A New Group of Two-Dimensional Non-van der Waals Materials with Ultra Low Exfoliation Energies

The exfoliation energy - quantifying the energy required to extract a two-dimensional (2D) sheet from the surface of a bulk material - is a key parameter determining the synthesizability of 2D compounds. Here, using ab initio calculations, we present a new group of non-van der Waals 2D materials derived from non-layered crystals which exhibit ultra low exfoliation energies. In particular for sulfides, surface relaxations are essential to correctly describe the associated energy gain needed to obtain reliable results. Taking into account long-range dispersive interactions has only a minor effect on the energetics and ultimately proves that the exfoliation energies are close to the ones of traditional van der Waals bound 2D compounds. The candidates with the lowest energies, 2D SbTlO$_3$ and MnNaCl$_3$, exhibit appealing electronic, potential topological, and magnetic features as evident from the calculated band structures making these systems an attractive platform for fundamental and applied nanoscience.

The discovery of new two-dimensional (2D) materials -traditionally derived from bulk layered compounds held together by weak van der Waals (vdW) forces -outlined over the last two decades a diverse zoo of representatives showcasing unique topological, 1 electronic, 2-4 magnetic, [5][6][7] and superconducting 8,9 properties.The weak interaction between the structural units in their bulk counterparts leads to a natural geometric separation of the 2D subunits in the crystals, giving rise to the possibility of mechanical 10 and liquid-phase 11 exfoliation.This class of nanostructures thus opens up prospects for fundamental research in reduced dimensions as well as for various applications in the energy sector, [12][13][14] catalysis, 15,16 and opto-electronics. 2,17,18The large scale deployment of these vdW 2D materials in modern technologies is, however, still very limited. 19 light of the ubiquitous use of many standard non-layered materials in research and technology for which handling and processing is well established, the search for non-vdW 2D materials is appealing. 20Recently and somewhat unexpectedly, atomically thin 2D sheets derived from non-vdW bonded oxides were indeed manufactured.4][25][26][27][28][29][30][31][32][33][34][35][36] Unlike, e.g.silicene or borophene, 37 these materials do not need to strongly interact with a substrate to be stable.From the computational side, a recent data-driven search on non-vdW 2D systems outlined 28 candidates with a variety of appealing electronic and magnetic properties. 38First application-oriented studies indicate promising perspectives for opto-electronics, 39 photo-catalytic activity for water splitting, 21,22 and photoconductivity. 28Despite these early successes, the non-vdW 2D materials space is still very narrow and it remains to be understood what promotes the exfoliability of non-vdW bulk systems into 2D sheets.
A key quantity determining the synthesizability of 2D materials is the exfoliation energy ∆E exf .It represents the energy needed to peel off one 2D sheet from the surface of the bulk parent material.It can be computed accurately according to the method of Jung et al. 40 at minimal computational cost, who proved that the exfoliation energy is the energy difference between an isolated 2D sheet and one such facet in the bulk, also known as inter-layer binding energy.For the standard reference material graphene, the exfoliation energy is known to be ∼20 meV/ Å2 from both experiment and advanced density functional theory (DFT). 41,42It was even found that this value appears to be universal for many layered systems largely independent of their electronic structure.However, later on a rather complete screening of the vdW 2D materials space outlined a much wider distribution of exfoliation energies over several ten meV/ Å2 , 43 likely due to contributions from other interactions than just vdW.Based on detailed energetic and structural considerations, upper bounds of ∼130 meV/ Å2 and 200 meV/atom for the exfoliation energy have been proposed to consider a material as (classically) exfoliable. 43,44As such, the calculated value of ∼140 meV/ Å2 (∼310 meV/atom) for hematene 38,45 seems surprisingly high, yet successful liquid-phase isolation of the 2D sheet was reported. 21 the recent data-driven search for non-vdW 2D systems, it was indicated that for several of the 28 oxidic candidates, the exfoliation energy is significantly smaller.For a few systems, it even comes close to the one of graphene if the surface cations are in a low (+1) oxidation state 38 suggesting that mechanical peel-off might be feasible.This effect was rationalized through the minimization of electrostatic interactions between the 2D sheets in case of small surface charges.While the consideration of the systems in vacuum is an idealization with respect to experiment, it allows for an assessment of the fundamental exfoliation energetics of these materials.Here, we focus explicitly on novel candidates with surface cations in low oxidation states and generalize the previous efforts by investigating also non-oxides.The goal is to try to find an answer to the question: How small can the exfoliation energy of a non-vdW 2D material get?
The input structures are retrieved via the AFLOW APIs [46][47][48] and web interfaces 49,50 as well as the library of crystallographic prototypes. 51The structures of Al 2 S 3 and MnNaCl 3 are depicted in as follows: First a 2×2 in-plane supercell is constructed from the relaxed structures.Then, the atomic coordinates are randomized (gaussian distribution, standard deviation 50 m Å) 52,53 and the structures are reoptimized.In all cases, the slabs relax back to the previous geometry.Although they also exhibit the same structure, Yb 2 S 3 and Lu 2 S 3 are not considered here as the corresponding 2D sheets were found to be dynamically unstable during the test.
