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过渡金属二硫化物MX2/三卤化铬CrX3组成的范德瓦耳斯异质结能有效操控MX2的谷极化, 在能谷电子学中有广泛的应用前景. 本文结合第一性原理和k投影能带反折叠方法比较研究了MoSe2/CrI3, MoSe2/CrBr3和WS2/CrBr3三种磁性范德瓦耳斯异质结的堆垛和电子结构, 探索了体系谷极化产生的物理机理. 计算了异质结不同堆垛的势能面, 确定了稳定的堆垛构型, 阐明了时间/空间反演对称破缺对体系电子结构的影响. 由于轨道杂化, 磁性异质结的导带情况复杂, 且MoSe2/CrI3体系价带顶发生明显变化, 不能与单层MX2直接对比. 而借助于反折叠能带, 计算清晰揭示了CrX3对MX2电子结构的影响, 定量地获得了MX2的能谷劈裂, 并发现层间距和应变可以有效调控能谷劈裂. 当层间距减小到2.6 Å 时, AB堆垛的MoSe2/CrI3谷劈裂值可达到10.713 meV, 是平衡结构的8.8倍, 相当于施加约53 T的外磁场. 通过k投影能带反折叠方法克服了异质结超胞电子结构不易分析的局限性, 对其他磁性范德瓦耳斯体系的研究具有重要的借鉴意义.The transition metal dichalcogenides MX2/Chromium Trihalides CrX3 van der Waals heterostructures can control the valley polarization of of MX2 effectively, which makes them possess promising potential applications in valleytronics. In the present work, the stacking order and electronic structure of MoSe2/CrI3, MoSe2/CrBr3 and WS2/CrBr3 are investigated based on the first-principle calculation and k-projection band unfolding method. The underlying mechanism of valley splitting is also explored. The stacking energy surfaces are calculated and the stable stacking configurations are determined. The effects of the breaking of time-symmetry and spatial-symmetry on electronic structure are also revealed. Because of the orbital hybridization, the conduction band of heterostructure becomes complicated and the valence band maximum changes drastically. It is thus difficult to compare the electronic structure of vdW heterostructure with that of free-standing MX2 directly. Through the unfolding energy band, the electronic structure change of MX2 induced by CrX3 is revealed clearly, and the valley splitting of MX2 is obtained quantitatively. Moreover, the interlayer distance and strain are found to be able to tune the valley splitting effectively. When the interlayer distance reduces to 2.6 Å, the valley splitting of MoSe2/CrI3 is enhanced to 10.713 meV with the increase of AB stacking, which is 8.8 times as large as the value of equilibrium structure. This work breaks through the limit of the complex electronic structure in supercell, providing an important reference for studying other magnetic vdW heterostructure.
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Keywords:
- valleytronics /
- heterostructure /
- stacking order /
- band unfolding
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图 1 MoSe2/CrI3异质结的结构和布里渊区 (a), (b)对应于AB和AB' 堆垛的MoSe2/CrI3结构; (c)大六边形和小六边形分别对应MoSe2原胞布里渊区和异质结布里渊区
Fig. 1. Structure and Brillouin zone of MoSe2/CrI3 heterostructure: (a), (b) MoSe2/CrI3 heterostructure with the AB and AB' stacking, respectively; (c) the large and small hexagon correspond to the MoSe2 protocell Brillouin region and the heterojunction Brillouin region, respectively
图 2 (a) MoSe2/CrI3, (b) MoSe2/CrBr3和(c) WS2/CrBr3体系的堆垛势能面图
Fig. 2. Stacking potential energy surfaces of (a) MoSe2/CrI3, (b) MoSe2/CrBr3 and (c) WS2/CrBr3
图 3 MoSe2/CrI3的自旋电荷密度分布 ((a)和(b))以及层间
${1}/{d} $ 随堆垛的变化图(c)Fig. 3. The spin density distribution ((a) and (b)) of MoSe2/CrI3, and the variation of
$ {1}/{d} $ with different stacking (c)
图 4 MoSe2, CrI3和AB堆垛 MoSe2/CrI3的能带图 (a)—(c)分别对应单层MoSe2不考虑自旋极化且不考虑自旋轨道耦合时、考虑自旋极化以及考虑自旋极化和自旋轨道耦合时的能带; (d)—(f)对应CrI3的情况; (g)—(i)对应 AB堆垛 MoSe2/CrI3
Fig. 4. Band structure of MoSe2, CrI3 and AB-stacking MoSe2/CrI3 vdW heterostructure: (a)–(c) The energy band obtained without spin polarization and spin orbital coupling, with spin polarization and with spin polarization and spin orbital coupling; (d)–(f) the case of CrI3; (g)–(i) the corresponding energy band of AB-stacking MoSe2/CrI3
