Measuring bulk entropy in bilayer graphene fractional quantum Hall states using integrated graphite-aluminum single electron transistors

Single-electron transistors (SETs) can be used as high-sensitivity charge sensors, allowing direct and local measurement of a two dimensional system's electronic chemical potential. Maxwell relations then relate this chemical potential to other thermodynamic properties such as the entropy, magnetization, or electrical polarization. Previous experiments to measure the chemical potential in van der Waals heterostructures have relied on SETs constructed of non-van der Waals materials, such as aluminum, limiting measurements to samples without a van der Waals top gate and as a result limiting sample quality. Here, we demonstrate a fabrication technique that incorporates a planar graphite layer directly into the on-stack SET, allowing the SET to function as a high-quality top gate, providing access to phases which depend on the applied out-of-plane displacement field and rely on ultra-high sample quality found only in all-van der Waals heterostructures. We apply this technique to probe the bulk entropy of dual-gated bilayer graphene at high magnetic field, where the `topological degeneracy' of the non-Abelian quasiparticles expected near certain even-denominator fractional quantum Hall states is expected to contribute 2 kB ln(2) of entropy per electron charge, well within our detection limit.