Electronic structures, charge transfer and charge orders in twisted transition metal dichalcogenide bilayers
EElectrically tunable charge transfer and charge orders in twisted transition metaldichalcogenide bilayers
Yang Zhang, Tongtong Liu, and Liang Fu
Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
Moir´e superlattices of transition metal dichalcogenide (TMD) bilayers have been shown to hostcorrelated electronic states, which arises from the interplay of emergent moir´e potential and long-range Coulomb interactions. Here we theoretically investigate structural relaxation and single-particle electronic properties in moir´e superlattices of transition metal dichalcogenide homobilayerand study the ground state charge orders in the effective honeycomb lattice of MX and XM region.From the large-scale density functional theory calculation and continuum model with layer degreesof freedom, we find that the out of plane gating field creates a tunable charge transfer gap andintroduces a mass term in the Dirac spectrum. At the flat band limit, we observe a series of charge-ordered insulating states at various filling n = 1 / , / , / , / , n = 1 / , /
3. Our work demonstrates that transition metal dichalcogenide homobilayer provides apowerful platform for the investigation of tunable charge transfer insulator and charge orders.
Moir´e superlattices are a fruitful platform for re-alizing and controlling correlated electron states, asevidenced by the remarkable success in twisted bi-layer graphene (TBG) [1–12] and trilayer graphene-hBN heterostructure[13–16]. Recently a new familyof moir´e materials based on transition metal dichalco-genides (TMD) [17–29] has attracted great interest.They host an abundance of correlated insulating statesat a series of fractional fillings [30–34].In TMD bilayers, moir´e bands are formed fromparabolic bands of individual layers. In twisted TMDhomobilayers, the moir´e bandwidth can be made arbi-trarily small by reducing the twist angle, which gives riseto strong correlation without fine tuning. Electrons orholes in these moir´e bands are tightly localized in high-symmetry stacking regions, which can be well describedby a simple effective tight binding model. This descrip-tion offers a convenient starting point for investigatinginteraction-induced states at finite density. Despite theconceptual simplicity, a quantitative modeling of moir´ebands in TMD is highly nontrivial. For example, themoir´e bandwidth of TMD heterobilayer WSe /WS isonly on the order of 10 meV, and depends highly on thelattice relaxation [30, 31, 35, 36].In this work, using the large scale density functionaltheory, continuum model approach and Monte Carlo sim-ulation, we study the effect of structural relaxation andelectric field on the moir´e band structure in twisted TMDhomobilayers and predict novel charge orders at frac-tional fillings in the strong-coupling regime. We focus onthe moir´e valence bands originating from the Γ pocket[37 ? –40]. Due to interlayer tunneling and lattice re-laxation, these moir´e bands are derived from localizedorbitals in MX and XM stacking regions that form ahoneycomb lattice. We find a pair of massless Diracfermions at K, K (cid:48) points of the mini Brillouin zone (BZ),which is protected by the D point group symmetry ofthe moir´e superlattice. Applying an out-of-plane electricfield breaks the sublattice symmetry of the honeycomb lattice and opens a tunable gap ∆ at the Dirac point. Weintroduce a new continuum model for twisted TMD ho-mobilayers, which captures the layer degrees of freedomand the electrically tunable gap.We further use an extended Hubbard model on thehoneycomb lattice and perform Monte Carlo simulationsto study the insulating electron crystals in the flat bandlimit. We find a distinctive set of charge orders at fill-ing n = 1 / , / , / , / , n = 1 / / /WS