Session L30: Graphene Devices: Fabrication, Characterization and Modeling: Nanomechanics

Mechanical Properties of Graphene on Surfaces Having Patterned Pyramid Arrays

There is wide interest in the science and applications of strain-engineering graphene's physical properties. To explore the mechanical behavior of graphene under strain with triangular symmetry, we deposited graphene on fabricated arrays of pyramid-shaped protrusions that were patterned on both polydimethylsiloxane (PDMS) and SiO$_{\mathrm{2}}$ surfaces. Using atomic force microscopy (AFM), we studied the morphological and adhesion changes of graphene on pyramid arrays having different spacing. The strain was also examined using Raman spectroscopy. We show that the mechanical properties of graphene vary with the spacing of pyramids in an array.   

Unconventional superconducting quantum interference in a suspended graphene resonator

In a coherent electron cavity, quantum interference of electron waves replaces classical diffusion as a key feature of electronic transport. Here we report novel behavior that emerges by coupling superconducting reservoirs to a Fabry-Perot resonator in bilayer graphene. In this device, a pair of superconducting electrodes is coupled to a suspended graphene membrane and defines a ballistic cavity between the two graphene-electrode interfaces. Tuning the Fermi wavelength in the cavity with a gate electrode moves the system on and off resonance, thus inducing an oscillatory critical current whose period satisfies the Fabry-Perot interference conditions. By varying the magnetic flux through the junction, we explore the rich interplay between superconducting quantum interference and resonant cavity states and demonstrate a non-trivial correspondence between the supercurrent and normal state resistance. To describe our findings, we use a numerical model based on the tight-binding approach and Landauer-Buttiker scattering formalism. These results constitute a departure from the conventional Josephson effect in graphene and motivate exploration of new effects at the intersection of superconductivity and optics-like phenomena.   

Tuning graphene properties by shear

Graphene being the thinnest possible membrane made out of carbon atoms is prone to deformations under slight external forcing. Here, we take advantage of this proneness to deformations to manipulate transport properties of graphene ribbons. We analyze the effect on conductance and LDOS of the spontaneous pattern produced when a wide ribbon is subject to shear. The deformation of the ribbon produces pseudo-magnetic fields, scalar potentials, and Fermi velocity renormalization resulting in the modification of transmission properties without the need of an external gate potential. Our proposal paves the way for producing electronic waveguides by using an elastic instability that spans from the nano to macro-scales.   

Graphene Nanoelectromechanical Systems as Stochastic-Frequency Oscillators

We measure the quality factor $Q$ of electrically-driven few-layer graphene drumhead resonators, providing an experimental demonstration that $Q$ $\sim$ $1/T$, where $T$ is the temperature. We develop a model that includes intermodal coupling and tensioned graphene resonators, yielding good quantitative agreement to experiment. Because the resonators are atomically thin, out-of-plane fluctuations are large. As a result $Q$ is mainly determined by stochastic frequency broadening rather than frictional damping, in analogy to nuclear magnetic resonance. Additionally, at larger drives the resonance linewidth is enhanced by nonlinear damping, in qualitative agreement with recent theory of damping by radiation of in-plane phonons. Parametric amplification produced by periodic thermal expansion from the ac drive voltage yields an anomalously large linewidth at the largest drives. Our results contribute towards a general framework for understanding the mechanisms of dissipation and spectral line broadening in atomically thin membrane resonators.   

Focused Ion Beam patterning of suspended graphene for cantilever and kirigami devices

We have developed techniques that use a Focused Ion Beam (FIB) to cut and manipulate suspended graphene. Using a dual-beam FIB, we can make cuts with a resolution of tens of nanometers, manipulate and pick up finished devices using a micromanipulator, and remove device and micromanipulator from the vacuum chamber. Remarkably, we have demonstrated that singly clamped graphene cantilevers can be fabricated reliably and are robust enough to be freely manipulated in air. This gives us the potential to perform novel electrostatic and mechanical measurements of graphene. Using the FIB's direct writing capabilities, we are also able to cut out more complex shapes, drawing inspiration from kirigami, the art of paper cutting. Using specific cuts, we can create soft in-plane springs, which might be used to study tension. This exploration of the fabrication and manipulation of graphene in three dimensions is a promising new avenue toward harnessing graphene's unique properties, and also holds promise for other 2D materials.   

