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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.-
Keywords:
- two-dimensional semiconductor /
- transition metal dichalcogenides /
- moiré superlattice /
- interlayer coupling
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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$\left|{f}_{0}\left({{{\boldsymbol{r}}}}_{0}\right)\right|$ ,$\left|{f}_{+}\left({{{\boldsymbol{r}}}}_{0}\right)\right|$ 和$\left|{f}_{-}\left({{{\boldsymbol{r}}}}_{0}\right)\right|$ 随层间平移${{{\boldsymbol{r}}}}_{0}$ 的变化Figure 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 $ ,${\boldsymbol{r}}_{0}\left(\boldsymbol{B}\right)= ({{\boldsymbol{a}}_{1}+{\boldsymbol{a}}_{2}})/{3}$ 和${\boldsymbol{r}}_{0}\left(\boldsymbol{C}\right)= {2\left({\boldsymbol{a}}_{1}+{\boldsymbol{a}}_{2}\right)}/{3}$ 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Figure 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Figure 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.表 1 导带和价带布洛赫函数在K点的C3量子数在不同旋转中心下的取值, 其中M为过渡金属, X为硫族原子, h为M和X组成的正六边形中心
Table 1. The
$ {C}_{3} $ quantum numbers of the conduction and valence band Bloch functions at K for different rotation centers. Here M is the transition-metal site, X is the chalcogen site, and h is the hollow center of the hexagon formed by M and X.M X h $ {\psi }_{\rm{c}, \mathit{K}} $ 0 –1 +1 $ {\psi }_{\rm{v}, \mathit{K}} $ –1 +1 0 
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表 2 局域在图2(a)中
$ \boldsymbol{A} $ ,$ \boldsymbol{B} $ 和$ \boldsymbol{C} $ 区域的$ \rm{S} $ 和$ {\rm{P}}^{\pm } $ 型波包的$ {C}_{3} $ 量子数, 其中旋转中心始终取为上层TMDs晶格的过渡金属M'. 这里只显示了谷指标为$ \tau =+ $ 和${\tau }{'}=+$ 的情况, 而$ \tau =- $ 和${\tau }{'}=-$ 可通过时间反演得到Table 2. The
$ {C}_{3} $ quantum numbers of$ \rm{S} $ - and$ {\rm{P}}^{\pm } $ -type wavepackets localized at A, B and C in Fig. 2(a). Here we set the rotation center as a transition-metal site M' in the upper TMDs layer. Only those with the valley indices$ \tau =+ $ and$\tau' =+ $ are shown, while the cases for$ \tau =- $ and$ \tau' =- $ can be obtained by a time reversal.能带 波包位置 上层($ {\tau }^{'}=+ $) 下层($ \tau =+ $) $ {\rm{S}}^{\rm{'}} $ $ {\rm{P}}^{\rm{'}+} $ $ {\rm{P}}^{\rm{'}-} $ $ \rm{S} $ $ {\rm{P}}^{+} $ $ {\rm{P}}^{-} $ $ \rm{c} $ $ \mathit{A} $ 0 +1 –1 0 +1 –1 $ \mathit{B} $ 0 +1 –1 +1 –1 0 $ \mathit{C} $ 0 +1 –1 –1 0 +1 $ \rm{v} $ $ \mathit{A} $ –1 0 +1 –1 0 +1 $ \mathit{B} $ –1 0 +1 0 +1 –1 $ \mathit{C} $ –1 0 +1 +1 –1 0
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[1] Wang Q H, Kalantar-Zadeh K, Kis A, Coleman J N, Strano M S 2012 Nat. Nanotech. 7 699
Google Scholar
[2] Mak K F, Shan J 2016 Nat. Photon. 10 216
Google Scholar
[3] Jariwala D, Sangwan V K, Lauhon L J, Marks T J, Hersam M C 2014 ACS Nano 8 1102
Google Scholar
[4] Schaibley J R, Yu H, Clark G, Rivera P, Ross J S, Seyler K L, Yao W, Xu X 2016 Nat. Rev. Mater. 1 16055
Google Scholar
[5] Xiao D, Liu G-B, Feng W, Xu X, Yao W 2012 Phys. Rev. Lett. 108 196802
Google Scholar
[6] Geim A K, Grigorieva I V 2013 Nature 499 419
Google Scholar
[7] Mak K F, Shan J 2022 Nat. Nanotech. 17 686
Google Scholar
[8] Huang D, Choi J, Shih C-K, Li X 2022 Nat. Nanotech. 17 227
Google Scholar
[9] Wang Y, Wang Z, Yao W, Liu G-B, Yu H 2017 Phys. Rev. B 95 115429
Google Scholar
[10] Zhang C, Chuu C-P, Ren X, Li M-Y, Li L-J, Jin C, Chou M-Y, Shih C-K 2017 Sci. Adv. 3 e1601459
Google Scholar
[11] Yu H, Liu G-B, Tang J, Xu X, Yao W 2017 Sci. Adv. 3 e1701696
Google Scholar
[12] Yu H, Liu G-B, Yao W 2018 2 D Mater. 5 035021
Google Scholar
[13] Wu F, Lovorn T, MacDonald A H 2018 Phys. Rev. B 97 035306
Google Scholar
[14] Wu F, Lovorn T, Tutuc E, Martin I, MacDonald A H 2019 Phys. Rev. Lett. 122 086402
Google Scholar
[15] Yu H, Chen M, Yao W 2020 Natl. Sci. Rev. 7 12
Google Scholar
[16] Seyler K L, Rivera P, Yu H, Wilson N P, Ray E L, Mandrus D G, Yan J, Yao W, Xu X 2019 Nature 567 66
