First-principles study of magnetic structures of triangular antiferromagnets NaYbS$_2$ and NaYbO$_2$

We investigate the magnetic interactions in triangular rare-earth delafossites materials NaYbO$_2$ and NaYbS$_2$ via first-principles calculations. The calculated Curie-Weiss temperatures are in good agreement with experiments. We perform classical Monte Carlo simulations of the two compounds using the extracted exchange parameters. We find that if only the nearest neighbor interactions are considered, the magnetic ground states of NaYbO$_2$ and NaYbS$_2$ are a stripe and a planar 120\degree~ N\'{e}el state, respectively. The simulated transition temperatures are much higher than the lowest experimental temperatures, where no magnetic ordering was observed. However, we show by adding suitable second neighbor interactions, the {\it classical} magnetic ground state of NaYbO$_2$ becomes to the $Z_2$ vortex phase, and the simulated specific heat $C_v$ are very similar to the experimental observations, with no obvious phase transition down to the extremely low temperature.