Towards tunable graphene phononic crystals

Our design of a mechanically tunable PnC requires a suspended graphene membrane so that the tension can be controlled by applying a voltage. This puts the natural limitation on device size as it is hard to create free-standing graphene with radii larger than 10 μm [27]. In order to improve the uniformity and to make the sample less sensitive to surface contaminations, we use a trilayer graphene. This comes at the cost of a reduced tunability compared to a monolayer sample but is necessary to clearly define a phononic band gap. To create the PnC structure, we pattern a regular honeycomb lattice of holes (diameter d) in the suspended graphene membrane. The unit cell and the corresponding first Brillouin zone of this lattice are shown in figure 1(a). The position and size of the phononic band gap in the structure depend on lattice parameter a, the ratio d/a, and the initial tension $T$ in the membrane. We chose d/a= 0.522 which maximizes the band gap size. For choosing the lattice parameter, we need to find a compromise. Ideally, we want to make a as small as possible, to fit a large number of unit cells into the finite suspended area. However, small lattice constants result in a high frequency range for the band gap, which will make the detection challenging, in terms of suitable detectors and electronics. Therefore, we use a lattice parameter of a = 1.36 μm, which corresponds to a PnC with approximately 7 unit-cells across in the armchair direction and 13 in the zigzag direction. Our fabrication protocol allows us to reproducibly pattern PnCs down to a lattice parameter a ∼100 nm. For smaller parameter values, the bridge separating the holes becomes narrower than 10 nm, and samples tend to break at this point. The final parameter of our PnC—the built-in tension in the sample—is beyond our control. An annealing step before pattering helps to reduce the tension as far as possible and cleans the device from fabrication-related residues. We calculate the phononic band structure of the PnC with these parameters and a built-in tension of 0.44 N m⁻¹ (obtained from experiments shown below) by matching the vibrational frequencies to a simulation in COMSOL (figure 1(b)). For this phononic lattice, we find a band gap between 28 and 34 MHz for out-of-plane modes. We note that only out-of-plane acoustic phonon modes are considered here, as they are most relevant in PnCs made from 2D materials because they are easy to excite and detect.

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To fabricate this structure, we use electron beam lithography and metallization on a SiO₂/Si wafer to define a hard mask for a subsequent wet-etching step (details in SI). After the etching, we have a ∼1 μm deep pit in the substrates on which we transfer an exfoliated trilayer graphene flake using the polymer (Polydimethylsiloxane, PDMS) stamp dry transfer [28–31]. This leaves us with a suspended graphene membrane, which is electrically contacted via the gold layer. We chose thin multilayer exfoliated graphene to reduce wrinkling in the device, for band gap uniformity and sample stability. Finally, the phononic lattice discussed above shown in figure 1(a) is cut in the graphene using a focus ion beam (figure 1(c)) [32]. In designing the device, our goal was to obtain high-uniformity devices with predictable boundary conditions yielding a well-defined band gap.

The PnC is measured inside a homebuilt vacuum chamber with base pressure better than 1.3 × 10⁻⁵ mbar (figure 1(d)). To drive the device, we use electrical actuation (an AC + DC voltage applied between the graphene and Si backgate, figure 1(d)) for its high efficiency and because it induces less heating and photodoping than e.g., optothermal actuation. By adjusting the DC component (V_g) of the applied voltage, we can tune the applied electrostatic pressure on the PnC and thus the tension in the suspended area. For detection, we use a 632.8 nm HeNe laser that is focused on the PnC. The suspended membrane and the backgate form a Fabry-Pérot interferometer along the z-direction and the reflected signal is thus modulated by the out-of-plane oscillation of the membrane [33]. We detect this signal in an avalanche photodetector (PD in figure 1(c)). To increase signal-to-noise ratio, we insert a 5 mm aperture before the PD.

The measurements of PnCs require resolving high-order vibrational modes which in general have very low vibrational amplitude. Therefore, multiple approaches to optimize the sensitivity of our detection scheme are needed. We adjust the cavity length in the etching step to maximize the interferometric signal (see SI), we also use a polarized laser beam in combination with a polarization beam splitter and a quarter wave plate that allows us to guide almost 100% of the reflected light into the PD. Finally, we implement a pinhole, which cuts out unwanted reflected light, and thereby reduces the laser noise in our detection scheme (details in SI). Overall, we obtain a displacement sensitivity better than 0.5 pm/$\sqrt {Hz}$ (see SI).

