Ghost anti-crossings caused by interlayer umklapp hybridization of bands in 2D heterostructures

In two-dimensional heterostructures, crystalline atomic layers with differing lattice parameters can stack directly one on another. The resultant close proximity of atomic lattices with differing periodicity can lead to new phenomena. For umklapp processes, this opens the possibility for interlayer umklapp scattering, where interactions are mediated by the transfer of momenta to or from the lattice in the neighbouring layer. Using angle-resolved photoemission spectroscopy to study a graphene on InSe heterostructure, we present evidence that interlayer umklapp processes can cause hybridization between bands from neighbouring layers in regions of the Brillouin zone where bands from only one layer are expected, despite no evidence for moir/'e-induced replica bands. This phenomenon manifests itself as 'ghost' anti-crossings in the InSe electronic dispersion. Applied to a range of suitable 2DM pairs, this phenomenon of interlayer umklapp hybridization can be used to create strong mixing of their electronic states, giving a new tool for twist-controlled band structure engineering.


Introduction
Crystalline periodicity modifies the interpretation of the momentum conservation law for electronic and optical processes in solids. It gives rise to a periodicity of the electronic dispersion in momentum space, so that, according to Bloch's theorem [1], the bandstructure is uniquely defined within one (the first) Brillouin zone. All processes in a crystal can then be divided into two types: those with small momentum differences that can be described within the first Brillouin zone and those where a large momentum transfer requires the involvement of other Brillouin zones. In the latter case, a momentum transfer ħ to the crystalline lattice, where is one of the reciprocal lattice vectors, satisfies the conservation of momentum and was dubbed Umklapprozesse (Umklapp processes) by Peierls [2]. When applied to heterostructures of two-dimensional materials (2DM), umklapp scattering from Moiré superlatttices has been shown to open new channels for electron kinetics [3] and optical transitions [4].
2DMs represent a broad class of compounds where atomic planes formed by strong inplane covalent bonding are held together by a weak van der Waals interaction. These weak outof-plane forces enable the stacked assembly of 2DM heterostructures (2DHS), where consecutive layers may involve atomic planes of different compounds with arbitrary lattice constants and orientation, with atomically clean interfaces [5][6][7] which allow neighbouring layers in the heterostructure to influence each other, in particular through tunneling. Tunneling across clean interfaces is subject to momentum conservation [8,9], so that it is resonantly enhanced in the part of momentum space where the bands of two 2DM intersect, causing resonant interlayer hybridization. Dramatic bandstructure modifications through resonant interlayer hybridization have been studied in twisted bilayers of graphene [10][11][12], graphene on single-crystal metal substrates [13,14], and graphene with other 2DM [15,16], leading to band anti-crossings and, potentially, to van Hove singularities in the density of states. Here, we demonstrate that interlayer umklapp processes in resonant tunneling lead to the appearance of additional features in the hybridized band structures of 2DHS.

