Evolution of Critical Current Density in CaKFe$_4$As$_4$ with La-doping

Single crystals of (Ca$_{1-x}$La$_x$)KFe$_4$As$_4$ (0 <= x <=0.16) have been grown by using the self-flux method, and the evolution of physical properties including the critical current density (Jc) with La-doping has been investigated. Tc decreases monotonically with increasing x, while Jc at the same temperature and magnetic field increases initially and reach its maximum at x = 0.082. The increase in Jc is more obvious at low temperatures and high fields. At T = 5 K and H = 40 kOe, Jc reaches 0.34 MA/cm$^2$, which is ~4 times larger than that for pure crystals. It is also found that anomalous temperature dependence of Jc in CaKFe$_4$As$_4$ is wiped away as the La content is increased. However, Jc shows non-monotonic field dependence (peak effect) at high fields in crystals with large x. In addition, we found that despite weak anisotropy of H$_{c2}$, there is extremely large anisotropy of Jc up to ~15, which is most likely caused by novel planar defects in the crystal, similar to CaKFe$_4$As$_4$. Jc characteristics in (Ca$_{1-x}$La$_x$)KFe$_4$As$_4$ with disorder outside FeAs planes is compared with that in CaK(Fe$_{1-x}$Co$_x$)$_4$As$_4$ with disorder within FeAs planes.