Selective directed self-assembly of coexisting morphologies using block copolymer blends

Directed self-assembly (DSA) of block copolymers is an emergent technique for nano-lithography, but is limited in the range of structures possible in a single fabrication step. Here we expand on traditional DSA chemical patterning. A blend of lamellar- and cylinder-forming block copolymers assembles on specially designed surface chemical line gratings, leading to the simultaneous formation of coexisting ordered morphologies in separate areas of the substrate. The competing energetics of polymer chain distortions and chemical mismatch with the substrate grating bias the system towards either line/space or dot array patterns, depending on the pitch and linewidth of the prepattern. This is in contrast to the typical DSA, wherein assembly of a single-component block copolymer on chemical templates generates patterns of either lines/spaces (lamellar) or hexagonal dot arrays (cylinders). In our approach, the chemical template encodes desired local spatial arrangements of coexisting design motifs, self-assembled from a single, sophisticated resist.

Supplementary Figure 9: Relative mismatch area (A m ) for dot and line morphologies sitting on stripe patterns. The line-patterns reach a minimum when their width exactly matches the dutycycle (w pat /L pat ). For dots, the energy penalty is generally larger, since the chemical stripes never exactly match the hexagonally-arranged dots.
Supplementary Figure 10: Energy difference between dot and line-patterns for a BCP blend on a chemical stripe pattern. The y-axis is the duty-cycle of the pattern. The x-axis is the pitch of the pattern. The color scale is as follows: purple when line-patterns are lower energy, green when dot-patterns are lower energy, and white when the two phases have equal energy. Thus, in the central region, one expects to observe line-patterns. In the white regions, mixed morphologies (lines and dots) are likely. In the outer green regions, dot-patterns should be strongly preferred.
Supplementary Figure 11: Experimentally, chemical patterns with different dose are nonlinear traces through the phase diagram (left). Three different possible traces are shown (red dashed lines), representing different doses. The corresponding prediction for the areal fraction of linepatterns is also shown (right).
Supplementary Figure 12: SEM of a non-trivial chemical pattern, used to locally direct the registry and morphology of a BCP blend. The inset (upper right) shows the logo used as the design target. The lower SEM is color-coded to emphasize the design of the guiding chemical template.
Supplementary Figure 13: SEM analysis of a BCP blend ordering on chemical line pattern, with regions of different template spacing giving rise to different BCP morphology. Image analysis is used to sum the image intensity along the columns (black curve), which is then converted into a measure of the local distance between neighboring rows of BCP morphology (blue line). As can be seen, regions of larger local pitch give rise to line patterns, whereas regions of smaller pitch form dot patterns.

Supplementary Discussion
We present a simple predictive model for the ordering of BCP Blends on chemical stripe patterns.

BCP Blends
Consider a blend of two block-copolymers: a cylinder-forming material (CYL) and a lamellarforming material (LAM). For concreteness, we consider the case where the matrix material is polystyrene (PS, denoted S), and the minority material is poly(methyl methacrylate) (PMMA, denoted M). If we blend the materials in a ratio of φ CYL = 1 − φ LAM , then we expect the final volume-fractions to be:  (1) is not identical to the unblended LAM phase, and the cylinder-like pattern (2) is not identical to the CYL phase. In particular, the blended phases must have a different volume-fraction of the matrix and minority components, to satisfy mass conservation. Thus, we instead refer to (1) as 'lines' and (2) as 'dots'. For the blending suggested above, the material can thus choose between 'thin lines' (i.e. a lamellar-pattern where the PMMA regions are thinner than the PS regions) or 'large dots' (i.e. hexagonally-arranged cylinders, where the cylinder-cores are larger than in the pure CYL phase). Lines: For a line-pattern of repeat-spacing L lines , the width of the PMMA regions must be w lines = L lines × f Mb .
Dots: For a dot-pattern where the center-to-center distance of cylinders is d dots , the distance between subsequent rows of dots is L dots = d dots × √ 3/2. The radius of each PMMA dot is related to the volume fraction:

Chemical stripe pattern
Consider a simple chemical line-pattern, with repeat-spacing (full period) given by L pat . The chemical pattern consists of alternating stripes of oxide (which attracts PMMA) of width w pat , and stripes of polystyrene brush (which of course attracts PS) of width L pat − w pat ; the duty-cycle is thus w pat /L pat .

