Abstract
Lateral superlattices in 2D materials provide a powerful platform for exploring intriguing quantum phenomena, which can be realized through the proximity coupling in forming moiré pattern with another layer. This approach, however, is invasive, materialspecific, and requires small lattice mismatch and suitable band alignment, largely limited to graphene and transition metal dichalcogenides (TMDs). Hexagonal boron nitride (hBN) of antiparallel (AA′) stacking has been an indispensable building block, as dielectric substrates and capping layers for realizing highquality van der Waals devices. There is also emerging interest on parallelly aligned hBN of Bernal (AB) stacking, where the broken inversion and mirror symmetries lead to outofplane electrical polarization. Here we show the that laterally patterned electrical polarization at a nearly parallel interface within the hBN substrate can be exploited to create noninvasively a universal superlattice potential in general 2D materials. The feasibility is demonstrated by first principle calculations for monolayer MoSe_{2}, black phosphorus, and antiferromagnetic MnPSe_{3} on such hBN. The potential strength can reach 200 meV, customizable in this range through choice of distance of target material from the interface in hBN. We also find sizable outofplane electric field at the hBN surface, which can realize superlattice potential for interlayer excitons in TMD bilayers as well as dipolar molecules. The idea is further generalized to ABstacked hBN subject to torsion with adjacent layers all twisted with an angle, which allows the potential and field strength to be scaled up with film thickness, saturating to a quasiperiodic one with chiral structure.
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Introduction
Lateral superlattices in graphene and transition metal dichalcogenides (TMDs) are drawing remarkable interest as fascinating arena to explore emergent quantum transport^{1,2,3,4}, quantum optical^{5,6,7,8}, and manybody phenomena^{9,10,11,12,13,14,15,16,17,18,19,20,21,22,23}. Such superlattices have been realized through forming longwave moiré pattern, where the registrydependent proximity coupling from adjacent layers periodically modulate the electronic structures. This approach, however, requires the building blocks to have small lattice mismatch and suitable band alignment, and its implementation has so far been limited to graphene and TMDs. The resultant moiré superlattices are materialspecific, and also dependent on the carrier species and internal degrees of freedom. Taking TMDs heterobilayers as example, the electrons at K and Q valleys and holes at K and Γ points all experience distinct moiré superlattice potentials, and the energy landscape depend on choices of compounds^{24,25}. The proximity coupling at the moiré interface also degrade or change other material properties of the target layer. For example, the TMD heterointerface for realizing superlattice potential of electrons^{18,19,20,21,22,23} also results in ultrafast electronhole separation that completely alters the optical properties^{5,6,7,8}.
The realization of the elegant moiré superlattices and highquality van der Waals devices has relied on the use of hexagonal boron nitride (hBN) as substrate and capping layer^{26,27}, for its remarkable thermal and chemical stability, and insulating band structures. Commonly used hBN has the antiparallel alignment between adjacent layers, known as the AA′ stacking, where boron atoms are vertically aligned with nitrogen atoms of neighboring layers. Another energetically favorable hBN structure is to have the layers parallelly aligned with the AB (Bernal) stacking, and growth of such AB hBN has been achieved using chemical vapor deposition^{28}. Unlike the AA′ one, the AB stacking allows the emergence of perpendicular electric polarization since the restriction from inversion symmetry and outofplane mirror symmetry are both lifted^{29}. Observations of ferroelectricity in ABstacked hBN are reported very recently: under marginally twisting where structure relaxation leads to micron or submicron scale alternating AB and BA stacking domains, the opposite outofplane electrical polarizations are sensed by Kelvin probe force microscope^{30,31}, and by the resistance of adjacent graphene^{32}.
