Abstract
Stacking of two-dimensional (2D) materials has emerged as a facile strategy for realising exotic quantum states of matter and engineering electronic properties. Yet, developments beyond the proof-of-principle level are impeded by the vast size of the configuration space defined by layer combinations and stacking orders. Here we employ a density functional theory (DFT) workflow to calculate interlayer binding energies of 8451 homobilayers created by stacking 1052 different monolayers in various configurations. Analysis of the stacking orders in 247 experimentally known van der Waals crystals is used to validate the workflow and determine the criteria for realisable bilayers. For the 2586 most stable bilayer systems, we calculate a range of electronic, magnetic, and vibrational properties, and explore general trends and anomalies. We identify an abundance of bistable bilayers with stacking order-dependent magnetic or electrical polarisation states making them candidates for slidetronics applications.
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Introduction
The field of 2D materials has evolved at a tremendous pace over the past decade and is currently impacting many areas of contemporary physics including spintronics1,2, valleytronics3, polaritonics4, unconventional superconductivity5, multiferroics6, and quantum light sources7. While numerous 2D monolayers have been extensively scrutinised in experiments and their properties systematically organised in computational databases8,9, studies of 2D multilayer structures have been much more sporadic.
The unit cell commensurate homobilayers (from hereon referred to as natural bilayers or simply bilayers) represent a well-defined and highly interesting class of 2D multilayer materials. Despite sharing the same Bravais lattice as the monolayer, their point group symmetry can differ depending on the stacking order. Such qualitative differences can influence physical properties profoundly with direct consequences for the material’s utilisation potential10,11,12. For example, non-volatile ferroelectric memories13, the valley Hall effect14, and spontaneous valley polarisation15, require materials with broken inversion symmetry. This may be achieved in a homobilayer even if the monolayer is centrosymmetric. Furthermore, due to the presence of the van der Waals (vdW) gap, the properties of bilayers can be tuned more effectively as exemplified by the giant Stark effect of interlayer excitons in bilayer MoS216,17, switching of magnetic states in bilayer CrI3 by either electrical gating18,19 or pressure20, and coupled ferroelectricity-superconductivity in bilayer MoTe221.
It has recently been proposed that the layer degree of freedom in vdW bilayers could form the basis for a distinct type of 2D interfacial ferroelectrics22,23,24. Subsequently, interfacial ferroelectricity has been demonstrated in bilayers of hexagonal boron nitride25,26 and transition metal dichalcogenides (TMDs)27,28,29. In these experiments, two stacking configurations with different out-of-plane polarisations are electrically switched via in-plane sliding of the layers. Beyond out-of-plane ferroelectricity, one might envision slide-induced switching of other physical quantities such as in-plane polarisation, conductivity, magnetism, or band topology30,31. Further progress in the emerging field of slidetronics32,\({N}_{{{{{{{{\rm{atoms}}}}}}}}}^{2}\) structures are generated by translating layer 1 by a vector tij given by the difference between the lateral positions of atoms i and j in the monolayer unit cell. Duplicate bilayers are subsequently removed. The interlayer binding energy, Eb, of the 8451 unique bilayers is then obtained by scanning the PBE-D3 energy of the frozen monolayers as a function of the interlayer distance (z-scan approach). Bilayers with Eb within 3 meV/Å2 of the most stable stacking are considered thermodynamically stable. The 2976 thermodynamically stable bilayers are then validated by checking their stability against lateral sliding. This is done by verifying that the energy as a function of the lateral displacement (x − x0) of the frozen layers, is a quadratic function with positive definite Hessian (A). The reliability of the z-scan approach is also verified by ensuring that the binding energy resulting from the z-scan approach (Eb) does not deviate from the binding energy obtained from a full relaxation of the bilayer (\({E}_{{{{{{{{\rm{b}}}}}}}}}^{{{{{{{{\rm{relax}}}}}}}}}\)) by more than 5 meV/Å2. The final 2586 bilayers are run through the property workflow.
