Geometrical scaling, i.e. the relation of indent depth to diameter, is of fundamental interest in both data storage and metrology. In data storage, the indent depth determines the amplitude of the read-back signal, whereas the indent width governs the achievable areal density. Therefore, geometric scaling is directly linked to the scaling of the technology.

In macroscale indentation, the ratio between indent depth and diameter is given by a faithful replication of the indenter geometry into the material surface. For conical and pyramidal indenters, for example, the indent depth is proportional to the indent diameter. This geometric scaling therefore allows an accurate measurement of the material’s hardness.

Surprisingly, simple linear scaling laws apply down to indent depths of 1 nm. However, in contrast to a simple proportionality as found macroscopically, here the indent diameter is always larger than a certain minimal indent diameter. This offset is even found to be independent of film thickness and indentation temperature.

To interpret these intriguing findings, three schools of thinking had to be combined. First, it is observed that the finite deformations blur the distinction between plastic and rubber-like deformation. Second, there is the engineering concept that plastic deformation is a thermally activated process. Finally, and most relevantly, the mobility of polymers is understood in terms of fundamental processes that involve the cooperative rearrangement of polymer-chain segments in a finite volume. Ultimately, the findings are rationalized in terms of a finite-strain criterion. This criterion implies that even stresses exceeding the yield stress will not suffice to induce a plastic deformation of the polymer, unless a critical strain is exceeded on a volume larger than that defined by cooperativity.

This conclusions help to pinpoint the questions that are of relevance for the scaling of probe-based data storage, but that can also readily be applied to nanoimprint lithography: rather than focusing solely on the size of the indenter, the cooperativity of the polymer has to be understood and controlled to exploit the ultimate resolution of this replication technique.