Higher-dimensional Fermiology in bulk moiré metals

Abstract

In the past decade, moiré materials have revolutionized how we engineer and control quantum phases of matter. Among incommensurate materials, moiré materials are aperiodic composite crystals whose long-wavelength moiré superlattices enable tunable properties without chemically modifying their layers. To date, nearly all reports of moiré materials have investigated van der Waals heterostructures assembled far from thermodynamic equilibrium. 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 (Sr$_6$TaS$_8$)$_{1+δ}$(TaS$_2$)$_8$ that are exfoliatable van der Waals crystals with atomically incommensurate lattices. Lattice mismatches between alternating layers generate moiré superlattices, analogous to those of 2D moiré heterobilayers, that are coherent throughout these crystals and are tunable through their synthesis conditions without altering their chemical composition. High-field quantum oscillation measurements map the complex Fermiology of these moiré metals, which can be tuned via the moiré superlattice structure. We find that the Fermi surface of the structurally simplest moiré metal is comprised of over 40 distinct cross-sectional areas, the most observed in any material to our knowledge. This can be naturally understood by postulating that bulk moiré materials can encode electronic properties of higher-dimensional superspace crystals in ways that parallel well-established crystallographic methods used for incommensurate lattices. More broadly, our work demonstrates a scalable synthesis approach potentially capable of producing moiré materials for electronics applications and evidences a novel material design concept for accessing a broad range of physical phenomena proposed in higher dimensions.

Figures (5)