Intrinsic mode-coupling and thermalization in nanomechanical graphene drums

Nanomechanical graphene resonators display strong nonlinear behavior, which leads to coupling between normal modes. This coupling allows for intermodal energy-transfer, which facilitates the redistribution of energy initially localized in a single mode. Further, the mode-coupling intrinsically limits the quality factor of the device. We study the mode-coupling in a circular graphene resonator using molecular dynamics and continuum mechanics. Mimicking a ring-down setup, the fundamental mode is excited with a given energy, and the time-evolution of this energy is computed. At $T>0$, we find a relaxation rate independent of system size and proportional to $T^*/\epsilon_{\rm pre}^2$, where $T^*$ is the effective temperature and $\epsilon_{\rm pre}$ is the pre-strain of the system \footnote{D. Midtvedt, Z. Qi, A. Croy, H. S. Park, A. Isacsson,arXiv:1309.1622}. At low temperatures, the system enters a metastable state where only very few low-frequency modes are excited, the life-time of which increases exponentially with decreasing excitation energy. This is similar to what is seen in the much studied Fermi-Pasta-Ulam (FPU) problem. We make a detailed comparison between the dynamics of a graphene drum and the FPU system, and propose to use graphene drums as test beds for FPU physics.   

Mechanical Nonlinearity in Graphene Resonators

Electromechanical resonators made from suspended graphene sheets show promise for many potential applications, including mass sensing, optomechanics, and tunable radio frequency electronics.[1,2,3] However, fundamental properties of these resonators, such as the strongly temperature dependent resonant frequency and quality factor, have yet to be understood. Measurements of mechanical nonlinearity can provide additional information about the properties of graphene membranes, including the modulus for stretching.[2,4] Here, we present careful studies of the nonlinear response of fully-clamped graphene resonators. We measure the coefficients of the cubic (Duffing) nonlinearity and nonlinear damping terms. We also discuss how these terms compare to expectations from elastic and entropic theories. These measurements increase our physical understanding of the mechanics of atomically thin membranes, and can help improve the performance of these novel electromechanical devices. [1] J. S. Bunch, et al., Science 315, 490 (2007) [2] C. Chen, et al., Nature Nanotechnology 4, 861-867 (2009) [3] R. A. Barton, et al., Nano Letters 12, 4681--4686 (2012) [4] A. Eichler, et al., Nature Nanotechnology 6, 339-342 (2011)   

Fabrication and Characterization of Graphene Nano-Mechanical Oscillators

Copper foil is the most commonly used substrate for chemical vapor deposition (CVD) growth of graphene, despite the impact of its surface roughness and polycrystalline structure on the resulting graphene. We instead grow graphene grown on large-domain thick ultra-flat copper discs, using LPCVD and APCVD. Compared to copper foil, graphene grown on these thick ultra-flat copper substrates by APCVD results in 50 times smoother graphene on copper. The grown graphene is transferred from copper using Poly (methyl methacrylate) (PMMA), and is patterned into nano-mechanical oscillators (NMO) of different geometrical shapes, using deep-UV lithography of PMMA. A study of the phase noise in the resonant frequency of the NMO, exhibits the characteristic `1/f' noise, which seems to depend on the number of layers of graphene.   