Next, the slide stability is checked for all thermodynamically stable bilayers by fitting the local potential energy surface to a second-order polynomial (see Supplementary section C for details). Unstable configurations are pushed along a negative gradient (or curvature for saddle points), and a constrained energy minimisation is performed kee** the monolayers frozen. The procedure is repeated until a slide stable configuration is obtained. After removing duplicate structures, we obtain 2586 unique thermodynamically and slide stable bilayers, from hereon referred to as stable bilayers.
We now address the question: when can a stacking configuration be expected to be experimentally realisable? Clearly, the energy of such configurations should not be too high above that of the most stable stacking configuration. The binding energy of the latter is denoted \({E}_{{{{{{{{\rm{b}}}}}}}},\max }\). To determine a reasonable bound on \(\Delta {E}_{{{{{{{{\rm{b}}}}}}}},\max }\), we analyse its value for stacking orders observed in naturally occurring vdW crystals. Figure 2a shows the distribution of \({E}_{{{{{{{{\rm{b}}}}}}}},\max }-{E}_{{{{{{{{\rm{b}}}}}}}},\exp }\), i.e., the binding energy of bilayers in the experimentally observed stacking relative to the most stable stacking found by our workflow (note the logarithmic scale). The analysis includes 247 experimental bulk structures and 226 monolayers (21 monolayers appear twice in different stacking patterns). From the 247 bulk structures, we extract 314 bilayers. Note that bulk crystals with more than two monolayers per unit cell can result in more than one unique bilayer.
a Histogram showing the difference between the interlayer binding energy of bilayers in an experimentally observed stacking configuration \(({E}_{{{{{{{{\rm{b}}}}}}}},\exp })\) and the most stable stacking configuration predicted by the stacking workflow (\({E}_{{{{{{{{\rm{b}}}}}}}},\max }\)). Stackings with a binding energy within the cutoff of 3 meV/Å2 from \({E}_{{{{{{{{\rm{b}}}}}}}},\max }\) (green shaded region) are considered thermodynamically stable in this work (note the logarithmic scale). The inset shows the distribution of the binding energies calculated for the experimentally observed stackings (orange) and all the predicted stable bilayers (green). The green distribution has been reduced by a factor 5. b Exfoliation force (Fex) versus interlayer binding energy: Exfoliation forces are shown for 20% of the stable bilayers for which we specifically calculated more data points on the Eb(z) curve. The exfoliation force is the minimum force per area required to pull the bilayer apart by vertical lift-off. A number of known exfoliable 2D materials are indicated by orange dots (Graphene, Phosphorene, BN, MoS2, NbSe2, PtSe2, WTe2, HfSe2). The inset shows the number of stable bilayers obtained per monolayer. c Matrix representation of the layer group of stable bilayers versus the layer group of the constituent monolayer. The layer groups with inversion symmetry are marked in bold font. The colour bars in panels b and c represent the number of bilayers.
For 73% of the 226 monolayers that make up the set of experimentally known bulk crystals, the most stable bilayer found by the stacking workflow, coincides with an experimentally observed stacking (leftmost bar in Fig. 2). However, there are also cases where an experimentally observed stacking has an energy above the predicted most stable stacking. This could be due to several reasons, e.g., differences in interlayer interactions for bulk and bilayer (in Supplementary Fig. 1 we show that this effect is very small), limited accuracy of the calculations, finite temperature effects not accounted for in the calculations, or the occurrence of meta-stable stacking orders in the natural bulk crystals. To encompass such effects while avoiding to include too many unphysical/unstable structures, we require that \(\Delta {E}_{{{{{{{{\rm{b}}}}}}}},\max } \, < \, 3\) meV/Å2. Out of the 314 bilayer stackings found in the experimental bulk crystals, 74% satisfy this condition.
The distributions of Eb for all the stable bilayers and the bilayers in experimentally observed stacking configurations are shown in the inset of Fig. 2a. The similarity of the two distributions supports our approach for selecting stable, experimentally realisable bilayers.
