Thickness of Elemental and Binary Single Atomic Monolayers Cite This: Nanoscale Horiz., 2020, 5,385 Peter Hess

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Thickness of elemental and binary single atomic monolayers Cite this: Nanoscale Horiz., 2020, 5,385 Peter Hess

The thickness of monolayers is a fundamental property of two-dimensional (2D) materials that has not found the necessary attention. It plays a crucial role in their mechanical behavior, the determination of related physical properties such as heat transfer, and especially the properties of multilayer systems. Measurements of the thickness of free-standing monolayers are widely lacking and notoriously too large. Consistent thicknesses have been reported for single layers of graphene, boronitrene, and SiC derived from interlayer spacing measured by X-ray diffraction in multilayer systems, first-principles calculations of the interlayer spacing, and tabulated van der Waals (vdW) diameters. Furthermore, the electron density-based volume model agrees with the geometric slab model for graphene and boronitrene. For other single-atom monolayers DFT calculations and molecular dynamics (MD) simulations deliver interlayer distances that are often much smaller than the vdW diameter, owing to further electrostatic and (weak) covalent interlayer interaction. Monolayers strongly bonded to a surface Received 14th October 2019, also show this effect. If only weak vdW forces exist, the vdW diameter delivers a reasonable thickness not Accepted 18th November 2019 only for free-standing monolayers but also for few-layer systems and adsorbed monolayers. Adding the DOI: 10.1039/c9nh00658c usually known corrugation effect of buckled or puckered monolayers to the vdW diameter delivers an upper limit of the monolayer thickness. The study presents a reference database of thickness values for rsc.li/nanoscale-horizons elemental and binary group-IV and group-V monolayers, as well as binary III–V and IV–VI compounds.

1. Introduction physical properties with the number of layers.5 Since for most two-dimensional (2D) materials accurate experimental thickness

Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. With the realization of single atomic layers the ultimate scaling values of free-standing monolayers are still lacking, first-principles in thickness has been reached. However, to date the actual theory plays a dominant role in estimating thicknesses, mainly by thickness of most monolayers has been neither measured nor calculating the thickness of bi-, tri-, and few-layer systems. If the calculated with the necessary accuracy. The enormous diﬃculties corresponding layered crystal exists, the interlayer spacing can encountered in an accurate direct measurement of the thickness been determined experimentally, for example, by X-ray diﬀraction of free-standing monolayers, e.g., by nanoindentation using (XRD)6 and high-resolution transmission electron microscopy atomic force microscopy (AFM), can be seen in the range of (HRTEM).7 values reported for the best studied compound graphene, varying A presentation of the mechanical properties of 2D materials between 0.4 and 1.7 nm.1 Only by applying sophisticated sample is possible by either 2D or 3D quantities. The 3D description uses preparation and accurate imaging techniques a reproducible bulk units such as N m2 for the Young’s modulus and ultimate thickness of 0.43 nm could be realized, which approaches the strength and implies the knowledge of the layer thickness or layer currently accepted value of 3.35 nm, derived for graphite.2 One volume, which usually is not rigorously known. The 3D properties reason for the systematic overestimation of thickness originates allow a comparison with those of well-known bulk materials, for from overlayers of the solvent in solution-processed monolayer example, graphene versus diamond. However, one has always to preparation.3 keep in mind that monolayers are not real 3D entities, owing to The layer thickness controls not only relevant properties the restriction of one dimension to an atomic-scale length. of monolayers such as their mechanical behavior and heat Moreover, it is not possible to measure 3D quantities. conductivity4 but to an even larger extend the optical and By using the mechanical 2D model we take 2D units, for electronic properties of few-layer and multilayer systems such example, N m1 for the Young’s modulus and critical strength. Of as the band gap. The latter find increasing attention for tuning course, 2D solids are also not 2D objects in the strict mathematical sense. Formally, however, the simple model of a continuum plane Institute of Physical Chemistry, University of Heidelberg, Im Neuenheimer Feld 253, can be employed for a useful description of layers with atomic D-69120 Heidelberg, Germany. E-mail: [email protected] thickness. Properties with 2D units are accessible by experiment