Mirror buckling of freestanding graphene membranes induced by local heating due to a scanning tunneling microscope tip

Scanning tunneling microscopy has been an invaluable tool in the study of graphene at the atomic scale. Several STM groups have managed to obtain atomic scale images of freestanding graphene membranes providing insight into the behavior of the stabilized ripple geometry. However, we found that the interaction between the STM tip and the freestanding graphene sample may induce additional effects. By varying the tunneling parameters, we can tune the position of the sample, in either a smooth or step like fashion. These phenomena were investigated by STM experiments, continuum elasticity theory and large scale molecular dynamics simulations. These results confirm that by increasing the tip bias, the electrostatic attraction between the tip and sample increases. When applied on a concave surface, this can result in mirror buckling which leads to a large scale movement of the sample. Interestingly, due in part to the negative coefficient of thermal expansion of graphene, buckling transitions can also be induced through local heating of the surface using the STM tip.   

Graphene Kirigami

We have developed a powerful new approach to working with graphene by applying the principles of kirigami, the sculptural art of paper cutting. We have release graphene from the surface, allowing us to treat it like a sheet of atom-thick paper. Working in water, we can pull the graphene along the surface or peel it up entirely. Combining this technique with lithographic patterning, we have created a variety of graphene kirigami devices including three-dimensional structures and resilient, atomically-thin hinges. We have also created soft in-plane springs by patterning a series of cuts into the graphene. The spring constants of these devices depend on the pattern of cuts, so the patterned graphene becomes an adjustable mechanical metamaterial. With possible spring constants ranging from 1 N/m to 10$^{\mathrm{-9}}$ N/m, these springs could be used as sensitive force measurement devices. Such kirigami patterning techniques could also be applied to flexible and stretchable electronics, including soft electrodes for biological experiments. This unusual way of interacting with graphene opens up a world of potential applications that we are just beginning to explore.   

Parametric stabilization of levitated graphene microparticles

After graphene was discovered research on it grew rapidly because of a variety of possible applications arising from its unique properties. Since graphene is strongly affected by its environment, it is desirable to decouple graphene from any kind of substrate when studying it. One of the ways to achieve this is to levitate charged graphene microparticles in a quadrupole trap [1]. Though graphene particles can stay in such trap for weeks under low vacuum conditions ($\sim$1 mTorr), their trapping time is significantly reduced down to a few hours when trap chamber is pumped to ultra-high vacuum ($\sim$10$^{-8}$ Torr). In this talk we investigate the possibility of increasing trapping time of such particles as well as for their cooling by means of parametric feedback similar to the feedback scheme used for cooling down laser-trapped nanoparticles [2]. In our system, motion of the graphene is used to provide a feedback signal which in turn is used to modulate a trap frequency at twice the frequency of graphene oscillations. Current progress and possible improvements and applications of such feedback scheme are discussed in the talk. \\[0pt][1] B. Kane, Phys. Rev. B \textbf{82}, 115441 (2010) \\[0pt][2] J. Gieseler et al., Phys. Rev. Lett. \textbf{109}, 103603 (2012)   

Measuring the Strength of Single Crystal and Polycrystalline Graphene

The mechanical properties of materials depend strongly on their crystallinity. In our work, we measure the yield strength of suspended single crystal and bicrystal graphene membranes fabricated from chemical vapor deposition grown graphene. Membranes are characterized structurally by transmission electron microscopy and mechanically tested using atomic force microscopy. A single crystal diamond tip with a large indentation radius is used to measure the intrinsic strength of suspended membranes for mechanical measurements. Single crystal membranes prepared by chemical vapor deposition retain strengths that are comparable to previous results of single crystal membranes prepared by mechanical exfoliation. Bicrystal grain boundary membranes with large mismatch angles have enhanced strengths when compared to their low angle counterparts. These boundaries show strengths that are comparable to single crystal graphene. To investigate this enhanced strength, we use aberration corrected high resolution transmission electron microscopy to map the atomic scale strain fields in suspended graphene. The enhanced strength of large angle bicrystal membranes is attributed to the presence of low atomic-scale strain at the boundaries.