The transport properties of a topological Josephson junction fabricated from a magnetically doped topological insulator (TI) were investigated. The conductance spectra of the Nb/Fe-Bi$_2$Te$_2$Se/Nb junction below 1 K showed an unusual trident-shaped zero-bias conductance peak with a tiny peak width of $sim$ 6 $mu$V. The central peak of the trident peak presents the dc-Josephson current, and the side peaks may reflect an induced unconventional Cooper pairing. Additionally, the critical currents followed inverse to temperature, which may also reflect the presence of an unconventional proximity effect. Furthermore, microwave irradiation derived a drastic change in the conductance spectra from the peak structure into oscillatory ones, a hallmark of the ac-Josephson supercurrent. The current-phase relation of the ac-Josephson effect under high power radiofrequency-irradiation was found to be 4$pi$-periodic. The results suggest that the junction based on magnetically doped 3D TIs may realize an unconventional Cooper pairing, thus enabling access to the basic physics of Majorana bound states and unconventional superconductivity.
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