Nanomechanical resonators (NRs) are physical systems hosting mesoscopic vibrational modes. Owing to their small masses and their high mechanical quality factors, they have found applications as ultrasensitive detectors, especially for mass and force sensing. They are also ideal systems to study the physics of mesoscopic vibrations, which is the topic of this paper. Of interest to us is the fact that (i) NRs respond nonlinearly to moderately strong driving forces, and (ii) their resonant frequencies are usually tunable on chip. Here we consider the dynamics of NRs where regimes (i) and (ii) are combined. While NRs in regime (i) can host vibrations with up to three possible amplitudes and phases, NRs in the regime combining (i) and (ii) can host up to five possible vibrational states. This added complexity can be exploited to encode information in the vibrations. Some of five states are stable while the others are unstable. Their existence depends on the large number of parameters of the vibrations, therefore, identifying the hierarchy of these states in terms of their amplitude, phase and stability is not straightforward. Such identification is however important to properly encode information in the vibrations. Here, we present an algorithm to identify the vibrational states and their hierarchy automatically. The input consists of the parameters of the vibrations and the range of drive frequencies. Solutions to the equation of motion are computed and stored in a N-by-M matrix, where N is the number of drive frequencies and M=5 is the maximum number of solutions. By scanning two chosen parameters of the vibrations, we use these matrices to build an image of the mechanical response in the chosen parameter space. Our algorithm recognizes features in the image that we identify as the emergence or the disappearance of new states. The order in which these states evolve as the drive frequency is changed dictates our information encoding protocol. Therefore, our algorithm facilitates the optimization of the vibration parameters and helps us design the resonators that are best suited for our type of information encoding.
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 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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