## Abstract

By introducing the heterogeneous nucleation concept to directed self-assembly of block copolymers, the ordering of dynamical process and defect pattern design in thin films of binary blend, AB diblock/C homopolymer (AB/C), are investigated by the time-dependent Ginzburg-Landau theory and simulated by the cell dynamics simulations. The detailed annealing process of a few isolated defects occurring in AB/C blend under triangular and hexagonal confinements is presented, and it indicates that angle-matched confinement of triangular and hexagonal potential well is favorable conditions for generating defect-free ordered structures. Meanwhile, we gave a model which composed of many double-spot potentials with controllable position and orientation to investigate the relationship between defect spacing and mismatched angle, and we found the relationship is similar to hard crystals. Additionally, as an example, the design of defect pattern of “NXU” for abbreviation of Ningxia University is proposed and tested. In this chapter, the feasibility of directed self-assembly of block copolymers based on the heterogeneous nucleation process is systematically confirmed.

### Keywords

- block copolymer
- directed self-assembly
- heterogeneous nucleation
- defect pattern
- ordered pattern

## 1. Introduction

Moore’s law is the observation that the number of transistors in a dense integrated circuit doubles approximately every 2 years [1, 2]. It means the semiconductor industry continuously pursues approaches to fabricating nanostructures with higher resolution and higher throughput at lower cost. As an important manufacturing process of semiconductor, conventional lithography is approaching its physical limit for fabricating large-scale defect-free geometrically simple patterns or device-orientated, irregular structures with sub-10 nm node [3, 4, 5]. New strategy of fabricating large-scale perfectly ordered patterns and device-orientated, interesting defect patterns via directed self-assembly of block copolymers becomes the most realistic and possible technology that is feasible to fabricate nanoimprint template to manufacture integrated circuits of semiconductor industry [6, 7, 8, 9, 10]. Block copolymers, a kind of macromolecules which is jointed via covalent bonds by two or more homopolymers, can self-assemble into nanoscale periodic morphologies (5–100 nm). However, the structure self-assembled by block copolymers in bulk or in thin films on a uniform substrate is difficult to form the desired structures [11, 12, 13, 14]. The reason is that block copolymers are typical system of soft matter; it is often interfered by thermal fluctuations resulting in the formation of uncontrollable defective morphologies. Theoretically, the defective morphologies are metastable states, and the defects would be removed as the annealing process that proceeds in a long time [15, 16]. The system would transform from metastable (disordered) state into a more stable state (ordered), that is, the system ends up in thermodynamic equilibrium state. But in fact, those defective morphologies are long-lived relative to experimentally accessible time, and the annealing time is so long that it is rare to obtain the target structure in the experiment [17, 18, 19, 20]. Note that the system would be readily trapped kinetically into one of the many possible metastable defective states, so this process is more studied from the perspective of dynamic evolution [15, 16, 18, 21, 22, 23, 24]. Therefore, how to overcome these shortcomings and obtain the desired target structure via self-assembly of block copolymers has become a hot topic in this field. Among the many ideas proposed by researchers, directed self-assembly (DSA) of block copolymers [6, 7, 8, 9, 11, 12, 15, 16, 23, 24] is the most attractive scheme which has been intensively studied, and it is also regarded as a promising new-generation lithography technique [3, 4, 5, 10].