heterobilayer. These sym-metry breaking charge orders can be directly probedby the optical anisotropy experiments [34, 41]. More-over, we predict that phase transitions between distinctcharge-ordered states at the same filling can be inducedby the electric field, which tunes the charge-transfer gap∆. Our work shows that twisted homobilayer MoS pro-vides an ideal platform for investigating electrically tun-able charge transfer gap and charge orders.We study TMD homobilayers with a small twist anglestarting from AA stacking, where every metal (M) orchalcogen (X) atom on the top layer is aligned with thesame type of atom on the bottom layer [51]. Within alocal region of a twisted bilayer, the atom configurationis identical to that of an untwisted bilayer, where onelayer is laterally shifted relative to the other layer by acorresponding displacement vector d . For this reason,the moir´e band structure of twisted TMD bilayers can beconstructed from a family of untwisted bilayers at various d , all having 1 × d = 0 , − ( a + a ) / , ( a + a ) / a , is the primitive lattice vector for untwisted bi-layers, correspond to three high-symmetry stacking con-figurations of untwisted TMD bilayers, which we refer toas MM, XM, MX. In MM (MX) stacking, the M atomon the top layer is locally aligned with the M (X) atomon the bottom layer, see Fig. 1a. Likewise for XM. The a r X i v : . [ c ond - m a t . s t r- e l ] S e p bilayer structure in these stacking configurations is in-variant under three-fold rotation around the z axis.In homobilayer TMD, the spin degenerate Γ pockets inthe valence band arises from electron tunneling betweenthe two layers. The k · p Hamiltonian takes the form: H ( d ) = (cid:32) − (cid:126) k m ∗ + (cid:15) b ( d ) ∆ T ( d )∆ † T ( d ) − (cid:126) k m ∗ + (cid:15) t ( d ) (cid:33) . (1)Here m ∗ = 1 . m e is the effective mass for the valenceband. ∆ T ( d ) is the interlayer tunnelling amplitudewhich depends on the in-plane displacement between thetwo layers. In contrast to the complex tunneling am-plitude for the K pockets [43], here the time reversalsymmetry at Γ pocket enforces ∆ T ( d ) to be real. Thepotential term (cid:15) b,t ( d ) denotes the energy of the valenceband maximum in the absence of tunneling, which arisesfrom the unequal layer weight of the wavefunction at MXand XM stacking configuration.We expand ∆ T ( d ) in Fourier components up to thesecond harmonic:∆ T ( d ) = w +2 w (cid:88) j =1 cos( G j · d )+2 w (cid:88) j =1 cos( G j · d ) , (2)where G i ( i = 1 , ,
3) is the three reciprocal lattice vectorin monolayer TMD. Due to three-fold rotation symmetry,∆ T is a local extreme for MM, MX and MX stackings,with ∆ T = w + 6 w + 6 w for d =0 (MM) and w − w − w for d = ± ( a + a ) / w is responsiblefor the large bonding and antibonding energy splitting forall d , while w , w capture the variation of the tunnelingamplitude at different lateral displacements.The interlayer tunneling strength depends significantlyon the layer spacing d . From the DFT calculation, wefind the equilibrium layer spacing of untwisted TMD bi-layers in MM, MX and XM stacking: d MM = 6 .
63A and d MX = d XM = 5 . E = 0 chosen to be the absolute vacuumlevel. Using the relaxed layer spacings, we find the energysplitting in MX (or XM) stacking to be stronger thanin M M , as a result of its smaller layer distance. Fromthe different energy splitting at Fig. 1c, we obtain thetunnelling parameter as w = 338meV, w w − d that varies slowly in space: d = θ ˆ z × r . Therefore we construct the following con-tinuum Hamiltonian for the moir´e bands from Γ pocket MM MX(XM) 𝐸 ( 𝑒 𝑉 ) 𝐸 ( 𝑒 𝑉 ) (b)(c) MM XM MX(a) M X FIG. 1: (a) Lattice structure of MM, MX, XM spots forAA stacking heterobilayer, M stands for metal atom and Xstands for chalcogen atom (Green for the top layer, yellow forthe bottom layer). DFT band structures of MM and MX(XM)stacking homobilayer in (b) MoS /MoS with identical layerspacing; (c) MoS /MoS with relaxed layer spacing. two band kp model: H = (cid:32) − (cid:126) k m ∗ + (cid:15) b ( r ) ∆ T ( r )∆ † T ( r ) − (cid:126) k m ∗ + (cid:15) t ( r ) (cid:33) (3)The position dependent tunneling term is obtained byreplacing d with θ ˆ z × r in Eq.(2):∆ T ( r ) = w +2 w (cid:88) j =1 cos( G mj · r )+2 w (cid:88) j =1 cos( G mj · r )(4)Where G mi = G i θ × ˆ z ( i = 1 , ,
