  Asymmetric two-junction SQUIDs with different current-phase relations in the
two Josephson junctions, involving higher Josephson harmonics, demonstrate a
flux-tunable Josephson diode effect (asymmetry between currents flowing in the
opposite directions, which can be tuned by the magnetic flux through the
interferometer loop). We theoretically investigate influence of junction
capacitance and thermal fluctuations on performance of such Josephson diodes.
Our main focus is on the "minimal model" with one junction in the SQUID loop
possessing the sinusoidal current-phase relation and the other one featuring
additional second harmonic. Capacitance generally weakens the diode effect in
the resistive branch (R state) of the current-voltage characteristic (CVC) both
in the absence and in the presence of external ac irradiation. At the same
time, it leads to qualitatively new features of the Josephson diode effect such
as asymmetry of the retrapping currents (which are a manifestation of
hysteretic CVC). In particular, the limiting case of the single-sided
hysteresis becomes accessible. In its turn, thermal fluctuations are known to
lead to nonzero average voltages at any finite current, even below the critical
value. We demonstrate that in the diode regime, the fluctuation-induced voltage
can become strongly (exponentially) asymmetric. In addition, we find asymmetry
of the switching currents arising both due to thermal activation and due to
Josephson plasma resonances in the presence of ac irradiation.
