Trigonal Quasicrystalline States in 30^\circ Rotated Double Moiré Superlattices
TTrigonal Quasicrystalline States in ◦ RotatedDouble Moiré Superlattices
J. A. Crosse and Pilkyung Moon New York University Shanghai, Arts and Sciences, Shanghai, 200122, China NYU-ECNU Institute of Physics at NYU Shanghai, Shanghai, 200062, China New York University, Department of Physics, New York, 10003, USA * [email protected] ABSTRACT
We study the lattice configuration and electronic structure of a double moiré superlattice, which is composed of a graphenelayer encapsulated by two other layers in a way such that the two hexagonal moiré patterns are arranged in a dodecagonalquasicrystalline configuration. We show that there are between 0 and 4 such configurations depending on the lattice mismatchbetween graphene and the encapsulating layer. We then reveal the resonant interaction, which is distinct from the conventional2-, 3-, 4-wave mixing of moiré superlattices, that brings together and hybridizes twelve degenerate Bloch states of monolayergraphene. These states do not fully satisfy the dodecagonal quasicrystalline rotational symmetry due to the symmetry of thewave vectors involved. Instead, their wave functions exhibit trigonal quasicrystalline order, which lacks inversion symmetry, atthe energies much closer to the charge neutrality point of graphene.
Introduction
When two or more two-dimensional atomic layers which do not share a common periodicity are overlaid, an additionalperiodicity in the form of moiré interference pattern emerges . The electronic structures of such systems - for example twistedbilayer graphenes , graphene on hexagonal boron nitride (hBN) , and twisted bilayer transition metal dichalcogenides with small twist angles θ ≈ ◦ - have been investigated extensively. These materials have very long moir’e superlattice vectors L M i ( i = , ) , and, hence, exhibit many exotic properties such as the Fermi velocity renormalization , mini Dirac pointsformation , Hofstadter’s butterfly , the emergence of superconductivity , correlated phases , and orbital magneticmoment .A special case occurs when two hexagonal lattices are overlapped at θ = ◦ [Fig. 1(a)]. In this instance the atomicarrangement is mapped on to a quasicrystalline lattice, which is ordered but not periodic, with a 12-fold rotational symmetry .Owing to the momentum mismatch , quasicrystalline twisted bilayer graphene exhibits the electronic structures of almostdecoupled bilayer graphene at most energy ranges. Nevertheless, it also hosts unique electronic states which satisfy the 12-foldrotational symmetry . Such quasicrystalline states arise from the resonant interaction between the states at specific wavevectors via the rotational symmetry of the quasicrystal as well as the translational symmetry of the constituent atomic layers .The red and blue hexagons in Fig. 1(b) show the first Brillouin zones of the two lattices. The numbered points and dashedlines show the wave vectors of the constituent monolayer states and the interlayer interaction which form the quasicrystallineresonant states. Such quasicrystalline states exhibit a wave amplitude distributed selectively on a limited number of sites in acharacteristic 12-fold rotationally symmetric pattern [Fig. 1(c)]. These states, however, appear at the energies (about ± . ) -far from the charge neutrality point of graphene. Similar quasicrystalline resonant states also arise in any bilayer stacked in aquasicrystalline configuration if all the dominant interlayer interactions occur between the atomic orbitals that have the samemagnetic quantum number . Thus, even transition metal dichalcogenides or square lattices can show the quasicrystallinestates.Recently, rapid progress has been made in stacking more than two incommensurate atomic layers and a number of studieshave investigated the effects of multiple moiré superlattice potentials on the electronic structure. The most notable exampleamong them is a double moiré system, which is composed of a graphene layer encapsulated by hBN layers (BN/G/BN) .The lattice mismatch between graphene and hBN results in a hexagonal moiré superlattice potential with a superlattice period L M i ( i = ,
2) that can be as long as 14 nm [Fig. 2(a)]. Such a long period [which results in short superlattice reciprocal latticevectors, Fig. 2(b)] carves the graphene electronic structures into superlattice bands with an energy scale much smaller thanthat of pristine graphene [Fig. 2(c)] . Recently, Leconte and Jung show that BN/G/BN at specific configurations can hosttwo hexagonal moiré patterns overlaid at a twist angle of 30 ◦ , and claimed that the system hosts quasicrystalline electronic a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b tructures . However, the interaction mechanism responsible for such unique electronic states in BN/G/BN, as well as theactual electronic band structures, and whether the wave functions actually satisfy the symmetry of the quasicrystal have not yetbeen investigated.Here, we investigate the conditions where the two hexagonal moiré patterns in double moiré superlattice are arranged in adodecagonal (12-fold) quasicrystalline configuration. Then we reveal the resonant interactions that bring together and hybridizetwelve degenerate Bloch states of monolayer graphene and show that such interactions reconstruct the band dispersion ofpristine graphene at these wave vectors. Compared to the resonant states of quasicrystalline twisted bilayer graphene wherethe actual atomic lattices are arranged in a dodecagonal configuration , the resonant states of BN/G/BN appear at theenergies much closer to the charge neutrality point of graphene. However, their wave functions show the quasicrystalline orderwith a 3-fold rotational symmetry rather than fully satisfying the 12-fold rotational symmetry of the double moiré pattern. Figure 1. (a) Lattice structures of quasicrystalline twisted bilayer graphene . The red and blue hexagons represent theunit cells of each layer. (b) The wave vectors of the twelve monolayer states C n ( n = , , · · · ,
11) which hybridize toquasicrystalline resonant states. The red and blue hexagons, and the red and blue arrows [ a ∗ i and ˜ a ∗ i ( i = , Γ point. (c) Local density of states of the quasicrystalline resonant states. The areaof the circle is proportional to the squared wave amplitude, and red and blue circles represent the states in the upper and thelower layers, respectively. Methods
Hexagonal moiré superlattices stacked at ◦ We consider a trilayer system composed of graphene sandwiched by hBN. Both graphene and hBN are two-dimensionalhoneycomb lattice whose unit cell comprises of two ( A and B ) sublattices. Graphene has carbon atoms in both sublattices,while hBN has nitrogen atom on A site and boron atom on B site. The lattice constant of hBN, ˜ a ≈ . , is slightlylarger than that of graphene, a ≈ .