Here we propose a noninvasive approach to realize universal superlattice potential on a general 2D material, exploiting the electrical polarization at a nearly parallel interface inside the hBN substrate or capping layer. Such interface can be readily created by stacking a nlayer AA′ hBN on another AA′ film (Fig. 1), which hosts electrical polarization patterned with periodicity b = a/δ, a the lattice constant of hBN and δ the twist angle from parallel. A triangular superlattice potential with period b is then electrostatically generated in the target 2D material placed on hBN surface, regardless of its crystal structure and orientation. This is corroborated by our first principle calculations for monolayers of MoSe_{2}, black phosphorus, and antiferromagnetic MnPSe_{3} on hBN substrates. The potential strength is proportional to \(\exp \left( {  nd\frac{{4\pi }}{{\sqrt 3 b}}} \right)\), nd the vertical distance of target material from the interface in hBN, and can be customized in the range 0–200 meV. This rapid exponential dependence corresponds to sizable outofplane electric field ~0.6V/b, which can realize superlattice potential for dipolar objects at the hBN surface, such as interlayer excitons in TMDs bilayers, or adsorbed dipolar molecules. Using AB hBN film subject to torsion where adjacent layers are all twisted with a common angle δ, the strengths of potential and electric field on the surface can be further scaled up with film thickness N, which saturates to a quasiperiodic form with a chiral structure when Nδ becomes sufficiently large.
Results
Electrostatic potential from a twisted interface in hBN
We first examine the electrical polarization at a nearly parallel interface in hBN, with twist angle δ << π but beyond the marginally twisting regime addressed in refs. ^{30,31,32} such that structural relaxation is not significant. This allows us to focus on a moiré pattern between rigid lattices at the nearly parallel interface. The local stacking registry can be characterized by the inplane displacement r between two nearneighbor B atoms across the interface (Fig. 2a), which is a linear function of the lateral position R. Having a fewlayer hBN to separate this interface from the surface may also help to stabilize the moiré from structural relaxation.
The nearly parallel interface is now described by a longwavelength moiré pattern r(R), with period b >> a. There exists an intermediate length scale (l), large compared to lattice constant a and small compared to b, such that the atomic registry in any region of size l closely resembles a latticematched stacking, so the local electronic structures in moiré can be wellapproximated by band structures of the latter^{33,34}. Firstprinciples calculations of various latticematched stackings can give the electrical polarization P as function of registry r. Combined with the mapping between r and location R in the longwavelength moiré, the laterally modulated electrical polarization can then be given as P(R) ≡ P(r(R)).
We start with first principle calculations of n + n multilayers where the top nlayer AA′ hBN is stacked on the bottom nlayer AA′ hBN, with a latticematched parallel interface of various registry r. Figure 2b shows an example of the charge redistribution caused by the interlayer coupling at the parallel interface with \(r = a/\sqrt 3\) (i.e., an AB interface). The presented quantity is the differential charge \({\mathrm{{\Delta}}}\rho \equiv \rho _{n + n}  \rho _{n,t}  \rho _{n,b}\), where ρ_{n + n} is the DFTcalculated planeaveraged charge density of the n + n multilayer, and ρ_{n,t/b} is that of the pristine top (bottom) nlayer. Remarkably, for various thickness n considered, Δρ is predominantly distributed with nearly the same profile on the two layers constituting the parallel interface. This shows that the charge redistribution is occurring at the parallel interface only, and the outer layers in antiparallel alignment only has very small effect on the resultant electrical polarization.
In the latticematched stacking, the spatially uniform electrical polarization corresponds to a voltage drop: \({\Delta}\varphi = \frac{1}{{\varepsilon _0}}P\), which can be directly extracted from the DFT calculations as the difference between the vacuum levels on the two side of the multilayer. The electrical polarization determined from this vacuum level difference, i.e., \(\varepsilon _0{\mathrm{{\Delta}}}\varphi\), is shown in Fig. 2c (triangle symbols). The magnitude of P is ~2 pC m^{−1}, consistent with other calculations for AB bilayers^{29}, while the modest dependence on the layer thickness n can be attributed to the small screening effect by the outer layers. Through the induced carrier density change in an adjacent graphene layer, the experiment in ref. ^{32} has determined an electrical polarization of P ~2.25 pC m^{−1} in ABstacked bilayer hBN, in good agreement with the calculations. Figure 2c also shows the electrical polarization evaluated from the differential charge, \(P = {\int} {z{\mathrm{{\Delta}}}\rho (z)dz}\) (star symbols), which is consistent with the \(\varepsilon _0{\mathrm{{\Delta}}}\varphi\) values.
