We study the low-energy electronic structure of three-dimensional Dirac semimetal, Cd$_3$(As$_{1-x}$P$_x$)$_2$ [$x$ = 0 and 0.34(3)], by employing the angle-resolved photoemission spectroscopy (ARPES). We observe that the bulk Dirac states in Cd$_3$(As$_{0.66}$P$_{0.34}$)$_2$ are gapped out with an energy of 0.23 eV, contrary to the parent Cd$_3$As$_2$ in which the gapless Dirac states have been observed. Thus, our results confirm the earlier predicted topological phase transition in Cd$_3$As$_2$ with perturbation. We further notice that the critical P substitution concentration, at which the two Dirac points that are spread along the $c$-axis in Cd$_3$As$_2$ form a single Dirac point at $Gamma$, is much lower [x$_c$(P)$<$ 0.34(3)] than the predicted value of x$_c$(P)=0.9. Therefore, our results suggest that the nontrivial band topology of Cd$_3$As$_2$ is remarkably sensitive to the P substitution and can only survive over a narrow substitution range, i.e., 0 $leq$ x (P) $<$ 0.34(3).
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