The transparent interface in epitaxial Al-InAs heterostructures provides an excellent platform for potential advances in mesoscopic and topological superconductivity. Semiconductor-based Josephson Junction Field Effect Transistors (JJ-FETs) fabricated on these heterostructures have a metallic gate that tunes the supercurrent. Here we report the fabrication and measurement of gate-tunable Al-InAs JJ-FETs in which the gate dielectric in contact with the InAs is produced by mechanically exfoliated hexagonal boron nitride (h-BN) followed by dry transfer using a van der Waals-mediated pick up process. We discuss the fabrication process that enables compatibility between layered material transfer and Al-InAs heterostructures to avoid chemical reactions and unintentional doping that could affect the characteristics of the JJ-FET. We achieve full gate-tunablity of supercurrent by using only 5~nm thick h-BN flakes. We contrast our process with devices fabricated using a conventional AlO$_{rm x}$ gate dielectric and show that h-BN could be an excellent competing dielectric for JJ-FET devices. We observe that the product of normal resistance and critical current, I$_{rm c}$R$_{rm n}$, is comparable for both types of devices, but strikingly higher R$_{rm n}$ for the h-BN-based devices indicating that the surface is doped less compared to AlO$_{rm x}$ gate dielectric.
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