图 5 磁性异质结轨道分辨的能带图 (a)—(d)分别是MoSe2/CrI3中Mo-d, Se-p, Cr-d, I-p的轨道投影能带图; (e)—(h)分别是MoSe2/CrBr3中Mo-d, Se-p, Cr-d, Br-p的轨道投影能带图; (i)—(l)对应于WS2/CrBr3体系
Fig. 5. Orbital resolved energy band of magnetic heterostructure: (a)–(d) The Mo-d, Se-p, Cr-d, I-p resolved energy band of MoSe2/CrI3; (e)–(h) the correspongding energy band of MoSe2/CrBr3; (i)–(l) the correspongding energy band of WS2/CrBr3
图 6 (a)自旋劈裂和谷劈裂的定义; (b), (c) AB堆垛的MoSe2/CrI3中MoSe2在
$1 \times1 $ 的布里渊区反折叠能带与单独存在的MoSe2比较图, 其中子图为K+/K–附近的价带和导带情况; (d)—(f)${{AB}}'$ 堆垛 MoSe2/CrI3, AB堆垛 MoSe2/CrBr3 和WS2/CrBr3的反折叠能带图Fig. 6. (a) The definition of spin splitting and valley splitting; (b), (c) the comparison between the unfolding band of MoSe2 in AB-stacking MoSe2/CrI3 heterostructure and free-standing MoSe2, the VBM (b) and CBM (c); (d)–(f) the unfolding energy band of
${{AB}}'$ stacking MoSe2/CrI3, AB stacking MoSe2/CrBr3 and WS2/${\rm{CrBr}}_3 $ , respectively
图 7 AB堆垛 MoSe2/CrI3能谷劈裂和能量随层间距和应力的变化 (a), (b)能谷劈裂和能量随层间距的变化关系; (c), (d) AB堆垛MoSe2/CrI3的能谷劈裂和能量随应变的变化关系
Fig. 7. The change of valley splitting and energy with interlayer distance and strain of MoSe2/CrI3: (a), (b) The change of valley splitting and energy with interlayer distance; (c), (d) change with strain
表 1 三种异质结不同堆垛的自旋和能谷劈裂(单位: meV)
Table 1. The spin splitting and valley splitting in unit of meV of three vdW heterostructure with different stacking order.
异质结 堆垛 a/Å 失配率/% $\varDelta_{{\rm{spin}}}^{{\rm{VBM}}/K_+}$ $\varDelta_{{\rm{spin}}}^{{\rm{VBM}}/K_-}$ $\varDelta_{{\rm{spin}}}^{{\rm{CBM}}/K_+}$ $\varDelta_{{\rm{spin}}}^{{\rm{CBM}}/K_-}$ $\varDelta_{{\rm{val}}}^{{\rm{VBM}}}$ $\varDelta_{{\rm{val}}}^{{\rm{CBM}}}$ MoSe2/CrI3 AB 6.65 4.9% 184.64 187.057 22.629 19.14 1.215 1.756 AB' 6.65 186.02 184.8 29.598 25.192 0.596 0.94 MoSe2/CrBr3 AB 6.65 3.1% 183.756 187.815 26.08 18.043 1.852 3.887 AB' 6.65 185.474 184.022 25.11 18.237 2.028 3.096 WS2/CrBr3 AB 6.4 0.7% 434.922 436.345 24.459 27.308 0.666 0.756 AB' 6.4 435.474 435.392 25.019 24.823 0.27 0.439 
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[1] Xu X, Yao W, Xiao D, Heinz T F 2014 Nat. Phys. 10 343
Google Scholar
[2] 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
[3] Liu G B, Xiao D, Yao Y, Xu X, Yao W 2015 Chem. Soc. Rev. 44 2643
Google Scholar
[4] Chu J, Wang Y, Wang X, Hu K, Rao G, Gong C, Wu C, Hong H, Wang X, Liu K, Gao C, Xiong J 2021 Adv. Mater. 33 2004469
Google Scholar
[5] Du L, Hasan T, Castellanos-Gomez A, Liu G B, Yao Y, Lau C N, Sun Z 2021 Nat. Rev. Phys. 3 193
Google Scholar
[6] 孙真昊, 管鸿明, 付雷, 沈波, 唐宁 2021 物理学报 70 027302
Google Scholar
Sun Z H, Guan H M, Fu L, Shen B, Tang N 2021 Acta Phys. Sin. 70 027302
Google Scholar
[7] Zhao S, Li X, Dong B, Wang H, Wang H, Zhang Y, Han Z, Zhang H 2021 Rep. Prog. Phys. 84 026401
Google Scholar
[8] Vitale S A, Nezich D, Varghese J O, Kim P, Gedik N, Jarillo-Herrero P, Xiao D, Rothschild M 2018 Small 14 1801483
Google Scholar
[9] Lu H Z, Yao W, Xiao D, Shen S Q 2013 Phys. Rev. Lett. 110 016806
Google Scholar
[10] Xiao D, Yao W, Niu Q 2007 Phys. Rev. Lett. 99 236809
Google Scholar
[11] Xiao D, Liu G B, Feng W, Xu X, Yao W 2012 Phys. Rev. Lett. 108 196802
Google Scholar
[12] Yao W, Xiao D, Niu Q 2008 Phys. Rev. B 77 235406
Google Scholar
[13] Zeng H, Dai J, Yao W, Xiao D, Cui X 2012 Nat. Nanotechnol. 7 490