• Figure 1: Introduction to Bulk Moiré Materials.a, Schematic of a hypothetical hexagonal crystal (left panel) and its expected diffraction pattern (right panel). b, Same as a, for a monoclinic crystal. c, Same as a, for a hypothetical aperiodic composite crystal (top left panel). X-ray and electron diffraction patterns (right panel) show a lattice-mismatch of Bragg vectors between neighboring layers (bottom left panel) and a long-wavelength moiré superlattice represented by a short $q$-vector (bottom center panel). The relevant features of this diffraction pattern can be captured by three quantities: the commensurate Bragg plane order $n$, and the moiré wavevector angle $\phi$, and the moiré wavevector length $|\boldsymbol{q}|$. d, Single-crystal diffraction (SCXRD) measurement of (Sr$_6$TaS$_8$)$_{1+\delta}$(TaS$_2$)$_8$, a new aperiodic composite compound. e-f, Zoomed-in wide-angle X-ray scattering (WAXS) showing the moiré superlattice and lattice-mismatch between Sr$_6$TaS$_8$ and TaS$_2$ layers. g, Proposed highest symmetry model structure consistent with all structural and stoichiometric characterization measurements. Alternating Sr$_6$TaS$_8$ and TaS$_2$ layers are atomically incommensurate with one another along the a-axis, and they share commensurate Bragg planes along the b-axis. The black parallelogram outline denotes the intralayer unit cell of the Sr$_6$TaS$_8$ layer, which is incommensurate with TaS$_2$ along the a-axis. [hkl] crystallographic directions are defined with respect to the TaS$_2$ lattice.
• Figure 2: Tunable 1D Moiré Superlattices and 2D Moiré Approximants.a-e, Optical images (left panels) and wide-angle X-ray scattering measurements (right panels) of five related families of bulk moiré materials. The spacer Sr$_6$TaS$_8$ layer in each family "snaps" into $q_y$-oriented commensurate Bragg planes that 3- (a), 5- (b), 8- (c), 11- (d), and 13-tuple (e) the TaS$_2$ unit cell in this direction. Scale bars are 0.2 mm and 2 nm$^{-1}$. f, High-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) cross-sectional image of STS(8,81$^\circ$) (c), which directly visualizes how neighboring TaS$_2$ and Sr$_6$TaS$_8$ monolayers are mutually incommensurate with each other and are moiré-modulated in the z-direction. g, Line cuts of the HAADF-STEM image in f. Neighboring Sr$_6$TaS$_8$ and TaS$_2$ layers are atomically incommensurate with one another. h, Simplified visualizations of Sr$_6$TaS$_8$ and TaS$_2$ lattices. i-k, Large field-of-view, simplified visualizations of representative Sr$_6$TaS$_8$ / TaS$_2$ moiré patterns, realized via tunable interlayer couplings between Sr$_6$TaS$_8$ and TaS$_2$ layers. Commensurate Bragg plane conditions ($a_0$-multiple right labels), the moiré wavelengths $\lambda_m$ (top labels), and their rotational alignment $\phi$ with respect to the TaS$_2$ lattice (bottom left labels) are all extracted quantitatively from X-ray diffraction measurements to construct these visualizations. Spacer unit cells are shown by black-outlined red-filled parallelograms. All percentages represent measured deviations of the lattice compression (red numbers), expansion (green numbers), and row-sliding shear (black numbers) of the Sr$_6$TaS$_8$ layer lattice relative to an average spacer lattice model. STS(3,90$^\circ$) represents the simplest structural member of the family with a purely 1D moiré superlattice, while STS(5,72$^\circ$) and STS(8,81$^\circ$) are representative of structures that are strictly 1D moiré superlattices, but are akin to 2D moiré approximants due to additional lattice aliasing in the commensurate b-axis.
• Figure 3: Tunable Fermiology in Moiré Cognate Crystals.a, High-field torque magnetization measurements of quantum oscillations in STS(8,81$^\circ$) for the field $\mu_0 H$ rotated by a polar angle $\theta$ within the crystal's bc-plane and scaled by the out-of-plane component $\mu_0 H \cos{\theta}$. The moiré $q$-vector in STS(8,81$^\circ$) points along a non-crystallographic, low-symmetry direction, 81$^\circ$ from its b-axis. b, Same as a, for STS(8,90$^\circ$), which is related to STS(8,81$^\circ$) via different degrees of interlayer hetero-shear. The moiré $q$-vector in STS(8,90$^\circ$) points along a crystallographic, high-symmetry direction, 90$^\circ$ from its b-axis (i.e., along the a-axis). c-d, FFTs of the temperature-dependent background-subtracted torque magnetization $\Delta M_\tau$ of STS(8,81$^\circ$) (c) and STS(8,90$^\circ$) (d), showing cascades of low-frequency oscillations. Careful comparisons of these frequencies suggest that nearby Fermi pockets merge with one another (c) and split away from one another (d) as the layers of these bulk moiré materials shear against one another to sweep the moiré superlattice from a low-symmetry direction (81$^\circ$) to a high-symmetry direction (90$^\circ$) with respect to the a-axis. Insets: Zoomed-in plots to show additional peaks in the FFT with lower FFT amplitudes.
• Figure 4: de Haas-van Alphen Frequency Comb in the Fermiology of a Simple Moiré Metal.a-b, High-field torque magnetization measurements of quantum oscillations in two crystals of STS(3,90$^\circ$) for the field $\mu_0 H$ rotated by a polar angle $\theta$ within the crystal's bc-plane and scaled by the out-of-plane component $\mu_0 H \cos{\theta}$. STS(3,90$^\circ$) is a simple, purely 1D moiré metal with its moiré $q$-vector pointed along its incommensurate a-axis (right inset of d). c-d, FFTs of the temperature-dependent background-subtracted torque magnetization $\Delta M_\tau$ of each sample of STS(3,90$^\circ$), showing an abundance of oscillations that are all roughly linearly spaced in frequency. FFT amp. scaled in right panels by 12.2$\times$ (c) and 8$\times$ (d) for clarity. Black triangle markers identify FFT peaks that are closely shared between data sets. Grey triangle markers identify FFT peaks that are either at slightly different frequencies between data sets or are only observed in only one data set. Left insets: Zoomed-in plots of the ultra-low frequency regime of the FFT. Right inset of c: Extracted FFT peak frequencies as a function of peak index, showing a series of linearly spaced frequencies spaced by roughly $\Delta F \cos{\theta} \approx 40 \text{ T}$, a value not represented by any observed peak in the low-frequency limit. Fitted peak index regions (red shaded regions) identify subsequences of FFT peaks that appear as clear combs of FFT peaks with relatively similar amplitudes. A linear fit of the entire domain similarly yields linear peak spacings of $\Delta F \cos{\theta} \approx 44.6 \text{ T}$ with $R^2 = 0.996$, a value again not represented by any observed peak in the low-frequency limit.
• Figure 5: Moiré Metals as Quantum Materials with Synthetic Superspace Dimensions.a-b, Schematic of the periodic lattice (a) and the quasi-2D Fermi surface (b) of a quasi-2D crystalline metal. c, Schematic a quasi-2D aperiodic composite / moiré metal lattice. d, Schematic of the violated translation symmetry of the aperiodic composite / moiré lattice, which prohibits a complete description of this metal’s Fermiology in 2D. e, Schematic of the emergent, higher-dimensional, periodic superspace lattice described by a lower-dimensional aperiodic composite lattice. By elevating the quasi-2D aperiodic lattice into a new superspace dimension $\zeta$, an emergent translation symmetry is restored in ($2 + 1$)D. f, Schematic of the resulting ($2 + 1$)D Fermi surface, leveraging the restored translation symmetry. Stacks of quasi-2D Fermi surfaces are linked along a new discrete synthetic dimension $k_\zeta$. g, Schematic of an extremal Fermi surface orbit in a crystalline metal, whose associated frequency is given by Onsager's relation. The wavy line depicts the phase of the electronic wavefunction, which is $2 \pi$-periodic. h, Same as g, for a moiré metal. Superspace cyclotron orbits have phase discontinuities in 3D, but are continuous and $2 \pi$-periodic in 4D. A linear sequence of frequencies stem from superspace Fermi orbits that coherently hop into the synthetic dimension. i-j, Schematics of the oscillations frequencies and cyclotron orbits in a quasi-2D crystalline metal (g) and a ($2 + 1$)D superspace moiré metal (h). Cyclotron orbits in the ($2 + 1$)D superspace moiré metal propagate by discrete momentum quanta in the synthetic dimension, producing a dense sequence of roughly linearly spaced de Haas-van Alphen oscillations frequencies.