Applicable Simulation Methods for Directed Self-Assembly -Advantages and Disadvantages of These Methods

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1 Review Applicable Simulation Methods for Directed Self-Assembly -Advantages and Disadvantages of These Methods Hiroshi Morita Journal of Photopolymer Science and Technology Volume 26, Number 6 (2013) SPST Nanosystem Research Inst., National Institute of Advanced Industrial Science and Technology (AIST), Central2-1, Umezono, Tsukuba, Ibaraki, , Japan h.morita@aist.go.jp In the past 40 years, many kinds of simulation methods of micro phase separation of block copolymer were proposed. Each technic has some features derived from the original theory, and has strong and weak points in each simulation. We have developed the simulation software OCTA and performed the application study of OCTA to analyze the micro phase separated structure. In this paper, these simulation technics for micro phase separation of block copolymer are explained as a showcase of those technics from the point of view of DSA analysis. The advantages and disadvantages of these methods in the application study of DSA are indicated. I hope that this paper becomes the compass to select the suitable simulation method to analyze each DSA problem. eywords: Self consistent field, Dissipative particle dynamics, OCTA 1. Introduction Block copolymers contain two or more different parts of polymer chain linked together, and are widely used as materials for plastics, rubbers, and many kinds of high functional devices. [1-3] If the components in block copolymer become less miscible, it takes micro phase separated structure in nano-scale. Several kinds of micro phase separated structures are observed, such as lamellar, cylinder, gyroid, and sphere, and these structures are ranged using three parameters, Flory-Huggins interaction parameter (χ), length of polymer chain (N), and fraction of one of the blocks (f). The phase diagram of diblock copolymer was investigated using self consistent field theory. [4, 5] Along previous studies, the phase separated structures can be controlled using the parameters of χ, N and f. The phase separated structures can also be studied using theory and/or the simulations. Random phase approximation (RPA),[6] self consistent field (SCF) method,[7-9] dissipative particle dynamics (DPD) method,[10] and so on, these methods were developed in the past 40 years. Not just only developing the simulation method, the softwares to simulate these methods are also developed. OCTA [11] system is one of the software tools to simulate the polymeric materials, and micro phase separated structures can also be simulated by several kinds of methods. Directed self-assembly (DSA) method will be one of the important technics to shrink the line width and many researchers studied DSA to introduce this method to the lithography. To design the lithography process, the simulation technics must be needed and DSA is not an exception. There are many kinds of simulation methods for micro phase separation and each method has the characteristics originated from the original theory, and is also limited its usage by the original theory. If we use one simulation method, we must understand its method which is the most suitable and best approach to solve the problem concerned with DSA under the theoretical limitation. This review indicates not only the simulation methods for the phase separated structures of diblock copolymer to analyze the DSA but also the advantage and the disadvantage points for each method to take the best approach for the analysis of DSA. 2. Coarse-grained models of polymer and OCTA system In the simulation of polymeric materials, the coarse-grained models of polymer are used frequently. In the simulation of polymeric materials, though polymer is larger molecules, a lot of number of atoms must be included in the simulation system. If the atomistic model is used, the number of particles in the system becomes larger and it is difficult to simulate this system. To reduce the system size, many kinds of coarse-grained (CG) models for polymer are proposed. Figure 1 shows the examples of CG models for polymer. Atomistic model is the model having the smallest unit among the models in this figure. Next CG model in which one particle covers several monomers Is a bead-spring model. Bead-spring (BS) model is Received November 13, 2013 Accepted December 10,