Google Scholar
[14] Cao T, Wang G, Han W, Ye H, Zhu C, Shi J, Niu Q, Tan P, Wang E, Liu B, Feng J 2012 Nat. Commun. 3 887
Google Scholar
[15] Cheng Y C, Zhang Q Y, Schwingenschlögl U 2014 Phys. Rev. B 89 155429
Google Scholar
[16] Li Y, Ludwig J, Low T, Chernikov A, Cui X, Arefe G, Kim Y D, van der Zande A M, Rigosi A, Hill H M, Kim S H, Hone J, Li Z, Smirnov D, Heinz T F 2014 Phys. Rev. Lett. 113 266804
Google Scholar
[17] MacNeill D, Heikes C, Mak K F, Anderson Z, Kormányos A, Zólyomi V, Park J, Ralph D C 2015 Phys. Rev. Lett. 114 037401
Google Scholar
[18] Aivazian G, Gong Z, Jones A M, Chu R L, Yan J, Mandrus D G, Zhang C, Cobden D, Yao W, Xu X 2015 Nat. Phys. 11 148
Google Scholar
[19] Srivastava A, Sidler M, Allain A V, Lembke D S, Kis A, Imamoğlu A 2015 Nat. Phys. 11 141
Google Scholar
[20] Stier A V, McCreary K M, Jonker B T, Kono J, Crooker S A 2016 Nat. Commun. 7 10643
Google Scholar
[21] Qi J, Li X, Niu Q, Feng J 2015 Phys. Rev. B 92 121403
Google Scholar
[22] Norden T, Zhao C, Zhang P, Sabirianov R, Petrou A, Zeng H 2019 Nat. Commun. 10 4163
Google Scholar
[23] Zhang W B, Qu Q, Zhu P, Lam C H 2015 J. Mater. Chem. C 3 12457
Google Scholar
[24] Huang B, Clark G, Navarro-Moratalla E, Klein D R, Cheng R, Seyler K L, Zhong D, Schmidgall E, McGuire M A, Cobden D H, Yao W, Xiao D, Jarillo-Herrero P, Xu X 2017 Nature 546 270
Google Scholar
[25] Gong C, Li L, Li Z, Ji H, Stern A, Xia Y, Cao T, Bao W, Wang C, Wang Y, Qiu Z Q, Cava R J, Louie S G, Xia J, Zhang X 2017 Nature 546 265
Google Scholar
[26] Seyler K L, Zhong D, Huang B, Linpeng X, Wilson N P, Taniguchi T, Watanabe K, Yao W, Xiao D, McGuire M A, Fu K M C, Xu X 2018 Nano Lett. 18 3823
Google Scholar
[27] Zhong D, Seyler K L, Linpeng X, Cheng R, Sivadas N, Huang B, Schmidgall E, Taniguchi T, Watanabe K, McGuire M A, Yao W, Xiao D, Fu K M C, Xu X 2017 Sci. Adv. 3 e1603113
Google Scholar
[28] Hu T, Zhao G, Gao H, Wu Y, Hong J, Stroppa A, Ren W 2020 Phys. Rev. B 101 125401
Google Scholar
[29] Zhang H, Yang W, Ning Y, Xu X 2020 Phys. Rev. B 101 205404
Google Scholar
[30] Zhang Z, Ni X, Huang H, Hu L, Liu F 2019 Phys. Rev. B 99 115441
Google Scholar
[31] Kresse G, Furthmüller J 1996 Comp. Mater. Sci. 6 15
Google Scholar
[32] Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169
Google Scholar
[33] Blöchl P E 1994 Phys. Rev. B 50 17953
Google Scholar
[34] Kresse G, Joubert D 1999 Phys. Rev. B 59 1758
Google Scholar
[35] Perdew J P, Burke K, Ernzerhof M 1996 Phys. Rev. Lett. 77 3865
Google Scholar
[36] Dudarev S L, Botton G A, Savrasov S Y, Humphreys C J, Sutton A P 1998 Phys. Rev. B 57 1505
Google Scholar
[37] Grimme S 2006 J. Comp. Chem. 27 1787
Google Scholar
[38] Chen M X, Weinert M 2014 Nano Lett. 14 5189
Google Scholar
[39] Chen M, Weinert M 2018 Phys. Rev. B 98 245421
Google Scholar
[40] Chen M X, Chen W, Zhang Z, Weinert M 2017 Phys. Rev. B 96 245111
Google Scholar
[41] Si J S, Li H, He B G, Cheng Z P, Zhang W B 2021 J. Phys. Chem. C 125 7314
Google Scholar
[42] Zollner K, Faria Junior P E, Fabian J 2019 Phys. Rev. B 100 085128
Google Scholar
[43] Ciorciaro L, Kroner M, Watanabe K, Taniguchi T, Imamoglu A 2020 Phys. Rev. Lett. 124 197401
Google Scholar
[44] Xie J, Jia L, Shi H, Yang D, Si M 2018 JPN J. Appl. Phys. 58 010906
Google Scholar
[45] Ge M, Wang H, Wu J, Si C, Zhang J, Zhang S 2022 NPJ Comp. Mater. 8 32
Google Scholar
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