Figure 2d plots the electrical polarization as function of r in a 2 + 2 configuration. \(r = 2a/\sqrt 3\) corresponds to an BA stacking interface, where P is exactly the opposite of the AB stacking one. For the AA stacking (r = 0), the outofplane mirror symmetry forbids any polarization. In general, r stacking registry is related to the −r one by a spatial inversion, so P is an odd function of r. Like many other quantities involving the interlayer processes^{25,35}, the registry dependence can be wellfitted by the lowest Fourier harmonics observing the rotational and translational symmetry,
where K is wavevector at the Brillouin zone corner. In Fig. 2d, the dots are the polarization calculated from Δρ under various r, while the curve is a plot of Eq. (1), showing excellent agreement.
For the twisted interface, Δρ and P as functions of r can then be mapped to the dependences of these quantities on location R, by the local approximation. We examine the electrostatic potential at the hBN surface produced by such a spatially modulated electrical polarization. Setting the interface to be the z = 0 plane, the Coulomb potential is given from the differential charge Δρ,
where \({\mathrm{{\Delta}}}\rho ({\mathbf{R}},z) \equiv {\mathrm{{\Delta}}}\rho ({\mathbf{r}}({\mathbf{R}}),z)\), \({P} ({\mathbf{R}}) = {\int}{{z}{^\prime}\mathrm{{\Delta}}}\rho ({\mathbf{R}},z^\prime)dz^\prime\), and \(G = \frac{{4\pi }}{{\sqrt 3 b}}\). The approximation in Eq. (2) holds well under the condition Gd << 1 (see Supplementary Note 1). While the uniform polarization in latticematched stacking can only produce voltage drop between the two sides of the layer, the spatially modulated polarization at the twisted interface can introduce inplane variation of electrostatic potential, with a strength that decays with the vertical distance z from the interface. In the limit of Gz << 1, the potential strength \(V_{{\mathrm{max}}}  V_{{\mathrm{min}}} = \frac{1}{{\varepsilon _0}}P_{{\mathrm{max}}}\), where P_{max} is the electrical polarization at \(r = a/\sqrt 3\), i.e., AB stacking (c.f. Fig. 2d).
Figure 2e plot the spatial profile of electrical polarization P(R), assuming the interface has a rigid twist from parallel, and the resultant electrostatic potential at three values of the vertical distance z/b. Taking b = 9.6 nm, which corresponding to twisting angle δ = 1.5°, these corresponds to the thickness of n = 2, 4, and 6 layers, respectively. The choice of twisting angle and the thickness of the top hBN layer together allows the control of both the periodicity and the strength of the superlattice potential. At the thin limit Gz << 1, the potential strength measured by V_{max}−V_{min} reaches 203 meV.
When a general 2D material is placed on the hBN surface, the electrons/holes experience the electrostatic potential through their charge, which shall produce a universal superlattice energy landscape determined by the hBN interface, regardless of the crystal type and orientation of the target materials. This is corroborated by our first principle calculations of the bandedge energies in three distinct examples, i.e., MoSe_{2}, black phosphorus (BP), and the antiferromagnetic MnPSe_{3} monolayers.
Figure 3a, b shows the DFTcalculated band edges, in a monolayer MoSe_{2} placed on an hBN bilayer, as functions of stacking registries. In these calculations, the MoSe_{2} has the crystal axes aligned with those of hBN with a 4:3 ratio in their lattice constants. The periodic cell used in the calculations is shown as inset of Fig. 3b, where the interlayer translation r_{t} characterizes the registry between MoSe_{2} and hBN. The two hBN layers have parallel alignment characterized by registry r_{bn} (inset of Fig. 3a). Figure 3a plots the energies of conduction band minima (CBM) and valence band maxima (VBM) as functions of r_{bn}, which shows the expected modulation by the r_{bn}dependent electrical polarization in hBN. For comparison, the energies of Q and Γ valleys are also shown. Indeed, all these energy extrema are modulated with the same profile as functions of r_{bn}. In the presence of a small twist of the hBN interface, this dependence, together with the mapping between the local registry r_{bn} and location R, produces the superlattice energy landscape for electrons in MoSe_{2}, which has the same profile for K, Q, and Γ valleys.
