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## Position- and momentum-dependent interlayer couplings in two-dimensional semiconductor moiré superlattices

Guo Rui-Ping, Yu Hong-Yi
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• #### 摘要

近些年来引起广泛关注的二维半导体莫尔超晶格系统中存在着莫尔激子、强关联电子态和面外铁电性等新奇物理现象, 电子的层间耦合对于理解这些现象至关重要. 本文研究了二维半导体双层莫尔超晶格中的层间耦合随位置和动量的变化. 外势场导致的局域布洛赫波包的层间耦合与波包宽度以及中心位置处的层间平移有着密切关系. 同时, 层间耦合随动量的变化使得基态$\rm{S}$型波包和激发态${\rm{P}}^{\pm }$型波包有着截然不同的随中心位置变化的层间耦合形式: 在两个$\rm{S}$型波包的层间耦合消失的位置, $\rm{S} $${\rm{P}}^{+} 型(或 \rm{S}$$ {\rm{P}}^{-}$型)波包之间的层间耦合达到最强. 利用该性质, 可以通过外加光电场来调控特定谷的基态波包的层间输运. 此外, 双层系统中发现的面外铁电性可以归结为不同层导带和价带间的耦合导致的电子在两层中的再分配现象. 将本文得到的层间耦合形式与单层紧束缚模型相结合, 可计算出垂直平面的电偶极密度, 其随层间平移的变化形式和数量级与实验观测相符.

#### Abstract

In recent years, various novel phenomena have been observed in two-dimensional semiconductor moiré systems, including the moiré excitons, strongly-correlated electronic states and vertical ferroelectricity. To gain an insight into the underlying physical mechanisms of these intriguing phenomena, it is essential to understand the interlayer coupling form of the electrons in moiré systems. In this work, the position- and momentum-dependent interlayer coupling effects in two-dimensional semiconductor moiré superlattices are investigated. Starting from the monolayer Bloch basis, the interlayer coupling between two Bloch states are treated as a perturbation, and the coupling matrix elements in commensurate and incommensurate bilayer structures are obtained, which are found to depend on the momentum and the interlayer translation between the two layers. Under the effect of an external potential, the Bloch states form localized wavepackets, and their interlayer couplings are found to depend on the wavepacket width as well as the interlayer translation at the wavepacket center position. Meanwhile the momentum-dependence results in very different interlayer coupling forms for the ground-state $\rm{S}$-type and the excited-state ${\rm{P}}^{\pm }$-type wavepackets. It is shown that at a position where the interlayer coupling between two $\rm{S}$-type wavepackets vanishes, the coupling between an $\rm{S}$-type wavepacket and a ${\rm{P}}^{+}$-type wavepacket (or between an $\rm{S}$- type wavepacket and a ${\rm{P}}^{-}$-type wavepacket) reaches a maximum strength. This can be used to manipulate the valley-selective interlayer transport of the ground-state wavepackets through external electric and optical fields. Besides, the vertical ferroelectricity recently discovered in bilayer systems can be attributed to the charge redistribution induced by the coupling between conduction and valence bands in different layers. Using the obtained interlayer coupling form combined with a simplified tight-binding model for the monolayer, the vertical electric dipole density can be calculated whose form and order of magnitude accord with the experimental observations.

#### 作者及机构信息

###### 通信作者: 俞弘毅, yuhy33@mail.sysu.edu.cn
• 基金项目: 国家自然科学基金 (批准号: 12274477) 资助的课题

#### Authors and contacts

###### Corresponding author: Yu Hong-Yi, yuhy33@mail.sysu.edu.cn
• Funds: Project supported by the National Natural Science Foundation of China (Grant No. 12274477)

