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By using high pressure and high temperature (3.7 GPa, 1120 $^{\circ}$C)synthesis technique, we have grown (Cu,C)Ba$_2$Ca$_3$Cu$_4$O$_{11+\delta}$single crystals. X-ray diffraction, scanning electron microscopy, resistivityand magnetization measurements are carried out and all show that the sampleshave good quality. The single crystal has onset and zero-resistance transitiontemperatures of about 111 K and 109.6 K, indicating a very narrow transitionwidth, which is consistent with a rather sharp magnetization transition.Magnetization hysteresis loops (MHLs) are also measured, showing a pronouncedsecond peak effect in the intermediate temperature region. The magneticcritical current density calculated from the MHLs at 77 K and 1.5 T is about6.4$\times$10$^4$ A/cm$^2$. By using a criterion of 1$\%$ normal stateresistivity, we have determined the irreversibility line which exhibits anirreversibility field of about 8 T at 77 K. Compared with other layeredsystems, it is easy to find that the irreversibility line is rather high andcould be further improved with the optimized transition temperature of about118 K as previously discovered in polycrystalline samples.

Characterization of the (Cu,C)Ba$_2$Ca$_3$Cu$_4$O$_{11+\delta}$ single crystals grown under high pressure