  Hybrid superconductor(S)-semiconductor(Sm) devices bring a range of new
functionalities into superconducting circuits. In particular, hybrid
parity-protected qubits and Josephson diodes were recently proposed and
experimentally demonstrated. Such devices leverage the non-sinusoidal character
of the Josephson current-phase relation (CPR) in highly transparent S-Sm-S
junctions. Here we report an experimental study of superconducting
quantum-interference devices (SQUIDs) embedding Josephson field-effect
transistors fabricated from a SiGe/Ge/SiGe heterostructure grown on a 200-mm
silicon wafer. The single-junction CPR shows up to three harmonics with gate
tunable amplitude. In the presence of microwave irradiation, the ratio of the
first two dominant harmonics, corresponding to single and double Cooper-pair
transport processes, is consistently reflected in relative weight of integer
and half-integer Shapiro steps. A combination of magnetic-flux and gate-voltage
control enables tuning the SQUID functionality from a nonreciprocal
Josephson-diode regime with 27% asymmetry to a $\pi$-periodic Josephson regime
suitable for the implementation of parity-protected superconducting qubits.
These results illustrate the potential of Ge-based hybrid devices as versatile
and scalable building blocks of novel superconducting quantum circuits.