It is generally expected that the heterointerface between hBN and target material such as MoSe_{2} does not change the energy landscape of the latter, which underlies the extensive use of hBN as substrate and capping layer. This is confirmed in our calculations, as shown in Fig. 3b, where the bandedge energies show no visible dependences on the stacking registry r_{t} between MoSe_{2} and hBN. It is worth noting that there is a charge transfer between MoSe_{2} and hBN (c.f. Supplementary Note 3), but its dependence on the registry r_{t} is smoothed out because of the large mismatch in their lattices. As the residue effect, we observe a small variation upper bounded by a few meV in MoSe_{2}’s bandedge energies as functions of r_{t} (Supplementary Fig. 2). We have also performed calculations for a different angular alignment between MoSe_{2} and hBN that forms \(\sqrt 7 \times \sqrt 7\) periodic cell, which points to the same conclusion (Supplementary Fig. 3).
Figure 3c, d shows the DFTcalculated CBM and VBM energies in monolayer MnPSe_{3} with the Neel type antiferromagnetic order, placed on the bilayer hBN. Figure 3e, f is for BP monolayer on the bilayer hBN substrate. We find that the band energies in these materials are also modulated with the same profile as functions of the stacking registry r_{bn} between the two parallel hBN layers, while no dependence on the registry r_{t} (between hBN and target material) is visible in this energy scale. The periodic cells used and more details of the calculations are given in the Supplementary Note 2.
These results are in full support of the expectation that the laterally modulated electrical polarization at the twisted interface inside hBN substrate or capping layer can generate universal superlattice energy landscape for carriers in a general 2D material. For the various band edges in the three different monolayers, the magnitude of the energy modulation falls in the range between 193 and 233 meV (c.f. Fig. 3a, c, e). The finite differences in the energy modulation strength can be attributed to the different extension of the bandedge wavefunction in the z direction, as well as the dielectric screening effect in the heterostructures.
Multiple twisted parallel interfaces in hBN
We first consider an AB stacking trilayer, which has two parallel interfaces both with stacking registry \(r = a/\sqrt 3\). The bottom panel of Fig. 4a presents the charge redistribution due to the interlayer coupling at both interfaces, \({\mathrm{{\Delta}}}\rho \equiv \rho _{3{\mathrm{L}}}  \rho _{\mathrm{t}}  \rho _{\mathrm{m}}  \rho _{\mathrm{b}}\), where ρ_{3L} is trilayer charge density from DFT, and ρ_{t/m/b} is that of the pristine top/middle/bottom monolayer. Through the comparison with the charge redistribution due to a single interface (c.f. top and middle panels of Fig. 4a), it is remarkable to note that Δρ in the trilayer is simply the sum of that due to each individual interface, showing that charge redistribution at adjacent interfaces can be treated as independent of each other. Figure 4b plots the electrical polarization P in trilayers where the two parallel interfaces have identical registry r. Under the various r, the trilayer P is all twice of that in the corresponding bilayer.
The above findings point to the possibility to design superlattices from the interference of electrostatic potentials from the electrical polarizations at multiple twisted parallel interfaces. For example, creating two independently configured twisted interfaces in hBN can allow the generation of bichromatic superlattice potentials^{36}. Here we consider the scenario of staring from an Nlayer film of AB stacking^{28}, and apply torsion such that adjacent layers are all twisted from each other by a common angle δ. This creates N − 1 nearly parallel interfaces with the same periodicity, but with a small rotation by δ from each other. With the charge redistribution at different interfaces being independent of each other, the electrostatic potentials contributed by adjacent interfaces are related by: \(V_{l + 1}\left( {{\mathbf{R}},z} \right) = V_l({\Bbb R}{\mathbf{R}},z + d)\), where \({\Bbb R}\) denotes the inplane rotation by δ, with V_{l}(R, z) described by Eq. (2). The overall electrostatic potential is then,
In the limit Nδ << 1, \(V_{{\mathrm{sum}}}\left( {\mathbf{R}} \right) \approx \mathop {\sum}\nolimits_l {V_0\left( {{\mathbf{R}},z + ld} \right)}\), one obtains a superlattice potential with the strength scaled up with the thickness N. In Fig. 4d, we show a plot of Eq. (3) at the surface of the hBN film for N = 10 and δ = 0.5°. The superlattice potential strength reaches ~1.3 eV. Figure 4c characterizes the potential strength with the increase of N. For larger δ, the potential strength saturates at smaller N. It is interesting to note that V_{sum}(R) always saturates with a common spatial profile with a chiral structure, as shown in Fig. 4e for the example of N = 20 and δ = 1°.