Ab initio Exfoliation Energies.In Fig. 1(c), the calculated exfoliation energies are depicted also including, for comparison to the considered sulfides, the previously obtained values for the corresponding 2D oxides Al 2 O 3 and In 2 O 3 . 38For the ternaries, the sheets can be terminated by either of the The exfoliable [001] facet (monolayer) is indicated in the orange dashed box.The black line denotes the conventional unit cell.The compass indicating crystal directions applies to both figures.(c) Calculated exfoliation energies from SCAN without (as sliced) and with (relaxed) structural optimization of the 2D sheets.As a reference, the exfoliation energy of graphene (Gr) 41,43 as well as for hematene and ilmenene 38 are indicated by the dashed horizontal black lines.For the ternaries, the data for the slabs with the energetically favorable termination are plotted and for all systems, the terminating element is underlined at the bottom axis.The dashed lines connecting the data points are visual guides.
two cation species.Here, only the results for the energetically favorable termination are plotted and a comparison to the unpreferred termination is presented in Fig. S1 in the Supporting Information (SI).
The ∆E exf of the newly considered systems extend over a large range of more than an order of magnitude.Ultra low values are achieved for the systems with surface cations in +1 oxidation states (such as Ag + , Li + , K + , Na + , and Tl + ) being as small as 7 meV/ Å2 for SbTlO 3 and 26 meV/ Å2 for MnNaCl 3 .
These numbers are comparable to or even below the graphene reference value of ∼20 meV/ Å2 .They are significantly smaller than for any other non-vdW 2D system considered before.
The blue curve denotes the results obtained when omitting structural relaxations of the extracted 2D facets, i.e. keeping them "as sliced" from the bulk, as is common reliable practice for traditional vdW 2D systems. 43These values reduce significantly for almost all candidates when including structural optimization as denoted by the red curve pinpointing at the crucial role of relaxations for the energetics of these materials.For the binary sulfides, this effect is particularly pronounced.Not only is the absolute value of the "relaxed" exfoliation energy of the sulfides a factor of four lower than  Contribution of Long-range vdW Interactions.When exfoliation energies get as small as for graphene or even lower, the question of the importance of long-range vdW interactions naturally arises.While the employed SCAN functional has been pointed out to capture intermediate-range vdW interactions relevant for e.g.hydrogen bonds, 55 it does not include long-range dispersion contributions.
These can be accounted for within the SCAN+rVV10 scheme. 56We have thus also calculated the exfoliation energies for all systems with this approach and compare them to the plane SCAN results in Fig. 3(a).As expected, inclusion of the long-range interactions increases the exfoliation energies but only by a rather constant shift of 10-15 meV/ Å2 -the typical order of magnitude for dispersive interactions.Hence no qualitative change in the exfoliation behavior is anticipated from this.
As a further comparison, there are seven materials (CdPS 3 , CdPSe 3 , CrGeTe 3 , CrSiTe 3 , FePSe 3 , MgPSe 3 , and MnPSe 3 ) in the AFLOW database with the same structural prototype that are clearly layered, i.e. traditional vdW bonded materials as evident from visual inspection of the bulk geometries.
Thus, the same structural prototype can host both vdW and non-vdW bonded 2D materials.The exfoliation energies for these well established 2D systems were also computed with the same methods and are provided in Fig. 3(b).Here, dispersive interactions are essential to obtain reliable absolute values as the constant shift by ∼10 meV/ Å2 increases the SCAN results by a factor of three.For the non-vdW systems in Fig. 3(a), the SCAN values account already for 70-80% of the SCAN+rVV10 result since here the largest contribution to the bonding does not derive from long-range interactions.This consideration of the different interaction contributions therefore provides a clear distinction between non-vdW and vdW 2D materials.SbTlO 3 is a special case at the intersection between the vdW and non-vdW 2D materials spaces.
The optimized bulk (layered) structure with the exfoliable facet highlighted is depicted in Fig. 3(c).
It is structurally equivalent to the other non-vdW systems as the Tl cations are at the surface of the 2D sheets in contrast to traditional 2D systems such as the ones from Fig. 3(b) which are terminated by the anions (see the side view of CdPS 3 in the inset as an example).Yet, the exfoliation energy of SbTlO 3 appears to be largely governed by long-range vdW contributions (see Fig. 3(a)).However, the intermediate range vdW interaction captured by the SCAN exfoliation energy accounts with ∼40% for a larger portion of the total exfoliation energy (from SCAN+rVV10) compared to the traditional vdW 2D compounds of Fig. 3(b) where this amounts on average to only ∼30%.As a result of these characteristics, this compound was already identified as a potential 2D material by Mounet et al. 43 although not discussed in detail.
An important general remark regarding the exfoliation energies must be made.While it is well known that the standard DFT functional PBE(+U ) tends to underestimate binding energies (overestimating bond lengths also known as underbinding, see also the comparison including the PBE+U exfoliation energies in Fig. SI2), there are several indications that the employed SCAN functional tends to overestimate covalent and ionic binding energy contributions (for oxides).Firstly, it has been shown that the SCAN results are usually very close to the exfoliation energies computed from LDA 38 for which overbinding effects are well established.Secondly, it has been demonstrated that SCAN systematically overestimates oxide formation enthalpies similar to LDA, 57 again indicating binding effects to be on the high side.SCAN+rVV10 was shown to compute long-range vdW contributions reliably with no particular bias with respect to standard reference results from the random phase approximation. 56,58Thus, we expect the SCAN+rVV10 results to provide an upper bound for the exfoliation energies of non-vdW 2D materials.all compounds in comparison to the respective bulk bands in the SI.

