Quantum Oscillations in Graphene Using Surface Acoustic Wave Resonators

Surface acoustic waves (SAWs) provide a contactless method for measuring wave-vector-dependent conductivity. This technique has been used to discover emergent length scales in the fractional quantum Hall regime of traditional, semiconductor-based heterostructures. SAWs would appear to be an ideal match for van der Waals heterostructures, but the right combination of substrate and experimental geometry to allow access to the quantum transport regime has not yet been found. We demonstrate that SAW resonant cavities fabricated on ${\mathrm{LiNbO}}_3$ substrates can be used to access the quantum Hall regime of high-mobility, hexagonal boron nitride encapsulated, graphene heterostructures. Our work establishes SAW resonant cavities as a viable platform for performing contactless conductivity measurements in the quantum transport regime of van der Waals materials.

Figure 1

Combining vdW materials with surface acoustic waves. (a) Schematic of a SAW resonator driven by a radio frequency (rf) source across the interdigital transducers (IDTs). A vdW heterostructure is placed in the center of the cavity and connected to gate electrodes. (b) The change in SAW velocity ($\delta v/v$) and attenuation ($\mathrm{\Gamma }$) as a function of the longitudinal sheet conductivity of a material placed in contact with the resonator. The transition region, where the velocity change is largest and there is a peak in the attenuation, is determined by the parameters of the particular piezoelectric substrate (see Supplemental Material [15]). (c) Simulation of a SAW standing-wave resonator on lithium niobate. A conductive sample in the middle of the cavity is shown in gray. Black arrows indicate the direction and magnitude of the electric field at the surface of the substrate. In the sample region, the conducting sample screens the transverse component of the electric field: this interaction with the electric field feeds back into the stiffness of the substrate lattice, modifying the speed of sound and sound attenuation as shown in (b).

Figure 2

Quantum transport in graphene using a surface acoustic wave resonator. (a) Optical microscope images of the device. The lower image shows an enlarged view of the full SAW resonator cavity. The image on the top left shows an enlarged view of the center of the cavity with the detail of the prepatterned gold electrodes without the graphene sample (the six transport electrodes were not used in this study). The image on the top right is a false-color image showing the graphene device in detail. The monolayer graphene is colored in light purple and the graphite top gate is colored in light green. (b) Amplitude and phase of the transmitted voltage through SAW resonant cavity as a function of frequency. Two resonances are indicated as $f_1$ and $f_2$. (c) The change in the cavity phase at fixed frequency $f_1$, as a function of magnetic field, at two different gate voltages, and at 1.7 K. Quantum oscillations are clearly visible above 0.6 T, and oscillations related to degeneracy-broken Landau levels are observed above 3 T. The data have been offset for clarity. (d) Landau fan diagram at 1.7 K, from 1 to 8 T. The color scale corresponds to the shift in SAW velocity, proportional to the shift in cavity phase through Eq. (3).