Free-standing graphene membranes are the archetypes of two-dimensional nanomechanical resonators. Because of their small size, ultra-low mass, and high elastic modulus, these resonators typically vibrate at frequencies ranging from a few megahertz to a few hundreds of megahertz, and their resonant frequencies can be widely tuned by electrical means. However, because of fabrication subtleties, it has been found that actual resonant frequencies and mode shapes vary greatly from device to device. To address it, here we simulate graphene nanomechanical drum resonators with COMSOL Multiphysics finite element software. We investigate the static and dynamic characteristics of the resonator in detail. In the static case, we analyze the bent shape of the graphene membrane induced by a static gate voltage for different sizes of a local gate electrode. In the dynamic case, we simulate the influence of device geometry and built-in strain on the resonant frequency of the first vibrational mode and on its tunability. Further, because folds may form in the membrane during exfoliation and transfer to the substrate, we investigate how this imperfection affects resonant frequencies and mode shapes. To this end, we introduce a nonuniform tension along a line in our finite element model. Our study may offer guidelines to design graphene resonators, with applications in components for radio frequency signal processing and communications and for nanomechanical sensing.
Two-dimensional (2-D) nanomechanical resonators are based on thin layers of graphene, black phosphorus, transition metal dichalcogenides and van der Waals heterostructures. Detection of nanomechanical vibrations can be done using optical reflectometry, whereby vibrations modulate the optical reflectance of the resonator. For this type of detection to work, it is essential to fabricate cavities with a precise depth. Here we report on the fabrication of 2-D nanomechanical resonators in which the cavity is made using focused ion beam (FIB) lithography. We mill down an array of cylindrical cavities with the same diameter but different depths. We drive vibrations electrically and detect vibrations optically. At the resonant frequency of vibrations, we observe that the measured signal, which is proportional to the vibrational amplitude and to a transduction factor, is different for different cavity depths. Since all resonators have the same diameter, are made of the same graphene flake and are actuated the same way, our observation implies that the transduction factor changes with cavity depth. Using principles of thin film optics, we show that each estimated transduction factor is indeed consistent with the dimensions of the resonator, including the cavity depth, the thickness of the patterned substrate, and the number of graphene layers. This result supports the idea of using FIB to fabricate cavities for 2-D nanomechanical resonators, instead of using standard wet etching or reactive ion etching which require additional lithography steps and cannot easily be used to pattern cavities with different depths on the same substrate.
Two-dimensional (2-D) nanomechanical resonators are interesting for the tunability of their resonant frequencies over wide frequency ranges using electrical means. These resonators are often made by transferring thin membranes of layered materials onto cavities fabricated in oxidized silicon wafers. The resonant frequency of vibrational modes is tuned by applying a dc voltage between the membrane and the silicon substrate acting as a global gate, which creates an electrostatic force that pulls the membrane towards the global gate and changes the strain within the membrane. Here, we measure the frequency response of 2-D resonators based on few-layer graphene transferred onto cavities milled in silicon oxide using focused ion beam (FIB) lithography. In response to a step in gate voltage, we find that resonant frequencies of vibrational modes decay in time. To explain this phenomenon, we propose that residual gallium ions from the ion beam form a floating gate at the bottom of the cavity and create a weak link between this floating gate and the graphene membrane. Leakage of charges between graphene and the floating gate lowers the strain induced by the voltage applied between graphene and the gate electrode, making the resonant frequency of the graphene membrane decay. We present a model based on a floating gate structure to effectively explain the decay of graphene resonant frequency in our device.
Nanomechanical resonators based on two-dimensional materials offer opportunities to study the mechanical properties of atomically thin membranes and to develop sensitive detection schemes. However, these applications are limited by problems with nanofabrication. In addition, graphene is a pure surface that is sensitive to contamination. It is challenging to keep graphene clean during fabrication. Here we present our graphene resonator fabrication process. We control the geometry of the cavity over which graphene is suspended to prevent the membrane from collapsing. Then we minimize the occurrence of fabrication residues on the supporting substrate and optimize the cleanliness and flatness of the interface between graphene and electrodes used for electrostatic actuation. After optimizing the fabrication of the graphene resonator, we measure the frequency response of our resonators using an optical interferometry setup. We control the resonant frequency of vibrational modes by applying a dc voltage between the membrane and an electrode patterned at the bottom of the cavity and verify that the response of our resonators is tunable over a wide frequency range.
We present a simple method to measure the width of a focused laser beam, where we define the width as the radius of the beam at which the intensity is 1/e2 of its on-axis value. Our method is based on measuring the power of light reflected off a metallized microstructure patterned on a oxidized silicon substrate that is placed on a tri-dimensional positioner. As the boundary between the microstructure and the substrate is scanned across the beam, changes in reflected power provide a quantitative measurement of the width of the beam. Our method is a useful alternative to the knife edge technique if optical measurements in transmission are not possible.
Mechanical resonators based on two-dimensional materials have gained attention for their interesting optical and mechanical properties, which translate into versatile applications such as ultrasensitive force detection and pressure sensing. Optical reflectometry is a technique of choice to measure the flexural vibrations of these resonators. The latter consists in sending normally incident monochromatic light on the resonator and measuring the intensity of reflected light, which varies as the distance d between the resonator and a nearby mirror varies. In this work we consider resonators based on suspended membranes of graphene, molybdenum disulfide and tungsten diselenide, and theoretically investigate the dependence of the reflectance R(d) of the resonator on the angle of incidence θ of the probing light. The optical response of these membranes is accounted for by their complex refractive indices. For s-polarized light, we find that R oscillates as a function of d with an amplitude that increases as theta increases. These results may help enhance the optical readout accuracy of these two-dimensional resonators.
Mechanical resonators based on suspended two-dimensional membranes are promising systems for developing sensitive detectors of mass, charge and force. To measure the flexural vibrations of the membrane, it is important to employ a technique capable of resolving tiny fluctuations of vibration amplitude. To this end, researchers have been developing optical detection methods based on Fabry-Perot interferences of light between the membrane and a mirror-like substrate, which relate the intensity of light reflected by the device to the distance between the membrane and the substrate. In this work, we calculate the membrane-to-substrate distances that maximize the optical responsivity of the resonator, which we define as the derivative of the resonator’s reflectivity with respect to membrane’s displacement. In addition, we examine how various substrates with different refractive indices affect this optical responsivity, including bare silicon, silicon coated with silicon oxide, dissipative metal mirrors, and non-dissipative Bragg reflectors. Our calculation method is based on the transfer matrix method for propagating electromagnetic fields. Our results are consistent with earlier theoretical and experimental results, and offer perspectives to enhance the optical responsivity of these mechanical resonators.
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