The z-scan approach allows us to fit Eb(z) by an analytical Buckingham potential for the vdW interaction between two 2D planes, and thereby determine the exfoliation force, Fex (see Methods and Supplementary section B). Figure 2b shows, not unexpectedly, that Fex is correlated with Eb. However, there are also deviations from a linear relationship implying that Eb is not always an accurate descriptor for exfoliability.
The pie chart in the inset of Fig. 2b shows the number of stable stackings found per monolayer. Interestingly, for about 2/3 of the monolayers, our workflow predicts the existence of multiple stable stacking configurations. Such bilayers could form the basis for novel switchable systems as will be discussed later.
As mentioned, the crystal symmetry is of key importance to many applications of 2D materials. In Fig. 2c we show the layer group type of the monolayer versus the layer group type of the bilayer. It can be seen that the crystal symmetry often changes upon stacking. In particular, the breaking/emergence of inversion symmetry (indicated by bold font) occurs frequently when monolayers are stacked and could lead to qualitatively different physical properties.
Vibrational properties
Raman spectroscopy is one of the most important and widespread techniques for characterising 2D materials. In first-order Raman spectroscopy, the Γ-point phonons are probed via inelastic light scattering yielding detailed structural and electronic information from nondestructive measurements. While most Raman studies of layered vdW materials have focused on high-frequency intralayer phonons, low-frequency Raman spectroscopy is emerging as a means to probe interlayer couplings with higher precision38. To support these developments we systematically explore the sensitivity of Raman spectroscopy to interlayer coupling by calculating the full Raman tensor of 481 non-magnetic monolayers (with up to 8 atoms per monolayer cell) and their 1244 stable bilayers. The computational methodology is described in Methods and extensive benchmarks against experiments are presented in Supplementary section F.
Any vdW bilayer supports three low-frequency interlayer modes: Two in-plane shear modes (degenerate for isotropic materials) and one out-of-plane breathing mode. As an example, Fig. 3a shows the calculated and experimental low-frequency Raman spectrum of bilayer MoS2 in AB2 (2H) and the AB (3R) stacking configurations (our bilayer notation is explained in Methods). Both the experimental and theoretical spectra show a significant difference in the relative peak amplitude in the two stacking configurations as well as a shift in frequency of the breathing mode of 2.0 cm−1 (calculation) and 3.5 cm−1 (experiment). Figure 3b compares the Raman spectra of bilayer Ge2S2 in two stacking configurations. Again, the low-frequency modes show significant frequency shifts, but even more drastic changes occur for the peak intensities due to different crystal symmetries. The high-frequency spectrum also shows differences, although to a lesser extent. These examples clearly illustrate the sensitivity of the interlayer modes to the interfacial vdW interactions and the potential of low-frequency Raman spectroscopy for identifying stacking orders.
a Low-frequency part of the Raman spectrum of bilayer MoS2 in the AB6 and AB stacking configurations, respectively. Side views of the atomic structure of the bilayers are shown as insets. The calculated spectra (full lines) are compared to the experimental spectra of ref. 60 (dashed lines). The broadening of the peaks in the theoretical spectra has been fitted to match the experiments. The in-plane shear and the out-of-plane breathing phonon modes are shown next to the respective Raman peaks. b Calculated Raman spectra of bilayer Ge2S2 in the two stacking configurations shown as insets. It can be seen that both the low and high-frequency parts of the Raman spectrum are sensitive to the stacking order. c Distributions of shear and breathing interlayer modes (illustrated by the blue bars and black arrows) as a function of the square of the phonon frequency, ω2, and inverse layer mass density, ρ−1. Only bilayers for which the two types of modes do not couple are included. The black lines have slopes 〈K⊥〉 = 9.42 and 〈K∥〉 = 2.64, corresponding to the mean value of the effective spring constants for the two distributions (shown in the inset). d Change in the monolayer optical phonon frequencies when stacked into bilayers. The plot shows the splitting (Δωsplit) versus the shift (Δωshift) of the monolayer phonon frequencies (see inset). The in-phase vibrations (BL1) are essentially unperturbed while the out-of-phase vibrations (BL2) are blue-shifted. The colour bar represents the number of bilayers on a logarithmic scale. The data points tend to cluster around the black dashed line corresponding to Δωsplit = 2Δshift (see main text).