246 nm, and we use a constant interlayer distance of d = .
322 nm between the adjacent twolayers . Here, we do not consider the lattice relaxation between graphene and hBN , since the effects of such relaxation onthe electronic structures is an order of a few meV. Nevertheless, our effective theory that respects the lattice symmetry is able toproperly describe both the gap at the primary Dirac point and the asymmetric gap opening at the two inequivalent mini Diracpoints, as well as the orbital magnetism of the structure .We define the atomic structure of the double moiré superlattices by starting from a nonrotated arrangement, where thehexagon center of the three layers share the same in-plane position ( x , y ) = ( , ) , and the A - B bonds are parallel to each other.We choose a = a ( , ) and a = a ( / , √ / ) ( a = .
246 nm) as the primitive lattice vectors of graphene, and τ A = − τ and τ B = τ [ τ = − ( / )( a − a ) ] as the coordinates of the A and B sublattices in the unit cell. The primitive lattice vectors ofthe top ( l = t ) and the bottom ( l = b ) hBN layers become ˜ a ( l ) i = M a i ( i = , M = ( + ε ) I represents the isotropicexpansion by the factor 1 + ε = ˜ a / a ≈ . τ ( l ) N = − τ ( l ) ± d e z and τ ( l ) B = τ ( l ) ± d e z [ τ ( l ) = − ( / )( ˜ a ( l ) − a ( l ) ) ], wherethe upper and lower signs are for the top and bottom layers, respectively, represent the coordinates of the nitrogen and boronatoms in the unit cell . We define the reciprocal lattice vectors a ∗ i and ˜ a ∗ i for graphene and hBN, respectively, so as to satisfy a i · a ∗ j = ˜ a i · ˜ a ∗ j = πδ i j . We then rotate the top and bottom hBN layers with respect to graphene by arbitrary angles θ ( t ) and igure 2. (a) Lattice structure of graphene on hBN . The black and red hexagons represent the unit cells of graphene andhBN, respectively, and θ shows the relative orientation. The green hexagons and the vectors L M i ( i = ,
2) represent the unitcells and superlattice vectors of the moiré superlattice, respectively. Here, the lattice constant of hBN is drawn 15% larger thanthat of graphene to enhance the visibility of the moiré pattern (actual difference is about 1.79%). Inset shows the latticeconfiguration of graphene; the black and white circles represent the A and B sublattices, a i and τ X show the primitive latticevectors and the coordinates of the sublattices, respectively. (b) Superlattice Brillouin zone (blue hexagon) near the Dirac point(the region surrounded by blue lines in the inset) and the reciprocal lattice vectors G M i of graphene on hBN. X and Y show theBrillouin zone corners where mini Dirac point appear, and φ shows the relative orientation of G M1 to the reciprocal latticevector a ∗ of pristine graphene. Inset shows the first Brillouin zone of graphene, where the black and white circles represent thethree equivalent Dirac points, K and K (cid:48) , respectively. (c) The band dispersion of the first two bands in the conduction andvalence bands of graphene on hBN with θ = ◦ , which show the band opening at the primary and the mini Dirac points. θ ( b ) around the origin, respectively. From now on, we use "BN/G/BN" for this configuration only. Due to the symmetry of thelattice, 0 ≤ θ ( l ) ≤ ◦ ( l ∈ t , b ) spans all the independent configurations.Figure 3(a) shows the moiré interference patterns which arise from the lattice mismatch between the top hBN and graphene(left side), and also that from the bottom hBN and graphene (right side), respectively, and Fig. 3(b) shows the atomicconfiguration of the three layers. The lattice vectors L M , ( l ) i and the reciprocal lattice period G M , ( l ) i ( i = ,
2) of each moirésuperlattice are L M , ( l ) i = cR ( φ ( l ) ) a i , G M , ( l ) i = c − R ( φ ( l ) ) a ∗ i , (1)respectively, where c = ( + ε ) / (cid:112) ε + ( + ε )( − cos θ ( l ) ) , φ ( l ) = arctan [ − sin θ ( l ) / ( + ε − cos θ ( l ) )] , and R ( φ ) is a rotationby φ . We plot | L M i | and | G M i | against θ in Fig. 3(c) in red and blue lines, respectively.Now, we will find the configuration where the unit cells of the two hexagonal moiré superlattices have the same size and areoverlaid with a relative twist angle of 30 ◦ . In such a configuration, the overlaid two hexagonal superlattices are mapped onto a12-fold rotationally symmetric quasicrystalline lattice without any translational symmetry, as first shown by Stampfli . FromEq. (1), the former and the latter conditions give | θ ( t ) | = | θ ( b ) | and φ ( t ) − φ ( b ) ≡ ◦ ( mod 60 ◦ ) , which can be simultaneouslysatisfied by θ ( t ) = − θ ( b ) and φ ( t ) = − φ ( b ) ≡ ◦ ( mod 30 ◦ ) . Figure 3(d) shows φ as a function of θ for various ε . Thered, green, blue lines correspond to ε > ε = ε <
