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The occurrence of the proton bubble-like structure has been studied withinthe relativistic Hartree-Fock-Bogoliubov (RHFB) and relativisticHartree-Bogoliubov (RHB) theories by exploring the bulk properties, the chargedensity profiles and single proton spectra of argon isotopes and $N = 28$isotones. It is found that the RHFB calculations with PKA1 effectiveinteraction, which can properly reproduce the charge radii of argon isotopesand the $Z=16$ proton shell nearby, do not support the occurrence of the protonbubble-like structure in argon isotopes due to the prediction of deeper boundproton orbit $\pi2s_{1/2}$ than $\pi1d_{3/2}$. For $N = 28$ isotones, $^{42}$Siand $^{40}$Mg are predicted by both RHFB and RHB models to have the protonbubble-like structure, owing to the large gap between the proton $\pi2s_{1/2}$and $\pi1d_{5/2}$ orbits, namely the $Z=14$ proton shell. Therefore, $^{42}$Siis proposed as the potential candidate of proton bubble nucleus, which haslonger life-time than $^{40}$Mg.

Central depressions in the charge density profiles of the nuclei around $^{46}$Ar