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1 In the format provided by the authors and unedited. DOI: /NPHYS3968 Topological mosaic in Moiré superlattices of van der Waals heterobilayers Qingjun Tong 1, Hongyi Yu 1, Qizhong Zhu 1, Yong Wang 2,1, Xiaodong Xu 3,4, Wang Yao 1* 1 Department of Physics and Center of Theoretical and Computational Physics, University of Hong Kong, Hong Kong, China 2 School of Physics, Nankai University, Tianjin , China 3 Department of Physics, University of Washington, Seattle, WA 98195, USA 4 Department of Materials Science and Engineering, University of Washington, Seattle, WA 98195, USA The supplementary materials include: Supplementary Figures S1-S6 Supplementary Text I-V Supplementary Text I. Interlayer hopping in commensurate bilayer TMDs In the following we first discuss the interlayer hopping at ± valleys for TMD bilayers with rotation symmetry. We show that certain interlayer hopping channels must vanish for these symmetry configurations. Then we extend our results to the TMD bilayers with a general translation using two-center approximation. a. TMD bilayers with rotation symmetry The interlayer hopping matrix element is = () (), where is interlayer hopping Hamiltonian between the two layers and is invariant under rotation for the six symmetry configurations of R- and H-ype. (), () denote the Bloch states at the high symmetry ± point in the upper (lower) monolayer, which are eigenstates of, (), () = () (), (), (), (S1) where () = is the eigenvalue of operating on the atomic orbits. The phase () depends on rotation centers at M atom (M), X atom (X) or hollow center (H) 1, 0, for M () =, for X. (S2), for H NATURE PHYSICS 1

2 Because = = ( ),,, a non-zero interlayer hopping requires that ( ),, =1. The allowed interlayer hopping channels for the six symmetry configurations are listed in Table 1 in the main text. b. TMD bilayers with a general translation For commensurate TMD bilayers with a general translation, rotation symmetry is broken. In this case, all the hopping channels are allowed. Below we formulate the interlayer hopping in such a commensurate TMD bilayer at the band edges ± under the two-centered approximation. The Bloch states at ± valleys are predominantly contributed from transition metal and ± ± orbitals. The Bloch functions at the conduction and valence band edges are,, () = () = ( ), ( ), where the summation is over all the metal sites in the upper layer ( in the lower layer), is the lattice site number of each layer,, = {c, v} are the band indexes of upper and lower layer respectively, and, = {0, ±2} are quantum numbers of atomic orbits. By setting a metal atom in the upper layer as the xy coordinate origin, we can write = + and = +, with, the two unit lattice vectors and,,, any integers, and is the layer-layer translation. The interlayer hopping matrix element between Bloch states right at + in the lower layer and + in the upper layer is =,, = 1, ( ) ( ). The corresponding orbital magnetic quantum number and in different interlayer hopping matrix element are summarized below: (S3) (S4) Table S1. The values of the magnetic quantum numbers and in different. Here = +1 ( 1) corresponds to the R-type (H-type) stacking. Under the two-center approximation, the hopping integral depends only on the relative position of two metal atoms ( ) = ( ) ( ), whose Fourier transformation is () =Ω () (Ω is the area of the unit cell). The interlayer hopping matrix element then has a form = 1, () = δ, δ,, () NATURE PHYSICS 2

3 = ( + ) = (). (S5) Here +, and is the reciprocal lattice vector. Below we shall drop the unimportant global phase factor We expect () to decay fast with the increase of (Ref 2 ), as the hopping integral ( ) = ( ) ( ) is generally a smooth function of. () is most pronounced at the smallest, that is, =,0, or ( is the in-plane -rotation operator) which are the three equivalent corners of the first Brillouin zone (c.f. Fig. S2b). The other terms with 2, corresponding to the Umklapp terms, are exponentially smaller and can be dropped 2. We note that under rotation: = = (), which leads to = = = ( ). (S6) Together with =, we find that =Ω () =Ω = (). Thus the hopping integral has a form, () + + = + +, (S7) where (). From the and values in Table S1, we get ( ) + +, ( ) + +, ( ) + +, ( ) + () + (), which is the Eq. (3) in the maintext. At high symmetry points ( = 0,, ( ) ), the interference of the phase factors in Eq. (S8) results in the vanishing of certain interlayer hopping channels, which agrees with the symmetry requirement discussed above (c.f. Table 1 in main text). c. Magnitude of the interlayer hopping matrix elements The magnitude of can be estimated from first principle band structures of R/H-type commensurate TMD homobilayers in the absence of spin-orbit coupling. If interlayer hopping is absent, the Bloch states,/ in the upper layer and,/ in the lower layer are degenerate (in absence of spin-orbit coupling). The switch on of interlayer hopping in a TMD homobilayer couples these degenerate states and the resulted conduction (valence) band splitting at K-point gives the magnitude of the hopping integral 2 (2 ). Specifically, for an (AA-stacking) TMD homobilayer the conduction (valence) band splitting corresponds to 2 6 (S8) NATURE PHYSICS 3

