Bismuth chalcogenides Bi$_2$Se$_3$ and Bi$_2$Te$_3$ are semiconductors, which can be both thermoelectric materials (TE) and topological insulators (TI). Lattice defects arising from vacancies, impurities, or dopants in these materials are important in that they provide the charge carriers in TE applications and compromise the performance of these materials as TIs. We present the first solid-state nuclear magnetic resonance (NMR) study of the $^{77}$Se and $^{125}$Te NMR resonances in polycrystalline powders of Bi$_2$Se$_3$ and Bi$_2$Te$_3$, respectively. The spin-lattice ($T_1$) relaxation is modeled by at most two exponentials. Within the framework of this model, the NMR measurement is sensitive to the distribution of native defects within these materials. One component corresponds to a stoichiometric fraction, an insulator with a very long $T_1$, whereas the other component is attributed to a sample fraction with high defect content with a short $T_1$ resulting from interaction with the conduction carriers. The absence of a very long $T_1$ in the bismuth telluride suggests defects throughout the sample. For the bismuth selenide, defect regions segregate into domains. We also find a substantial difference in the short $T_1$ component for $^{125}$Te nuclei (76 ms) and $^{77}$Se (0.63 s) in spite of the fact that these materials have nearly identical lattice structures, chemical and physical properties. Investigations of the NMR shift and Korringa law indicate that the coupling to the conduction band electrons at the chalcogenide sites is much stronger in the telluride. The results are consistent with a stronger spin-orbit coupling (SOC) to the $p$-band electrons in the telluride. If most parameters of a given material are kept equal, this type of experiment could provide a useful probe of SOC in engineered TI materials.
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