All nanoscale objects undergo vibrations due to random thermal noise. The vibrational modes can be modeled and predicted using the equipartition theorem , with atomic force microscopy (AFM) making use of this for a vibrating cantilever. For a vibrating cantilever beam, the mean square displacement of the tip is given by Equation 1, where h is the height of the structure, kB is the Boltzmann constant, T is the absolute temperature, E is the Young’s modulus, and I is the second moment of inertia of the cross-section of the cantilever. As objects become smaller with large aspect ratios, the magnitude of these vibrations will become significant compared to the structure’s CD and gap width. Eventually, the point will be reached where the amplitude of the vibrations is enough for the features to begin touching each other. A typical vibrating cylinder structure and rectangular structure are shown in Figures 1(a) and (b) respectively, and a plot of calculated displacement for cylinder and rectangular structures as a function of their aspect ratio is shown in Figure 2.
Using the mean square displacement of the tip z2 and the fact that the tip displacement follows a Gaussian probability distribution, we can calculate the statistical likelihood that the tips of two identical adjacent structures will touch, which is shown in Equation 2. Here P is the probability that the structures will be touching, erf is the error function, and a is the spacing between the structures. This equation can also be inverted to find the maximum allowable feature height or aspect ratio for a given touching probability tolerance. A plot of calculated probability for cylinder and rectangular structures as a function of their aspect ratio is shown in Figure 3.
When touching occurs, there are several possible outcomes, depending on the balance between thermal, elastic, and van der Waals forces. For two structures with rounded tips that are touching, the energy of the system associated with the van der Waals force is given by Equation 3, where A is the Hamaker constant for the material of the structure, r is the radius of curvature of the tips, and ε is the separation distance of the surfaces when touching, around 3Å. When more than just the tips of the structures are touching, the model of Koide et al.  can be used as well.
If the van der Waals forces dominate, when vibrations are strong enough for the features to touch then the features will in contact and behave similarly to the classic capillary pattern collapse. If the elastic forces dominate, then touching will not cause immediate and total collapse, but they may continue to touch regularly, possibly causing damage to pattern tips. If instead thermal forces dominate over both elastic and van der Waals forces, then overall pattern integrity is compromised, and the features will behave similar to randomly vibrating strings or membranes, depending on the feature geometry.
In our paper, we will show the analysis to look at the balance between these three forces, and determine the layout of the phase map between these different failure modes based on feature composition and geometry.
H.-J. Butt and M. Jaschke, "Calculation of thermal noise in atomic force microscopy," Nanotechnology, vol. 6, pp. 1-7, 1995.
T. Koide, S. Kimara, H. Iimori, T. Sugia, K. Sato, Y. Sato and Y. Ogawa, "Effect of Surface Energy Reduction for Nano-structure Stiction," ECS Transactions, vol. 69, no. 8, pp. 131-138, 2015.