  Rare-earth iron garnets $R_{3}$Fe$_{5}$O$_{12}$ are fascinating insulators
with very diverse magnetic phases. Their strong potential in spintronic devices
has encouraged a renewal of interest in the study of their low temperature spin
structures and spin wave dynamics. A striking feature of rare-earth garnets
with $R$-ions exhibiting strong crystal-field effects like Tb-, Dy-, Ho-, and
Er-ions is the observation of low-temperature non-collinear magnetic structures
featuring "umbrella-like" arrangements of the rare-earth magnetic moments. In
this study, we demonstrate that such umbrella magnetic states are naturally
emerging from the crystal-field anisotropies of the rare-earth ions. By means
of a general model endowed with only the necessary elements from the crystal
structure, we show how umbrella-like spin structures can take place and
calculate the canting-angle as a result of the competition between the
exchange-interaction of the rare-earth and the iron ions as well as the
crystal-field anisotropy. Our results are compared to experimental data, and a
study of the polarised spin wave dynamics is presented. Our study improves the
understanding of umbrella-like spin structures and paves the way for more
complex spin wave calculations in rare-earth iron garnets with non-collinear
magnetic phases.