0, respectively, and the thick black line corresponds to hBN. The twohexagonal moiré superlattices form a dodecagonal quasicrystalline configuration at θ where the line and the dashed horizontallines cross. If the lattice constant of the top and bottom layers is the same as that of the middle, graphene layer, i.e., ε = ◦ . On the other hands, the systems with ε < < ε < . ε ≥ . θ which satisfy the conditions. When the top and bottom layersare hBN, i.e., ε ≈ . θ = . ◦ , θ = . ◦ , θ = . ◦ , θ = . ◦ , and the corresponding | G M i | are 0.0182, 0.0251, 0.0795, 0.438 times the | a ∗ i | . Note that θ gives very long | G M i | , and accordingly very short | L M i | , whichcompetes with the length scale of monolayer graphene. By choosing θ ( t ) = − θ i and θ ( b ) = θ i ( i = , , , φ ( t ) = − φ i and φ ( b ) = φ i , where φ = − ◦ , φ = − ◦ , and φ = φ = − ◦ . Then, the twelve moiré reciprocal lattice vectors {± G M , ( l ) i | i = , , , l = t , b } , (2) igure 3. Lattice structure of graphene (gray hexagons) encapsulated by the top (red) and bottom (blue) hBN layers with twistangles of θ ( t ) and θ ( b ) , respectively. Green hexagons show the unit cells of the moiré superlattices which are formed betweenthe graphene and each of the hBN layer. We draw only the top (bottom) hBN layer at the left (right) side to enhance thevisibility of the pattern. The two hexagonal superlattice unit cells are arranged at a relative angle of 30 ◦ , and form adodecagonal quasicrystalline pattern when overlaid. (b) Atomic configuration of the three layers which shows the D pointgroup symmetry. (c) The lengths of the moiré lattice vectors (red line) and reciprocal lattice vectors (blue line) plotted against θ . The circles correspond to the values at the configuration shown in Fig. 3(a). (d) The angle φ between the primitive vectors(both the real-space and the reciprocal lattice vectors) of graphene and moiré superlattice plotted against the twist angle θ between the two lattices. The red, green and blue lines show the plot for the systems with ε > ε = ε <
0, respectively.The black line corresponds to that between graphene and hBN ( ε = . − θ and θ from graphene,respectively.where G M , ( l ) = − G M , ( l ) − G M , ( l ) , are arranged in 12-fold rotational symmetry [Fig. 4(a)], just like the reciprocal lattice vectorsin quasicrystalline twisted bilayer graphene that give rise to the resonant states [Fig. 1(b)] .It should be noted that, however, although the overlap of the two moiré interference patterns are mapped onto a quasicrys-talline tiling with 12-fold rotational symmetry, the actual lattice structure belongs to the symmetry group D ; it is invariantunder C rotation about the axis perpendicular to the xy -plane and under three C rotation about the axes in the plane, butlacks inversion symmetry. If we replace the top and bottom hBN layers by a material having the same types of atoms in bothsublattices, then the lattice has the symmetry group D which is still lower than the 12-fold rotational symmetry. Hamiltonian of double moiré superlattices
The total tight-binding Hamiltonian of the double moiré superlattice is expressed as H = H G + H ( t ) hBN + H ( b ) hBN + U ( t ) + U ( l ) , (3)where H G and H ( t ) hBN ( H ( b ) hBN ) represent the Hamiltonian for the intrinsic monolayer graphene and the top (bottom) hBN,respectively, U ( t ) ( U ( b ) ) is for the interlayer coupling between the graphene and the top (bottom) hBN. However, since the hBN igure 4. (a) Relative orientation of the moiré reciprocal lattice vectors of the lower moiré superlattice ( G M , ( b ) i , blue arrows)for θ i ( i = , , ,
4) with respect to the direction of the vectors of the upper moiré superlattice ( G M , ( t ) i , red arrows). (b) Relativedirection of the wave vectors involved in the resonant coupling [showing the number n of C n , Eq. (14)] with respect to thedirection of G M , ( t ) i . In both figures, note that the actual direction of G M , ( t ) is φ rotated from a ∗ [Eq. (1)].electronic bands are far from the charge neutrality point of graphene, we can project the total Hamiltonian on to the Blochbases of graphene p z orbitals at each sublattice, | k , X (cid:105) = √ N ∑ R X e i k · R X | R X (cid:105) (4)where | R X (cid:105) is the atomic orbital at the site R X = n a + n a + τ X ( n i ∈ Z , X = A , B ), k is the two-dimensional Bloch wavevectors and N = S tot / S is the number of the graphene unit cells with an area S = ( √ / ) a in the total system area S tot . Then,the Hamiltonian near the Dirac point K ξ = − ξ ( a ∗ + a ∗ ) / ξ = ± K and K (cid:48) , respectively, is reducedto a 2 × ,˜ H = H G + U ( t ) † ( − H ( t ) hBN ) − U ( t ) + U ( b ) † ( − H ( b ) hBN ) − U ( b ) ≡ H G + V ( t ) hBN + V ( b ) hBN . (5)The intralayer matrix elements of graphene are given by H G = (cid:18) h AA h AB h BA h BB (cid:19) , h X , X (cid:48) ( k ) = ∑ L − T ( L + τ X (cid:48) X ) e − i k · ( L + τ X (cid:48) X ) , (6)where L = n a + n a , τ X (cid:48) X = τ X (cid:48) − τ X , and − T ( R ) = V pp π (cid:34) − (cid:18) R · e z R (cid:19) (cid:35) + V pp σ (cid:18) R · e z R (cid:19) , V pp π = V pp π e − ( R − a / √ ) / r , V pp σ = V pp σ e − ( R − d ) / r , (7)is the transfer integral between two p z orbitals at a relative vector R , V pp π ≈ − .
