NanoVelcro: Theory of Guided Folding in Atomically Thin Sheets with Regions of Complementary Doping.

Folding has been commonly observed in two-dimensional materials such as graphene and monolayer transition metal dichalcogenides. Although interlayer coupling stabilizes these folds, it provides no control over the placement of the fold, let alone the final folded shape. Lacking nanoscale "fingers" to externally guide folding, control requires interactions engineered into the sheets that guide them toward a desired final folded structure. Here we provide a theoretical framework for a general methodology toward this end: atomically thin 2D sheets are doped with patterns of complementary n-type and p-type regions whose preferential adhesion favors folding into desired shapes. The two-colorable theorem in flat-foldable origami ensures that arbitrary folding patterns are in principle accessible to this method. This complementary doping method can be combined with nanoscale crumpling (by, for example, passage of 2D sheets through holes) to obtain not only control over fold placements but also the ability to distinguish between degenerate folded states, thus attaining nontrivial shapes inaccessible to sequential folding.