DSA is one of the most effective methods to form long-range desired patterns of block copolymer domains. The basic idea of DSA is that the small-scale pattern information is encoded into the molecular structure of block copolymers. Then, the short-range guiding patterns on the substrate from the chemical (chemoepitaxy) or topographical (graphoepitaxy) field [14, 19, 20, 25, 26, 27, 28] could direct the orientation on long-range order and control the registration of block copolymer structures with external boundaries. Based on this concept, different geometrically confined systems, and various topographical or chemical guiding patterns on the substrate, have been devised to guide the self-assembly of block copolymers to form the desired nanostructures. For example, in AB diblock copolymer, as the macromolecular architecture changes from symmetric to asymmetric, the minor component can self-assemble into several ordered structures, such as lamellar, gyroid, Fddd network, and hexagonally arranged cylindrical, and body-centered cubic (bcc) spherical phases in the bulk [29, 30, 31]. Among those order structures, the hexagonally arranged cylindrical and lamellar phases are extensively used to prepare the samples of standing cylinders arranged in triangular lattice and geometrically stripe with line and space patterns, respectively [12, 19, 21, 22, 23, 24, 25, 26, 32, 33]. Clearly, the guiding patterns must be sparse and have similar symmetries and integer multiple of periods to the domain period of the bulk phase. The density multiplication (DM) is introduced to evaluate the directing efficiency, which is defined as the ratio between the number of self-assembled domains and the number of directing substrate domains in a given area or volume of the sample. Intuitively, the sparser the guiding pattern, the lower the manufacturing cost. However, it has been proved that there exists the bottleneck in directing efficiency. The upper limit of DM in experiment for hexagonally arranged cylinders/spheres is small than 25 in the literature [12, 21, 22]. This means that the patterns on substrate could not greatly increase DM. Therefore, it remains a challenge to satisfy the rigorous demands on defect concentration and precise control required by the fabrication of integrated circuits.

Theoretically, the reason for efficiency limit is that the phase separation kinetics are spinodal microphase separation or spontaneous nucleation [34, 35]. It has an extremely high nucleation rate which induces nucleated domain grains with random positions and orientations [23]. With every domain area that grows and merges each other, the system leads to a large number of clustered defects derived from the incommensurate orientations between neighboring domain areas [15, 16]. In other words, structures formed via spinodal microphase separation cannot keep long-range order because thermal fluctuations at any locations simultaneously produce multiple grains that do not have coherent locations and orientations. Once the defects are formed, the system enters into the metastable state. As we have known, the transition from the metastable state to the stable state in block copolymer system is an extremely long process. In order to eliminate this disadvantage and increase the DM, a new DSA method, the heterogeneous nucleation, was proposed to suppress the spinodal decomposition and precisely control the domain location and orientation [15, 16, 23, 24], that is, the thermodynamics of the system must be regulated, such that the phase separation kinetics is dictated by a controlled nucleation process. These characteristics are very similar as crystals [36], so the classical nucleation theory may also be valid in block copolymers.

According to classical nucleation theory, the edges and corners of a confining system and the external nuclei can serve as heterogeneous nucleation sites with different nucleation rate [23, 35, 36, 37]. Defining the nucleation rate

In this chapter, we select frequently used model which is thin films of binary blend, AB diblock copolymer and C homopolymer (AB/C) [15, 16, 23, 24], with a lateral confinement and neutral top and bottom surfaces. In this way, the hexagonally packed cylindrical A domains are aligned perpendicular to the substrate. Homopolymer C is often added into block copolymers to regulate the segregation degree and the domain spacing. In other words, homopolymer can adjust the homogeneous nucleation rate effectively. The kinetics of micro-/macrophase separation in this system can be described by the time-dependent Ginzburg-Landau (TDGL) theory. According to previous work, we adopt cell dynamics simulation (CDS) to solve the equations of TDGL model [15, 16, 23, 24, 38, 39].

## 2. Model and theory

In this system, AB/C blend, a number of parameters such as the chain lengths and interaction parameters are expressed as a series of phenomenological parameters. The volume fraction of C homopolymer is defined as

where

Here

The long-range interaction term

where

The phase separation kinetics of AB/C blend is dominated by following two conserved Cahn-Hilliard equations:

where

where “nn” denotes nearest neighbors and “nnn” next-nearest neighbors.