38 eV , V pp σ ≈ .
48 eV, and r ≈ . . The effective potentials by hBN to graphene, V ( l ) hBN , are explicitly written as V ( l ) hBN = W + { W ξ e i ξ G M , ( l ) · r + W ξ e i ξ G M , ( l ) · r + W ξ e i ξ G M , ( l ) · r + h . c . } , (8)where we truncated much weaker terms O ( u ) which are associated with longer momentum difference. Here, W = V (cid:18) (cid:19) , W ξ = V e i ξψ (cid:18) ω − ξ ω − ξ (cid:19) , W ξ = V e i ξψ (cid:18) ω ξ ω ξ ω − ξ (cid:19) , W ξ = V e i ξψ (cid:18) ω − ξ ω − ξ (cid:19) , (9) nd V = − u (cid:18) V N + V B (cid:19) , V e i ξψ = − u (cid:18) V N + ω ξ V B (cid:19) , (10)where ω = e π i / , and u ≈ − t ( K ξ ) ≈ .
152 eV is the in-plane Fourier transformation of the transfer integral between two p z orbitals [Eq. (7)] t ( q ) = S (cid:90) T ( r + z ˜ XX e z ) e − i q · r d r (11)at q near the Dirac point . We will discuss more about u later. By using V C = V N = − .
40 eV, and V B = .
34 eV, as theon-site potential of carbon, nitrogen, and boron atoms, respectively , we get V ≈ . V ≈ . ψ ≈ − . V N = V B , then ψ ≡ π / ( mod π ) . The symmetry of such a structure increases to D , and the reduced Hamiltonian ˜ H gains theinversion symmetry˜ H ( − ξ ) ( k , r ) = σ x [ ˜ H ( ξ ) ( − k , − r )] σ x . (12)It is straightforward to show that the reduced Hamiltonian ˜ H spans the subspace {| k , X (cid:105) | k = k + ∑ l = t , b ∑ i = , , m ( l ) i G M , ( l ) i , m ( l ) i ∈ Z } , (13)for any k in the momentum space. To investigate the electronic structures near k , for any practical calculation, we only needa limited number of bases around k inside a certain cut-off circle k c , because the interaction with the states far from k is veryweak due to multiple scattering. We can, then, obtain the energy eigenvalues at all the wave vectors in Eq. (13) by diagonalizingthe Hamiltonian matrix within the finite bases, and the quasiband dispersion of the system by plotting the energy levels against k . Here the wave number k works like the crystal momentum for the periodic system, and so it can be called the quasicrystalmomentum. Quasicrystalline resonant interaction by two moiré superlattice potentials
In addition to the typical 2- and 3-wave interaction by each moiré superlattice [Fig. 2(c)] , the 12-fold rotational symmetryof the wave vectors which couple the monolayer states [Eq. (2)] as well as the translational symmetries of the two moirésuperlattices enables a unique interaction between twelve degenerate monolayer states. Such a resonant coupling occurs at thetwelve symmetric points C n = K ξ + | G M , ( l ) i | sin ( π / )( cos θ n , sin θ n ) , (14)shown in Fig. 4(b), where θ n = π + n π − φ i ( n = , , , · · · , Γ point , the waves involved in the resonant interaction inBN/G/BN are centered around the Dirac point K ξ . These twelve states are degenerate if we ignore the small trigonal warpingin this low energy regime. We see that the states at C i strongly interact with the states at C i − and C i + by the reciprocal latticevectors of the top and bottom moiré superlattices. The interaction to the states at any other k can be safely neglected since theinteraction strength is much less or the two states are not degenerate in most cases. Hence, these states form one-dimensionalmonatomic chain with twelve sites and two pseudospins whose interaction is described by the moiré potential [Eq. (5)].It should be noted that this is not the only resonant coupling in this system. As shown in previous work on quasicrystallinetwisted bilayer graphene (e.g., Appendix A in Ref. ), there are more sets of states, with different wave numbers, that show theresonant interaction between the constituent monolayer states. However, the set in Fig. 4(b) is associated with the strongestinteraction | t ( q ) | and, hence, gives the largest energy separation between the hybridized states. esults and Discussion By using the Bloch bases ( | k ( ) (cid:105) , | k ( ) (cid:105) , · · · , | k ( ) (cid:105) ) near the twelve wave vectors k ( n ) = C n + k , where | k ( n ) (cid:105) is ( | k ( n ) , A (cid:105) , η | k ( n ) , B (cid:105) ) with η = ω ξ × floor ( n / ) , we can express the Hamiltonian of the resonant ring H ξ ring by a 24 ×