Band Structures and Magnetic Properties of SbTlO
According to the band structures in Fig. 4, both systems are large gap insulators (calculated band gaps: 2.94 eV and 4.01 eV) with valence band maximum and conduction band minimum at the Kand Γ-points, respectively.An appealing feature for SbTlO 3 is the Dirac cone like linear band crossing at the high-symmetry K-point at about 4.2 eV above the Fermi level which might be readily accessible via moderate doping.This feature might hint at interesting topological properties calling for further investigations for instance addressing the explicit calculation of topological invariants.MnNaCl 3 on the other hand shows ferromagnetic coupling of the Mn moments amounting to ∼4.6 µ B .The energy difference to the antiferromagnetic configuration is ∼14 meV/formula unit.This magnetic behavior is also reflected in the spin polarized bandstructure in Fig. 4(b) where both bands at the edges of the gap are derived from majority spin while minority spin states are separated by several hundred meV.In contrast to the previously reported magnetic non-vdW 2D candidates in Ref., 38 the magnetic Mn ions are not at the surface of the slabs but in the interior (see also Fig. 1(b)).This is an important difference as the magnetic properties can be expected to be structurally better protected from environmental influences such as adsorbates.Based on the outlined electronic and magnetic characteristics, these systems can thus reveal potential for e.g.optoelectronic and/or spintronic applications.
Conclusions.We have outlined a new group of non-vdW 2D materials exhibiting ultra low exfoliation energies -ultimately getting as small as the one of graphene.The investigated sulfides Al 2 S 3 and In 2 S 3 have exfoliation energies a factor of four smaller than the corresponding oxides, which can be traced back to exceptionally strong surface relaxations allowing for a significant energy gain.The smallest values close to the ones of traditional 2D systems are found for SbTlO 3 and MnNaCl 3 , as evident from the comparison to several vdW materials.The computed band structures of these most easily exfoliable compounds exhibit appealing electronic, possibly topological, and magnetic properties.Our results may thus be an important guide for extending the family of non-vdW 2D exfoliable systems representing a new class of low dimensional compounds and for studying their characteristics as well as applications.

