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Spatially Resolved Optical Excitation of Mechanical Modes in Graphene NEMS

Emerging applications in nanoelectromechanical systems (NEMS) made from two-dimensional (2D) materials demand simultaneous imaging and selective actuation of the mechanical modes. Focused optical probes to measure and actuate motion offer a possible solution, but their lateral spatial resolution must be better than the size of the resonator. While optical interferometry is known to have excellent spatial resolution, the spatial resolution of the focused, laser-based optical driving is not currently known. Here, we combine separately scanned interferometry and optical drive probes to map the motion and forces on a suspended graphene nanomechanical resonator. By analyzing these maps with a force density model, we determine that the optical drive force has a spatial resolution on the order of the size of the focused laser spot. Using the optical force probe, we demonstrate the selective actuation and suppression of a pair of orthogonal, antisymmetric mechanical modes of the graphene resonator. Our results offer a powerful approach to image and actuate any arbitrary high-order mode of a 2D NEMS.

modes in bulk micromechanical beams because the resulting radiation pressure and photothermal bending forces are localized to the immediate vicinity of the laser spot 23 . When applied to 2D NEMS, however, it is unknown if optical driving can still achieve the same spatial resolution, as there are several reasons that the resolution could exceed the size of the resonator. For instance, 2D NEMS thermalize rapidly because of their ultralow intrinsic heat capacity (mass) and exceptionally high thermal conductivity 24 , which can make thermomechanical bending less local. 2D materials also poorly absorb and reflect light, which significantly decreases photon pressure, and they commonly have lateral dimensions (~2 − 5 μm) close the size of the laser spot. Nevertheless, optical driving has been employed to actuate 2D NEMS with both defocused (compared to the size of the membrane) and focused lasers 1-3,10,18,24-26 . The defocused drive laser, like gating, exerts a symmetric force and is therefore inefficient at driving higher-order modes. By using a narrowly focused drive laser and scanning it over the resonator, it is possible to infer the spatial resolution of the optical drive. However, experiments to date have either used static lasers, which can only measure mechanical spectra, or co-localized probe and drive lasers 1,10 , which convolve spatially resolved motion with actuation and therefore prevent an assessment of the spatial resolution.
To determine the spatial localization of the optical force of a focused laser on a 2D NEMS, we measure the vibrational amplitude of a suspended graphene membrane, which is proportional to the driving force, while we scan the position of a focused, driving laser across the membrane. By comparing the resulting force images to the mechanical mode shapes obtained by scanning optical interferometry, we find that the resolution of optical drive force is limited by the spot size of the laser, and this resolution is sufficient to efficiently and selectively actuate higher-order modes of the graphene membrane.
The graphene NEMS devices we study in this work are few μm diameter nanomechanical drumheads (a 3 μm device is pictured in Fig. 1(a)). We fabricate the drumheads by suspending single-layer graphene over cavities etched into SiO2 on Si using a semi-dry transfer process 27 . The devices are actuated using an amplitude modulated 445 nm laser 1-3,10,24-26 , and the amplitude and phase are measured using an interferometer operating at 532 nm and standard lock-in amplifier techniques, similar to previous work 13,15 . A schematic of our experimental setup is shown in (Fig. 1(b)). Both the 445 nm drive and 532 nm probe lasers are scanned using independent dual-axis galvo mirrors and coupled into the same optical path. This allows us to map the mechanical mode shape while driving the drumhead at a specific location. Conversely, we can probe the motion at a specific location, typically an antinode, while scanning the drive across the drumhead. In the following, we present results for a 3 μm diameter device (device 1) and a 5 μm device (device 2), however, we observe similar, reproducible results for other devices and across a range of drumhead sizes. We create two types of spatial maps of the membrane: mode maps and force maps. To obtain these maps, first we measure the frequency response spectrum of the graphene drumheads to find the mechanical resonance of the mode of interest. Then, we set the driving frequency well below the mechanical resonance to minimize the variation of the amplitude and phase response due to the position of the drive 15 . Typical amplitude and phase curves with the off-resonant drive frequency indicated are shown in Fig. 1(c). These off-resonant, fixed frequency approach allows scans to be completed in ~1 minute, which is 10-100 times faster than measuring the full frequency response at each point 15  The mode and force maps for the fundamental mode of device 1 (U01) are shown in Fig. 1(d) and 1(e), respectively. The graphene drumhead used to obtain this data has a resonance frequency of 15.7 MHz and a quality factor of Q ~120. The mode map amplitude ( Fig. 1(d), upper) exhibits azimuthal symmetry and it oscillates with a constant phase ( Fig. 1(d), bottom) of ~50° across the entire drum ( Fig. 1(d), bottom), consistent with theory. The constant phase implies that the relative displacement between the probe and drive lasers does not impact the speed of the mechanical response. The drive force map resembles the measured mode shape (Fig. 1(e), top); it has azimuthal symmetry and its amplitude falls off radially from the center, providing strong evidence that the position of the drive laser does impact the magnitude of the driving force. From the force map, we also see that the membrane responds even when the drive laser is well off the membrane; this has been observed previously 28 and is ascribed to a propagating heat wave. The force phase map is also azimuthally symmetric, but, in contrast to the mode phase map, the phase varies smoothly from 0~− 45° near membrane center to 0~− 100° at the edge of the drum.