3) is the three reciprocallattice vector in moir´e superlattice. Likewise, the intra-lyer potential (cid:15) t,b ( t, b stand for top and bottom layer,respectively) can be expressed as the first order Fourierexpansion over moir´e reciprocal lattice vector: (cid:15) t,b ( r ) = 2 V (cid:88) j =1 , , cos (cid:16) G mj · r ± φ (cid:17) (5)The sign of phase factor φ changes under layer exchange,enforced by C y symmetry as shown in Fig. 2a. Thepotential term is crucial for the later modelling with outof plane gating field.We now compare the band structure from continuummodel with the large scale density functional theory. Themoir´e superlattice is fully relaxed with van der Waals cor-rection incorporated by the vdW-DF (optB86) function- 𝑑 " 𝑑 %& 𝜃 = 3.15 ∘ (b) Twist angle ( 𝜃 ) (a) 𝐶 /0 L a y e r d i s t a n c e ( A ) FIG. 2: (a) Real-space moir´e pattern of heterobilayer TMDheterobilayer, where MM, MX, XM spots within one supercellare labeled, and diagram for layer distance in 2 × d far and d near , and out of plane corrugation diagram path in (b). als [45] as implemented in the Vienna Ab initio Simu-lation Package[46]. We plot the twist-angle dependentlayer distance, d far at MM region, and d near in MX(XM) region, in Fig. 2b. At small twist angle θ ∼ C y symmetry. As a result, low-energy moir´ebands are formed from layer-hybridized orbitals in MXand XM regions, which form a honeycomb lattice withidentical on-site potential.We perform the large scale DFT simulation to calcu-late the band structures for various twist angles, shownin Fig. 3. We find that above a small moir´e period L m ∼ . θ = 3 . ◦ , the two top-most moir´e s bands are well separated from the remainingbands. Similar band structures are also found in large-scale DFT calculation with fully relaxed lattice structurefor homobilayer MoS [37, 38] and WS [39]. Fitting theDFT moir´e band structure to continuum model, we ob-tain the parameters as w = 338 meV, w = −
16 meVand w = − V = 6 meV, φ = 121 ◦ at twistangle θ = 2 . ◦ . These values are consistent with theestimation from untwisted structures.As shown in Fig. 3(a,c), the moir´e bands exhibit Diracpoints at K and K (cid:48) points of the moir´e Brillouin zone.These Dirac points are protected by the D point group oftwisted TMD homobilayer: the doublet at K or K (cid:48) forma two-dimensional E representation. The bandwidth ofDirac bands changes monotonously from 250 meV to 10meV when twist angle θ ranges from 6 ◦ to 2 ◦ as shown (a)(c) (d)(b) 𝐸 ( 𝑒 𝑉 ) 𝐸 ( 𝑒 𝑉 ) FIG. 3: (a)DFT Band structure for θ = 3 . ◦ ; (b)Twist angledependent bandwidth for the first two moir´e bands of the hon-eycomb lattice; DFT (black cross) and continuum model(blueline) band structures for (c) θ = 2 . ◦ ; (d) θ = 2 . ◦ with0.5 V /nm out of plane gating field. in Fig. 3b. This provides an ideal platform to studythe tunable correlation physics of Dirac electrons at thefilling of n = 2 holes per moir´e unit cell.In the case of twisted bilayer graphene [47], the lowenergy Dirac fermion is protected by the C z symmetry,which can not be broken by out of plane field. How-ever, in MX and XM region of the twisted homobilayerMoS , the wavefunctions have unequal layer weight as in-dicated from the untwisted calculation. Thus the out ofplane gating field breaks the C y symmetry and gaps outthe Dirac fermion. A simplified continuum model tar-geting at antibonding orbitals well captures