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Magnetic ordering and spin dynamics in $S=5/2$ staggered triangular lattice antiferromagnet Ba$_2$MnTeO$_6$

We report studies of the magnetic properties of a staggered stacked triangular lattice Ba$_2$MnTeO$_6$ using magnetic susceptibility, specific heat, neutron powder diffraction and inelastic neutron scattering measurements, as well as first principles density functional theory calculations. Neutron diffraction measurements reveal an antiferromagnetic order with a propagated vector $\textbf{\emph{k}}=(0.5, 0.5, 0)$ and N{é}el transition temperature of $T_\text{N}\approx20$ K. The dominant interaction derived from the Curie-Weiss fitting to the inverse DC susceptibility is antiferromagnetic. Through modelling the INS spectrum with the linear spin wave theory, the magnetic exchange interactions for the nearest intralayer, nearest interlayer, and next nearest interlayer are determined to be $J_1=0.27(3),J_2=0.27(3),$ and $J_3=-0.05(1)$ meV, respectively, and a small value of easy-axis anisotropy of $D_{zz}=-0.01$ meV is introduced. We derive a magnetic phase diagram that reveals that it is the competition between $J_1, J_2$, and $J_3$ that stabilizes the collinear stripe-type antiferromagnetic order.