Results and Discussion
An example of such an effect is illustrated in Figure 1, where angle-resolved photoemission spectroscopy with submicrometre spatial resolution (μARPES) has been used to probe the valence band structure in a graphene on InSe 2DHS. In Figure 1b we sketch the valence band dispersion of monolayer InSe (unfolded over the second Brillouin zone replicas marked by = 0,1 … 5) and the -bands of graphene. Notably, no interlayer band crossing occurs in the first Brillouin zone of InSe so one would not expect any resonant hybridization of electronic states (and hence anti-crossing features) in the InSe monolayer spectrum. We find no evidence for replica bands due to a Moiré superlattice potential. Nonetheless, the measured μARPES spectrum features a pronounced anti-crossing anomaly near the edge of the valence band, as highlighted by a black box in Figure 1a, showing the photoemitted intensity in an energy-momentum slice taken along the Γ to KGr direction. This "ghost" anti-crossing occurs due to interlayer umklapp hybridization where resonance conditions are achieved by the band crossing between graphene and InSe dispersions in the second Brillouin zone of InSe, also present in the measured spectra of graphene bands (see the purple box in Figure 1a). Overlaid on the left are theoretical predictions for the isolated InSe bands (blue dashed) and graphene band (red dashed). The black dashed line indicates the position of the graphene band when folded into the 1 st Brillouin zone of InSe. Right: mirrored, the double-differential of the same spectra shown on the left. InSe is at a twist angle of 22.3 ± 0.6° with respect to graphene. An anti-crossing is highlighted by the purple box on the right and a ghost anti-crossing by the black box. (b) 3D schematic of the uppermost valence band of InSe and graphene band; red/yellow lines highlight the position of overlap. Blue and black hexagons represent InSe and graphene Brillouin zones respectively. The grey plane marks the slice of energy-momentum space covered by the μARPES data in (a). The red arrow indicates folding of an anti-crossing in the second Brillouin zone of InSe to a ghost anti-crossing in the first Brillouin zone. Inset: atomic schematic cross-section of the graphene/1L InSe/graphite 2D heterostructure.
The 2DHSs studied in this work were assembled by dry transfer in an inert environment, where exfoliated crystals of InSe and GaSe were deposited on thick graphite and encapsulated with monolayer graphene (see Methods and schematic inset in Figure 1b). This method allows for ARPES probing of buried layers through graphene (as graphene's ARPES spectrum is already well known [17,18]) while allowing for surface charge dissipation into a conductive substrate (platinum-coated n-Si wafer) [19]. Several samples were fabricated using different thicknesses of InSe and GaSe crystals, and different twist angles with respect to the graphene lattice.
The ghost anti-crossings, and their origin, are more apparent when looking across reciprocal space. The photoemission intensity at a constant energy near the top of the InSe upper valence band (UVB), in a region around Γ, is shown in Figure 2a and reveals the twisted lines in reciprocal space at which the ghost anti-crossings occur. The measured data (black dashed rectangle) have been averaged and rotated to form the complete image, as described in Supplementary Material, Section 1. Low energy electron diffraction from a sub-micrometre spot (μ-LEED), taken at the same position on the sample as the μARPES measurements (see low energy electron microscopy image in Supplementary Material, Section 2), gives diffraction peaks from both the graphene and InSe layers. In-plane, InSe has a hexagonal lattice with lattice parameter, aInSe = 4.00 Å [20], 60% larger than that of graphene, agr = 2.46 Å. When stacked with a twist angle θ between the layers, they form an incommensurate structure where the Brillouin zone corners in the graphene layer, KGr, lie in the second Brillouin zone of the InSe layer. By identifying the LEED peak positions for both materials we find the twist angle between their crystalline lattices θ = 22.3 ± 0.6°. In Figure 2d we plot a contour map of the InSe UVB energy in its first and second Brillouin zone, with the first Brillouin zone of graphene overlaid (black hexagon). As shown in the 3-dimensional band schematic, Figure 1b, in monolayers of InSe the UVB dispersion takes the shape of an inverted 'Mexican hat' [21] around the zone centre, Γ , with the valence band maximum (VBM) close to but not at Γ, and the band disperses to a minimum at the zone corner KInSe. By contrast, the upper valence band in graphene forms the characteristic Dirac cones, meeting the conduction band at the six Dirac points at the zone corners, KGr. Band anti-crossings occur where the graphene and InSe bands would have been coincident. Their position is shown on the contour map by red lines, drawn using an interpolation formula [22], which map out distorted-triangular closed curves around the Dirac cones, in the second Brillouin zone of InSe. Umklapp scattering by an InSe reciprocal lattice vector, InSe , replicates these anti-crossings in the first Brillouin zone of InSe. This emphasizes the difference between this interlayer Umklapp process and the Moiré phenomena previously reported in systems such as twisted bilayer graphene where a Moiré superlattice potential creates replica bands shifted by the Moiré wave vector M = InSe − ℎ and interaction between the primary and replica bands creates flat bands, as previously observed by μARPES [11,12]. Here, no replica bands are apparent, suggesting negligible Moiré superlattice potential, and the ghost anti-crossings are found by mapping the band anticrossings by InSe not M . However, to reproduce the experimentally measured pattern, the angular dependence of the interlayer hybridization must also be considered. To do this, we employ a method developed [17,23] for the description of ARPES intensity maps of graphene [24] and tunnelling between 2D crystals [9,25]

Conclusion
In summary, our data present evidence for interlayer umklapp scattering in 2D heterostructures. The ghost anti-crossings created near the valence band maximum of monolayer InSe demonstrate points of strong coupling with the adjacent graphene layer, with their position controlled by the relative orientation between the layers. Further control could be gained through changing band-alignments, using chemical doping or a perpendicular electric field [29]. By selecting suitable 2DMs pairs, it should thus be possible to engineer strong mixing of their electronic states at or near the band edges of many 2D semiconductors, or near the Fermi-level of metals and semi-metals, giving a new tool for band structure engineering. This interlayer umklapp scattering should not be limited to band hybridisation, and we expect it also to manifest in further novel electron, phonon and photon interactions in 2D heterostructures.