the topmostmoir´e bands, but can not describe the band structure andcharge distribution involving layer degrees of freedom.We further calculate the band structure of the fullyrelaxed moir´e superlattice of homobilayer MoS with theapplied gating field. As shown in Fig. 3d, an out ofplane gating field 0.5 V /nm creates a 2.4 meV gap at Kpoint, while the bandwidth of the first energy-separablemoir´e band is 12 meV. At K point of the band edge, thewavefunction of the first band is localized at MX region,while the second band at XM region. For small twistangle θ = 2 ◦ with wavelength L m = 9 . E d = 1 V /nm induces a charge transfer gap ∆ up to5 meV, even larger than the bandwidth for the topmostmoir´e band (see supplementary material). A larger field-induced ∆ can be achieved in twisted TMD homobilayerswith reduced interlayer tunneling (which competes withthe layer potential asymmetry). This can be realized byinserting an hBN layer in between the top and bottomTMD layers [24].In the TMD superlattice, the local minimums of theperiodic moir´e potential can be viewed as the effectivemoir´e atoms to host charge. Under the harmonic approx-imation, the size of Wannier orbital for the topmost moir´e (a)(b)n=1/2n=2/3 Increseaing Δ Increseaing Δ FIG. 4: Ground state charge order at filling (a) n=1/2 withincreasing charge transfer gap ∆, (b) n=2/3 with increasingcharge transfer gap ∆. band is given by ξ = (cid:113) (cid:126) m ∗ ω = 2 ( π ) − √ L m ( (cid:126) m ∗ V m ) ( V m is the moir´e potential integrated to antibondingorbitals). In homobilayer system without lattice mis-match, the kinetic energy over nearest neighbor inter-action strength ( t/V ) can be tuned to arbitrarily small,so that the classical model is well justified at sufficientlysmall twist angle. The effective extended Hubbard modelwithout kinetic energy is given by: H = (cid:80) j ∈ B ∆ n j + (cid:80) i U n i ↑ n i ↓ + (cid:80) i (cid:54) = j V ij n i n j (6)Here ∆ is the charge transfer gap between two sublatticesites A and B, and V ij is the extended interaction within i and j sites.In twisted homobilayer MoS , the gating field intro-duces a charge transfer gap ∆. We first discuss the sit-uation with large ∆. At filling n <
1, the effective tightbinding model reduces to a triangular lattice model, asin the case of WSe /WS , and exhibits similar chargeorders. Various insulating states have been observed atfractional filling n = 1 / , / , / , / , / , / U >> ∆,the system at n = 1 should be regarded as a charge trans-fer insulator [35]. When doping to a higher filling n > ×
120 sites with periodic boundary condition for theextended Hubbard model with different gate screeningfrom L m / L m . The distance dependent interac-tion strength is plotted in Fig. S2 up to V , and theinteraction cutoff is chosen as 0 . V . We identify aseries of charge orders at n = 1 / , / , / , / ,
1. For n < /
2, moir´e electrons are all filled to one sublattice,exhibiting similar charge orders as the reduced triangularlattice in WSe2 /WS heterobilayer (see supplementarymaterials).Interestingly, for small ∆, charge transfer involving twosublattices already takes place for filling n ≥ /
2. At filling factor n = 1 /
2, we find an emerging rectangu-lar lattice with √ × / × c = 2( V − V − V + 2 V − V + V + ... ). For d = L m = 9 . c = 0 . e (cid:15)L m ∼ . c can be further lowered by increasing moir´e wavelength.At filling factor n = 2 /
