Josephson effect of massive pseudospin-1 fermions in the ferromagnetic dice lattice

We theoretically study the Josephson effect of massive pseudospin-1 fermions in a superconductor/ferromagnet/superconductor (S/F/S) junction based on the dice lattice, including $S_z$- and $+U$-type mass terms with the magnitude $m$. For the $S_z$-type fermions, it is found that, in the absence of ferromagnetic exchange field $h$, the supercurrent is effectively inhibited by $m$ or the junction length $L$. In the presence of $h$, $0-\pi$ state transition can be realized by modulating $h$, $m$, or $L$. The $m$-induced state transition occurs more frequently for a larger $h$ or $L$. Particularly, the inhibition of supercurrent caused by $m$ still remains for a large $m$. In the doping case, corresponding to a nonzero $E_F$, the $m$-induced state transition may disappear for a large $h$ or $L$. With $E_F$ increased, the inhibition caused by $m$ diminishes for a small $h$ whereas intensifies for a large $h$. In addition, the supercurrent features for the $+U$-type fermions distinct from those for the $S_z$-type fermions are demonstrated.