Quantum phase diagram of the $S=1\mathrm{/}2$ triangular-lattice antiferromagnet Ba${}_3$CoSb${}_2$O${}_9$

The magnetic phases of the ideal spin-1/2 triangular-lattice antiferromagnet Ba${}_3$CoSb${}_2$O${}_9$ are identified and studied using ${}^{135,137}$Ba nuclear magnetic resonance (NMR) spectroscopy in magnetic fields ranging to 30T, oriented parallel and near perpendicular to the crystallographic $ab$-plane. For both directions, the saturation field is approximately 33T. Notably, the NMR spectra provide microscopic evidence for the stabilization of an up-up-down spin configuration for in-plane fields, giving rise to an one-third magnetization plateau ($M_{\text{sat}}\mathrm{/}3$), as well as for a higher field phase transition near to $\sim (3\mathrm{/}5)M_{\text{sat}}$ for both field orientations. Phase transitions are signaled by the evolution of the NMR spectra, and in some cases through spin-lattice relaxation measurements. The results are compared with expectations obtained from a semi-classical energy density modeling, in which quantum effects are incorporated by effective interactions extracted from the spin-wave analysis of the two-dimensional model. The interlayer coupling also plays a significant role in the outcome. Good agreement between the model and the experimental results is achieved, except for the case of fields approaching the saturation value applied along the c-axis.