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and a direct comparison with other monolayers is possible since Owing to the lack of measurements, theoretical methods there is no dependence on an assumed monolayer thickness. For such as DFT and molecular dynamics (MD) currently play an this reason, 2D properties are preferable in a comparative study of essential role in elucidating the layered structure of few-layer monolayers. The boundaries of a monolayer do not allow a systems. This includes the stability of different stacking con- straightforward derivation of the mechanically active volume or figurations of bi-, tri-, and multi-layer systems. Since in layered layer thickness. The introduction of simple geometrical volume materials strong intralayer covalent bonding and weak inter- models, such as a uniform slab with defined thickness, solve layer dispersion forces are simultaneously in action present the problem to perform the dimensional transformation from theoretical approaches have serious problems to accurately measured 2D to quasi 3D units. describe these soft materials. Therefore, it is not surprising As mentioned above, the knowledge of the layer thickness is that even for the best studied compounds graphite and boron not only an essential prerequisite to understand the behavior of nitride the suggested interlayer binding energies vary between monolayersbutitisalsocrucialforthedeterminationofphysical 24 and 86 meV per atom.9,10 These values have been calculated properties of layered materials. van der Waals (vdW) forces have a for the most stable stacking configuration of bilayers and bulk minor influence on the bond length and bonding energy of 2D crystals, which to some extent have different interlayer inter- covalent networks, however, they control the cohesion of the action. In most cases, however, the main uncertainties originate whole layered assembly and thus key properties of few-layer from the theoretical approximations employed. systems. Not only vdW forces, but also electrostatic interaction An alternative concept for an estimate of the monolayer and covalent bonding of the stacked structure, determine the thickness is the use of tabulated vdW radii, which describe the interlayer spacing. Note that the bottom-up synthesis of layered size of an atom. More accurately, the vdW diameter defines the compounds, using monolayers as building blocks and weak vdW distance of closest approach of atoms (‘hard-sphere model’). interaction, opens a huge new field of artificially assembled 2D Note that different sets of such compilations of atom sizes, with materials. Combining monolayers with specific properties may slightly different vdW radii, are available in the literature for lead to vdW superlattices and heterostructures of technological the elements of the periodic table. The vdW diameters are importance.8 employed here to estimate the thickness of planar elemental An often-used procedure to determine the thickness is the and binary monolayers. application of the geometric volume model of a rectangular slab The use of vdW radii may have a limited scope of applic- with a thickness taken from the interlayer spacing of the corres- ability to nano-objects because simple geometrical models may ponding layered crystal. If such a layered three-dimensional (3D) not describe the reality of arbitrary nanoscale objects. In fact, crystal exists, it is possible to measure the spacing by X-ray the size and shape of atoms and their bonding configurations experiments or HRTEM. The vdW diameter delivers comparable are determined by the electron density.11 Therefore, a more thicknesses if the interlayer interaction is based on weak vdW accurate but also more elaborate geometry-independent approach forces. However, depending on hybridization interlayer chemical based on DFT calculations of electron density has been applied to bonding may occur and a strong reduction of the interlayer graphene and boronitrene, resulting in thickness values in very

Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. separation makes the interlayer spacing no longer suitable to good agreement with those obtained by the simple geometrical estimate the thickness of the free-standing monolayer. Unfortu- slab model.12 nately, the information on the interlayer interaction energy This finding reveals that the slab model is a valuable approxi- needed for such a judgement of the layer spacing is quite limited. mation for planar monolayers. Unfortunately, up to now no DFT Of course, these experimental approaches are not applicable in the calculations on the electron density-based volume are available for case of entirely synthetic monolayers without layered structure. buckled and puckered monolayers. It would be of great interest to Ab initio calculations are the first choice to compute the compare electron density-based volumes with those obtained by pristine structure as well as chemical and physical properties of merely adding the corrugation effect to the vdW diameter, to find a 2D materials. Besides the prediction of new stable 2D materi- formal thickness of corrugated monolayers. One goal of this study als, routes of synthesis could be proposed. From the two main is to compare systematically monolayer thicknesses derived for theoretical approaches, namely the wave-function-based meth- planar and corrugated monolayers with those of other existing ods and density functional theory (DFT) calculations, the latter methods, to explore the suitability and limits of the geometrical one is more often employed. Here the many-body system of slab model as simple tool of thickness evaluation. electrons is treated by considering non-interacting electrons In the following, we compare the thickness values extracted moving in an effective potential. To take the exchange and experimentally and theoretically from the interlayer spacing of correlation effects into account in the calculation of electron density, layered systems with vdW diameters of elemental and binary the generalized gradient approximation (GGA) is employed that group-IV and group-V single atomic monolayers with planar, most often uses the Perdew–Burke–Ernzerhof (PBE) functions. buckled, or puckered structure. Furthermore, the thickness Alternatively, but seldom for 2D materials the local density analysis includes isoelectronic binary group III–V and group IV–VI approximation (LDA), in the form of Ceperley–Alder (CA) and monolayers. The goal is to compare the information provided by parameterized by Perdew–Zunger (PZ), is employed. In the various sources on the thickness of monolayers and to clarify the treatment of few-layer systems larger differences between the reasons for the observed deviations. The recommended definition GGA and LDA approach may be observed. of thickness allows a consistent transformation of measured

