Interface-Governed Deformation of Nanobubbles and Nanotents Formed by Two-Dimensional Materials

Nanoblisters such as nanobubbles and nanotents formed by two-dimensional (2D) materials have been extensively exploited for strain engineering purposes as they can produce self-sustained, nonuniform in-plane strains through out-of-plane deformation. However, deterministic measure and control of strain fields in these systems are challenging because of the atomic thinness and unconventional interface behaviors of 2D materials. Here, we experimentally characterize a simple and unified power law for the profiles of a variety of nanobubbles and nanotents formed by 2D materials such as graphene and ${\mathrm{MoS}}_2$ layers. Using membrane theory, we analytically unveil what sets the in-plane strains of these blisters regarding their shape and interface characteristics. Our analytical solutions are validated by Raman spectroscopy measured strain distributions in bulged graphene bubbles supported by strong and weak shear interfaces. We advocate that both the strain magnitudes and distributions can be tuned by 2D material-substrate interface adhesion and friction properties.

Figure 1

From top to bottom: Atomic force microscopy (AFM) phase and height images of spontaneously formed graphene bubbles on ${\mathrm{SiO}}_2$ (a), a multilayer graphene tent on ${\mathrm{SiO}}_2$ (b), and a CVD-${\mathrm{MoS}}_2$ tent on gold film (c). (d) From left to right: optical image of graphene flakes exfoliated on prepatterned ${\mathrm{SiO}}_2$ with microcavities, AFM height images of a monolayer graphene bubble, and a four-layer ${\mathrm{MoS}}_2$ bubble. Note that ($S$) represents bubbles or tents formed spontaneously while ($P$) represents those formed by controllable air pressurization.

Figure 2

Universal shape characteristics of 2D material bubbles and tents. (a) Normalized bubble profiles measured by our experiments and collected from literature. Note that samples from Ref. [17] feature atomically smooth interfaces, are labeled by *. (b) Normalized tent profiles measured by our experiments and simulation results in the literature. The simulation data about graphene and ${\mathrm{MoS}}_2$ is from Refs. [36, 24], respectively.

Figure 3

Normalized strain distribution curves predicted by our analytical solution (solid lines) and solved by numerical analysis (markers) in bubbles (a) and tents (b), subjected to both clamped (strong interface) and frictionless (sliding interfaces) boundary conditions. The strain is normalized by $h^2/a^2$, giving rise to deflection-independent curves. The numerical results are solved for a monolayer graphene with aspect ratios ranging from $0.02<h/a<0.2$.

Figure 4

Schematics of the graphene drumheads formed on a ${\mathrm{SiO}}_2$ substrate (a) and on a graphene-covered ${\mathrm{SiO}}_2$ substrate (b). (c) Raman shifts of the $G$ band at the center of graphene bubbles predicted by our analytical solution (solid curves) and measured by our experiments (markers). (d) Normalized Raman shifts of the $G$ band ($\mathrm{\Delta }{\omega }_G{a}^2/{\omega }_0{h}^2$) as functions of the normalized radial position ($r/a$) for monolayer graphene bubbles.