Within the harmonic approximation, the frequency of an interlayer vibration fulfils ω2 = K/ρ, where ρ is the monolayer mass density and K is the effective spring constant quantifying the strength of the interfacial vdW interaction. Figure 3c shows the distributions of shear modes (orange) and breathing modes (green) of all bilayers as a function of ω2 and ρ−1. Lines with slopes equal to the mean of the two K-distributions (see inset) are shown. The shear modes have strikingly similar spring constants while a significantly larger spread is found for the breathing modes. The shown distributions include all bilayers where the two types of interlayer modes decouple. For the remaining ca. 250 bilayers, the shear and breathing modes mix revealing an intrinsic mechanical coupling between out-of-plane uniaxial strain and interlayer shear.
We now turn to the intralayer phonons. Each intralayer phonon of the monolayer will give rise to two phonons in the bilayer. Due to the interfacial vdW interactions, the phonons of the bilayer will split (Davydov splitting) and shift relative to the monolayer phonons, see inset of Fig. 3d. By projecting the phonon modes of the bilayer onto the modes of the isolated monolayers, we are able to track the phonon hybridisation and unambiguously determine the frequency split and shift for each individual mode. The enhanced intensity of the data points along the line Δωsplit = 2Δshift (note the log scale) shows that in many cases the softening of the in-phase vibrations is weaker than the hardening of the out-of-phase vibrations. The extensive data set of Raman tensors provided in this work will advance the understanding and interpretation of vibrational spectroscopy of 2D materials.
Electronic properties
The idea of modifying electronic band gaps via layer stacking has been paradigmatic in the field of 2D materials. Well-known examples include the indirect to direct band gap transition in group-VI TMDs39 and the metal-semiconductor transition in PtSe240,41. In general, a qualitative change in the gap type has dramatic consequences for the material’s properties and possible applications. To systematically explore the opportunities and limitations of band gap stacking engineering, we have calculated the electronic band structure including the spin-orbit coupling for all bilayers (only the stable ones are included in the figure). Note that unstable stacking configurations are relevant as reference structures when building models for twisted bilayers.
One mechanism affecting the band gap upon stacking is interlayer hybridisation. This effect is controlled by the overlap of wave functions across the vdW gap and as such it is sensitive to both the interlayer distance and lateral stacking configuration. In addition to interlayer hybridisation, the band gap can be affected by the presence of an out-of-plane polarisation in polar bilayers. The latter effect can be quantified by the potential step across the bilayer created by the polarisation, ΦP.
Figure 4a shows the change in band gap, ΔEg, from monolayer to bilayer as a function of ΦP. Except for 10 bilayers, the gap is always reduced upon stacking. The common feature of the 10 anomalous monolayers is the presence of hydrogen atoms on the surface, which upon stacking interpenetrates leading to a strong and complex hybridisation. Bilayers with large ΦP consist of monolayers with broken mirror symmetry (Janus monolayers) stacked with aligned dipoles. Except for the metals, characterised by ΔEg = 0, these bilayers undergo a band gap reduction of ≈ 0.5ΦP. This occurs due to the offset of the bands in the two layers produced by the polarisation potential. For bilayers with ΦP = 0, the gap change is solely driven by hybridisation. For this subset, the gap change shows a decreasing trend as a function of the monolayer gap (see inset). This can be understood as a reduced strength of valence-conduction band hybridisation across the vdW gap. The change in electronic type upon stacking is shown in panel (c). A total of 349 bilayers undergo either a change in band gap type or a semiconductor-metal transition upon stacking. In particular, we find 126 bilayers with an emergent direct gap (composed of indirect gap monolayers) of interest for applications in nanophotonics and opto-electronics.
