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The link between crystal and electronic structure is crucial for understanding structure-property relations in solid-state chemistry. In particular, it has been instrumental in understanding topological materials, where electrons behave differently than they would in conventional solids. Herein, we identify 1D Bi chains as a structural motif of interest for topological materials. We focus on Sm$_3$ZrBi$_5$, a new quasi-one-dimensional (1D) compound in the Ln$_3$MPn$_5$ (Ln = lanthanide; M = metal; Pn = pnictide) family that crystallizes in the P$6_{3}$/mcm space group. Density functional theory calculations indicate a complex, topologically non-trivial electronic structure that changes significantly in the presence of spin-orbit coupling. Magnetic measurements show a quasi-1D antiferromagnetic structure with two magnetic transitions at 11.7 and 10.7 K that are invariant to applied field up to 9 T, indicating magnetically frustrated spins. Heat capacity, electrical, and thermal transport measurements support this claim and suggest complex scattering behavior in Sm$_3$ZrBi$_5$. This work highlights 1D chains as an unexplored structural motif for identifying topological materials, as well as the potential for rich physical phenomena in the Ln$_3$MPn$_5$ family.