4 (2 6 ), whereas for an (AB-stacking) TMD homobilayer valence band splitting corresponds to 2 6. Considering the similarity of the d-orbitals of Mo and W atoms, from the TMD homobilayers shall provide a reasonable estimation of that in the TMD heterobilayers. From the first principle calculations of band structure of an (AA-stacking) homobilayer MoS2 3, we extract a K-point conduction (valence) band splitting 2 50 mev (2 70 mev), which gives 8 mev ( 11 mev). Ref 4 shows the K-point valence band splitting 2 in the absence of spin-orbit coupling for (AB-stacking) homobilayer TMDs, the resulted is summarized in Table S2. For and, no first principle calculations are available. The values of and are chosen to be in the same order as and ± in our calculations (c.f. main text). These hopping values can be enhanced if the interlayer distance is reduced under pressure. We note that the qualitative feature of the topological mosaic (c.f. Figure 2 and 3 in main text) is already determined by the symmetry dictated behaviors of the hopping channels (i.e. Table 1 in main text), regardless of the strengths of these hopping channels and their detailed dependences as long as they observe symmetry requirement. The relative strengths of these hopping channels and their dependence can only affect the shapes and areas of the TI phase regions, while the magnitude of the hybridization gap is proportional to the overall strength of these hopping channels. MoS2 WS2 MoSe2 WSe2 14 mev 18 mev 18 mev 22 mev Table S2. The values of for the four kinds of (AB-stacking) bilayer TMDs, taken from Ref 4. Ref 4 also showed that the interlayer hopping strength depends sensitively on the interlayer distance. Since there lacks direct measurements on the latter quantity for the bilayer, in the various calculations of the interlayer hopping strength, different values for interlayer distance have been used 4, including the experimentally measured value from the bulk crystal, and the ones from different ab initio calculations, which can differ by a significant amount (10%). So here we examine the effect on the interlayer hopping strength when the interlayer distance varies. We take the example of MoS2, and again extract the interlayer hopping strength from the splitting at the K point calculated in the absence of spin-orbit coupling. The calculation is performed with the projector-agumented wave (PAW) method implemented in the Quantum Espresso package 5. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional 6 and scalar relativistic pseudopotential without including the spin-orbit coupling has been exploited, and the cutoff energy for plane wave basis is set as 80 Ry. A k-point sample is generated by the Monkhorst-Pack (MP) approach, and the self-consistent ground state is achieved with the total energy converge criteria Ry. The structure parameters other than the interlayer distance are taken from Ref. 7. Fig. S1 plots the dependence of the valence band interlayer hopping ( ± ) on the interlayer distance d (between metal planes), for and homobilayers. The interlayer hopping strength is found to be an exponential function of the interlayer distance. NATURE PHYSICS 4

5 Fig. S1 The interlayer coupling strength as a function of interlayer distance for (a) -type and (b) -type bilayer MoS 2. Blue dots are results from our ab initio calculations, and the solid curves are the fits with exponential decay. The red dot is a calculation in Ref 3. The dashed vertical lines denote the values of interlayer distance from ab initio calculations using PBE+vdW or LDA 3, using PBE only 7, and bulk experimental value in the 2H stacking 8. ab initio calculations for commensurate TMDs bilayers 3 also find interlayer distance depends on the atomic registry in the stacking (c.f. Table S3 below). It is interesting to note both topological nontrivial and configurations (c.f. Fig. 2 in main text) correspond to the smallest interlayer distance of 6.2 A, and hence larger interlayer hopping strength. In the Moire superlattices formed in the CVD grown MoS2/WSe2 heterobilayer, STM/S measurement has also shown the modulation of the interlayer distance in the Moire pattern 9, and the correlation of with the local interlayer atomic registry agrees with the ab initio calculations in Table S3, but exhibits an even larger variation magnitude (the difference between the smallest and largest is ~1.5 A in MoS2/WSe2 heterobilayer, compared to the variation range of ~0.6 A for the ab initio results of bilayer MoS2 3 ). Table S3 Ab initio results (PBE+vdW or LDA) of bilayer MoS 2 interlayer distance for different stacking pattern. Adopted from Ref. 3. / 6.8 A 6.2 A 6.2 A 6.2 A 6.8 A NATURE PHYSICS 5