24 matrix H ξ ring ( k ) = H ( ) W ξ Y ξ †1 W ξ †2 H ( ) X ξ †2 X ξ H ( ) W ξ †1 W ξ H ( ) Y ξ Y ξ †1 H ( ) W ξ W ξ †2 H ( ) X ξ †2 X ξ H ( ) W ξ †1 W ξ H ( ) Y ξ Y ξ †1 H ( ) W ξ W ξ †2 H ( ) X ξ †2 X ξ H ( ) W ξ †1 Y †1 W ξ H ( ) . (15)Here H ( n ) i = (cid:32) h ( n , i ) AA h ( n , i ) AB h ( n , i ) BA h ( n , i ) BB (cid:33) + W , h ( n , i ) X (cid:48) X ( k ) = h X (cid:48) X [ R ( − n π / ) k + C i ] , (16)and X ξ = W ξ †1 for θ W ξ for θ W ξ †2 for θ and θ Y ξ = V e − i ξψ (cid:18) ω − ξ ω − ξ (cid:19) for θ V e i ξψ (cid:18) ω − ξ ω ξ (cid:19) for θ V e − i ξψ (cid:18) ω ξ ω ξ ω − ξ (cid:19) for θ and θ (17)for θ , θ , and both θ and θ , respectively.Obviously, Eq. (15) is symmetric under rotation by four span of the ring (i.e., moving C n to C n + ), which actuallycorresponds to the [ R ( π / )] (120 ◦ rotation) of the entire system. This means that the resonant states of the BN/G/BN havea 3-fold rotational symmetry, which is much lower than the 12-fold rotational symmetry of the double moiré patterns of thesystem. This Hamiltonian cannot obtain a 6-fold rotational symmetry, even if we replace the top and bottom hBN layers bya material having the same types of atoms in both sublattices (e.g., V N = V B ), since (i) atomic structures lacks the 12-foldrotational symmetry and (ii) the 12 wave vectors involved are centered at K ξ which has a 3-fold rotational symmetry [Fig. 2(b)].Thus, the relevant terms cannot be gauged out by a similarity transformation. There is, however, an exception in the systemswith V N = V B ; the wave functions at k = , and only at this k , show a 6-fold rotational symmetry [Figs. 6(d)-(f)].The interlayer interaction strength u [Eq. (10)] deviates from − t ( K ξ ) as the distance between k ( n ) and K ξ increases. Thiseffect becomes obvious in the system with a longer | G M , ( l ) i | , e.g., in BN/G/BN with θ . However, what is more importantis that, the u associated with the interaction between the neighboring Bloch states | k ( n ) (cid:105) ( n = , , · · · ,
11) [dashed lines inFig. 4(b)] are not the same. Since the Fourier transformation of the transfer integral between p z orbitals, t ( q ) [Eq. (11)], areisotropic along the in-plane direction, t ( q ) depends only on | q | . If any electronic state is mainly comprised of three monolayerstates of which waves vectors are arranged in a 3-fold rotationally symmetric way around K ξ , e.g., K ξ + k , K ξ + R ( π / ) k , K ξ + R ( π / ) k , then the | q | associated with the interaction between the monolayer states are identical since we can freelychoose one among the three equivalent K ξ , i.e., K ξ , R ( π / ) K ξ , R ( π / ) K ξ , for each state. This is the case that happens atthe mini Dirac point (the hexagonal Brillouin zone corners) of graphene on hBN . On the contrary, t ( q ) for each interactionof the resonant coupling in BN/G/BN are not identical, since there are more than three states involved. As a result, u in q. (10) varies ± ,
3% for θ , θ , θ , and ±
11% for θ . Nevertheless, since the twelve wave vectors are arranged in a 12-foldrotationally symmetric way, equally spaced triplets (i.e., n ∈ { , , } , n ∈ { , , } , n ∈ { , , } , n ∈ { , , } ) satisfy the3-fold rotational symmetry with K ξ . As a result, the oscillation of u is consistent with the rotational symmetry of H ξ ring and does not reduce the 3-fold symmetry further. Hereafter, we will consider the structures with θ , θ , θ only, and use u = − t ( K ξ ) ≈ .
152 eV, i.e., ignore the variation of u .Since a hBN layer lacks the inversion symmetry, it is natural to ask whether the band structures change if we rotate oneof the hBN layers by 180 ◦ (We label this structure BN/G/NB). The BN/G/NB also belongs to the symmetry group D , whilethe three C axes are 60 ◦ rotated from those of BN/G/BN. It is well known that twisted double bilayer graphene andBN/G/BN at general angles show the change of electronic structures with respect to such a change. The 180 ◦ rotation ofhBN corresponds to the swap of the boron and nitrogen atoms. Thus, we can get the effective potential Eq. (8) of the moirésuperlattice from such a layer by replacing ψ by − ψ + π / e i ξψ ω − ξ ( e − i ξψ ω ξ ) to the Bloch bases | k ( n ) (cid:105) with n ≡ , ( mod 4 ) for θ , θ , θ ( θ ). As a result, theresonant states are invariant with respect to the replacement of BN to NB. Figure 5. (a) Electronic structure in the valence band side of the 30 ◦ rotated double moiré superlattice at θ . (b) Energycontours of the third (top panel) and the sixth (bottom panel) valence bands which clearly show the 3-fold rotational symmetry.Figure 5(a) shows the valence band structures of BN/G/BN at θ near the resonant states at C n plotted as a function of k . The twelve Dirac cones are arranged on a circle with a radius ∆ k = | G M , ( l ) i | sin ( π / ) and they are strongly hybridizednear k = to exhibit the characteristic dispersion including parabolic bottoms, a frilled band edge, and new Dirac points at − .