2 Fig.1 Coarse-grained models of polymers. severally used in the coarse-grained molecular dynamics (CGMD) simulations. [12] Next larger CG model is the dissipative particle dynamics (DPD) method. In this model, several BS particles were coarse-grained to one DPD particle. The coarse-graining level of both DPD and mesh models is almost the same. On the other hand, the simulation using these CG models can be performed using OCTA system which has been developed by Prof. Masao Doi and co-workers. Since 2002, OCTA has been applied to many kinds of problems concerned with the polymeric materials, such as expansion, adhesion, friction, coating and so on. Using several simulators in OCTA, phase separated structures can also be simulated. For example, macro phase separated structures of polymer blend film can be simulated using dynamic SCF method. [13] The spinodal wave from both surface and substrate can be found. Using the DPD method, the evaporation induced phase separation can be simulated. [14] As time goes on, the film thickness decreases due to the evaporation of common solvent and the phase separation proceeds. DPD methods can also be applied to the development of the lithography process. From these example studies, SCF and DPD methods are used, and these methods can also be applicable to DSA simulations. In the following section, the overview of these methods will be explained. 3. Particle-based Models CGMD method is one of the candidates of DSA simulation. Typical example of CGMD method is the method using bead-spring model. CGMD method is classified as the particle-based model, and the most famous CGMD model, remer-grest model, is referred in reference 12. Using CGMD method, mechanical properties can be simulated. If we apply CGMD method, the strength in the bended or sheared pattern can be simulated. The disadvantage of this method is that CGMD method can only be applied to the simulation which can treat the phenomena in the time range of much less than millisecond order. The phase separation of block copolymer takes much more time and CGMD is not recommended as the simulation method for DSA. On the other hand, DPD is also the important candidate for DSA simulation. DPD proposed by Groot and Warren [10] is classified to the particle-based model. In DPD method, a set of particles, each having mass m i, is used. The position r i and the velocity v i of the particle i is determined by the equation of motion, dr i = v (1), i dt d mi dt v i = f (2). i The characteristic of the DPD is the force f i. f i consists of F C, F D, and F R called as the conservative force, dissipative force, and drag force, respectively. C D R f = F + F + F. (3) i ( ) j i F C stands for the force arising from the interaction potential. There are two types of interaction potential, the bonding potential and the non-bonding potential as written below; f r F )] C bond non bond = [ f ( r ) + f ( r (4), r bond f r ) = non bond ( Cr (5), r ( ) a 1 r < rc ( r ) = r (6), c 0 ( r rc ) where r C is the cut off distance, a is a maximum repulsion between particles i and j, and r is equal to r i - r j. Noted that the a is related to χ parameter used in SCF. The dissipative force and the drag force are given by 802

3 F D = ) r r D γω ( r v (7), r r R R r = σω θ (8), r F where θ is a randomly fluctuating variable with D R Gaussian statistics, ω, ω, and σ are written by D ω r ( ) [ ( )] ( ) R 2 1 r < rc r = ω r = rc 0 ( r r ) 2 c (9), σ 2 = 2γk B T (10). In DPD method, polymer chain is modeled as the connected beads by spring, and molecular length and its distribution can be controlled. DPD simulation can cover the time range of second unit. This simulation can be accelerated by the weak interaction potential in eq. (6), and the DPD particles can permitted to overlap. The phantom polymer chain dynamics can be treated by DPD method. Therefore, mechanical strength by entanglement of polymer chains cannot be represented by DPD method. As a real lithography process, we need the smaller pattern size such as half-pitch less than 10 nm in the near future, and the same size of the periodic length of lamellar structure which can be realized by high χ block copolymer is needed. Using DPD, high χ system, which is difficult to simulate using RPA-based method, can be stably-treated. This is one of the advantages of DPD method. If we introduce the boundaries for guide made by particles, we can also introduce both chemical and physical guide. From these features, DPD is recommended as the candidate of the simulation method for DSA. Other candidate of DSA simulation is Monte-Carlo (MC) method. There are many studies of the micro phase separated structures using MC method, and we cannot cover too many studies in this review. Details can be found in reference 15. One of the advantages of MC methods is the flexibility of the model. Both field-based mesh and particle-based model can be treated and the annealing to stable structure, dynamical process, and transient structures can be simulated. Unfortunately, there is no specified simulator of MC method in OCTA, and in the near future we want to add the MC simulator in OCTA. 4. Models using field on mesh One of the candidates of DSA simulation classified to the mesh-based model is SCF method. There are several kinds of procedures to solve the SCF methods. In the solving technique, there are two kinds of solving models, which are continuum [7] and Scheutjens -Fleer models. [9] In the point of view of the expansion space of SCF equations, we can solve in the both real and Fourier spaces. In this review paper, the method using continuum model in the real space is explained. Dynamical process of phase separation can be simulated by dynamic SCF method which was proposed by Fraae, [16] and it is applied to the simulation of micro phase separated structure of thin film. [17] In this paper, dynamical SCF method is explained. In solving the SCF method, mean field approximation is applied and followings are assumed. 1) The chain conformations are represented by the Gaussian-chain statistics, and are described using the path integral formalism. 2) The short-range interactions between different types of segments are introduced and are described by the Flory-Huggins χ parameters. 3) The dynamics is introduced by assuming a simple local diffusion of the segments driven by the local gradient of the chemical potential. Here we introduce an index denoting the indices of the species type (A or B). Denoting the density of the -type segments as φ, the time evolution of the set of density fields { φ } is assumed to obey the following equation 2 φ ( r, t) = L µ ( r, t) (11) t where t is the time, µ ( r,t) and L are the chemical potential and the segment mobility of the - type segments of the chain. Here, we neglected the dependence of the segment mobility L on the segment density φ for simplicity. In order to solve the time evolution described by eq.(11), we have to evaluate the chemical potential µ ( r,t) for the given profiles of the segment density fields φ. In polymeric systems, the conformational entropy of the chains contributes to the chemical potential. In order to evaluate it, we use the mean-field approximation where we consider a Gaussian chain under the influence of external mean fields. We denote the mean fields acting on the -type segments as { V }. Then the partition function of such a Gaussian chain in the external fields are given by the path integrals that obey the following equation 2 b 2 Q (, r ; s, r ) = + V Q ( 0, r ; s, r ) (12) s 0 s 0 where Q ( i r ; j, r ) 0 6, is the statistical weight of a i j -type subchain between i-th and j-th segments, b is s 803