#### 施引文献

• 图 1  双层TMDs中布洛赫电子之间的层间耦合　(a)层间转角为$\theta$的双层TMDs示意图, 其中蓝(橙)色大圆圈代表上(下)层的过渡金属原子M' (M), 蓝(橙)色小圆圈代表上(下)层的硫族原子X' (X); 下半部分为局部放大后得到的不同层的原胞, 平面坐标原点选在上层的某个过渡金属原子上; (b)上下层的六角布里渊区示意图; (c)在两层TMDs晶格完全匹配的情况下, (14)式中的$\left|{f}_{0}\left({{{\boldsymbol{r}}}}_{0}\right)\right|$, $\left|{f}_{+}\left({{{\boldsymbol{r}}}}_{0}\right)\right|$$\left|{f}_{-}\left({{{\boldsymbol{r}}}}_{0}\right)\right|随层间平移{{{\boldsymbol{r}}}}_{0}的变化 Fig. 1. Interlayer coupling in bilayer TMDs: (a) Schematic illustration of a bilayer TMDs with a twist angle \theta , where large blue (orange) circles stand for the transition-metal atoms M' (M) in the upper (lower) layer, small blue (orange) circles are the chalcogen atoms X' (X) in the upper (lower) layer. The lower part indicates an enlarged view of unit cells in two layers. The xy -coordinate origin is set on a transition-metal atom of the upper layer, and a nearby transition-metal atom in the lower layer has the spatial coordinate {{{\boldsymbol{r}}}}_{0}. (b) The upper- and lower-layer Brillouin zones. (c) The values of \left|{f}_{0}\left({{{\boldsymbol{r}}}}_{0}\right)\right|, \left|{f}_{+}\left({{{\boldsymbol{r}}}}_{0}\right)\right| and \left|{f}_{-}\left({{{\boldsymbol{r}}}}_{0}\right)\right| in Eq. (14) as functions of {{{\boldsymbol{r}}}}_{0} when the two TMDs lattices are fully commensurate. 图 2 莫尔超晶格中局域化的布洛赫波包之间的层间耦合 (a) 层间转角接近0°下周期为 \lambda 的莫尔超晶格示意图, 菱形为一个超原胞, \boldsymbol{A} , \boldsymbol{B} , \boldsymbol{C} 为三个范围小于 \lambda 同时又大于单层晶格常数a的局部区域, 其内部的原子排布与层间平移分别为 {\boldsymbol{r}}_{0}\left(\boldsymbol{A}\right)=0 , {\boldsymbol{r}}_{0}\left(\boldsymbol{B}\right)= ({{\boldsymbol{a}}_{1}+{\boldsymbol{a}}_{2}})/{3}$${\boldsymbol{r}}_{0}\left(\boldsymbol{C}\right)= {2\left({\boldsymbol{a}}_{1}+{\boldsymbol{a}}_{2}\right)}/{3}$的晶格匹配的双层结构接近. (b) 层间平移${\boldsymbol{r}}_{0}\left({\boldsymbol{R}}_{\rm{c}}\right)$随区域中心位置${\boldsymbol{R}}_{\rm{c}}$的变化, 其满足${\boldsymbol{r}}_{0}\left({\boldsymbol{R}}_{\rm{c}}\right)={\boldsymbol{r}}_{0}+\boldsymbol{R}-{\boldsymbol{R}}_{\rm{c}}$. (c) 通过外场调控两个局域在$\boldsymbol{B}$处的$\rm{S}$型波包之间的层间耦合, 其中黑色直线箭头代表与$S$型波包有关的非零层间耦合, 红(蓝)色波浪箭头代表用${\sigma}^{+}$(${\sigma}^{-}$)圆偏振光场实现同一层内部的$\rm{S}$型到${\rm{P}}^{+}$(${\rm{P}}^{-}$)型波包的激发. 上下层波包的相对能量可通过施加层间电压来调节. (d) 在双层系统中, 可通过施加${\sigma }^{+}$圆偏振光将下层中$-\boldsymbol{K}$谷的$\rm{S}$型波包通过层间跃迁移动到上层, 而$+\boldsymbol{K}$谷则被留在下层. 波包的局域化来源于上下层接触区域形成的莫尔超晶格势, 我们设下层感受到的势阱较深因此基态波包无法自由运动, 而上层感受到的势阱较浅因此基态波包可在层内运动, 它们在一个面内电场的作用下可产生谷流

