Non-collinear magnetic structures of TbCoO$_3$ and DyCoO$_3$

The orthoperovskites TbCoO$_3$ and DyCoO$_3$ with Co$^{3+}$ in a non-magnetic low-spin state have been investigated by neutron diffraction down to 0.25 K. Magnetic ordering is evidenced below $T_N=3.3$ K and 3.6 K, respectively, and the ordered arrangements are of canted type, A$_x$G$_y$ for TbCoO$_3$ and G$_x$A$_y$ for DyCoO$_3$ in Bertaut's notation. The experiments are confronted with the first-principle calculations of the crystal field and magnetism of Tb$^{3+}$ and Dy$^{3+}$ ions, located in the $Pbnm$ structure on sites of $C_s$ point symmetry. Both these ions exhibit an Ising behavior, which originates in the lowest energy levels, in particular in accidental doublet of non-Kramers Tb$^{3+}$ ($4f^8$ configuration) and in ground Kramers doublet of Dy$^{3+}$ ($4f^9$) and it is the actual reason for the non-collinear AFM structures. Very good agreement between the experiment and theory is found. For comparison, calculations of the crystal field and magnetism for other systems with Kramers ions, NdCoO$_3$ and SmCoO$_3$, are also included.