164 eV and − .
203 eV. We have similar resonant states also in the conduction band, while the energy spacing between theresonant states is much smaller than in the valence band, just like the cases of graphene on hBN and quasicrystalline twistedbilayer graphene . As predicted by the symmetry of H ξ ring , the band structures exhibit three-fold rotational symmetry around k = , as we can clearly see from the energy contours in Fig. 5(b). The structures with the other angles, θ i ( i = , , | K ξ − C n | . The energy splitting between the resonant states in BN/G/BN, O ( | V | ) , is muchsmaller than that in the quasicrystalline twisted bilayer graphene, O ( | u | ) , since the interaction here involves a second order cattering through the hBN layer. Nevertheless, the resonant states of BN/G/BN appear at the energies much closer to thecharge neutrality point of graphene than those of quasicrystalline twisted bilayer graphene (about ± . ) , since the wavevectors responsible for the interaction in BN/G/BN ( O ( | G M , ( l ) i | ) ) are much shorter than those in quasicrystalline twisted bilayergraphene ( O ( | a ∗ i | ) ). Thus, the resonant states of BN/G/BN appear at much smaller, experimentally feasible, electron densities.At k = , we can reduce H ξ ring to an 8 × H ( ξ , m ) ring ( k ) = H W ξ Y ξ †1 ω − m W ξ †2 H X ξ †2 X ξ H W ξ †1 Y ξ ω m W ξ H , (18)by using the Bloch condition along the one-dimensional chain. Here, H i = H ( ) i ( k = ) ( i = , , ,
3) and m = − , , H ( ξ , m ) ring exhibits a symmetry ( Σ K ) − H ( ξ , m ) ring Σ K = H ( ξ , m (cid:48) ) ring , (19)for m and m (cid:48) satisfying m + m (cid:48) ≡ − ξ ( mod 3 ) . Here, Σ is diag ( σ x χ , σ x , σ x χ ∗ , σ x ) for θ , θ , θ and diag ( σ x , σ x χ ∗ , σ x , σ x χ ) for θ , where χ is e i ξψ ω − ξ , and K stands for complex conjugation. Thus, the states with ( m , m (cid:48) ) = ( , − ξ ) form twofold doublets,and belong to two-dimensional E irreducible representation of D point group, while the states m = ξ is non-degenerate, andbelong to either of A or A . Figure 6.
The electron probability distribution of the wave functions at k = of the double moiré superlattice (a)-(c) in theabsence of the inversion symmetry, i.e., V N (cid:54) = V B and ψ (cid:54) = π / ( mod π ) , and (d)-(f) in the presence of the inversion symmetry,i.e., V N = V B and ψ ≡ π / ( mod π ) . (a) and (d) are for θ , (b) and (e) are for θ , and (c) and (f) are for θ . We show the thirdstate ( m = ξ ) in the valence band, since the first two states ( m = , − ξ ) are degenerate.Figure 6 shows the quasicrystalline wave functions of the third resonant state ( m = ξ ) in the hole side at k = on thegraphene lattice. Figures 6(a)-(c) show the wave functions in a system in the absence of the inversion symmetry, i.e., V N (cid:54) = V B nd ψ (cid:54) = π / ( mod π ) , such as graphene encapsulated by hBN layers, and Figs. 6(d)-(f) show those in a system with theinversion symmetry, i.e., V N = V B and ψ ≡ π / ( mod π ) , such as graphene encapsulated by a material having the same typesof atoms in both sublattices. In both systems, the wave amplitude show the quasicrystalline order which is distributed ona limited number of sites in a pattern which is incompatible with the periodicity. Unlike the rotational symmetry of thedouble moiré pattern (12-fold), however, the wave functions in the absence of the inversion symmetry show a 3-fold rotationalsymmetry, while those in the presence of the inversion symmetry show a 6-fold rotational symmetry, which slightly resemblesthe probability distribution of the system with a true 12-fold rotational symmetry . It should be noted that, however, thewave functions of the system with the inversion symmetry lose the 6-fold rotational symmetry at k (cid:54) = . Figures 6(a) and(d), (b) and (e), (c) and (f) show the wave functions at θ , θ , θ , respectively. The length scale of the patterns is much largerthan that of the quasicrystalline wave functions in quasicrystalline twisted bilayer graphene , since the difference betweenwave vectors involved is on the order of | G M , ( l ) i | in the dual moiré superlattice and on the order of | a ∗ i | in the twisted bilayergraphene ( | G M , ( l ) i | (cid:28) | a ∗ i | ). In addition, the systems with θ and θ show larger scale, since | G M , ( l ) i | is almost proportional to θ [Fig. 3(c)]. Such an electron distribution would be prominent at the energies where the band curvature of the resonant states arelarge enough to give high density of states. At the energies away from the resonant states, on the contrary, we will mainly seethe simple overlap of the periodic wave functions of the two typical graphene on hBN superlattice with a twist angle of 30 ◦ . Conclusions