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and calculated properties given in 2D units into those in 3D units. Furthermore, correlations are considered, which exist within groups of structurally and chemically related compounds.13

2. Results 2.1 Thickness of elemental group-IV monolayers The interlayer spacing of graphite of 0.341 nm, determined by XRD experiments about hundred years ago,6 agrees quite well with the currently accepted theoretical value for the layered graphite structure with AB stacking of 0.335 nm.2 Similarly, detailed calculations of the interlayer spacing, considering dispersive, electrostatic, and Pauli interactions, confirmed a spacing of 0.335 nm.10 Furthermore, these interlayer separations are in very good agreement with the thickness of a single graphene Fig. 1 Monolayer thickness values approximated by the vdW diameter, layer of 0.34 nm, as obtained from the standard vdW radii of the sum of vdW diameter and buckling effect, as well as measured and 14 elements. Since the interaction incorporates only vdW forces, the calculated interlayer spacing of elemental and binary group-IV compounds. spacing extracted from the multilayer graphite system agrees well with the thickness defined by the vdW diameter. Note that in the case of carbon atoms graphite has been chosen for the definition electron density of graphite. This volume has been employed to of the vdW radius and therefore such a good agreement cannot be find the equivalent slab thickness.12 Regardless of surface undula- expected for all elements. The interlayer interaction energies tions of the electron distribution in the real monolayer the model obtained by an improved description of soft layered materials, delivered a thickness of 0.331 nm, in surprisingly good agreement using vdW-DFT calculations, varied between 22 and 85 meV with the well-established values. The space group of the plane per atom, depending on the number of layers considered and the honeycomb monolayer of graphene is P6/mmm.15 approximation employed.9 Table 1 presents the relevant structural data of the group-IV The van der Waals radii used here that were partially introduced monolayers, including geometry or space group, vdW diameter, by Bondi and represent distances of closest approach. Therefore, corrugation, thickness values, and theoretical methods. Fig. 1 they are smaller than gas-phase equilibrium distances.14 More displays the different results obtained for the monolayer thick- precisely they may be defined as the distance, where the potential ness and interlayer spacing. Besides graphene, the other energy passes through zero in a head-on collision with negligible elemental group-IV monolayers (2D-Xenes) silicene, germanene, kinetic energy.14 Note that a potential change of the free atom size stanene, and plumbene are attracting increasing attention. occurring during the formation of the covalent monolayer may be is They are of great promise for future applications, e.g.,as Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. already included in the definition of the vdW radius. Since a precise potential 2D topological insulators (TIs), with the quantum determination of the vdW radius is not possible, the tables spin-Hall (QSH) effect, owing to strong spin–orbit coupling published by different authors may not agree completely in all (SOC). In comparison to graphene, in these 2D-Xenes sp2 and elements of the periodic table. sp3 hybridization states compete. In monolayers this leads to In the electron density-based volume model already mentioned low buckling in silicene with a weak p cloud because of a larger above, the monolayer volume enclosing 99.64% of the total sp2 contribution. From silicene to plumbene buckling increases electrons has an average electron density that agrees with the with an increasing contribution of sp3 hybrids in the mixed

Table 1 Geometry or space group, vdW diameter, buckling, monolayer thickness, theoretical methods, interlayer spacing, and interlayer bonding of elemental and binary group-IV compounds

Geometry vdW diam. Buckling Thickness Method Inter. spacing Inter. bonding space group dw (nm) Dz (nm) dw + Dz (nm) monolayer dsp (nm) (meV per atom) Graphene P6/mmm15 0.34014 016 0.340 DFT-PBE16 0.335,10 0.3416 (22–85)9,10 Silicene P3%m120 0.42014 0.04116 0.461 DFT-PBE16 0.321,19 0.3418 — Germanene hex., buckl.16 0.42214 0.06816 0.490 DFT-PBE16 0.33519 30021 Stanene hex., buckl.16 0.43414 0.09016 0.524 DFT-PBE16 0.326,24 0.3324 — Plumbene hex., buckl.27 0.40414 0.09327 0.497 First-princip.27 ——

SiC P6%m215 0.38014 028 0.380 DFT-LDA28 0.347,30 0.34019 (34–55)34 GeC hex., plane28 0.38114 028 0.381 DFT-LDA28 0.369,32 0.35019 — SnC hex., plane28 0.38714 028 0.387 DFT-LDA28 —— GeSi hex., buckl.19 0.42114 0.05719 0.478 DFT-CA-PZ19 0.32019 24033 SnSi hex., buckl.29 0.42714 0.07329 0.500 DFT-PBE29 —— SnGe hex., buckl.29 0.42814 0.08029 0.508 DFT-PBE29 ——

