Mapping strong electronic coupling in metavalent PbS moiré superlattices

13 Moir´e superlattices are twisted bilayer materials, in which the long-range superlattice potentials 14 from interlayer interactions can create quantum conﬁnement in each layer. The quantum 15 conﬁnement can slow down or localize electrons in moir´e superlattices, providing a tunable 16 platform for studying strongly correlated physics 1,2 , such as superconductivity 3,4 , Mott insulators 5 , 17 and interacting topological insulators 6,7 . Previously, moir´e superlattices were built exclusively 18 using materials with weak van der Waals interactions 8 and fabrication of moir´e superlattices with 19 strong interlayer chemical bonding (e.g., covalent, metavalent, or ionic bonding) was considered to 20 be impractical 9 . Here using lead sulﬁde (PbS) as an example, we report a strategy for synthesizing 21 moir´e superlattices coupled by strong metavalent bonding. We use water-soluble ligands as a 22 removable template to obtain free-standing ultra-thin PbS nanosheets and assemble them into 23 direct-contact bilayers with various twist angles. Atomic-resolution imaging shows the interlayer 24 distance is approximately equal to the Pb–S bond length, rendering a strong metavalent coupling. 25 mapping by electron energy loss spectroscopy reveals a strong localization of electronic 26 states in small-angle twisted bilayers, which agrees with our DFT calculations. This study opens a 27 new door to exploration of deep energy modulations within moir´e superlattices alternative to van 28 der Waals twistronics.


Introduction 30
Recently, moiré superlattices have been synthesized by stacking two layers of two-dimensional 31 (2D) materials with relative twist angles 9,10 . In the moiré superlattices, the twisting topology 32 determines the 2D quantum confinement and it offers an additional degree of freedom to 33 modulate the electronic structure, usually referred to as twistronics 11,12 . So far, all 2D moiré 34 superlattices are synthesized using van der Waals (vdW) materials, such as graphene and transition-35 metal dichalcogenide, where the two layers of materials are coupled through vdW interactions. 36 Twistronics based on these vdW materials has attracted great interest in various fields, ranging 37 from physics 13-20 to materials science 9,21-23 , and chemistry 24,25 . Different from materials coupled 38 by chemical bonding at an interface, such as conventional semiconductor heterostructures, vdW 39 twistronics have reduced strength in modulating the electronic structures due to the weak interlayer 40 coupling. Most of the experimental observations of exotic electronic properties, especially those 41 associated with electron transport, are realized at extremely low temperatures 1,2,5-7 . To increase 42 the electronic modulation imposed by moiré superlattice, one approach is to replace the vdW 43 interactions with strong chemical bonding such as covalent, ionic, or metavalent bonding. 44 Achieving strong quantum confinement in moiré superlattices by chemical bonding will pave 45 a way to fabricating a new class of materials beyond the current vdW twistronics and it may also 46 shed light on some challenging issues in other systems. For example, strongly coupled moiré 47 superlattices can be structurally and functionally similar to an array of quantum dots 26 , offering an 48 alternative route to super-crystals 27 by avoiding the notorious issue of connection defects formed 49 during quantum dot self-assembly 28 . Moreover, the energy bands near the Fermi level in moiré 50 patterns can be flattened due to the strong modulation, which may trap electrons in individual 51 "quantum-dot" potentials upon suitable doping, leading to Wigner crystallization 29 . Thus, creating 52 strongly coupled moiré superlattices through chemical bonding combines the strengths of two 53 fields: the tunable confinement of 2D moiré superlattices and the strong coupling in conventional 54 semiconductor heterostructures. Since moiré superlattices cannot be achieved using conventional 55 semiconductor synthesis methods, e.g., epitaxial growth 9,30 , it is unclear whether it is possible to 56 synthesize chemically bonded moiré superlattices. 57 Here, we use PbS as a model system to demonstrate a strategy for constructing moiré 58 superlattices with strong interlayer coupling through metavalent bonding. Such chemically 59 bonded moiré superlattices are obtained for the first time. The strong quantum confinement and 60 localization of electronic states in small-angle twisted moiré superlattices are validated through the 61 combination of electron energy loss spectroscopic mapping and theoretical calculations.