We investigated the lattice configuration and electronic structures of a double moiré superlattice of which two hexagonal moirépatterns are arranged in a dodecagonal quasicrystalline configuration. We first find the condition which gives a 30 ◦ stack of thetwo moiré patterns in graphene encapsulated by another layers, and show that there are 0 to 4 such configurations depending onthe lattice mismatch between graphene and the encapsulating layer. And we show that, although the moiré patterns satisfy a12-fold rotational symmetry, the actual atomic lattice has only a 3-fold rotational symmetry ( D ) if the encapsulating layershave different atomic species in the sublattices (e.g., hBN).We then reveal the resonant interaction which brings together and hybridize twelve degenerate Bloch states of monolayergraphene as well as the band dispersion around the resonant states. Compared to the resonant states of quasicrystalline twistedbilayer graphene, of which atomic lattices are arranged in a dodecagonal configuration, the resonant states of double moirésuperlattice lack the 12-fold rotational symmetry; they hexagonal quasicrystalline order at a specific point k = in the Brillouinzone if the encapsulating layers the same types of atoms in both sublattices, and trigonal quasicrystalline order otherwise. Theseunique states appear at the energies much closer to the charge neutrality point of graphene and experimentally feasible electrondensities. References Berger, C. et al.
Electronic confinement and coherence in patterned epitaxial graphene.
Science , 1191–1196 (2006). Lopes dos Santos, J. M. B., Peres, N. M. R. & Castro Neto, A. H. Graphene bilayer with a twist: Electronic structure.
Phys. Rev. Lett. , 256802 (2007). Shallcross, S., Sharma, S. & Pankratov, O. A. Quantum interference at the twist boundary in graphene.
Phys. Rev. Lett. , 056803 (2008). Bistritzer, R. & MacDonald, A. H. Moiré bands in twisted double-layer graphene.
Proc. Natl. Acad. Sci. , 12233–12237(2011). Moon, P. & Koshino, M. Energy spectrum and quantum hall effect in twisted bilayer graphene.
Phys. Rev. B , 195458(2012). Moon, P. & Koshino, M. Optical absorption in twisted bilayer graphene.
Phys. Rev. B , 205404 (2013). Dean, C. R. et al.
Boron nitride substrates for high-quality graphene electronics.
Nat. nanotechnology , 722–726 (2010). Hunt, B. et al.
Massive dirac fermions and hofstadter butterfly in a van der waals heterostructure.
Science , 1427–1430(2013). Ponomarenko, L. et al.
Cloning of dirac fermions in graphene superlattices.
Nature , 594–597 (2013).
Kindermann, M., Uchoa, B. & Miller, D. L. Zero-energy modes and gate-tunable gap in graphene on hexagonal boronnitride.
Phys. Rev. B , 115415 (2012). Ortix, C., Yang, L. & van den Brink, J. Graphene on incommensurate substrates: Trigonal warping and emerging diraccone replicas with halved group velocity.
Phys. Rev. B , 081405 (2012). Wallbank, J. R., Patel, A. A., Mucha-Kruczy´nski, M., Geim, A. K. & Fal’ko, V. I. Generic miniband structure of grapheneon a hexagonal substrate.
Phys. Rev. B , 245408 (2013). Moon, P. & Koshino, M. Electronic properties of graphene/hexagonal-boron-nitride moiré superlattice.
Phys. Rev. B ,155406 (2014). Hsu, W.-T. et al.
Second harmonic generation from artificially stacked transition metal dichalcogenide twisted bilayers.
ACS nano , 2951–2958 (2014). Fang, H. et al.
Strong interlayer coupling in van der waals heterostructures built from single-layer chalcogenides.
Proc.Natl. Acad. Sci. , 6198–6202 (2014).
Bistritzer, R. & MacDonald, A. H. Moiré butterflies in twisted bilayer graphene.
Phys. Rev. B , 035440 (2011). Cao, Y. et al.
Unconventional superconductivity in magic-angle graphene superlattices.
Nature , 43–50 (2018).
Cao, Y. et al.
Correlated insulator behaviour at half-filling in magic-angle graphene superlattices.
Nature , 80–84(2018).
Moriya, R. et al.
Emergence of orbital angular moment at van hove singularity in graphene/h-bn moiré superlattice.
Nat.communications , 5380 (2020). Yao, W. et al.
Quasicrystalline 30° twisted bilayer graphene as an incommensurate superlattice with strong interlayercoupling.
Proc. Natl. Acad. Sci. , 6928–6933 (2018).
Ahn, S. J. et al.
Dirac electrons in a dodecagonal graphene quasicrystal.
Science , 782–786 (2018).
Moon, P., Koshino, M. & Son, Y.-W. Quasicrystalline electronic states in 30 ◦ rotated twisted bilayer graphene. Phys. Rev.B , 165430 (2019). Spurrier, S. & Cooper, N. R. Theory of quantum oscillations in quasicrystals: Quantizing spiral fermi surfaces.