## 3. Results and discussion

### 3.1 Phase-ordering process

It has been proved that the large-scale perfectly ordered hexagonal patterns can be obtained by many block copolymer systems with different DM. Early experiments and theories indicated that the maximum DM would not exceed 25 [12, 21, 22], and later it was increased to 34 confirmed by self-consistent mean field theory [40]. Recently, a very innovative idea has been proposed by Xie et al. [23], which greatly improves the directing efficiency. The idea is based on the concept of heterogeneous nucleation. By careful adjustment of the sparse periodic array of several pairs of potential wells in AB/C blend, it realized perfect hexagonal pattern with DM values of at least 128. However, this scheme has serious shortcomings in manufacture process because the nucleation agent unit size is as small as the lattice constant

Most of the above work focuses on the implementation of perfectly ordered hexagonal patterns and rarely gives the ordering process after growth time

Since AB/C blend is laterally confined in triangular and hexagonal well, an external field,

where the shortest distance of the position

It is worth noting that in the process on DSA of block copolymer, the defects are inevitable. When angle-matched confinement is introduced, a number of isolated dislocations (or five-seven defect pairs) are occurred due to the thermal fluctuations. In theory and experiment, the dislocations would be annihilated in long time. In Figure 1, the defect annealing process which started as the growth time

As can be seen from Figure 1a, there are five dislocations when the phase separation just finished and the domains were filled in the sample at

Additionally, we also considered the long-time annealing behavior under the hexagonal confinement. In Figure 2a, there are four isolate five-seven dislocations and four isolate seven disclinations in sample. When the domain nucleated from each angle and the domain area expanded and intersected each other at the center of the edge, the distance between the cylinders is more than one time of

It is worth noting that the phase-ordering kinetics given in Figures 1 and 2 describes the process from a disordered metastable state to an ordered stable state. On the one hand, from the perspective of energy, the defect annealing process is also the process of releasing stress energy and reducing free energy of the system [42, 43]. On the other hand, the angle-matched confinement with triangular and hexagonal well is favorable conditions for generating defect-free ordered structures.

The other one is worth noting that the size commensurability or size tolerance of the directing effect to the confinement is very important. In general, the spacing of domains self-assembled by the block copolymer with a certain confinement is adjusted itself within a limited range to obtain ordered structure. However, when the size is incommensurate with the domain spacing, it is very difficult to form defect-free structure. The defect concentration is introduced to evaluate the direct efficiency, which is defined as

In previous work, it has been proved that there is a size tolerance window for manufacture of large-scale ordered structures via the heterogeneous nucleation process in the hexagonal and rectangle confinements. In other words, the directing effect is dependent on the geometric confinement size, even for the angle-matched hexagonal confinement. But it is amazing that there is no size dependent for regular triangular confinement systems when nucleation is corner-induced with a weak field

### 3.2 Spacing of dislocations

Based on the concept of heterogeneous nucleation [36], different boundary conditions acted as nucleating agents and can be introduced, such as the corners and sides of different geometric confinements. The defect-free hexagonal patterns can be realized. However, due to the common irregular structures in the semiconductor industry, the same attention should be paid to the defect as to ordered structures in block copolymers.

The large-scale regular/irregular soft crystal structures formed from DSA of block copolymers are fundamentally different from those conventional crystals. The defects or dislocations in this soft crystal may have distinctive features as to those in hard crystals. But we might study and understand some property of soft matter with the help of classical nucleation theory. According to classical nucleation theory, the average distance between dislocations is related to the mismatched angle between neighboring domain grains. As shown in Figure 4a, taking two hexagons, the axis of one hexagon is horizontal, and the other one is tilted

where

Figure 4b shows the relationship between the overlap area and mismatched angle. When

In order to validate this relationship, we propose a model in which two domain grains are induced by two pairs of double-spot potential. The schematic is shown in Figure 5. Using the double-spot potential as the nucleation agents, the domain grain with definite orientation can be produced. In Figure 5, each pair of the double well is tilted *θ* are presented in Figure 6.

