Probing complex stacking in a layered material via electron-nuclear quadrupolar coupling

For layered materials, the interlayer stacking is a critical degree of freedom tuning electronic properties, while its microscopic characterization faces great challenges. The transition-metal dichalcogenide 1T-TaS$_2$ represents a novel example, in which the stacking pattern is not only enriched by the spontaneous occurrence of the intralayer charge density wave, but also recognized as a key to understand the nature of the low-temperature insulating phase. We exploit the $^{33}\rm{S}$ nuclei in a 1T-TaS$_2$ single crystal as sensitive probes of the local stacking pattern via quadrupolar coupling to the electron density distribution nearby, by combining nuclear magnetic resonance (NMR) measurements with the state-of-the-art first-principles electric-field gradient calculations. The applicability of our proposal is analyzed through temperature, magnetic-field, and angle dependent NMR spectra. Systematic simulations of a single 1T-TaS$_2$ layer, bilayers with different stacking patterns, and typical stacking orders in three-dimensional (3D) structures unravel distinct NMR characteristics. Particularly, one 3D structure achieves a quantitative agreement with the experimental spectrum, which clearly rationalizes the coexistence of two types of interfacial environments. Our method may find general applications in the studies of layered materials.