4 the uhn segment size that is assumed to be the same for A and B segments. The density field can be obtained by following equation, ( ) ( ) ( ) dr0 drnq 0, r0 ; i, ri Q i, ri ; N, rn φ i, r = (13) dr dr Q ( 0, r ; N, r ) 0 N and the total density summation of φ ( i,r), ( r ) = φ ( i, r) i 0 N φ is calculated by the φ (14). Self consistent field V, where denotes the type of the subchains, is used, = χ φ ' γ (15) V + where χ is the interaction parameter for a - segment pair and γ is the Lagrange multiplier for the local incompressible condition. Note that an explicit expression for the constraining potential γ cannot be obtained. It can be obtained only through an iteration calculation using eqs.(12)-(15) as is described in the following. Equations (12)-(15) form a closed set of self-consistent equations. In order to solve it, we have to determine the constraining potential γ by adjusting it iteratively so that the density profiles calculated by eq.(14) coincide with the current density profiles obtained by time integration of eq.(11). Due to this reason, V s are called self-consistent fields (SCF). Once the set of self-consistent equations (12)-(15) are solved, the chemical potential for the -type segment is given by γ µ =. (16) Substituting the chemical potential µ into eq.(11), one can integrate the equation of motion, eq.(11), to update the density profiles. Here, we note that when solving eq.(11), it should be supplemented by the total incompressible condition φ ( r, t) = constant, (17) which produces an extra current of the segments that should be added to the right hand side of eq.(11). In the SCF simulation, the process solving eq.(12) is the time-consuming step. Although it takes much more costs to solve SCF method, it is called the quantitatively exact simulation method for the micro phase separated structure in the statistical point of view under the mean field approximation. There are many studies concerned with DSA using SCF and it is easily understood that SCF is one of the important simulation methods to study DSA. For example, Takahashi et al. have studied the defect in the line patterned structure using SCF. [18] They discussed the kinetic energy barrier from the defect structure to no defect structure. Using SCF, the free energy can be calculated and the energy barriers in the dynamical processes can be analyzed. This analysis is much important to inform us the energies to clean up the defect. From this point of view, SCF method is the recommended method. Using SCF method, further analysis can be done. For example, the information concerned with the single chain can also be obtained. The bended domain, which is also discussed by Stoykovich et al. [19], is analyzed by SCF method and the single chain distribution at the bended area can be simulated. [20] In the bended lamellar, one subchain which is directed to the outer layer is shrunk to the direction normal to the lamellar face as shown in Fig.2. Another subchain directed to the inner layer becomes expanding. The conformational change of polymer chain occurs due to the stabilization of the defect position. This conformational analysis of polymer chain at the defect will give you the important information to reduce the defect. Fig.2 Single chain distribution analysis at bended lamellar On the other hand, approximated method of SCF theory is also proposed. One of the typical method is Ohta-awasaki method [21,22] which is one of the derived methods of RPA and it is applied to DSA by Yoshimoto and Taniguchi. [23] In these methods, the periodic boundary condition can only be applied, and other boundary condition cannot be applied. Boundary condition is much important and sometimes much effective to the simulation result. If Ohta-awasaki method is applied to DSA, artificial effect to introduce the guide is needed though the error derived by the artificial effect becomes much larger. Though the simulations using RPA based methods are low cost 804