As a nucleation agent, the double-spot potential spaced at equal distance

for

In Figure 6, we select square confinement with fixed boundary conditions, and the size of sample is

Because the above setup can be used to investigate the defect distribution under continuous change of mismatched angles, we only give eight typical morphological snapshots of five-seven dislocations at

where

Although the above discussions focus on the properties of dislocations, they also provide us with a new idea that dislocations are controllable with heterogeneous nucleation technique. One on hand, nucleation agents with desired positions and uniform orientation can make single-crystal pattern efficiently because of no defect occurring at this condition. On the other hand, by tailoring the positions and orientations of nucleation agents, we can program the defect patterns in soft crystals and obtain the devise-oriented irregular structures which might have potential applications in metamaterials, semiconductor industry, etc.

### 3.3 Defect patterns

As well as we known, a significant feature of soft matter is sensitively responsive to thermal fluctuations as well as external fields. In these systems, the contribution of entropy is not negligible and even more important than the contribution of enthalpy. Therefore, the probability of formation of defects increases dramatically because of the responsiveness to thermal fluctuations. Conversely, the opportunities to design soft crystals are offered, such as fabrication of large-scale order structures and devise-oriented irregular structures, because of their sensitive responsiveness to external fields.

In Section 3.2, we have already discussed the properties of dislocations in detail and have proposed a feasible technique to obtain the desired pattern by controlling the nucleation process. The technique is originated from the concept of heterogeneous nucleation. Using this method, we give a design scheme to implement a defect pattern of “NXU” which is abbreviation for Ningxia University.

We choose the size of the two-dimensional sample as

The final defect pattern induced by the nucleus agent distributed in Figure 7 is shown in Figure 8. As we expected, the desired pattern “NXU” is obtained. It proves the feasibility of the technique for producing desired patterns via a heterogeneous nucleation process in block copolymers. In principle, most of the irregular patterns at the nanoscale can be obtained by this method.

## 4. Conclusions

In block copolymers, as a typical system of soft matter, the kinetic and thermodynamic behaviors are sensitively responsive to thermal fluctuations and external fields. The pattern self-assembled by block copolymers is inevitable to face this problem. In order to produce desired pattern, an effective technique of DSA via heterogeneous nucleation is adopted to investigate the behaviors of self-assembly. Two potential applications are considered by the method in this chapter. One is the phase-ordering process, and the other is the defect pattern design. The corresponding process of thin films of binary blend, AB diblock copolymer, and C homopolymer is simulated using the CDS based on the TDGL, demonstrating the feasibility and practicality of heterogeneous nucleation in fabrication of desired nanoscale patterns.

Because there are many factors that affect the self-assembly process, the phase-ordering process is very important in the sample preparation of lithography techniques. By adjusting the heterogeneous nucleation process, the path of phase-ordering kinetics from a metastable disordered phase to ordered phase is easier to achieve. We choose two angle-matched confinements, lateral triangular and hexagonal confinements, whose sides and corners are acted as nucleation agents, to demonstrate the phase-ordering process. Especially, the annealing process for a small number of defects was also recorded. It may be useful for the preparation of large-scale ordered structures.

On the contrary, irregular structures with various defect patterns in the semiconductor industry have received little attention. However, the study for defect in soft matter is of great significance both theoretically and experimentally, because of so many distinctive features for soft matter as to those in hard crystals. We gave a robust model which composed of many double-spot potentials with controllable position and orientation, to investigate the relationship between defect spacing and mismatched angle. Additionally, as an example, the design of defect pattern of “NXU” for abbreviation of Ningxia University is proposed.

With the development of DSA techniques via heterogeneous nucleation process, various defect-free patterns on large scales or irregular device-oriented structures are technically easy to implement. In order to obtain more complex structures or structures with smaller characteristic dimensions, more efforts should devote to optimizing the properties and extending the choice of materials in theory and experiment.

## Acknowledgments

Rui Lu thanks the funding support by the National College Students’ innovation and entrepreneurship training program (Grant No. 201810749005). T.Y. thanks the funding support by the Natural Science Foundation of Ningxia (Grant No. NZ1640).

## Conflict of interest

The authors have no conflict of interest to declare.