6 II. Tight-binding model in bilayer TMDs To confirm the topological phase transitions predicted by the bulk k.p models (Eq. 1 in main text), and to visualize the topologically protected helical states at edges or at TI/NI boundaries, we construct here a tight-binding (TB) model for R/H-type commensurate bilayer TMDs with a general relative translation. Our TMD bilayer model is constructed by adding interlayer hopping to the widely used three-band TB model for monolayer TMDs 10, which consists of three metal atomic d orbitals {, ± }. This three-band TB model describes well the low-energy physics in the ± valleys of monolayer TMDs. The TMD bilayer TB Hamiltonian is = () + () () () (S9) where H () is the three-band TB Hamiltonian for monolayer TMDs. ϵ= 1 is for the H- type stacking and ϵ =1 is for the R-type stacking. I is identify matrix, ( ) is the monolayer band gap in the upper (lower) layer, while is tunable by the interlayer bias. With this definition of, >0 corresponds to a TMD bilayer heterojunction of the normal type-ii band alignment where is the band gap, while <0 corresponds an inverted type-ii band alignment where the conduction band of lower layer energetically overlaps with the valence band of upper layer (c.f. Fig. 1a in main text). () is the interlayer hopping to be constructed below. In all of our numerical calculations, for the H part, we have used WSe2 TB parameters given in Ref. 10. a. Modeling the interlayer hopping In the lower (upper) layer, the three bands near are characterized by distinct orbital magnetic quantum numbers ( ), whose values are summarized below: Conduction band Valence band Remote band Table S4. The corresponding magnetic quantum numbers and for the three bands. Here = +1 ( 1) corresponds to the R-type (H-type) stacking. The remote band corresponds to that above the conduction band. In the {, ± } basis, () has its matrix elements () = = ( ) ( ), = () ( + ). (S10) NATURE PHYSICS 6

7 where ( ) and () ( - ) are the monolayer tight-binding basis functions in the upper and lower layers, respectively. ( + ) are in-plane position vectors of the metal atom in the upper (lower) layer, and ( + ) = () ( ) is the hopping integral between two metal d-orbitals from the two layers respectively with an in-plane displacement +. We consider here short-ranged interlayer hopping integrals () (i.e. decays fast with, here = ( cos, sin )), and its dependence on the angular coordinate has to follow the rotational symmetry of the metal -orbitals, which leads to = () ( means rotating the coordinate by angle). The angular and radial parts of () are then separable: () = () = ± (). For () to be analytical near =0, we have assumed the following function forms of (): () = e for = e for. (S11) where a is the lattice constant of the monolayer. The term in ensures the hopping between orthogonal d-orbitals will vanish when their lateral displacement =0, and it also ensures the analyticity of () near =0. The exact forms of () and the corresponding parameters adopted in the TB calculations are given in Fig. S2c. The () we used decays fast with a length scale of lattice constant a (c.f. Fig. S2c). With these hopping integrals, only the interlayer hopping between a metal site with its three nearestneighbor metal sites in the opposite layer need to be considered (c.f. Fig. S2a), and Eq. (S10) reduces to: () ( ) + ( ) + ( ). (S12) The resulted () values as functions of the interlayer-translation are shown in Fig. S3. Compared to Eq. (S8), we can see that the major features are in good agreement. Most importantly, with () observing the rotational symmetry, the resultant interlayer coupling fully observes the symmetry requirement, i.e. the vanishing of certain interlayer hopping matrix elements at high symmetry configurations (c.f. Table 1 in main text). Away from the points that correspond to the high symmetry configurations, the interlayer hopping matrix elements given by Eq. (S12) have some quantitative differences from Eq. (S8), due to the short-ranged nature of () (Umklapp terms () with 2 are not neglegible, see Fig. S2d). The TMD bilayer TB model adopting the hopping integrals in Fig. S2c has been used to calculate the phase diagram in Fig. 3i, and the band structures for the various TMD bilayer configurations in Fig. 3a-h. NATURE PHYSICS 7