5 simulation method, the simulations using RPA-based method has errors, which may derive the wrong results even if in a qualitative standpoint. Due to this reason, only the experts of RPA-based simulation technics, who can estimate the error of their simulations to check their simulation results carefully, can have a proper use of RPA-based method. To apply the RPA-based method to the DSA simulation, further improvements from the original theory would be expected. points in the advantage and disadvantage of these simulation methods are listed in Table 1. (a) (b) (c) (d) Fig.3 SCF simulation result including chemical guide by SUSHI/OCTA simulator. 5. Sample of DSA simulation using OCTA DSA simulation can also be done using OCTA system. In OCTA, SCF and DPD simulations can be performed using SUSHI and COGNAC simulators, respectively. Using SUSHI, DSA on the chemical guide can be simulated as shown in Fig. 3. This is the two dimensional SCF simulation result with the chemical guide. In the simulations, the width of the attractive part of chemical guide can be changed. This simulation imitates the simulations by Detcheverry et al. [24] The bottom chemical guide was modeled by the obstacle options, in which the interaction between polymer and obstacle surface can be added. The physical guide can also be done by obstacle option in SUSHI. Figure 4 shows the SCF simulation result in two dimensional with physical guide. In this simulation, one subchain is attractive to the both physical guides placed at the upper and lower layers, and this is similar result of reference 18. Using DPD method, DSA simulation can also be performed. Figure 5 shows the DPD simulation result using the chemical guide. Chemical guide is introduced as the bottom layer constituted by the particles. Interaction between bottom layer and polymer particles is controlled to induce the favorable interaction of one component of blocks, and the patterned structures are realized on OCTA system. 6. Concluding Remarks In this paper, the simulation methods for micro phase separated structure are reviewed and classified in the usage of DSA simulation. Finally, our discussed Fig.4 Two dimensional SCF simulation results for the top down images with defects by SUSHI/OCTA simulator. Fig.5 DPD simulation result using chemical guide by COGNAC/OCTA simulator. Each method has strong and weak points when these methods are applied to DSA simulations. We cannot specify that one method is only the best method for DSA simulation. Therefore, the researchers who perform the DSA simulation must evaluate the simulation method on a case-by-case basis. To help the application studies of DSA, the applicability is also indicated in Table 1, and OCTA system would become the standard DSA simulation tools including many kinds of methods which have applied to the DSA simulations. It may be used the DSA simulation in the purpose of making the low LER and no defect structure. To overcome these problems, the control of the polymer chain on the wafer is needed. Therefore, the 805

6 Table. 1 Applicable simulation methods for DSA and those features Dynamical Self-consistent Dissipative particle Monte Calro Generalized Random Ohta -awasaki approximation field method dynamics method method phase approximation (Uneyama s ) Mesh or Particle Mesh Particle Particle / Mesh Mesh Mesh Which simulator in OCTA? Boundary conditions SUSHI COGNAC Partially work in SUSHI and COGNAC Drops (OCTA friend software by Dr. Uneyama) Any Any Any Only periodic boundary MUFFIN Only periodic boundary Including Mw (each Mw (each Mw (each Block ratio Block ratio parameters for block), block), block), polymer chains Block ratio, Block ratio, Block ratio, Distribution of Distribution of Distribution of Mw Mw Mw Strong segregation (high χ) system Analysis for single polymer chain Advantage Disadvantage Recommendation in DSA simulation Specific applicability in DSA study Possible but heavy calculation is needed. O O With errors due to approximation With errors due to approximation O O O Difficult Difficult Obtain Easy to treat high Easy to treat high Fast for both 2D Fast for both 2D statistically χ system χ system and 3D system and 3D system correct result Expensive Need storage Need storage Difficult to Difficult to calculation, disk to save each disk to save each include the guide include the guide Need much more time step. time step. effect correctly. effect correctly. time for 3D Only obtaining Only obtaining Less info. for Less info. for calculation transient transient polymer chain. polymer chain. structure structure. Smaller range of Smaller range of parameters for parameters for polymers polymers. Highly Highly Highly Only experts can Only experts can recommend recommend recommend use due to errors. use due to errors. Defect, LER, Defect, LER, Defect, Non Defect Structure near Structure near Structure near guide, guide, guide, 806

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