The measured vibrational spectrum of our PnC is shown in figure 1(e). We observe the fundamental mode at 2.97 MHz and ∼35 subsequent modes above the noise level. We track the fundamental modes vs. applied gate voltage and extract the built-in tension T₀ as well as the mass density ${\rho _{2{\text{D}}}}\,$ of the device from fitting the capacitive softening regime (see SI). We find ${T_0} = 0.44{\text{ N}}\,{{\text{m}}^{ - 1}}$ and ${\rho _{2{\text{D}}}} = 5.2 \times {10^{ - 5}}{\text{kg}}\,{{\text{m}}^{ - {\text{2}}}}$, respectively. These parameters are used as inputs for the simulations in figures 1(b) and (f). We prefer to use tension rather than strain as our key parameter since the former ultimately determines the frequency of our resonator ($f\sim \sqrt T$) while the latter depends on the Young's modulus of the patterned device, which is hard to measure directly. The extracted value for mass density is about 23 times higher compared to what is expected for pristine graphene, likely due to the impurities always present on the surface of 2D materials.

While we find a densely populated 'forest' of vibrational modes at frequencies above the fundamental mode, we see a region of strong suppression between 28 and 33 MHz (blue in figure 1(e)). This region is exactly where the band gap is expected from our band structure calculation. To better understand this spectrum, we carry out a 'finite' simulation accounting for the finite size of the PnC and considering the boundary conditions of the device. We can directly compare our experimental vibrational spectra (figure 1(e)) to the outcome of the simulation (figure 1(f)). We see a matching phononic band gap in both graphs, as well as similar peaks on both sides of the band gap, comparable to the peaks in the phononic density of states associated with flat bands at both sides of the band gap (figure 1(b)).

We note that the simulation does not match the measurements perfectly. This is likely due to unavoidable stress non-uniformity in our devices caused by nonlinearity of the cut drum-head and variations in boundary conditions. While the frequencies of modes from circular-membrane systems are strongly affected by these effects, the position of the phononic band gap is relatively insensitive to them. It is also noteworthy that both the simulation and experiments show modes inside the band gap. These are localized states related to the finite size non-patterned edges of our system (details in SI). Finally, it should be noted that the measurements are limited by the size of the laser spot size preventing us from resolving most of the modes with frequency much higher than the band gap. When we compare our data to measurements of non-patterned devices (see SI), we find only a few (∼5–8) lowest frequency modes compared to N > 35 in the patterned PnC sample. Also, we do not find a region of suppressed displacement in the reference samples (see SI). Both points highlight the effects of the phononic order in our devices.

Our next goal is to explore changes in vibrational spectra with induced tension and show in-situ tunability of our phononic system. As we vary the DC component of the gate voltage, we observe a smooth evolution of the vibrational spectrum with the fundamental mode downshifting from 2.96 MHz to 2.74 MHz and then upshifting to 7.92 MHz (figure 2). The initial downshift is due to the so-called capacitive softening [34–38], while the upshift is caused by the increase of tension. The higher order modes show a variety in different tuning behavior, which we attribute to a combination of photothermal backaction effects [39–41] and different coupling to the electrostatic field for different modes [42–44]. While there are interesting interactions taking place, this is not the scope of the work. We assume that the application of DC voltage produces tension, $T = \frac{{{f^2}}}{{f_0^2}}{T_0}$, where f₀ and f are the values of the fundamental mode frequency before and after the application of gate voltage. We find that we generate up to T − T₀ =2.68 N m⁻¹ additional tension at our highest V_g = 140 V.

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We now analyze the evolution of the phononic band gap region, which spans from 28 to 33 MHz at zero V_g. Experimentally, we find that the region upshifts together with other modes for increasing V_g. Using an effective spring constant extracted from the fundamental modes, we can simulate the evolution of the phononic band gap (green line in figure 2). We find that that region upshifts by 9 MHz at our highest tension. The region preserves a low vibrational amplitude. At the same time, we observe the appearance of new vibrational peaks inside it. This behavior also seen in simulations likely corresponds to new defect modes activated by the gradual violation of the translational symmetry and stronger drive power at higher voltages. Overall, the data in figure 2 suggest that the phononic band gap in the graphene PnC is mechanically tunable by 9 MHz (29%). Comparing this to previous theoretical predictions [22], we find an order of magnitude lower tunability upon applying a comparable pressure. This is due to the larger stiffness and increased built-in tension in the trilayer devices compared to monolayer samples (T₀ = 0.01 N m⁻¹) used in [22]. Furthermore, we can compare our results to another tunable phononic system made from graphene [23]. In this approach, the graphene membrane is clamped by many pillars forming a regular array. While the system is more rigid, it can be tuned by approximately 10%–15%. The device does not feature a phononic band gap.