Methods
The ab-initio calculations are performed with AFLOW 59,60 and the Vienna Ab-initio Simulation Package (VASP) [61][62][63] employing the exchange-correlation functionals PBE, 64 SCAN, 65 SCAN+rVV10, 56 and PBE+U [66][67][68] with parameter choices in accordance with the AFLOW standard 69 as well as setting the internal VASPprecision to ACCURATE.For SCAN, projector-augmented-wave (PAW) pseudopotentials 70 of VASP version 5.4 are used and non-spherical contributions to the gradient of the density in the PAW spheres are included for SCAN and PBE+U .The [001] 2D facets are constructed from the bulk standard conventional unit cell with the respective AFLOW commands 71 resulting in structures with 10 atoms and including at least 20 Å of vacuum perpendicular to the slabs.For all facets, relaxation of both the ionic positions and the cell shape are carried out unless stated otherwise.The AFLOW internal automatic determination of k-point sets is used and for the calculations of the 2D facets, the setting for the number of k-points per reciprocal atom 69 is reduced to 1,000 resulting in Γ-centered 10 × 10 × 1 grids.The bandstructures are calculated for the optimized SCAN geometry using PBE(+U ) according to the AFLOW standard 69 as this functional has been successfully employed previously for the electronic properties of non-vdW 2D systems in Ref. 38 For computational efficiency, the dynamic stability check through the construction of 2×2 in-plane supercells was carried out with PBE(+U ).
The bulk and 2D candidate systems with expected magnetic ordering (MnNaCl 3 , CrGeTe 3 , CrSiTe 3 , FePSe 3 , and MnPSe 3 ) are rigorously checked for magnetism using the algorithm developed within the CCE method, 57,72 i.e. investigating all possible FM and AFM configurations in the structural unit cell for five different sizes of induced magnetic moments each.In each case, the lowest energy magnetic state is used for the further calculations.
The exfoliation energy is computed as: where E slab and E bulk indicate the total energies of the relaxed 2D material and bulk, respectively and A is the in-plane surface area according to the relaxed bulk unit cell.As proven in Ref., 40 the exfoliation energy from the surface of the material is exactly equal to the binding energy between layers/facets in the bulk.
Numerical data for the exfoliation energies are included in the Supporting Information.
A New Group of Two-Dimensional Non-van der Waals Materials with Ultra Low Exfoliation Energies Supporting Information Tom Barnowsky, 1, 2 Arkady V. Krasheninnikov, 1, 3 and Rico Friedrich

Figs 1 (
Figs 1(a) and (b) as examples to visualize the structural prototype of the investigated systems.The exfoliated [001] facets are also indicated.The dynamic stability of the outlined 2D systems is verified

Figure 3 :
Figure 3: Contribution of long-range vdW interactions.Comparison of the calculated exfoliation energies from SCAN and SCAN+rVV10 for (a) the eight non-vdW 2D candidates and (b) seven vdW 2D systems with the same structure.The dashed horizontal black line indicates the graphene reference value. 41,43The dashed lines connecting the data points are visual guides.Inset in (b): side view of CdPS 3 .The vertical black line in (a) separates binaries from ternaries.(c) Atomic structure of SbTlO 3 .The exfoliable [001] facet (monolayer) is indicated in the orange dashed box.The black line denotes the conventional unit cell.

Figure 4 :
Figure 4: Band structures and densities of states.Band structure and density of states of (a) 2D SbTlO 3 and (b) 2D MnNaCl 3 .The energies are aligned at the respective Fermi energy E F .For 2D SbTlO 3 , a linear band crossing at the K-point is highlighted by the circle.For the spin polarized band structure in (b), majority spin bands (positive DOS) are indicated in black while minority spin bands (negative DOS) are in red.

Fig. S1 showsFIG
Fig.S1shows the exfoliation energies for the energetically preferred (more stable) and unfavored (less stable) termination for ternaries.Due to the large difference in the oxidation states of the surface cations for the first five systems (+1 vs. +5) the values vary by a factor five to almost 30 for the different terminations.For MnNaCl 3 , the change is less pronounced since due to the Cl − anions, the inner Mn is assigned the oxidation state +2.
1, 2, * 1 Institute of Ion Beam Physics and Materials Research, Helmholtz-Zentrum Dresden-Rossendorf, 01328 Dresden, Germany 2 Theoretical Chemistry, Technische Universität Dresden, 01062 Dresden, Germany Exfoliation energies for different terminations for ternary systems.The terminating elements for the energetically more stable slabs are underlined.As a reference, the exfoliation energy of graphene[1, 2]is indicated by the dashed black line.The dashed lines connecting the data points are visual guides.

TABLE I :
Exfoliation energies for binaries.Exfoliation energies for the binaries calculated with different functionals from only a static ("as sliced") electronic calculation as well as when relaxing the ionic positions and (in-plane) cell parameters of the 2D materials.Note that in case of Al2S3 PBE+U reduces to PBE according to the standard workflow of AFLOW.

TABLE II :
Exfoliation energies for ternaries.Exfoliation energies for the ternary systems calculated with different functionals for fully relaxing all systems, i.e. the ionic positions and the (in-plane) cell parameters.The data for the energetically preferred (more stab.)cation termination are given.For SCAN, also the values from only a static ("as sliced") electronic calculation and the energetically less preffered (less stab.)cation termination are included.Note that in case of AsLiO3, KSbO3, NaSbO3, SbTlO3, and MgPSe3 PBE+U reduces to PBE according to the standard workflow of AFLOW.All values are in eV/ Å2 .compound PBE(+U ) SCAN SCAN+rVV10 compound PBE(+U ) SCAN SCAN+rVV10 more stab.more stab.more stab.less stab.more stab.