Snap-Through Instability of Graphene on Substrates
© to the authors 2009
Received: 6 August 2009
Accepted: 2 October 2009
Published: 17 October 2009
We determine the graphene morphology regulated by substrates with herringbone and checkerboard surface corrugations. As the graphene–substrate interfacial bonding energy and the substrate surface roughness vary, the graphene morphology snaps between two distinct states: (1) closely conforming to the substrate and (2) remaining nearly flat on the substrate. Since the graphene morphology is strongly tied to the electronic properties of graphene, such a snap-through instability of graphene morphology can lead to desirable graphene electronic properties that could potentially enable graphene-based functional electronic components (e.g. nano-switches).
Graphene is a monolayer of carbon atoms densely packed in a honeycomb crystal lattice. It exhibits extraordinary electrical and mechanical properties [1–5], and has inspired an array of tantalizing potential applications (e.g., transparent flexible displays and biochemical sensor arrays) [6–10]. Graphene is intrinsically non-flat and tends to be randomly corrugated [11, 12]. The random graphene morphology can lead to unstable performance of graphene devices as the corrugating physics of graphene is closely tied to its electronic properties [13, 14]. Future success of graphene-based applications hinges upon precise control of the graphene morphology over large areas, a significant challenge largely unexplored so far. Recent experiments show that, however, the morphology of graphene can be regulated by the surface of an underlying substrate [15–19]. In this paper, we quantitatively determine the regulated graphene morphology on substrates with various engineered surface patterns, using energy minimization. The results reveal the snap-through instability of graphene on substrates, a promising mechanism to enable functional components for graphene devices.
Recent experiments show that monolayer and few-layer graphene can partially follow the rough surface of the underlying substrates [15–19]. The resulting graphene morphology is regulated, rather than the intrinsic random corrugations in freestanding graphene. The substrate-regulated graphene morphology results from the interplay between the interfacial bonding energy and the strain energy of the graphene-substrate system [15, 17], which can be explained as follows.
When graphene is fabricated on a substrate surface via mechanical exfoliation  or transfer printing [10, 20], the graphene–substrate interfacial bonding energy is usually weak (e.g., van der Waals interaction). As the graphene corrugates to follow the substrate surface, the graphene–substrate interaction energy decreases due to the nature of van der Waals interaction; on the other hand, the strain energy in the system increases due to the intrinsic bending rigidity of graphene. At the equilibrium graphene morphology on the substrate, the sum of the interaction energy and the system strain energy reaches its minimum.
Since Lennard–Jones potential decays rapidly beyond equilibrium atomic pair distance, E int can be estimated by adding up the van der Waals forces between each graphene carbon atom and the substrate portion within a cut-off distance from this carbon atom. If the cut-off distance is large enough, such an estimate of interaction energy converges to the theoretical value of E int. In all simulations reported in this paper, a cut-off distance of 3 nm was used and shown to lead to variations in the estimated value of E int less than 1%.
As n and m become large enough, Eq. 3 converges to the theoretical value of E int. In all simulations in this paper, n = 106, m = 400.
where D and ν are the bending rigidity and the Poisson’s ratio of graphene, respectively.
As shown in Eq. 6, for a given substrate surface corrugation (i.e., A s, A y , λ x , and λ y ), E g increases monotonically as A g increases. On the other hand, the graphene–substrate interaction energy, E int, minimizes at finite values of A g and h, due to the nature of van der Waals interaction. As a result, there exists a minimum of (E g + E int) where A g and h reach their equilibrium values. The energy minimization was carried out by running a customized code on a high performance computation cluster. In all computations, D = 1.41 eV,l = 0.142 nm, ρs = 2.20 × 1028/m3,σ = 0.353 nm and A s = 0.5 nm, which are representative of a graphene-on-SiO2 structure [27, 28]. Various values of ɛ λ x , λ y , and A y were used to study the effects of interfacial bonding energy and substrate surface roughness on the regulated graphene morphology.
Results and Discussion
respectively, where λ is the wavelength of the out-of-plane corrugations for both the graphene and the substrate surface. The numerical strategy similar to that aforementioned was implemented to determine the equilibrium amplitude of the regulated graphene morphology.
In this paper we focus on graphene morphology spontaneously regulated by substrate surfaces via weak interaction. When a graphene/substrate structure is subject to external loading, the graphene strain energy due to stretching and the substrate strain energy may also need to be considered. In this sense, the present model overestimates the graphene corrugation amplitude. Also the graphene/substrate interaction can be enhanced by the possible chemical bondings or pinnings at the interface [26, 29, 30]. In this sense, the present model underestimates the graphene corrugation amplitude.
In summary, we investigate the graphene morphology regulated by substrates with herringbone and checkerboard surface corrugations. Depending on interfacial bonding energy and substrate surface roughness, the graphene morphology exhibits a sharp transition between two distinct states: (1) closely conforming to the substrate surface and (2) remaining nearly flat on the substrate surface. The quantitative results suggest a promising strategy to control the graphene morphology through substrate regulation. While it is difficult to directly manipulate freestanding graphene , it is feasible to pattern the substrate surface via lithography [21, 22] and strain engineering [23, 24]. The regulated graphene morphology on such engineered substrate surfaces may lead to new pathways to control the graphene electronic properties, introducing desirable properties such as band-gap, or p/n junction behavior. In particular, the results shown in this paper (e.g., Figs. 2 3 5) reveal a wide range of design tunability of the graphene snap-through instability on substrates through substrate surface patterning and interfacial adhesion tailoring, which offers abundant unexplored potential toward the design of functional graphene device components (e.g., nano-switches, nano-resonators). We then call for experimental demonstration of these envisioned concepts.
This work is supported by the Minta-Martin Foundation, a UMD General Research Board summer research award to T. L., and NSF CMMI 0856540. Z.Z. also thanks the support of the A. J. Clark Fellowship.
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