Spin waves near the edge of halogen substitution induced magnetic order in Ni(Cl$_{1-x}$Br$_x$)$_2$$\cdot$4SC(NH$_2$)$_2$

We report an inelastic neutron scattering study of magnetic excitations in a quantum paramagnet driven into a magnetically ordered state by chemical substitution, namely Ni(Cl$_{1-x}$Br$_x$)$_2$$\cdot$4SC(NH$_2$)$_2$ with $x=0.21(2)$. The measured spectrum is well accounted for by the generalized spin wave theory (GSWT) approach [M. Matsumoto and M. Koga, [J. Phys. Soc. Jap. 76, 073709 (2007)]. This analysis allows us to determine the effective Hamiltonian parameters for a direct comparison with those in the previously studied parent compound and `underdoped' system. The issue of magnon lifetimes due to structural disorder is also addressed.