Phys. Rev.B , 081405 (2019).
Yan, C. et al.
Scanning tunneling microscopy study of the quasicrystalline 30° twisted bilayer graphene.
2D Mater. ,045041 (2019). Yu, G., Wu, Z., Zhan, Z., Katsnelson, M. I. & Yuan, S. Dodecagonal bilayer graphene quasicrystal and its approximants. npj Comput. Mater. , 1–10 (2019). Deng, B. et al.
Interlayer decoupling in 30° twisted bilayer graphene quasicrystal.
ACS nano , 1656–1664 (2020). Yu, G., Katsnelson, M. I. & Yuan, S. Pressure and electric field dependence of quasicrystalline electronic states in 30 ◦ twisted bilayer graphene. Phys. Rev. B , 045113 (2020).
Crosse, J. A. & Moon, P. Quasicrystalline electronic states in twisted bilayers and the effects of interlayer and sublatticesymmetries.
Phys. Rev. B , 045408 (2021).
Wang, L. et al.
New generation of moiré superlattices in doubly aligned hbn/graphene/hbn heterostructures.
Nano letters , 2371–2376 (2019). An ¯delkovi´c, M., Milovanovi´c, S. P., Covaci, L. & Peeters, F. M. Double moiré with a twist: Supermoiré in encapsulatedgraphene.
Nano Lett. , 979–988 (2020). Leconte, N. & Jung, J. Commensurate and incommensurate double moire interference in graphene encapsulated byhexagonal boron nitride.
2D Mater. , 031005 (2020). Liu, L., Feng, Y. P. & Shen, Z. X. Structural and electronic properties of h-bn.
Phys. Rev. B , 104102 (2003). Giovannetti, G., Khomyakov, P. A., Brocks, G., Kelly, P. J. & van den Brink, J. Substrate-induced band gap in graphene onhexagonal boron nitride: Ab initio density functional calculations.
Phys. Rev. B , 073103 (2007). Woods, C. et al.
Commensurate–incommensurate transition in graphene on hexagonal boron nitride.
Nat. physics ,451–456 (2014). Jung, J., DaSilva, A. M., MacDonald, A. H. & Adam, S. Origin of band gaps in graphene on hexagonal boron nitride.
Nat.communications , 6308 (2015). Note that we defined the sublattice coordinates τ X ( X = A , B ) and τ ˜ X ( l ) ( ˜ X = N , B , l = t , b ) differently from those in ourprevious work , to make the points with the highest rotational symmetry, i.e., the hexagonal center, as the center of thesystem. However, both definitions keep the interaction matrices [Eq. (9)] the same. Yankowitz, M. et al.
Emergence of superlattice dirac points in graphene on hexagonal boron nitride.
Nat. physics ,382–386 (2012). Stampfli, P. A Dodecagonal Quasiperiodic Lattice in Two Dimensions.
Helv. Phys. Acta , 1260–1263 (1986). Note that the theoretical model can be easily expanded to the systems with the encapsulating layers other than hBN; e.g., itis straightforward to expand the model to the atomic layers of which energy bands are close to the charge neutrality pointof graphene by explicitly using the Bloch bases for those layers.
Slater, J. C. & Koster, G. F. Simplified lcao method for the periodic potential problem.
Phys. Rev. , 1498–1524 (1954). Note that the value of V pp π ≈ − .
38 eV used in this work is different from that used in the previous works ( V pp π ≈ − . and graphene on hexagonal boron nitride . In this work, we scaled V pp π by a factorof 1 .
25 to compensate the deviation of the Fermi velocity of a pristine graphene, which affects the entire energy scale, dueto the summation over sites in the hopping range.
Trambly de Laissardière, G., Mayou, D. & Magaud, L. Localization of dirac electrons in rotated graphene bilayers.
NanoLett. , 804–808 (2010). Sławi´nska, J., Zasada, I. & Klusek, Z. Energy gap tuning in graphene on hexagonal boron nitride bilayer system.
Phys.Rev. B , 155433 (2010). Koshino, M. Band structure and topological properties of twisted double bilayer graphene.
Phys. Rev. B , 235406 (2019). Chebrolu, N. R., Chittari, B. L. & Jung, J. Flat bands in twisted double bilayer graphene.
Phys. Rev. B , 235417 (2019). Liu, J., Ma, Z., Gao, J. & Dai, X. Quantum valley hall effect, orbital magnetism, and anomalous hall effect in twistedmultilayer graphene systems.
Phys. Rev. X , 031021 (2019). Crosse, J. A., Nakatsuji, N., Koshino, M. & Moon, P. Hofstadter butterfly and the quantum hall effect in twisted doublebilayer graphene.
Phys. Rev. B , 035421 (2020).
Acknowledgements
J.A.C was supported by the National Science Foundation of China (Grant No. 12050410228). P.M. acknowledges the supportby National Science Foundation of China (Grant No. 12074260) and Science and Technology Commission of ShanghaiMunicipality (Shanghai Natural Science Grants, Grant No. 19ZR1436400). J.A.C. and P.M. were supported by the NYU-ECNUInstitute of Physics at NYU Shanghai. This research was carried out on the High Performance Computing resources at NYUShanghai.
Author contributions statement
J.A.C. and P.M. conducted the calculation, analysis, and wrote the manuscript.
Additional information
Competing interests