  In this work, we investigate the microscopic nature of the magnetism in
honeycomb iridium-based systems by performing a systematic study of how the
effective magnetic interactions in these compounds depend on various electronic
microscopic parameters. We show that the minimal model describing the magnetism
in A$_2$IrO$_3$ includes both isotropic and anisotropic Kitaev-type
spin-exchange interactions between nearest and next-nearest neighbor Ir ions,
and that the magnitude of the Kitaev interaction between next-nearest neighbor
Ir magnetic moments is comparable with nearest neighbor interactions. We also
find that, while the Heisenberg and the Kitaev interactions between nearest
neighbors are correspondingly antiferro- and ferromagnetic, they both change
sign for the next-nearest neighbors. Using classical Monte Carlo simulations we
examine the magnetic phase diagram of the derived super-exchange model.
Zigzag-type antiferromagnetic order is found to occupy a large part of the
phase diagram of the model and, for ferromagnetic next-nearest neighbor
Heisenberg interaction relevant for Na$_2$IrO$_3$, it can be stabilized even in
the absence of third nearest neighbor coupling. Our results suggest that a
natural physical origin of the zigzag phase experimentally observed in
Na$_2$IrO$_3$ is due to the interplay of the Kitaev anisotropic interactions
between nearest and next-nearest neighbors.