3, the charges form a zigzagstripe order with 6 × C ro-tation symmetry. This zigzag type charge configurationis energetically favored compared to a linear stripe atscreening distances from d = 1 / L m to d = 10 L m . As∆ increases, the zigzag charge stripe transitions to the √ × √ c = V − V − V + V .... = 0 . e (cid:15)L m ∼ . d = L m = 9 . /hBN/MoSe het-erostructure [24], where the gating field induced chargetransfer between the top and bottom layers has alreadybeen observed at relatively high temperature.For n = 1, we find that even at ∆ = 0, the ground stateis fully sublattice polarized, which spontaneously breaksthe honeycomb lattice symmetry. For filling n > , the valence band maximum islocated at K with weak interlayer tunneling amplitudeand intralayer potential both on the order of 10 meV.The complex tunneling term between two layers bringsfurther complications for the theoretical and experimen-tal investigation of the insulating states [26, 28, 43]. Inthe case of heterobilayer TMD such as WSe /WS , theexistence of secondary moir´e potential minima is pro-posed [35]. However, due to the large band offset be-tween WSe and WS , charges are localized at the WS layer, which limits the tunability of potential differencebetween two moir´e regions.In conclusion, we present a combined study of lat-tice relaxation, single-particle electronic structure, andground state charge orders on the twisted homobilayerMoS . Unlike the previous moir´e charge transfer insula-tor in WSe2 /WS heterobilayer, here out of plane gat-ing field breaks C y symmetry and induces a control-lable charge transfer gap. With Monte Carlo simula-tion, we predict additional stripe type charge orders atfilling n = 1 / , / Acknowledgment
We thank Atac Imamoglu, Yuya Shimazaki, Cory R.Dean and Qianhui Shi for numerous discussions on ex-periments, and Zhen Bi, Lede Xian, and Angel Rubio forvaluable theoretical discussions. [1] Y. Cao, V. Fatemi, A. Demir, S. Fang, S. L. Tomarken,J. Y. Luo, J. D. Sanchez-Yamagishi, K. Watanabe,T. Taniguchi, E. Kaxiras, et al., Nature , 80 (2018).[2] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi,E. Kaxiras, and P. Jarillo-Herrero, Nature , 43(2018).[3] X. Lu, P. Stepanov, W. Yang, M. Xie, M. A. Aamir,I. Das, C. Urgell, K. Watanabe, T. Taniguchi, G. Zhang,et al., Nature , 653 (2019).[4] A. Kerelsky, L. J. McGilly, D. M. Kennes, L. Xian,M. Yankowitz, S. Chen, K. Watanabe, T. Taniguchi,J. Hone, C. Dean, et al., Nature , 95 (2019).[5] Y. Jiang, X. Lai, K. Watanabe, T. Taniguchi, K. Haule,J. Mao, and E. Y. Andrei, Nature , 91 (2019).[6] Y. Xie, B. Lian, B. J¨ack, X. Liu, C.-L. Chiu, K. Watan-abe, T. Taniguchi, B. A. Bernevig, and A. Yazdani, Na-ture , 101 (2019).[7] Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora,R. Polski, Y. Zhang, H. Ren, J. Alicea, G. Refael, et al.,Nature Physics , 1174 (2019).[8] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watan-abe, T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean,Science , 1059 (2019).[9] E. Codecido, Q. Wang, R. Koester, S. Che, H. Tian,R. Lv, S. Tran, K. Watanabe, T. Taniguchi, F. Zhang,et al., Science Advances , eaaw9770 (2019).[10] A. L. Sharpe, E. J. Fox, A. W. Barnard, J. Finney,K. Watanabe, T. Taniguchi, M. Kastner, andD. Goldhaber-Gordon, Science , 605 (2019).[11] S. Tomarken, Y. Cao, A. Demir, K. Watanabe,T. Taniguchi, P. Jarillo-Herrero, and R. Ashoori, Physi-cal review letters , 046601 (2019).[12] U. Zondiner, A. Rozen, D. Rodan-Legrain, Y. Cao,R. Queiroz, T. Taniguchi, K. Watanabe, Y. Oreg, F. vonOppen, A. Stern, et al., arXiv preprint arXiv:1912.06150(2019).[13] G. Chen, L. Jiang, S. Wu, B. Lyu, H. Li, B. L. Chittari,K. Watanabe, T. Taniguchi, Z. Shi, J. Jung, et al., NaturePhysics , 237 (2019).[14] G. Chen, A. L. Sharpe, P. Gallagher, I. T. Rosen, E. J.Fox, L. Jiang, B. Lyu, H. Li, K. Watanabe, T. Taniguchi,et al., Nature , 215 (2019), URL https://doi.org/10.1038/s41586-019-1393-y .[15] M. Serlin, C. Tschirhart, H. Polshyn, Y. Zhang, J. Zhu,K. Watanabe, T. Taniguchi, L. Balents, and A. Young,Science (2019).[16] X. Liu, Z. Hao, E. Khalaf, J. Y. Lee, K. Watanabe,T. Taniguchi, A. Vishwanath, and P. Kim, arXiv preprintarXiv:1903.08130 (2019).[17] C. Zhang, C.-P. Chuu, X. Ren, M.-Y. Li, L.-J. Li,C. Jin, M.-Y. Chou, and C.-K. Shih, Science advances , e1601459 (2017).[18] H. Yu, G.