Higher-dimensional Fermiology in bulk moiré metals

In the past decade, moiré materials have revolutionized how we engineer and control quantum phases of matter^1,2. They are versatile platforms for strongly correlated electronic phenomena^3,4 and support new ferroelectric^5,6, magnetic⁷ and superconducting states⁸. Among incommensurate materials⁹, moiré materials are aperiodic composite crystals^10,11 whose long-wavelength superlattices enable tunable properties without chemically modifying their layers. So far, nearly all reports of moiré materials have investigated van der Waals heterostructures assembled far from thermodynamic equilibrium (T < 150 °C)^1,2. Here we introduce a conceptually new approach to synthesizing high-mobility moiré materials in thermodynamic equilibrium. We report a new family of foliated superlattice materials (Sr₆TaS₈)_1+δ(TaS₂)₈ that are exfoliatable, incommensurate-lattice, van der Waals crystals. Lattice mismatches between alternating layers generate moiré superlattices, analogous to 2D moiré heterobilayer superlattices, which are coherent throughout these crystals and tunable through synthesis conditions without altering their chemical composition. Quantum oscillation measurements map the complex Fermiology of these moiré metals^12-14, showing that the Fermi surface of the structurally simplest moiré metal comprises more than 40 distinct cross-sectional areas. This is naturally understood by proposing that these bulk moiré metals encode electronic properties of higher-dimensional superspace crystals in ways paralleling well-established crystallographic methods for incommensurate lattices^15,16. More broadly, our work demonstrates a scalable synthesis approach potentially capable of producing large-area moiré materials for electronics applications and evidences a new material design concept for accessing phenomena proposed in higher dimensions^17-21.