  Recently, the Josephson diode effect (JDE), in which the superconducting
critical current magnitudes differ when the currents flow in opposite
directions, has attracted great interest. In particular, it was demonstrated
that gate-defined Josephson junctions based on magic-angle twisted bilayer
graphene showed a strong nonreciprocal effect when the weak-link region is
gated to a correlated insulating state at half-filling (two holes per moir\'e
cell). However, the mechanism behind such a phenomenon is not yet understood.
In this work, we show that the interaction-driven valley polarization, together
with the trigonal warping of the Fermi surface, induce the JDE. The valley
polarization, which lifts the degeneracy of the states in the two valleys,
induces a relative phase difference between the first and the second harmonics
of supercurrent and results in the JDE. We further show that the nontrivial
current phase relation, which is responsible for the JDE, also generates the
asymmetric Shapiro steps.
