Direct evidence for anisotropic three-dimensional magnetic excitations in a hole-doped antiferromagnet

We report a neutron-scattering study on ${\mathrm{Pr}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{NiO}}_{4}$ ($x\ensuremath{\approx}0.5$) with magnetic incommensurability $\ensuremath{\epsilon}=0.461$ at 10 K, showing direct evidence for out-of-plane magnetic excitations caused by a non-negligible interlayer spin coupling (${J}_{\ensuremath{\perp}}$). The magnetic spectrum is fully represented by linear spin wave theory considering an unconventional body-centered-like stacking of discommensurated spin stripes. A large easy-plane-type single-ion anisotropy ($\ensuremath{\approx}1.5$ meV) has been found to be responsible for a finite-energy ($\ensuremath{\approx}13$ meV) Goldstone mode at the magnetic zone center. Our results validate the presence of a sizable out-of-plane interaction resulting in an anisotropic three-dimensional nature of spin dynamics in ${\mathrm{Pr}}_{2\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{NiO}}_{4}$ near the half-doped region.