BCP on chemical stripe pattern
When a BCP material orders on a chemical pattern, the morphology is strongly influenced by the underlying pattern. The driving force for this is the same chemical incompatibility that drives the two materials of the BCP to phase-separate and form a nanoscale morphology: the immiscibility given by χ. The BCP material will organize itself to maximize favorable interactions (PMMA on top of oxide; PS on polystyrene brush) and minimize the less-favorable cross-interactions (PMMA on polystyrene; PS on oxide). A pure BCP can attempt to accommodate an underlying pattern in a few ways: reorienting the morphology, distorting the morphology (stretching or compressing the equilibrium repeat-spacing), or converting to an entirely different morphology. This last option is generally difficult for BCPs, since the block-fraction (f ) will define a single energetically-preferred morphology (the one that minimizes interfacial area between the blocks). A blend of BCP materials is demonstrably more responsive in this regard: it is able to adopt either of the parent morphologies, since either morphology requires some energy compromise (stretching/compression of BCP chains). Experimentally, we have established that a φ ≈ 0.5 blend forms, on a neutral substrate, a mixture of dots and lines; that is, there is a very small energy difference between these two morphologies. When a given morphology (dots or lines) assembles on a chemical pattern, the intrinsic BCP ordering will necessarily distort to maximize overlap with the chemical pattern. A BCP with equilibrium repeat-spacing L 0 will distort until its repeat-spacing L matches the underlying pattern L pat . This distortion, of course, involves chain compression/stretching. We can similarly analyze the BCP blend by treating it as a system having an equilibrium L 0 defined by the blending ratio.

Chain stretching
In a BCP blend being ordered by a chemical pattern, the BCP chains will not be in their equilibrium configuration. In particular, the chains will need to compress or stretch in order to accommodate the non-native periodicity (L) imposed by the chemical pattern. Consider a Gaussian chain of N repeat units (each of length a). The radius of gyration is: The stretching of such a chain to a distance R follows: 1 Where we have defined β = 1/k B T . Adding in various realistic conditions (self-avoidance, solvent effects) will modify the pre-factor and the exponent. E.g. for a real chain in a good solvent in 3D, the prefactor is 1.084 and the exponent 2.42. 1 Nevertheless, using βF ≈ (R/R g ) 2 is a reasonable approximation in the general case. Conversely, the compression of a chain induces an energy penalty for confinement of: This form is expected for 'weak confinement' (R/R g ≈ 1); 1,2 'strong confinement' (R/R g 1) would modify the exponent to ∼ 2/3. 3-6 As expected (Supplementary Figure 5, red line), the potential exhibits roughly spring-like behavior near R g . Combining the results (c is a constant): Within a bulk morphology, block-copolymer chains are not, however, found at their unperturbed R g . This is because of the constraints of the morphology. Phase-separation includes a driving force to minimize the interfacial area (due to the chemical-mismatch, χ). Minimizing the interfacial area can be accomplished by stretching the BCP chains (thereby reducing the average area per chain). A compromise is reached, counter-balancing chain-distortion and interfacial energies, whereupon chains are stretched to R eq . Experimentally, this chain-stretching is observed to be on the order of R eq /R g ≈ 1.2 to 2.0. 7,8 As a rough guide, we thus include a F χ ∼ 1/R energetic contribution, which accounts for interfacial effects (and, implicitly, other driving forces in BCP ordering). This contribution (Supplementary Figure 5, blue line) shifts the minimum in the total energy per chain (F total , black line) to R eq . This allows us to define L 0 as the equilibrium repeat-spacing in the BCP morphology. Any distortion of the morphology involves a corresponding distortion to the polymer chains.