Electric field on hBN and superlattice for dipolar objects
As shown in Eq. (2), the potential V(R, z) decays exponentially in the outofplane direction in a length scale \(G^{  1} = \frac{{\sqrt 3 }}{{4\pi }}b\). This is associated with an outofplane electric field on the hBN surface with magnitude inversely proportional to superlattice period b. Figure 5a shows the electric lines of forces, which have a common profile when plotted in the dimensionless coordinates (R/b,z/b). For the example of a single parallel interface with δ = 2°, which corresponding to b = 7.2 nm, the spatial profile of E_{z} and total field strength E are shown in Fig. 5b, along the long diagonal of a supercell, at a vertical distance of z = 2d from the interface. The field strength can reach ±6 mV/Å. This sizable electric field further enables the possibility to create electrostatic superlattice potential for dipolar objects such as interlayer excitons in bilayer TMDs, and dipolar molecules. Taking interlayer exciton as an example, it couples to E_{z} through its outofplane electric dipole, and the superlattice potential strength in the field of Fig. 5b can reach ~80 meV, assuming the interlayer distance of 7 Å in TMDs. Dipolar molecules adsorbed could have their orientations adjusted to the total field E, then the potential landscape is determined by E. The electric field strength can also be scaled up by having multiple twisted parallel interfaces in hBN, for creating a quasiperiodic disordered landscape for the dipolar object, as shown in Fig. 5c.
Discussion
We have shown the possibility to introduce a superlattice energy landscape on a general 2D material by exploiting twisted parallel interfaces inside the widely used hBN substrate. Unlike the existing moiré superlattices, this noninvasive approach can be applied to the hBN encapsulated highquality monolayers, bilayers, etc., not limited by the lattice type and periodicity of the target materials. It is also worth noting that the orientation of the superlattice potential is determined by the hBN interface only, and can be placed at an arbitrary angle to the crystalline axis of the target material, which is also a control parameter to tailor the minibands through various zone folding schemes. The approach can be further combined with the existing moiré superlattices in twisted graphene, twisted TMDs, and TMDs heterobilayers. In addition to the substrate, the twisted parallel interfaces can also be created in the hBN capping layer where similar effect is expected, or even in the hBN spacing layers exploited as tunneling barriers. The latter could point to interesting superlattice effects on the tunnelling processes.
In reality, structural relaxation can be relevant at the interfaces when the twisting angle is small^{30,31,32}. The structural relaxation can increase the area of certain energetically favorable stacking while shrink the unfavorable ones, at the cost of inhomogeneous strain. This changes the mapping function r(R) in a supercell, and can quantitatively change the spatial profile of P(R) from the rigid twist one shown in Fig. 2e. Correspondingly, the spatial profile of the electrostatic potential can be modulated by such structural relaxation. As long as the local approximation is still applicable, the electrical polarization profile can still be quantified using P(r(R)) from the relaxed texture r(R) of the interface. On the other hand, for the multilayers subject to torsion (Fig. 4 inset), the structural relaxation can be complicated with the large number of twisted interfaces. Compared to those shown in Fig. 4c–e for the rigid interfaces, the potential profile can get much more complex and the scaling of the potential strength with N can be significantly changed by such relaxation in the small δ limit.
Data availability
All data needed to evaluate the conclusions in the paper are included in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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Acknowledgements
We thank helpful discussions with Xiaodong Xu, Hongyi Yu, Mingxing Chen, and Qingjun Tong. We acknowledge support by the National Key R&D Program of China (2020YFA0309600), Croucher Senior Research Fellowship, and HKU Seed Funding for Strategic Interdisciplinary Research.
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W.Y. conceived and designed the research. P.Z. and C.X. performed the calculations. All authors analyzed the results and wrote the manuscript.
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Zhao, P., Xiao, C. & Yao, W. Universal superlattice potential for 2D materials from twisted interface inside hBN substrate. npj 2D Mater Appl 5, 38 (2021). https://doi.org/10.1038/s41699021002214
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DOI: https://doi.org/10.1038/s41699021002214
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