Random Walk Simulation for the Growth of Monolayer in Dip Pen Nanolithography

Dip-pen nanolithography (DPN) has emerged as a general tool for fabricating nanopatterns on a range of substrates, including metallic, insulating and semiconducting substrates. In DPN, an atomic force microscope (AFM) tip functions as a source to continuously create nanodroplets of molecules that then spread to form a self-assembled monolayer (SAM) on a substrate. Molecular transport from the tip to the substrate is mediated by the water meniscus that naturally forms between the tip and substrate under ambient conditions. To assess the ultimate resolution and limitation of DPN, it is important to understand the fundamentals of DPN, particularly at the molecular level. Little is known about the molecular mechanisms and time scale of SAM growth in DPN. In this respect, theory and modeling have proven to be useful. Central to modeling DPN is how a nascent droplet (deposited from an AFM tip) spreads out to form a SAM afterward. Given that molecules are designed to chemisorb to a substrate, Jang et al. proposed a diffusion model of DPN assuming that the molecules are trapped irreversibly once they have reached the chemisorption sites on the substrate. Molecules can diffuse on top of other molecules. This model gives isotropic SAM patterns for a tip fixed in position but fails to reproduce the dendritic SAM patterns observed in DPN using dodecylamine or polyethylene glycol on mica. Molecular dynamics (MD) simulations showed that a SAM pattern grows mainly via a serial pushing mechanism (Figure 1(a)), in which a molecule in the upper layer (open circle in the second layer in the figure) pushes a molecule on the substrate out of its original position, and the molecule pushed out in turn pushes out another molecule nearby. Such a consecutive push propagates and reaches the periphery, after which the pushing stops. This serial pushing was taken to have a finite directional coherence length, Nd, which is defined as the number of consecutive pushing events in the same direction (see Figure 1(b)). By simply varying Nd, this model can reproduce a variety of SAM patterns, ranging from isotropic to dendritic patterns. The serial pushing model also captured the essential features of the MD simulation. Previously, the serial pushing model was applied to cases where an AFM tip is fixed in position. On the other hand, the SAM patterns in DPN are generally made by moving the tip on a substrate. The outcome of DPN depends on the moving speed of the tip. In this study, various line-based SAM patterns were simulated to determine how DPN is affected by the tip scan speed and directional coherence length.