Abstract
We describe and analyze in detail the shapes of Fe islands encapsulated under the top graphene layers in graphite. Shapes are interrogated using scanning tunneling microscopy. The main outputs of the shape analysis are the slope of the graphene membrane around the perimeter of the island, and the aspect ratio of the central metal cluster. Modeling primarily uses a continuum elasticity (CE) model. As input to the CE model, we use density functional theory to calculate the surface energy of Fe, and the adhesion energies between Fe and graphene or graphite. We use the shaft-loaded blister test (SLBT) model to provide independent stretching and bending strain energies in the graphene membrane. We also introduce a model for the elastic strain in which stretching and bending are treated simultaneously. Measured side slopes agree very well with the CE model, both qualitatively and quantitatively. The fit is optimal for a graphene membrane consisting of 2–3 graphene monolayers, in agreement with experiment. Analysis of contributions to total energy shows that the side slope depends only on the properties of graphene/graphite. This reflects delamination of the graphene membrane from the underlying graphite, caused by upward pressure from the growing metal cluster. This insight leads us to evaluate the delamination geometry in the context of two related, classic models that give analytic results for the slope of a delaminated membrane. One of these, the point-loaded circular blister test model, reasonably predicts the delamination geometry at the edge of an Fe island. The aspect ratio also agrees well with the CE model in the limit of large island size, but not for small islands. Previously, we had speculated that this discrepancy was due to lack of coupling between bending and stretching in the SLBT model, but the new modeling shows that this explanation is not viable.
Highlights
There are numerous situations where it is important to understand or utilize the interaction between a metal, and a two-dimensional (2D) material (or by extension a heterostructure of 2D materials, or a three-dimensional (3D) van der Waals material)
It includes a schematic that defines the key dimensions of an iron cluster encapsulated by a graphene layer: h is the island height; d is the diameter of the island top; a is the width of the sloping perimeter of the graphene membrane; and t is the thickness of the graphene membrane, given by t = LC tGML
Where EL is the total energy of the slab in a supercell, NL is the total number of atoms in the slab, A is the area of the bottom or top face of the slab, and sbulk is the energy per atom in the bulk crystal
Summary
There are numerous situations where it is important to understand or utilize the interaction between a metal, and a two-dimensional (2D) material (or by extension a heterostructure of 2D materials, or a three-dimensional (3D) van der Waals material). One example is electrical connections or heat sinks for device applications, where metal architectures with stable and high-area contacts to the 2D material are needed [1, 2]. Another example is tuning the electronic properties of the 2D material, where dopants and intercalants can modify the Fermi level via charge transfer [3, 4]. In almost all of these applications, it is desirable to maximize the contact area between the metal and the 2D material, i.e. to achieve ‘flat’ growth of the
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