  Very recently, a 3$d$ based honeycomb cobaltate Na$_2$Co$_2$TeO$_6$ has
garnered tremendous attention due to the proposed proximity to the Kitaev
spin-liquid state as its 4$d$/5$d$ counterparts. Here, we use Zn to substitute
Co in a broad range and perform systematic studies on
Na$_2$Co$_{2-x}$Zn$_x$TeO$_6$ by structural, magnetic, and thermodynamic
measurements, and track the doping evolution of its magnetic ground states. Due
to the extremely close radii of Zn$^{2+}$ and high-spin Co$^{2+}$ ions, the
substitution can be easily achieved. X-ray diffractions reveal no structural
transition but only minor changes on the lattice parameter $c$ over a wide
substitution range $0 \leq x \leq 1.5$. Magnetic susceptibility and specific
heat measurements both suggest an antiferromagnetic ground state which is
gradually suppressed with doping. It can survive with $x$ up to $\sim1.0$. Then
it evolves into a spin-glass phase with short-range order that is rapidly
supplanted by a magnetically disordered state when $x \geq 1.3$. By summarizing
all these data, we construct a magnetic phase diagram of
Na$_2$Co$_{2-x}$Zn$_x$TeO$_6$. Our results demonstrate that the Zn doping can
effectively suppress the magnetic order and induce a possibe quantum
paramagnetic state. These may serve as a platform to investigate the Kitaev
physics in this system.
