Generalized Wigner crystallization in moiré materials

Abstract

Recent experiments on the twisted transition metal dichalcogenide (TMD) material, WSe2/WS2, have observed insulating states at fractional occupancy of the moiré bands. Such states were conceived as generalized Wigner crystals (GWCs). In this article, we investigate the problem of Wigner crystallization in the presence of an underlying (moiré) lattice. Based on the best estimates of the system parameters, we find a variety of homobilayer and heterobilayer TMDs to be excellent candidates for realizing GWCs. In particular, our analysis based on rs indicates that MoSe2 (among the homobilayers) and MoSe2/WSe2 or MoS2/WS2 (among the heterobilayers) are the best candidates for realizing GWCs. We also establish that due to larger effective mass of the valence bands, in general, hole-crystals are easier to realize that electron-crystals as seen experimentally. For completeness, we show that satisfying the Mott criterion n 1/2 Mott a * = 1 requires densities nearly three orders of magnitude larger than the maximal density for GWC formation. This indicates that for the typical density of operation, HoM or HeM systems are far from the Mott insulating regime. These crystals realized on a moiré lattice, unlike the conventional Wigner crystals, are incompressible due the gap arising from pinning with the lattice. Finally, we capture this many-body gap by variationally renormalizing the dispersion of the vibration modes. We show these low-energy modes, arising from coupling of the WC with the moiré lattice, can be effectively modeled as a Sine-Gordon theory of fluctuations.

Monolayer Cu₂X (X=S, Se): excellent thermoelectric material with low lattice thermal conductivity

“…中存在电阻随温度线性变化以及电阻随磁场线性变化的关系 [114] ，转角双层 WSe2 半填充附近的金属相中存在类似的行为 [32] 。在 WSe2/WS2 半填充处，谷 g 因子高达 270 [113] ，远超过单层 TMDs 的谷 g 因子，意味着摩尔平带增强了电子 间相互作用。通过光学与电输运实验证明，WSe2/WS2 半填充是具有 Curie-Weiss 行为的反铁磁态。AA 堆垛的 WSe2/MoTe2 异质结中的莫特绝缘态对平行磁场没 有任何响应，说明体系存在很强的 Ising 自旋轨道相互作用 [33] 。在 AB 堆垛的 WSe2/MoTe2 异质结中通过平行磁场的响应和非局域测量证明在全填充处存在一 个随电场变化的量子谷-自旋霍尔绝缘体 [34] 。TMDs 摩尔超晶格的磁输运是既丰 富又十分复杂的现象，受到电子相互作用、塞曼能变化以及不同散射机制的影 响。 许多强关联的现象需要样品保持低载流子浓度，在电输运测量中接触依然 是最重要的问题。为了解决这个问题，许多课题组采用非输运的方法来探测这 个体系中的强关联物理现象，例如利用扫描隧道显微镜直接观测电子态和态密 度，或者采用光学测量中的反射谱、光学探测电阻、微分电容、微波阻抗等探 测手段 [115][116][117][118][119] 。很多实验显示当电荷有序填充到 TMDs 的摩尔超晶格时，体系就 会出现关联的绝缘态，其中在整数填充处称为莫特绝缘态，在分数电荷填充处 称为广义的维格纳晶格 [117] 。Feng Wang 课题组利用扫描隧穿显微镜 (STM) 的 dI/dV 谱直接观测到 1/3、1/2、2/3、1 等电荷填充态 [118] 。值得注意的是魏格纳晶 格相的电输运信号在这个体系之前只在非常干净的 GaAs/AlGaAs 体系 [120] 以及石 墨烯体系 [121] 中观测到，同时还需要在极低温的条件测量 [122] [119] 。Jie Shan 等人巧妙地利用 WSe2 的 A 激子 1s 和 2s、3s 反射谱对临近样品绝缘性进行感应。WSe2 的 2s 反射谱可以探 测 WS2/WSe2 异质摩尔超晶格中非常丰富的关联绝缘态以及分数填充态 [117] 。 通过目前的实验结果，并结合二硫化钼导带独特的能带结构，我们在图 4 中预测转角双层二硫化钼中可能会出现许多丰富的强关联效应。 图 4 转角双层二硫化钼中可能存在的强关联现象 (a) 关联绝缘态。红线代表莫 特绝缘态。蓝色区域代表电荷局域态与费米液体共存的状态，称为轨道选择性 莫特态。黑色虚线内代表近藤晶格模型 [101] [123] ；(e) 量子 临界行为，红色和蓝色区域表示电阻随温度的依赖关系。其中α≠β (d) Wigner Crystal state. The figure shows representative 2/3 electron filling [123] ; (e) Quantum critical behaviors in which α≠β. The blue and red regions indicate the resistance dependence on temperature.…”

Charge-order-enhanced capacitance in semiconductor moiré superlattices

“…The charge-ordered insulating states at fractional fillings highlight the importance of the extended Coulomb interaction in addition to U (Ref. 10,15,28,29). To date, detection of these states has heavily relied on optical techniques, largely due to challenges in achieving good TMD-metal contacts for electrical studies, particularly at low charge density and low temperature.…”

Vortex Dynamics, Pinning, and Magic Angles on Moiré Patterns

“…They could also be applied to colloids interacting with optical traps, where it would be possible to change θ as a function of time. Additionally, there are proposals that the insulating state in some bilayer systems consists of a Wigner crystal that could undergo depinning transitions in which the threshold could exhibit dips at the commensurate angles [39,40].…”