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Conceptual discussion of metavalent moiré superlattices 64 We first describe theoretically why metavalent moiré superlattices can give rise to stronger coupling 65 effects than vdW moiré superlattices. As shown in Fig. 1a, bulk PbS has a rock-salt crystal 66 structure and features an unconventional metavalent bonding between Pb and S atoms, in which 67 the valence electrons are delocalized to an extent between covalent and metallic bonding 25 . This 68 metavalent Pb-S bonding is much stronger than vdW interactions 9 , and it can be used as the 69 interlayer interaction to construct strongly coupled PbS moiré superlattices if one can assemble 1c). Among these four stacking configurations, AA stacking has the largest interlayer distance, but 83 the smallest bandgap, reflecting distinctive structural and energetic variations stemming from the 84 moiré pattern. Energy modulation by moiré pattern is evaluated by moiré potential 17 , defined as the 85 maximal free energy fluctuation in the real space. We estimate the moiré potential by calculating 86 the largest free energy difference among all possible stacking configurations using approximate 87 small unit cells. Fig. 1d shows the calculated moiré potentials of various structures, including the 88 reported vdW superlattices, the metavalent PbS synthesized in this work, and our predictions of 89 other chemically bonded superlattices. The moiré potential of PbS is 40 meV per atom, more than 90 twice of the reported vdW superlattices. Generally, chemical bonding leads to much deeper energy 91 modulation compared to vdW interactions. The deep energy modulation can localize electrons in 92 the high-symmetry points with local energy extrema, providing an array of identical quantum-dot-93 like potentials 26,27 . Until now, the properties of the chemically bonded moiré superlattices and 94 their structural stability remains unknown due to the lack of a synthesis strategy.

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3 Synthesis and characterization of PbS moiré superlattices 96 We use metavalent PbS as a model system of non-vdW moiré superlattices to assess the feasibility 97 of achieving the predicted modulation of electronic structures. Ultra-thin PbS nanosheets have 98 been previously synthesized in organic low-polar solvents 34 , in which the interaction between 99 the solvent-phobic PbS core and the long-alkyl-chain ligands is designed to be strong to guide 100 asymmetric growth and stabilize the formed nanocrystals. However, the strong core-ligand 101 interaction also leads to difficulty in ligand removal, and the ligands prevent direct metavalent 102 bonding between two nanosheets 34 . To overcome this dilemma, we developed an aqueous 103 synthesis strategy employing two surfactant ligands that have adequate solubility and bind 104 moderately with the inorganic core. The schematic in Fig. 2a shows that Pb 2+ and S 2precursors 105 and two organic ligands (i.e., hexylamine and dodecyl sulfate) in an acidic aqueous solution at 106 80 • C for 20 min produce ligand-capped ultra-thin PbS nanosheets. The synthetic mechanism is 107 discussed in Supplementary Information Section 2. Due to the high polarity of the PbS surfaces, 108 high-polar solvents (such as water) are required to remove the ligands. In this synthesis method, 109 both ligands can be readily removed by washing with dilute basic and acidic aqueous solutions

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Mapping the localization of electronic states 163 We employ monochromated electron energy loss spectroscopy (EELS) in an aberration-corrected 164 scanning TEM (STEM) to map the local electronic excitation of moiré superstructures as a function 165 of twist angle. Features in low-loss EEL spectra arise due to inter-band excitation and intra-166 band transitions in a similar way to optical spectra 36,37 . Owing to the advantage of the latest 167 direct detection camera and the large absorption efficiency of PbS, the low-loss spectra in the 168 exciton region exhibit high signal-to-noise ratios (Extended Data Fig. 5). Moreover, using a disk-169 filter convolution method, we considerably enhance the signal-to-noise ratio for each spectrum

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Because the band edges are not at high-symmetric points, we calculated the bandgap based on self-321 consistent calculations using a dense k-mesh rather than reading the bandgap from band structure.

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The k-point mesh was sampled by the Monkhorst-Pack method with a separation smaller than 0.01 323Å −1 .

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Because most commensurate moiré supercells contain more than 1000 atoms (4360 in the 325 largest calculated supercell), we used the atomic basis set of single-Zeta functions and FHI 326 pseudopotentials in the direct calculations. This atomic basis set was tested to be accurate by 327 comparing with the plane-wave method in small systems.

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In all calculations, a vacuum layer thicker than 20Å was applied to avoid the periodic images.

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All geometry and ions were fully relaxed until the force of each atom decreases to 0.01 eV/Å.