Google Scholar
[17] Tran K, Moody G, Wu F, et al. 2019 Nature 567 71
Google Scholar
[18] Jin C, Regan E C, Yan A, Utama M I B, Wang D, Zhao S, Qin Y, Yang S, Zheng Z, Shi S, Watanabe K, Taniguchi T, Tongay S, Zettl A, Wang F 2019 Nature 567 76
Google Scholar
[19] Baek H, Brotons-Gisbert M, Koong Z X, Campbell A, Rambach M, Watanabe K, Taniguchi T, Gerardot B D 2020 Sci. Adv. 6 eaba8526
Google Scholar
[20] Zhang L, Zhang Z, Wu F, Wang D, Gogna R, Hou S, Watanabe K, Taniguchi T, Kulkarni K, Kuo T, Forrest S R, Deng H 2020 Nat. Commun. 11 5888
Google Scholar
[21] Brotons-Gisbert M, Baek H, Campbell A, Watanabe K, Taniguchi T, Gerardot B D 2021 Phys. Rev. X 11 031033
Google Scholar
[22] Tang Y, Li L, Li T, Xu Y, Liu S, Barmak K, Watanabe K, Taniguchi T, MacDonald A H, Shan J, Mak K F 2020 Nature 579 353
Google Scholar
[23] Regan E C, Wang D, Jin C, Utama M I B, Gao B, Wei X, Zhao S, Zhao W, Zhang Z, Yumigeta K, Blei M, Carlström J D, Watanabe K, Taniguchi T, Tongay S, Crommie M, Zettl A, Wang F 2020 Nature 579 359
Google Scholar
[24] Shimazaki Y, Schwartz I, Watanabe K, Taniguchi T, Kroner M, Imamoğlu A 2020 Nature 580 472
Google Scholar
[25] Wang L, Shih E M, Ghiotto A, et al. 2020 Nat. Mater. 19 861
Google Scholar
[26] Chu Z, Regan E C, Ma X, Wang D, Xu Z, Utama M I B, Yumigeta K, Blei M, Watanabe K, Taniguchi T, Tongay S, Wang F, Lai K 2020 Phys. Rev. Lett. 125 186803
Google Scholar
[27] Xu Y, Liu S, Rhodes D A, Watanabe K, Taniguchi T, Hone J, Elser V, Mak K F, Shan J 2020 Nature 587 214
Google Scholar
[28] Huang X, Wang T, Miao S, Wang C, Li Z, Lian Z, Taniguchi T, Watanabe K, Okamoto S, Xiao D, Shi S-F, Cui Y-T 2021 Nat. Phys. 17 715
Google Scholar
[29] Liu E, Taniguchi T, Watanabe K, Gabor N M, Cui Y-T, Lui C H 2021 Phys. Rev. Lett. 127 037402
Google Scholar
[30] Li H, Li S, Regan E C, Wang D, Zhao W, Kahn S, Yumigeta K, Blei M, Taniguchi T, Watanabe K, Tongay S, Zettl A, Crommie M F, Wang F 2021 Nature 597 650
Google Scholar
[31] Li H, Li S, Naik M H, Xie J, et al. 2021 Nat. Phys. 17 1114
Google Scholar
[32] Shimazaki Y, Kuhlenkamp C, Schwartz I, Smoleński T, Watanabe K, Taniguchi T, Kroner M, Schmidt R, Knap M, Imamoğlu A 2021 Phys. Rev. X 11 021027
Google Scholar
[33] Jin C, Tao Z, Li T, Xu Y, Tang Y, Zhu J, Liu S, Watanabe K, Taniguchi T, Hone J C, Fu L, Shan J, Mak K F 2021 Nat. Mater. 20 940
Google Scholar
[34] Rogée L, Wang L, Zhang Y, Cai S, Wang P, Chhowalla M, Ji W, Lau S P 2022 Science 376 973
Google Scholar
[35] Yasuda K, Wang X, Watanabe K, Taniguchi T, Jarillo-Herrero P 2021 Science 372 1458
Google Scholar
[36] Woods C R, Ares P, Nevison-Andrews H, Holwill M J, Fabregas R, Guinea F, Geim A K, Novoselov K S, Walet N R, Fumagalli L 2021 Nat. Commun. 12 347
Google Scholar
[37] Niu R, Li Z, Han X, et al. 2022 Nat. Commun. 13 6241
Google Scholar
[38] Stern M V, Waschitz Y, Cao W, Nevo I, Watanabe K, Taniguchi T, Sela E, Urbakh M, Hod O, Shalom M B 2021 Science 372 1462
Google Scholar
[39] Weston A, Castanon E G, Enaldiev V, et al. 2022 Nat. Nanotech. 17 390
Google Scholar
[40] Wang X, Yasuda K, Zhang Y, Liu S, Watanabe K, Taniguchi T, Hone J, Fu L, Jarillo-Herrero P 2022 Nat. Nanotech. 17 367
Google Scholar
[41] Liang J, Yang D, Wu J, Dadap J I, Watanabe K, Taniguchi T, Ye Z 2022 Phys. Rev. X 12 041005
Google Scholar
[42] Zheng Z, Ma Q, Bi Z, et al. 2020 Nature 588 71
Google Scholar
[43] Liu G-B, Xiao D, Yao Y, Xu X, Yao W 2015 Chem. Soc. Rev. 44 2643
Google Scholar
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Google Scholar
[45] Bistritzer R, MacDonald A H 2011 Proc. Natl. Acad. Sci. USA 108 12233
Google Scholar
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Google Scholar
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Google Scholar
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