a Change in electronic band gap from monolayer to bilayer as a function of the out-of-plane polarisation potential across the bilayer, ΦP (see inset of panel b, where CBM (VBM) stands for conduction band minimum (valence band maximum)). Only stable bilayers are shown in the plot. The dashed line indicates the situation where the bilayer gap is exactly 0.5ΦP smaller than the monolayer gap. For the bilayers with zero out-of-plane polarisation, the gap change is shown in the inset as a function of the monolayer band gap. The colour bars in panels a and b show the number of bilayers on a logarithmic scale. b The change in band gap between different stable stackings of the same monolayer plotted as a function of the difference in the out-of-plane dipole of the two bilayers, ΔΦP. c The distribution of bilayers according to the change in electronic type from monolayer to bilayer (M: Metal, SC: Semiconductor, D: Direct band gap, ID: Indirect band gap). There are 349 bilayers with an electronic type different from their monolayer. d The distribution of monolayers according to the electronic types of all its stable bilayers. There are 91 monolayers with stacking-dependent electronic types (D & ID or M & SC).
Figure 4b shows the difference in band gap between two stable stacking configurations of the same monolayer as a function of the difference in polarisation potential, ΔΦP. Despite the weakness of the vdW interaction, stacking-dependent gap changes of up to 0.5 eV can arise purely from interlayer hybridisation (ΔΦP = 0). For bilayer pairs with finite dipole differences (ΔΦP > 0), gap changes of up to 1 eV occur when Janus monolayers, such as MoSSe, are stacked with parallel and anti-parallel dipole orientations, respectively. As can be seen in panel (d), most bilayers exhibit the same electronic type in all (stable) stacking configurations. However, 91 monolayers (~10%) support stacking configurations with different, and potentially switchable, electronic types.
Magnetic properties
Magnetically ordered bilayers constitute a highly attractive platform for designing versatile building blocks in next-generation spintronic devices, such as spin filter magnetic tunnel junctions42 and spin tunnel field effect transistors43. The interfacial vdW coupling allows for possible ferromagnetic (FM) or antiferromagnetic (AFM) orders. We write the energy difference between the two magnetic states as
which defines the effective interlayer exchange coupling J. Here S is the average spin per magnetic atom and Na is the number of magnetic atoms in the monolayer unit cell, see Supplementary section H for details. Accurate determination of J from first principles is challenging as it depends sensitively on the stacking configuration, temperature, and the model used for exchange-correlation effects. In Supplementary Table 1 we show that our calculations correctly predict the experimentally observed magnetic order (FM or AFM) in seven out of nine magnetic homobilayers.
Figure 5a shows J versus Eb for 595 stable magnetic bilayers. The lack of any clear correlation between the two quantities suggests that they are governed by different physical mechanisms. Indeed, Eb is governed by vdW interactions, which is a non-local correlation effect, while J is an exchange effect, which depends on the spatial overlap of the orbitals carrying the magnetic moments in the two layers.
a Interlayer exchange coupling versus interlayer binding energy for the most stable, magnetic bilayer stackings. The orange (green) symbols represent bilayers with FM (AFM) order being lowest in energy. The crystal structure of the two experimentally observed stacking configurations (also predicted as the two most stable) of bilayer CrI3 are shown. The data points corresponding to the bilayer structures in panel b are highlighted with red and cyan symbols. b Fraction of magnetic monolayers for which the magnetic order of all stable stackings is either FM (green), AFM (orange), or mixed (yellow). From the latter group, AgVP2Se6 and FeTaTe3 are highlighted as examples of bilayers that may be switched between FM and AFM magnetic configurations by sliding. c Distribution of all stable bilayer stackings according to magnetic order (FM/AFM) and electronic type (metal/semiconductor).
