e. immobile particles in the 2D array placed at random locations)34-36. Hence, orientational order is long-range in the 2D solid phase that becomes quasi-long range in the hexatic phase and switches to short-range in the isotropic liquid. This can be quantified using the 6-fold bond orientation order parameter
r 1 nl i 6θ kl ψ 6 (rk ) = ∑ e nl l =1 where the sum goes over all nl nearest neighbors of microdomain k, and θkl is the angle between the bond of microdomains k and l and an arbitrary fixed reference axis. This is determined in real space through the Delaunay triangulation. An orientational correlation function is calculated by
r r r r C6 (r = rk − rl ) = ψ 6* (rk )ψ 6 (rl ) An algebraic decay of C6 ~ r
-η6
characterizes the hexatic phase where η6 is a decay exponent.
However, in the liquid state, the orientational order is perturbed leading to an exponential decay in C6 ~ e-r/δ6 where δ6 is the orientational correlation length.
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Figure 3. (Left) Orientation correlation function C6 calculated from the experimental results for both the ‹11›, and ‹20› phases at different noise levels. The hexatic-liquid transition at a slope of η6 = ¼ is marked by dashed line. (right) The fraction of microdomains with 6-fold coordination as a function of noise level. A strong reduction in 6-fold coordinated domains is observed above a noise level ζ of 0.1.
The log-log plots in Fig.3 show C6 for the ‹11› and ‹20› phases at different noise levels. At low noise levels (ζ < 0.1), the orientational correlation function C6 remains flat at a value very close to 1. A sharp decay is observed at values of r > 600 nm due to edge effects. This shows that longrange order is present despite the increased level of noise. Beyond ζ = 0.1, the C6 shifts to lower magnitudes and it essentially vanishes at ζ = 0.25 indicating the loss of both short and long range order. It is noted that reduction in the magnitude of C6 is more pronounced in the ‹11› compared to the ‹20› phase. The high density of template posts in the ‹11› system strongly couples to the nearby polymer domains and makes the BCP self-assembly more susceptible to template errors37 (see Fig.S3 for the C6 of post templates without the BCP in the Supplementary Materials).
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In the analysis of C6, no clear sign of algebraic decay is observed in the long distance limit and the loss of short range order quickly progresses as the noise increases above ζ = 0.15. The KTHNY theory predicts that C6 of a hexatic phase will algebraically decay with an exponent η6