The non-constant phase could be due to strain variations or other defects in the drumhead, which cause local changes to the thermal conductivity 26 . Our measurements of the fundamental mode verify the drumhead amplitude response depends on the position of the focused drive laser, and that spatial resolution is less than the membrane size.
To The non-zero we observe could be due to the thermal propagation time from one antinode to the other, similar to the nonuniform phase seen in Fig. 1(e). We note that the position dependence of the driving force phase could provide a new means to measure the position of a focused laser. force distribution will approximate any azimuthally symmetric force centered at the drive laser position, such as photothermal stress or photon pressure. To obtain ( ), we numerically integrate the overlap integral in Eq. 1 using ( ; ) and the theoretical mode shape of U11 for a circular membrane in two-dimensions. Cross-sections of ( ) perpendicular to the nodal line are shown in Fig 2(e) for various values of σ. For small values of σ (FWHM = 0.5 μm), we find that ( ) rises rapidly moving away from the nodal line and reaches a maximum before quickly falling off. As σ approaches the width of the membrane (FWHM = 1.2 μm), we find that ( ) increases slowly from the nodal line and reaches a nonzero value near the membrane boundary. In this case, the gaussian spot is large enough to have significant overlap with the mode shape, even when it is placed near the boundary. We can infer an experimental by adjusting in the simulated ( ) to match the measured force and mode maps for the two polarizations of the U11 mode. For the U11 V mode profile shown in Fig. 2(d), this process gives an experimental between 0.5 and 1 μm, a value approximately equal to the spot size of the focused laser in our setup.
Thus, this model predicts that the optical force is localized to the laser position within ≈ 1 μm. Though this procedure only yields an approximate value for the localization of the force density, the stark contrast between the measured force map and that predicted from a larger area force density (i.e. > 1 μm) strongly suggests that the drive force localized to a small region centered around the laser spot. Reducing the laser spot size by using shorter wavelengths or higher NA objectives could further enhance the control of the optomechanical drive efficiency, especially for smaller drumheads or beams which tend to vibrate at high frequencies 29 . The force map in this case is more complex than the mode map, unlike those shown for device 1. (Scale = 1 micron).
As further evidence of the local nature of the optical force in 2D NEMS, we demonstrate that a focused drive laser can selectively suppress or excite either polarization of the U11 mode. To show this, we position the probe laser at a location sensitive to the motion of U01 and both polarizations of U11, and then we measure the frequency response while the drive laser is positioned at four different locations on the membrane, all either on an antinode or node of the orthogonal U11 H and U11 V modes (see Fig. 3). The spectra show that the mode can be excited when the drive is placed on an antinode, or suppressed by ~75% when placed on a node. The suppression is sensitive to the beam shape and beam positioning-which we did not fully optimize-so it is possible to achieve a much higher degree of mode suppression. Suppressing individual modes of a degenerate pair is typically quite hard, since they overlap in frequency, making this technique useful for probing the motion of a single mechanical polarization 30 . Placing the drive laser at the point of a mode's maximum response also reduces the need for high laser powers, which can lead to irreversible changes in the device 25 . Although we only study the first three modes here, this technique could also be used at higher frequencies, where the dense spectrum of modes can overlap significantly 31 .
In some devices, we observe more complex behavior in the force map than the mode map would indicate. For device 2, the mode map appears to be highly symmetric, similar to theoretical predictions 15 as well as device 1. However, we see significant variation in the phase and amplitude response of the device in the force map and observe a "dead-zone", where the drive force disappears and a 180° phase change occurs ( Fig. 3(b)), much like crossing a nodal line. This is a likely due to a breakdown of our assumption of a gaussian force density and could be cause by various defects in the drumhead, such as adlayers or grain boundaries 26 . These observations indicate that the position of the drive force can drastically alter the expected amplitude and phase of a given mode. A better understanding of origin of this non-uniform phase response will be important for experiments which precisely measure the oscillation phase 24 .
In summary, we have combined spatially-resolved imaging with a force density model to infer the spatial resolution of the optical drive in a graphene nanomechanical resonator. Despite the fast thermalization, low reflectivity, and micrometer-scale size of the graphene resonator, we found that the optical force is localized to within 1 μm of the laser spot and can selectively and efficiently actuate high-order mechanical modes. The combination of high-spatial-resolution optical drive and read-out enables full multimodal control of suspended 2D nanomechanical resonators for future NEMS applications. Our highresolution, all-optical approach could be combined with optical beam shaping and spatial light modulation to selectively address an arbitrary subset of resonators within large arrays, a feat not easily achievable with electrostatic gating, or could serve as a point source of propagating mechanical waves for use in 2D nanomechanical circuits 32 and waveguides 33 .