8 Figure S2 Interlayer hopping integrals in the tight-binding model. (a) In-plane displacements of a metal atom in the upper layer (red ball) from its near-neighbor metal atoms in the lower layer (blue balls). With the function form of interlayer hopping integrals assumed in (c), a metal atom only hops to its three nearest-neighbor metal atoms in the opposite layer (i.e. from red atom to the three blue atoms at positions,, if the red atom is anywhere in the left half of the diamond, or blue atoms at,, if the red atom is in the right half of the diamond). (b) The extended Brillouin zone (BZ) with the first BZ marked orange. The yellow and light blue points denote the K and -K points respectively. (c) ) and (d) their Fourier transformations ) (c.f. supplementary text Ib). Figure S3 dependence of the interlayer hopping matrix elements at K point. (a-d) Upper row: () (Eq. (S12)) in R-type TMD bilayers based on the hopping integrals adopted in the TB model calculation for Fig. 3 of main text. Lower row: Eq. (S8) neglecting Umklapp terms, adopted in the k.p model calculation in Fig. 2 of main text. (e-h) H-type TMD bilayers. NATURE PHYSICS 8

9 b. The effect of remote bands In this three-band TB model, besides the lowest conduction and top valence bands that span the massive Dirac cones, there is also higher energy conduction bands (see Fig. S4a). So this model also allows us to analyze the effect of remote bands on the interlayer hybridization. The remote bands have two effects on the interlayer hybridization of the massive Dirac cones. First, the virtual intralayer and interlayer hoppings to these remote bands can correct the energy dispersions of the Dirac cones. The second effect is a more important one: there can be additional interlayer hopping quantum pathways through these remote bands to hybridize the Dirac cones at the inverted regimes, which can affect the topological property of the inverted band. Because the qualitative feature of topological phase diagram is determined by the topological property of the high symmetry configurations, in the following we focus on these TMD bilayers to illustrate the effect of remote bands. Expending the monolayer TB model at K points in the three-band basis, we have mon = + + +[ + ( + ) + ( + ) +h.. ], (S13) where,, denote respectively the Bloch states at the conduction band, valence band and the remote (higher conduction) band at K point, with,, being their energies. The intralayer couplings between,, are all linear in q with a chirality of 1 or -1. Figure S4 Quantum path way from upper layer valence band to lower layer conduction band. Allowed interlayer transitions at K point are shown by green arrows for six high symmetry configurations. The red (blue) arrows denote intralayer couplings with chirality -1 (1). NATURE PHYSICS 9

10 For the H-type and R-type TMD bilayer configurations with the rotation symmetry, the allowed interlayer hopping channels at K including that to the remote band are obtained from symmetry analysis 1 (c.f. supplementary text Ia), as shown in Fig. S4. In the inverted heterojunction, the hybridization happens between the upper Dirac cone (c-band) in lower layer and lower Dirac cone (v-band) in upper lower, highlighted by the shading in Fig. S4, where possible quantum pathways from the upper Dirac cone (c-band) in lower layer to lower Dirac cone (v-band) in upper lower can be identified: (i) For and configurations (Fig. S4a and d), the interlayer hybridization is still dominated by the direct hopping allowed between the upper Dirac cone (c-band) in lower layer to lower Dirac cone (v-band) in upper lower, since the hopping via other bands are higher order processes. Including the remote band contributions do not change the fact that and remain topologically trivial. (ii) For the and configurations (Fig. S4 e and f), there is a quantum pathway through a remote band which has the same chirality and is in the same order with the one within the Dirac cones in both layers, so the modification in the interlayer hybridization is only quantitative and does not change the quantum spin Hall conductance of the inverted band. There is also a higher order quantum pathway through a remote band which is much weaker and can be neglected. (iii) For the configuration (Fig. S4c), the quantum pathway through remote band is higher order compared to the one within the Dirac cones in both layers, so its effect is negligible and does not change the quantum spin Hall conductance of the inverted band. (iv) For the configuration (Fig. S4b), however, the quantum path ways through the remote bands have a chirality of 1, while the quantum path way within the Dirac cones is a weaker higher order process with a chirality of -2. In this case, the presence of the remote bands can change the quantum spin Hall conductance in the inverted band from -2 to 1. III. Tight-binding model for the 1D Moire superlattices in bilayer TMDs Here we consider the tight-binding modeling of TMD heterobilayers with 1D Moire superlattices, where the two layers have identical lattice period in y (zigzag) direction, while in x (armchair) direction they have a lattice mismatch. We assume one of the layers is unstrained, while the other layer is elongated in the x direction. In reality, such 1D Moire forms when a tensile uniaxial strain applies to a layer with a slightly larger lattice constant (c.f. Fig. 4 inset in the main text), which will enlarge the lattice period in one direction while reducing the period in the orthogonal direction, tuning a 2D Moire superlattice eventually to a 1D Moire superlattice (c.f. from Fig. 4b to 4d in the main text). The modeling of the interlayer hopping in this Moire pattern uses the function form of the interlayer hopping integral () given in supplementary text II. The local-to-local variation in the interlayer hopping is directly accounted through the variation in the atomic registry. In the strained layer, the intralayer hopping also changes as the distances and relative orientations between the pairs of atoms (c.f. Fig. S5). In our tight-binding model, this effect is also taken into account for the intralayer nearest hopping. Under strain, the positions of the NATURE PHYSICS 10

