Pressure-enhanced superconductivity in Eu$_3$Bi$_2$S$_4$F$_4$

The pressure effect on the newly discovered charge-transferred BiS$_2$-based superconductor, Eu$_3$Bi$_2$S$_4$F$_4$, with a $T_c$ of 1.5 K at ambient pressure, is investigated by transport and magnetic measurements. Accompanied with the enhancement of metallicity under pressures, the onset superconducting transition temperature increases abruptly around 1.0 GPa, reaching $\sim$10.0 K at 2.26 GPa. AC magnetic susceptibility measurements indicate that a new superconducting phase with a higher $T_c$ emerges and dominates at high pressures. In the broad pressure window of 0.68 GPa$\leq$$p$$\leq$2.00 GPa, the high-$T_c$ phase coexists with the low-$T_c$ phase. Hall effect measurements reveal a significant difference in electronic structures between the two superconducting phases. Our work devotes the effort to establish the commonality of pressure effect on the BiS$_2$-based superconductors, and also uncovers the importance of electron carrier density in the high-$T_c$ phase.