Distortion of BCP morphology
We define = L pat /L 0 as the distortion-factor for the BCP unit-cell. That is, when the BCP morphology assembles on a chemical stripe-pattern of spacing L pat , the unit-cell will expand/contract accordingly to match. Owing to volume conservation, any contraction of the unit-cell along one direction will result in a corresponding expansion along the orthogonal direction. For instance, compressing a line-pattern results in thinner lines (with corresponding expansion along the line longaxis), while compressing hexagonal dot-patterns results in ellipsoidal dots on a distorted hexagonal lattice. Supplementary Figure 6 emphasizes how distorting a BCP unit-cell leads to a corresponding distortion of the BCP chains. The chain distortion of course depends on the angle of the chain within the unit cell: This immediately points to a key difference between line-patterns and dot-patterns: in linepatterns (BCP lamellar phases), the polymer chains are all oriented orthogonal to the lines (i.e. along the repeat-spacing). We define this as θ = 0 • (of course, there are also chains at θ = ±180 • ). In a dot-pattern (BCP hexagonal cylinder phase), the polymer chains are again orthogonal to the domain interfaces, but the local cylindrical symmetry means that chains are oriented at all possible angles (−180 • to +180 • ). This modifies the response of the morphology to distortion. Specifically, we expect the total energy cost for the unit-cell to be: where f (θ) is the orientation-distribution for the chains in the given morphology. I.e. for lines f (θ) is a narrow peak (Gaussian or delta-function) near θ = 0 • , whereas for dots we expect a uniform distribution across all θ. (Note that we are ignoring that in an actual hexagonal morphology, chains at different angles have slightly different amounts of chain-extension, in order to space-fill the entire morphology.) The energy-penalties for chains at different orientations are shown in Supplementary Figure 7. The combined effect for the entire unit-cell (based on Equation 11) is shown in Supplementary Figure 8. As depicted in Supplementary Figure 8, the energies of the line and dot-patterns are equal for = 1, consistent with the experimental observation of coexisting morphologies on neutral substrates. This further implies that dot-patterns are lower-energy for all . However, on a chemicallypatterned substrate, there may be an energetic preference for one morphology vs. the other, depending on their relative overlap with the chemical pattern. Thus, we can imagine that the two energy curves are shifted. This is depicted in Supplementary Figure 8 (right), where different amounts of lead to different morphologies.

Lines on chemical stripe pattern
If a BCP line phase forms on top of a line-pattern, there will be a strong driving-force for the BCP repeat-spacing to be matched to the chemical pattern (L lines = L pat ). However, because the BCP line-width is dictated by the volume-fraction (f ), it is not guaranteed that this width will match the pattern (w lines = w pat ). As a result, there will be some surface area of energetic mismatch. In particular, the relative mismatched area will follow: This is depicted in Supplementary Figure 9. The free-energy cost associated with such a configuration will scale with this mismatch-area, A m (multiplied by χ).

Dots on chemical stripe pattern
Similarly, when a BCP dot phase forms on top of a line-pattern, there will be a driving force for the MMA cylinder cores to overlap (as much as possible) with the oxide stripes. This will involve a distortion of the morphology so that the rows of dots overlap with the underlying lines (L dots = L pat ). However, there will be an inevitable mismatched area between the MMA cylinder cores and the chemical pattern. When the dots are smaller than the chemical stripes (2r dots < w pat ), we expect: If the dots are larger than the chemical stripes (2r dots > w pat ), the mismatched area is more complex: The mismatch area for line and dots patterns is compared in Supplementary Figure 9.

Phase diagram
We can combine the above results to predict the expected response of a BCP blend across a range of conditions. The corresponding phase diagram is shown in Supplementary Figure 10, where the horizontal axis is the pitch of the underlying chemical pattern, the vertical axis is the duty cycle of the pattern (i.e. the ratio of the stripe width to the repeat distance), and the color-scale denotes the relative energies of the dot and line-patterns. In the central region of the diagram (purple lobe), lines are lower-energy; thus when the pitch of the chemical pattern is close to the repeatspacing of the blend, line-patterns are preferred. In the outer regions (green areas), substantial chain distortion causes dots to instead be lower-energy, and thus preferred. In the transition regions (white), one would expect to see mixtures of dots and lines. Note that this simple model is only applicable for modest distortions of the unit cell ( ∼ 1). More complex phenomena are likely to arise (new morphologies, BCP no longer tracking chemical pattern, etc.) under extreme distortions. Thus, the predicted reappearance of lines for low may not be physically-meaningful. This theoretical phase diagram is consistent with the experimental data. The data presented in the main text (at a given dose) are horizontal cuts through this diagram. At fixed electron-beam dose, the duty cycle is not strictly conserved; for simplicity, consider traces through the phase diagram that go as w pat /L pat ≈ e − /dose , and that the fraction of lines (f l ) and dots (f d ) are exponential in the energy difference (∆F = F dots − F lines ): The predicted area fraction for lines (Supplementary Figure 11) exhibits the experimentally-observed features: a plateau where lines are preferred, falling sharply into dot-patterns for smaller pitches, and decaying more slowly into dot-patterns for larger pitches. The dose dependence is also recovered, with mixed morphologies dominating for non-optimal doses. This also highlights that one can select between line and dot-patterns in two complementary ways: by changing the pitch (at fixed dose or duty cycle), or by changing the dose (at fixed pitch).