11 nearest neighboring metal atoms are changed from ~ to ~. For small η (c.f. Fig. S5), following reference 11, the intralayer hopping is modified according to h( ) =h( )(1 5 ) (S14) where h( ) is the interlayer hopping matrix element in the unstrained monolayer 10. Figure S5 The position of nearest neighboring atomic pairs within the strained layer (red balls), compared to the unstrained layer (blue balls). IV. Conduction band spin splitting Figure S6 (a) Schematics of spin splitting at the conduction and valence band edges of monolayer MX 2 at K valley. The ones at K valley are their time reversal. (b) and (c) Depending on the choices of the compounds and stacking in the TMD bilayer, the hybridization gap for the relevant spin species can be a true band gap (b), or overlap with the conduction band of the other spin species (c). TMDs have a strong spin-orbital coupling originated from the d orbitals of the metal atoms 1. The valence band edge splitting is about 0.15 ev for the MoX2 monolayer and about 0.45 ev for WX2 monolayer. The conduction band edge splitting is relatively smaller, about a few mev for MoS2 and tens of mev for WSe2. WS2 and MoSe2 1, as schematically shown in Fig. S6a. Because of the giant spin splitting in the valence band, only spin up (down) cone at the K (-K) valley is relevant in the interlayer hybridization. However, depending on the sign of the conduction band spin splitting, there could be two possible alignment between the hybridization gap for the relevant spin species with the conduction band of the other spin species, as shown in Fig. S6b and c. For a given choice for the TMD bilayer compounds, the R-type and H-type stacking would correspond respectively to these two different alignment scenarios. For device NATURE PHYSICS 11

12 applications of the helical modes in the hybridization gap, it is desired to have the hybridization gap being a true band gap for the inverted TMD bilayer (i.e. Fig. S6b), by choosing the TMD bilayer compounds and the stacking configurations. We note that the TMD bilayer TB model given in supplementary text II has taken into account the large spin-orbit splitting in the valence band, but not in the conduction band. So in plotting the band structures in Fig. 3a-h, we have added a valley dependent spin shift posteriorly to the conduction subbands that correspond to the scenario in Fig. S6b. V. Band alignment between monolayer semiconducting TMDs of hexagonal lattices (1H- MX2) C V CrS 2 CrSe 2 CrTe 2 CrS 2 CrSe 2 CrTe 2 MoS 2 MoSe 2 MoTe 2 WS 2 WSe 2 WTe a 0.91 d a 0.25 a 0.44 d 0.73 d MoS 2 MoSe 2 MoTe a 1.32 d 0.70 a a 0.80 d 0.24 a 1.09 d 0.45 a 0.88 b 0.95 a 0.96 e,g 1.34 f 1.79 d 0.39 b 0.49 a,g 0.74 f 1.27 d WS a 1.11 b 1.57 f 1.20 a,e 1.23 g WSe 2 WTe a 1.30 d 0.33 a 0.45 b 0.58 a,e 0.91 f 1.3 h 1.77 d 0.04 b 0.13 e 0.17 g 0.34 f 0.86 a,g 1.12 f 1.42 c 1.65 d 0.91 b 0.95 a,e 0.97 g 1.29 f 1.26 a 1.31 b,e 1.77 c 1.78 d 0.80 a,g 1.26 d 1.15 f 1.58 d 0.88 b 0.89 a 1.32 f 1.76 d 0.50 b,e 0.54 g 0.72 f 0.61 b,g 0.88 f 0.48 e,g 0.75 f 1.09 c Table S5. The numbers in row-i and column-j denotes the band gap of the type-ii heterojunction, i.e. the energy difference between the valence band-edge in compounds i and the conduction band-edge in compounds j. The sources of the numbers are denoted by the superscripts: a. DFT-LDA 12 ; b. DFT-PBE 13 c. DFT-HSE06 13 ; d. LDA-G 0W 0 12 ; e. DFT-PBE 14 ; f. DFT-HSE06 14 ; g. DFT-PBE 15 ; h. Experiment 16 (microbeam XPS and STM/STS). Negative values highlighted in blue font mean the TMD heterobilayer is in the inverted regime. NATURE PHYSICS 12

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