-B. Liu, J. Tang, X. Xu, and W. Yao, Scienceadvances , e1701696 (2017).[19] K. L. Seyler, P. Rivera, H. Yu, N. P. Wilson, E. L. Ray,D. G. Mandrus, J. Yan, W. Yao, and X. Xu, Nature ,66 (2019).[20] K. Tran, G. Moody, F. Wu, X. Lu, J. Choi, K. Kim,A. Rai, D. A. Sanchez, J. Quan, A. Singh, et al., Nature , 71 (2019).[21] L. Yuan, B. Zheng, J. Kunstmann, T. Brumme, A. B.Kuc, C. Ma, S. Deng, D. Blach, A. Pan, and L. Huang,Nature Materials pp. 1–7 (2020).[22] W. Li, X. Lu, S. Dubey, L. Devenica, and A. Srivastava,Nature Materials pp. 1–6 (2020).[23] M. Brotons-Gisbert, H. Baek, A. Molina-S´anchez,A. Campbell, E. Scerri, D. White, K. Watanabe,T. Taniguchi, C. Bonato, and B. D. Gerardot, NatureMaterials pp. 1–7 (2020).[24] Y. Shimazaki, I. Schwartz, K. Watanabe, T. Taniguchi,M. Kroner, and A. Imamo˘glu, Nature , 472 (2020).[25] Y. Bai, L. Zhou, J. Wang, W. Wu, L. J. McGilly, D. Hal-bertal, C. F. B. Lo, F. Liu, J. Ardelean, P. Rivera, et al.,Nature materials pp. 1–6 (2020).[26] L. Wang, E.-M. Shih, A. Ghiotto, L. Xian, D. A. Rhodes,C. Tan, M. Claassen, D. M. Kennes, Y. Bai, B. Kim,et al., Nature materials pp. 1–6 (2020).[27] L. J. McGilly, A. Kerelsky, N. R. Finney, K. Shapovalov,E.-M. Shih, A. Ghiotto, Y. Zeng, S. L. Moore, W. Wu,Y. Bai, et al., Nature Nanotechnology , 580 (2020).[28] Z. Zhang, Y. Wang, K. Watanabe, T. Taniguchi,K. Ueno, E. Tutuc, and B. J. LeRoy, Nature Physicspp. 1–4 (2020).[29] A. Weston, Y. Zou, V. Enaldiev, A. Summerfield,N. Clark, V. Z´olyomi, A. Graham, C. Yelgel, S. Magor-rian, M. Zhou, et al., Nature Nanotechnology pp. 1–6(2020).[30] E. C. Regan, D. Wang, C. Jin, M. I. B. Utama, B. Gao,X. Wei, S. Zhao, W. Zhao, Z. Zhang, K. Yumigeta, et al.,Nature , 359 (2020).[31] Y. Tang, L. Li, T. Li, Y. Xu, S. Liu, K. Barmak,K. Watanabe, T. Taniguchi, A. H. MacDonald, J. Shan,et al., Nature , 353 (2020).[32] Y. Xu, S. Liu, D. A. Rhodes, K. Watanabe, T. Taniguchi,J. Hone, V. Elser, K. F. Mak, and J. Shan, arXiv preprintarXiv:2007.11128 (2020).[33] X. Huang, T. Wang, S. Miao, C. Wang, Z. Li, Z. Lian,T. Taniguchi, K. Watanabe, S. Okamoto, D. Xiao, et al.,arXiv preprint arXiv:2007.11155 (2020).[34] C. Jin, Z. Tao, T. Li, Y. Xu, Y. Tang, J. Zhu, S. Liu,K. Watanabe, T. Taniguchi, J. C. Hone, et al., arXivpreprint arXiv:2007.12068 (2020). [35] Y. Zhang, N. F. Yuan, and L. Fu, arXiv preprintarXiv:1910.14061 (2019).[36] H. Li, S. Li, M. H. Naik, J. Xie, X. Li, J. Wang, E. Re-gan, D. Wang, W. Zhao, S. Zhao, et al., arXiv preprintarXiv:2007.06113 (2020).[37] M. H. Naik and M. Jain, Physical review letters ,266401 (2018).[38] L. Xian, M. Claassen, D. Kiese, M. M. Scherer,S. Trebst, D. M. Kennes, and A. Rubio, arXiv preprintarXiv:2004.02964 (2020).[39] M. Angeli and A. MacDonald, arXiv preprintarXiv:2008.01735 (2020).[40] S. Venkateswarlu, A. Honecker, and G. T. de Lais-sardi`ere, arXiv preprint arXiv:2005.13054 (2020).[41] Y. Shimazaki, C. Kuhlenkamp, I. Schwartz, T. Smolen-ski, K. Watanabe, T. Taniguchi, M. Kroner,R. Schmidt, M. Knap, and A. Imamoglu, arXivpreprint arXiv:2008.04156 (2020).[42] S. McDonnell, A. Azcatl, R. Addou, C. Gong, C. Battaglia, S. Chuang, K. Cho, A. Javey, and R. M.Wallace, ACS nano , 6265 (2014).[43] F. Wu, T. Lovorn, E. Tutuc, I. Martin, and A. MacDon-ald, Physical review letters , 086402 (2019).[44] K. Uchida, S. Furuya, J.-I. Iwata, and A. Oshiyama,Physical Review B , 155451 (2014).[45] J. Klimeˇs, D. R. Bowler, and A. Michaelides, PhysicalReview B , 195131 (2011).[46] G. Kresse and J. Furthm¨uller, Computational materialsscience , 15 (1996).[47] R. Bistritzer and A. H. MacDonald, Proceedings of theNational Academy of Sciences , 12233 (2011).[48] K. Slagle and L. Fu, arXiv preprint arXiv:2003.13690(2020).[49] J. Zaanen, G. Sawatzky, and J. Allen, Physical ReviewLetters , 418 (1985).[50] Z. Bi and L. Fu, arXiv preprint arXiv:1911.04493 (2019).[51] AB stacking can be viewed as a 180 ◦◦