  Motivated by recent experiments exploring the spin-orbit-coupled magnetism in
$4d$- and $5d$-band transition metal oxides, we study magnetic interactions in
Ir- and Rh-based compounds. In these systems, the comparable strength of
spin-orbit coupling (SOC), crystal field splitting (CF) and Coulomb and Hund's
coupling leads to a rich variety of magnetic exchange interactions, leading to
new types of ground states. Using a strong coupling approach, we derive
effective low-energy super-exchange Hamiltonians from the multi-orbital Hubbard
model by taking full account of the Coulomb and Hund's interactions in the
intermediate states. We find that in the presence of strong SOC and lattice
distortions the super-exchange Hamiltonian contains various kinds of magnetic
anisotropies.
  Here we are primarily interested in the magnetic properties of Sr$_2$IrO$_4$
and Sr$_2$Ir$_{1-x}$Rh$_x$O$_4$ compounds. We perform a systematic study of how
magnetic interactions in these systems depend on the microscopic parameters and
provide a thorough analysis of the resulting magnetic phase diagram. Comparison
of our results with experimental data shows good agreement. Finally, we discuss
the parameter space in which the spin-flop transition in Sr$_2$IrO$_4$,
experimentally observed under pressure, can be realized.