Methods
Sample Fabrication. Bulk rhombohedral γ-InSe crystals, purchased from 2D Semiconductors and grown using the vertical Bridgman method, were mechanically exfoliated down to thin (1L to 10L) crystals on a silicon oxide substrate. Using the PMMA dry peel transfer technique [30], monolayer graphene was used to pick-up and stamp InSe crystals onto either graphite or hBN, each of which was laterally large (>50 μm), thin (<50 nm) and positioned on a (3 nm) Ti/ (20 nm) Pt -coated highly n-doped silicon wafer. Both heterostructure samples were annealed to 150° for 1 hour in order to remove impurities via the self-cleaning mechanism [6]. All that is described above took place within an Ar glove box to prevent sample degradation. The same samples were used for both μARPES and μLEED measurements.
μARPES. μARPES spectra were acquired from the Spectromicroscopy beamline of the Elettra light source [31]. A low energy (27 eV is an average of multiple diffraction patterns collected over an incident electron energy range of 27-60 eV in steps of 1 eV. The diffraction pattern shown in Figure 2k was collected with an incident electron energy of 55 eV.

Modelling of sublattice effects on resonant hybridization between graphene and InSe.
We

Ab initio calculations. Linear-scaling DFT (LS-DFT) calculations in the Projector Augmented
Wave formalism [27,33] were used to model the InSe/Gr heterostructure, using the ONETEP code [26]. Further details are given in Supplementary Material, Section 3.  and 3b. Overlaid in blue are the Lorentzian peak fits, with the peak positions shown by purple triangles. Note that for clarity, alternate data lines are shown giving fewer peak positions than shown in the main text. c) Contour plots of the band structure for the InSe / graphene heterostructure of Figure 3. The blue hexagons mark the Brillouin zones of InSe and the black hexagon the first Brillouin zone of graphene. The solid black lines labelled a and b correspond to the positions in momentum space of the energy-momentum spectra shown in Figure 3a and 3b respectively.

Section 6: Band structure calculations.
The valence band structure for isolated 1L InSe, overlaid in Figure 1a, was calculated using first principles density functional theory (DFT) as implemented in the VASP code [1]. The inplane lattice parameter was taken to be 4.00 Å and the interlayer distance was taken to be 8.32 Å, found from experiments [2]. The atomic positions were relaxed until forces on the atoms were less than 0.005 eV/Å. The plane wave cut-off energy was 600 eV and the Brillouin zone was sampled with a 24 × 24 × 1 grid. Spin-orbit coupling was taken into account for the calculations. The local density approximation (LDA) exchange-correlation functional was used, with a scissor correction to correct for the underestimated band gap.
The valence band structure of the graphene π band overlaid in Figure 1a was calculated using the tight-binding model [3], with parameters t and t' equal to -3.2 eV and 0.0 eV respectively.
The contour plots and 3D schematics presenting the valence band energy surfaces for 1L InSe the tight-binding model of ref. [4] originally developed for few-layer InSe. The parameters refitted to scissor-corrected DFT data for few-layer GaSe [5] are shown in the table below.

0.052
Linear-scaling DFT (LS-DFT) calculations in the Projector Augmented Wave formalism [6,7] were used to model the InSe/Gr heterostructure, using the ONETEP code [8]. The optB88 van der Waals density functional [9] was used as it has been shown to give reliable in-plane and interlayer distances for this class of materials [10]. We used PAW datasets from the GBRV library [11], and applied Ensemble DFT [12], using an NGWF radii of 12 a0 and a psinc grid of 1200 eV, which are sufficient for well-converged band structures for these materials. We modelled the heterostructure at the same alignment angle, 23°, as in the experimental system in Figure 1 and