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To include the possible vdW interaction between different layers, the DFT-D2 functional was 331 implemented.   Fig. 2h. The boundaries of three nanosheets are indicated by green arrows, and one representative interlayer distance is indicated by red arrows. Local interlayer distances at each position (0 to 50Å) are evaluated by analysing the image intensity in corresponding slabs with a 4Å width. Four representative slabs at the positions of 2, 12, 34, and 46Å are boxed and labelled for illustration. b, Interlayer distance in each slab extracted from the image intensity analysis through a similar method shown in Fig. 2i. c, Fluctuation of the interlayer distance at the interface between L1 and L2 at different positions. Extended Data Fig. 5 | Data processing method for EELS mapping. a, Illustration of the areas for extracting an individual spectrum, the corresponding disk-filtered spectrum, and the normalized spectrum of the overall double-layer region. b, Detail of the 7 × 7 (pixel) disk kernel for filtering. c, Noisy spectrum extracted from the original data at the pixel labelled in panel a. d, Disk-filtered and further smoothed spectra corresponding to the same pixel, but with a much higher signal-tonoise ratio. Filtering is implemented by convolution of original 3D data with the disk kernel shown in panel b, and smoothing is based on Savitzky-Golay method. e, Processing of the normalized spectrum of the overall double-layer region. The integration of the original spectra shows an extremely high signal-to-noise ratio, therefore, the further processing methods (such as Savitzky-Golay smoothing, Richardson-Lucy deconvolution, and zero-loss-peak removal through powerlaw fitting) preserve the feature of the original spectrum. Energy dispersion of all spectra in panels c-e is 9 meV/pixel. Extended Data Fig. 6 | EELS mapping of moiré superlattices. Left and right columns of ad correspond to the moiré superlattices with twisted angles of 3.0 • and 4.6 • , respectively. All nanosheets measured in EELS mapping have a similar thickness with less than 10% difference based on the analysis of the scattering intensity in overview images. Scale bars are 20 nm (left column) and 10 nm (right column). a, Survey image of the moiré superlattices. Yellow box marks the scanning region for STEM-EELS. b, Scanned images with two representative spots in singleand double-layer regions, marked with triangle and square, respectively. c, Plot of low-loss spectra from two representative spots. For other scanned spots, the spectra are omitted for clarity, only their spectral peaks between 1.3 to 2.5 eV are illustrated by the scattered dots. Dot colour follows the colourmap in d. d, 2D mapping of the energy of spectral peaks of all scanned pixels. Colourmap is identical to that in Fig. 3d. e, Histogram of the peak positions of the absorption spectra in the EELS measurements of three moiré superlattices with different twist angles (labelled). Histograms correspond to the scatter plots in Fig. 3c and Extended Data Fig. 6c. .06°7 .63°6 .06°E xtended Data Fig. 7 | Band structures directly calculated from moiré cells with commensurate angles. Energy dispersion along the high symmetric path for moiré cells constructed from bilayer 2-atom-thick nanosheets (a) or bilayer 4-atom-thick nanosheets (b). A variety of commensurate angles are calculated for showing the influence of twist angles and nanosheet thickness. Figure 1 Structure and strong coupling of PbS moire superlattice. a, 3D and the cleaved 2D structures of PbS rocksalt crystal, emphasizing the metavalent interaction in all directions. b, Different local atomic alignments occur in a PbS moire superlattice with a twist angle of 8. Blue square marks the moire unit-cell. Four representative stacking con gurations are highlighted as Pb on Pb (AA), Pb on S (AB), middle point (MP), and diagonal point (DP). c, DFT calculations of AA, AB, MP, and DP con gurations, showing the structure, bandgap, and interfacial distance of each con guration. d, DFT calculations on varieties of moire superlattices, including the reported vdW superlattices, the metavalent PbS synthesized in this work, and our predictions of other chemically bonded superlattices. The results show their moire potential and the largest interfacial distance variation among different stacking con gurations.  (200) and twist angles. Inset shows the relationship between moire spatial frequencies (green) and two sets of individual rock-salt spatial frequencies (red and orange). g, Atomic resolution TEM images and corresponding simulated images (false coloured) of bilayer moire superlattices with a variety of twist angles. h, Side view of a moire superlattice composed of three layers, L1, L2, and L3. The region marked by the orange box is ipped and zoomed in panel i. i, Structural analysis of side-view details. From top to bottom are original TEM image, simulated image, image intensity and peak position of each atomic layer, and each interlayer spacing. Scale bar: b, 30 nm; c, 1 nm; d, 100 nm; e, 30 nm (top left), 5 nm−1 (top right), 0.5 nm−1 (bottom two); g, 2 nm; h, 5 nm.

Figure 3
Mapping of electronic states through STEM-EELS. a, Overview image of a moire superlattice with a twist angle of 1.3. Yellow box marks the scanning region for STEM-EELS. Scale bar, 20 nm. b, Scanned image with single-and double-layer regions false-coloured and three representative spots marked. c, Plot of lowloss spectra from three representative spots with the spectral peaks marked with opaque dots. For other scanned spots, the spectra are omitted for clarity, only their spectral peaks are illustrated by the semitransparent dots. Dot colour follows the colourmap in d. d, Spatial mapping of the energy of spectral peaks that are shown by scattered dots in c. e, Simulated pattern of AA, AB, and DP con gurations and single layer (SL) region in the moire superlattice with a twist angle of 1.3. f, Integrated spectra of the double-layer regions of moire superlattices with a variety of twist angles. A spectrum of single-layer PbS nanosheet is added for comparison. g, The change of the peak energy of double-layer regions upon the change of the twist angle. Hollow dots at 1.3 shows the position of two separated peaks (marked by red arrows in panel f), whereas the solid dot shows the average of the two peaks. Energy dispersion of all EEL spectra is 9 meV/pixel.