What Will That DSA Template Do, Anyway?

As directed self assembly techniques make the transition from line and space test patterns to the more complex structures seen in real devices, modeling is emerging as a significant issue. How will the co-polymers behave in the presence of a particular template pattern?

While several laboratory-scale modeling methods exist, most are too computationally expensive to be used for large area structures. Monte Carlo and self-consistent field models often involve hours or days of computation time with large workstation clusters.

In light of these challenges, the work of Kenji Yoshimoto and Takashi Taniguchi at Kyoto University deserves special attention. They took as their starting point the Ohta-Kawasaki model, derived from a self-consistent field solution near the critical point χN ~ 10.5. Rather than modeling individual polymer chains, this approach treats the two co-polymers as a combination of continuous fluids, and phase segregation as a diffusion process. The key parameters are all known properties of the particular co-polymer formulation:

• Χ, the Flory-Huggins interaction parameter.
• N, the degree of polymerization.
• f, the volume fraction of polymer A.
• a, the average distance between two monomers.

Additional parameters describe the dimensions and interaction characteristics of the underlying surface.

The Kyoto University work uses the above parameters and η, the local polymer concentration, to define the free energy at any point in the material. The difference between this energy and the energy of the template surface gives a chemical potential, μ. From there, standard diffusion equations lead to:

Where L is the diffusion constant and  is the flux due to thermal fluctuations. That is, the local polymer concentration at any point depends on the free energy of the polymer, the free energy of the template surface, and the diffusion rates of the co-polymers. The second term incorporates random fluctuations and can be used to calculate line-edge roughness. Nikon’s Stephen Renwick, however, set it to zero and obtained:

Where μ’ is obtained by simplifying the equation for μ and

Renwick’s result depends only on physical constants and an equation of motion, and as such can be iterated on standard desktop PCs. The model is limited to relatively small values of χN, around 10, but puts fast, accurate DSA simulations within reach of typical workstation PCs.