Fig. 2.  The interlayer coupling between two localized Bloch wavepackets in the moiré superlattice. (a) Schematic illustration of a moiré superlattice with a period λ when the twist angle is close to 0°. The diamond shape corresponds to a supercell. $\boldsymbol{A}$, $\boldsymbol{B}$ and $\boldsymbol{C}$ are three local regions with spatial extensions smaller than λ but larger than the monolayer lattice constant $a$, whose atomic registrations are close to lattice-matched bilayer structures with interlayer translations ${\boldsymbol{r}}_{0}\left(\boldsymbol{A}\right)=0$, ${\boldsymbol{r}}_{0}\left(\boldsymbol{B}\right)= ({{\boldsymbol{a}}_{1}+{\boldsymbol{a}}_{2}})/{3}$ and ${\boldsymbol{r}}_{0}\left(\boldsymbol{C}\right)= {2\left({\boldsymbol{a}}_{1}+{\boldsymbol{a}}_{2}\right)}/{3}$, respectively. (b) The variation of the interlayer translation ${\boldsymbol{r}}_{0}\left({\boldsymbol{R}}_{\rm{c}}\right)$ with the local region center ${\boldsymbol{R}}_{\rm{c}}$, which satisfies ${\boldsymbol{r}}_{0}\left({\boldsymbol{R}}_{\rm{c}}\right)={\boldsymbol{r}}_{0}+\boldsymbol{R}-{\boldsymbol{R}}_{\rm{c}}$. (c) Tuning the interlayer coupling between two $\rm{S}$-type wavepackets localized at $\boldsymbol{B}$ with external fields. Black straight arrows indicate the nonzero interlayer couplings related to the $\rm{S}$-type wavepackets, while red (blue) wavy arrows stand for ${\sigma }^{+}$ (${\sigma }^{-}$) circularly-polarized optical fields which can excite an $\rm{S}$-type to a ${\rm{P}}^{+}$-type (${\rm{P}}^{-}$-type) wavepacket. The relative energy position between the upper- and lower-layer wavepackets can be tuned by an interlayer bias.(d) A schematic illustration of how to move the $-\boldsymbol{K}$ valley $\rm{S}$-type wavepackets from the lower- to upper-layer by applying a ${\sigma }^{+}$ circularly-polarized optical field. Meanwhile the $+\boldsymbol{K}$ valley wavepackets remain in the lower-layer. The wavepacket is localized by the moiré potential at the region where two layers are in contact. We assume that the carriers in the lower-layer feel deep potential wells thus cannot move, while those in the upper-layer feel shallow potential wells thus can move under the effect of an in-plane electric field, which results in a valley current.

图 3  电子在层间的再分配效应　(a)双层半导体TMDs或绝缘体hBN的费米面${E}_{\rm{F}}$示意图; (b)单层hBN的pz轨道紧束缚能带, 插图中${\Delta }{{{\boldsymbol{R}}}}_{\rm{1, 2}, 3}$是hBN晶格的3个最近邻位移矢量, $t$是最近邻跃迁强度; (c)计算得到的双层hBN层极化密度随层间平移${{{\boldsymbol{r}}}}_{0}$的变化; (d)三种不同${{{\boldsymbol{r}}}}_{0}$时的双层hBN原子排布, 上半部为上方视角, 下半部为侧方视角, 虚线箭头代表从N向B原子的电子转移, 其中蓝(橙)色大圆圈代表上(下)层的N原子, 蓝(橙)色小圆圈代表上(下)层的B原子

Fig. 3.  The interlayer charge redistribution effect: (a) Diagram of the Fermi level ${E}_{\rm{F}}$ of the bilayer semiconductor TMDs or insulator hBN; (b) the conduction and valence bands obtained from the pz-orbital tight-binding model for the monolayer hBN, in the inset, ${\Delta }{{{\boldsymbol{R}}}}_{\rm{1, 2}, 3}$ corresponds to the three nearest-neighbor displacement vectors, and $t$ is the nearest-neighbor hopping; (c) the calculated layer-polarization density in bilayer hBN as a function of the interlayer translation ${{{\boldsymbol{r}}}}_{0}$; (d) the atomic registries of bilayer hBN under three different ${{{\boldsymbol{r}}}}_{0},$ the upper (lower) parts correspond to the top-view (side-view). The dashed arrows denote the electron redistribution from N to B atoms. Here the large blue (orange) circles stand for the N atoms in the upper (lower) layer, small blue (orange) circles are the B atoms in the upper (lower) layer.

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##### 出版历程
• 收稿日期:  2022-10-26
• 修回日期:  2022-11-29
• 上网日期:  2022-12-21
• 刊出日期:  2023-01-20

## 二维半导体莫尔超晶格中随位置与动量变化的层间耦合

• 1. 中山大学物理与天文学院, 广东省量子精密测量与传感重点实验室, 珠海　519082
• 2. 中山大学, 光电材料与技术国家重点实验室, 广州　510275
• ###### 通信作者: 俞弘毅, yuhy33@mail.sysu.edu.cn
基金项目: 国家自然科学基金 (批准号: 12274477) 资助的课题

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