We find 37 monolayers with stable bilayers exhibiting FM or AFM order depending on the stacking configurations (Fig. 5b). One of these materials is the well-known CrI3. Among the remaining bilayers, we highlight AgVP2Se6 and FeTaSe3. Both materials are experimentally known in bulk form and the relatively small Eb indicates that they should be exfoliable. According to the C2DB, monolayer FeTaTe3 is metallic with in-plane magnetic easy axis, while monolayer AgVP2Se6 is a semiconductor with FM order, out-of plane easy axis, and in-plane ferroelectric order44. For both materials, the interlayer FM and AFM stacking configurations are related by a pure translation implying that the magnetic state could be switched by sliding.
Interestingly, monolayers with broken mirror symmetry also form bilayers with stacking-dependent magnetic order. For example, bilayers of the Janus monolayer VTeS are metallic and the built-in dipole leads to charge transfer, which affects the magnetic properties of the two constituent monolayers. In particular, one stacking yields a ferrimagnetic bilayer due to the changes in the magnetic moments induced by the interlayer coupling. In general, magnetic Janus heterostructures are expected to comprise rich possibilities for controlling intrinsic magnetic properties.
Figure 5c shows the distribution of all the stable magnetic bilayers according to magnetic order (FM/AFM) and electronic type (metallic/semiconducting). It can be seen that among the semiconductors there is a strong tendency for AFM order while the metals have mainly FM order.
Ferroelectric switching
Ultrathin vdW materials with bistable stacking configurations could form the basis of ferroelectric devices such as fast non-volatile memories22,23,24. Recent experiments have demonstrated interfacial ferroelectric (IF) switching in bilayers of hBN, some group-VI TMDs, and a few more materials (see ref. 24). Beyond those systems IF switching in nine homobilayers of AB honeycomb structures was previously studied using first-principles calculations45.
Our database introduces over 1600 pairs of bilayers with at least two stable stacking configurations related by interlayer translation. To explore the phenomenon of IF more systematically, we have calculated the barrier height and change of out-of-plane polarisation, ΔΦP, for the subset of non-magnetic bistable bilayers composed of hexagonal AB, AB2, and ABC monolayers that support two stable and slide-equivalent stacking configurations, see Fig. 6. Within this class we identify 133 bilayer pairs with barriers below 3 meV/Å2. These include the experimentally known systems for which our calculated polarisation change, ΔΦP, are in good agreement with the measured values (in meV): 160/218 (hBN), 114/94 (MoS2), 108/114 (MoSe2), 109/112 (WS2), 113/112 (WSe2) for calculation/experiment.
Calculated energy barrier and polarisation change between pairs of stable stacking configurations related by a pure layer translation for bilayers composed of hexagonal AB, AB2, and ABC monolayers. Materials with a known bulk parent are shown in yellow. A cross on a circle indicates that the bilayers are metallic. For the materials for which ferroelectric switching has been experimentally demonstrated (hBN, MoS2, MoSe2, WS2, and WSe2), we indicate the calculated (measured) polarisation change with blue stars (triangles). Note that the barrier height values are always calculated as we do not have experimental values available for them. Inset: Sketch of potential energy surface for a bistable bilayer. The green arrows indicate the out-of-plane polarisation in the two stacking configurations.
As can be seen from Fig. 6, we find several candidates including some that were not studied prior to this work and some with experimentally known bulk phases (yellow symbols). We also highlight the switchable metallic bilayer pairs (cross markers). Interestingly, we find several bilayers composed of ABC Janus monolayers, indicating that the built-in dipole of the monolayer promotes IF. For example, the semiconductors, BiITe, which has been exfoliated in a few-layer form46, is bistable with ΔΦP = 90 meV. Moreover, MoSSe, which has been synthesised as monolayer47, exhibits IF in two distinct rotational phases corresponding to twist angles of 0 and 60 degrees and with ΔΦP of 120 and 146 meV, respectively. A complete overview of all stacking configurations of MoSSe and other selected bilayers can be found in Supplementary section I.
