Large moiré superstructure of stacked incommensurate charge density waves

TL;DR

The study demonstrates a giant moiré superstructure arising from stacking two incommensurate charge density waves (I-CDWs) in EuTe$_{4}$, enabling moiré engineering within a single crystal. Using high-flux, high-resolution X-ray diffraction on ultrathin EuTe$_{4}$ flakes, the authors resolve two coexisting CDWs with in-plane wavevectors $q_1 \ approx 0.644(5) b^*$ and $q_2 \ approx 0.678(5) b^* + 0.5 c^*$, producing an in-plane moiré period of about 13.6 nm. These CDWs adhere to a joint commensuration with the lattice, $q_1 + 2 q_{2, ext{in-plane}} = 2 b^*$, while interlayer coupling induces a reconstructed, partially commensurate moiré with $\ delta' = 0.037 b^*$ corresponding to a real-space period of 27 unit cells; the primary moiré component remains incommensurate with a period of 13.6 nm. The moiré pattern exhibits pronounced thermal hysteresis due to metastable out-of-plane CDW configurations, explaining the giant resistivity hysteresis and light/pulse-induced states, and suggesting that electronic gating could tune the moiré by shifting CDW wavevectors. Overall, this work reveals a new route to moiré engineering based on stacked incommensurate orders and highlights interlayer ordering as a key determinant of macroscopic properties in layered CDW materials.

Abstract

Recent advances in van der Waals heterostructures have opened the new frontier of moiré physics, whereby tuning the interlayer twist angle or adjusting lattice parameter mismatch have led to a plethora of exotic phenomena such as unconventional superconductivity and fractional quantum spin Hall effect. We extend the concept of moiré engineering to materials that host incommensurate orders, where we discovered a long-period, thermally-hysteretic moiré superlattice in a layered charge density wave (CDW) compound, EuTe$_\text{4}$. Using high-momentum-resolution X-ray diffraction performed on ultrathin flakes, we found two coexisting, incommensurate CDWs with slightly mismatched in-plane wavevectors. The interaction between these two CDWs leads to their joint commensuration with the high-symmetry lattice as well as a large moiré superstructure with an in-plane period of 13.6~nm. Due to different out-of-plane orders of the incommensurate CDWs, the moiré superstructure exhibits a clear thermal hysteresis, accounting for the large hysteresis observed in electrical resistivity and numerous metastable states induced by light or electrical pulses. Our findings pave the way for a new development in moiré engineering based on an incommensurate lattice. They further highlight the important role of interlayer ordering in determining the macroscopic properties of these stacked incommensurate structures.

Figures (4)

• Figure 1: Moiré superlattice formed by stacking two CDWs.a--c, Pathways towards the construction of a moiré superlattice: a, Stacking of twisted homo-bilayer, where two identical layers of crystals are stacked with a twist angle. b, Stacking of aligned hetero-bilayers, where two layers of different crystals with similar lattice parameters are stacked to form a moiré pattern. c, A new way of constructing moiré superlattice, where multiple charge density waves (CDWs) with slightly different wavevectors are aligned in parallel, forming a beating pattern and hence moiré superlattice. d, Comparison between incommensurate CDW (I-CDW) and commensurate CDW. For an I-CDW, the ratio between the wavevector and the lattice constant is an irrational number; for a commensurate CDW, it is a rational number. $m$ and $n$ are co-prime positive integers. e, Crystal structure of EuTe$_4$ and the schematic of the X-ray reciprocal space mapping experiment. The structure of EuTe$_4$ features alternating monolayer and bilayer Te square-net sheets. The synchrotron X-ray diffraction measurement was performed in transmission geometry. The crystal was rotated 360$^\circ$ continuously to cover the entire reciprocal space. The diffraction signals were detected by a large-area X-ray detector.
• Figure 2: X-ray diffraction pattern and refined CDW structure of EuTe$_\text{4}$.a, $(2,K,L)$ cut of x-ray diffraction (XRD) reciprocal space mapping at 300 K in the heating branch, featuring two sets CDW satellite peaks at integer and half-integer $L$ respectively. b, line-cut of the diffraction image in a along $L=0$ and $L=-0.5$. c, Schematic of charge density wave distortion configuration in EuTe$_4$ refined from a combination of powder XRD and reciprocal space mapping data. The integer $L$ and half-integer $L$ CDW satellites, respectively, correspond to CDW distortions in monolayer and bilayer Te sheets. The phase of bilayer CDW shifts by 180$^\circ$ in adjacent unit cells, forming a two-unit-cell superlattice and hence manifesting the half-integer CDW peaks in the out-of-plane direction.
• Figure 3: Jointly commensurate CDW orders induced by charge transfer.a, Schematics of charge transfer in real space (left) and consequent band structure shifts in reciprocal space, indicated in the schematic Fermi surface (top right) and schematic band dispersion along the Y-$\Gamma$-Y direction (bottom right) of the non-CDW state of EuTe$_4$ from tight-binding calculation. The charge transfer process can be understood as doping the nominally charge-neutral square-like sheet in the Te bilayer with 1 hole while doping monolayer sheets with 2 electrons. This results in an energy shift in monolayer (solid blue curve) and bilayer (solid red curve) bands from the nominally charge-neutral band position (dotted black curve) in opposite directions. Taking commensurate wavevector $\mathbf{q}=2/3\mathbf{b}^*$ as a starting point, the in-plane components of wavevectors in monolayer and bilayer will respectively change by $-2\Delta$ and $\Delta$ (in units of $\mathbf{b}^*$), respectively, due to the change in the geometry of electronic bands. b, CDW in-plane wavevector amplitudes of monolayer ($q_1$) and bilayer ($q_2$) as a function of temperature, showing temperature invariance across a thermal loop spanning from 100 K to 400 K. c, Temperature dependence of the joint commensuration condition $q_1+2q_2$ and the reconstructed moiré superlattice wavevector amplitude ($\delta'$), both showing negligible temperature-dependent changes within experimental uncertainties.
• Figure 4: Giant thermal hysteresis in EuTe$_\text{4}$ due to metastable domains of CDW stacking disorder.a, $(2, K, L)$ cuts of reciprocal space mapping at different temperatures in a thermal cycle: 350 K → 100 K → 400 K → 300 K [see (i)--(iv)]. b--d, Ratio of intensity in the region of interest (ROI) (dashed over solid for each pair) indicated in a for different $K$ values, labeled as C2, C1, and C3, respectively. The dashed and solid ROIs represent regions where satellite peaks are forbidden and allowed respectively in the ground state without disorder. These ratios reflect the degree of stacking disorder due to thermal excitations. e, Schematics of CDW configurations for ground state (G) and metastable states (M). The up and down arrows indicate the phase of CDWs with respect to the lattice. The yellow shade highlights the difference in the monolayer CDW phase between the two configurations. Only one type of metastable state is demonstrated here for simplicity. A more detailed discussion of metastable state configurations can be found in Supplementary Information. f, Real-space schematics of ground state (G, red) and metastable states (M, black) domains, which evolve over the temperature cycle; the schematics are inferred from the diffraction data. The white boundaries indicate domain walls. The specific configuration of domain distribution may vary throughout the thickness of the sample while the ratio between the area of G and M domains remains similar through out the sample.