Recent advances in van der Waals heterostructures have opened the new frontier of moir\'{e} 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 moir\'{e} engineering to materials that host incommensurate orders, where we discovered a long-period, thermally-hysteretic moir\'{e} superlattice in a layered charge density wave (CDW) compound, EuTe4. 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 moir\'{e} superstructure with an in-plane period of 13.6~nm. Due to different out-of-plane orders of the incommensurate CDWs, the moir\'{e} 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 moir\'{e} 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.
@article{arxiv.2501.09715,
title = {Large moir\'{e} superstructure of stacked incommensurate charge density waves},
author = {B. Q. Lv and Yifan Su and Alfred Zong and Qiaomei Liu and Dong Wu and Noah F. Q. Yuan and Zhengwei Nie and Jiarui Li and Suchismita Sarker and Sheng Meng and Jacob P. C. Ruff and N. L. Wang and Nuh Gedik},
journal= {arXiv preprint arXiv:2501.09715},
year = {2025}
}