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hybridization states. This is the reason for decreasing p bonding Inspection of Fig. 1 reveals large deviations between the 16,17 by pz orbitals. thickness derived from the spacing of bi- and multi-layers of AFM measurements on silicene monolayers prepared by the elemental group-IV compounds and the vdW diameter, with liquid-phase method resulted in a thickness of 0.6 nm.18 graphene as exception. As mentioned already, the interlayer Measurements of the interlayer distance of few-layer silicene interaction is weak for graphene, while the increasing contri- nanosheets by HRTEM and selected area electron diﬀraction bution of sp3 hybrids for the other homologues causes buckling (SAED) resulted in an interlayer spacing of only 0.34 nm.18 In of the 2D sheets that finally leads to 3D covalent bonding of comparison with the latter thickness, which is comparable to the whole layered network. Experiments approved the calculated that of graphene, the vdW diameter of 0.420 nm yields a value low interlayer distances reported for such layered systems. Thus, of 0.461 nm, if the buckling eﬀect of 0.041 nm is included. It the reason why these interlayer distances are not suitable to should be mentioned that by a first-principles study a much estimate the thickness of free-standing monolayers is partial lower interlayer distance of 0.312 nm was found for the lowest covalent interlayer bonding. The sudden expansion of the energy configuration of the buckled bilayer.19 Owing to buckling orbital size and thus thickness of silicene and its followers, in the symmetry of silicene is reduced to the space group P3%m1.20 comparison to graphene, causes an increasing weakening of p

Remarkably, under tension, the interlayer distance of the bilayer bonding between pz orbitals in comparison to graphene. decreases further, because the interlayer interaction is intensified Consequently, the interlayer spacing may not be suitable to by covalent s bonds and the resulting interlayer distance approaches calculate mechanical properties of monolayers in the case of the intralayer bond length of 0.241 nm of the honeycomb network.21 2D-Xenes. Nonetheless, the established Griﬃth’s rule for the Unfortunately, no measurements of the interlayer spacing fracture strength of s E E/9,13 where s is the ultimate strength are available for germanene multilayers. The already discussed and E the Young’s modulus, has been questioned for stanene first-principles study on optimized stacking of bilayers reports by using 0.326 nm as thickness.24 Conversely, independent DFT an interlayer spacing of 0.335 nm for buckled germanene.19 calculations of the Young’s modulus and ultimate strength This small interlayer distance is consistent with the large confirm the rule. The reason for this inconsistency could be interlayer interaction energy of 300 meV per atom calculated the too small thickness used by the authors that is much for germanene.22 Similar to silicene, a flat phase with the same smaller than the value of 0.524 nm suggested here (see Table 1). interlayer and intralayer bond length of 0.241 nm of the bilayer is energetically more stable than its stacked buckled phase.21 The 2.2 Thickness of binary group-IV monolayers monolayer thickness estimated by adding the vdW diameter of From the class of binary group-IV monolayers we consider here 0.422 nm and buckling eﬀect of 0.068 nm16 amounts to 0.490 nm. the three plane sp2-bonded compounds SiC, GeC, and SnC and By AFM experiments a similar thickness of 0.33 nm has been the three buckled compounds GeSi, SnSi, and SnGe. They have measured for stanene,23 in surprisingly good agreement with found interest owing to promising applications in optoelectronics, the presented first-principles-based DFT calculation of 0.326 nm based on the capability of indirect-to-direct bandgap transitions, in for free-standing stanene.24 As discussed before, the buckled going from multilayer to monolayer systems.28,29 Table 1 presents

Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. bilayer of stanene can also be converted to the stable flat phase the geometry or space group, vdW diameter, buckling height, with a reduced interlayer separation that essentially agrees with theoretical method, and multilayer data, while Fig. 1 offers a the intralayer bond length of 0.294 nm.21 Contrary to these comparison of vdW diameters with the calculated interlayer values, a much larger thickness of 0.546 nm has been proposed spacings of the binary group-IV compounds. As vdW diameter of the for the buckled monolayer.25 This value is consistent with the binary compounds the sum of the vdW radii of the two constituents thickness of 0.524 nm obtained with the vdW diameter of is employed. Therefore, the corresponding thicknesses are in the 0.434 nm and the buckling distance of 0.090 nm.16 same range as those of the elements (see Table 1). Plumbene, as the heaviest element of the group has found The interlayer spacing of hexagonal SiC multilayers of 0.347 nm attention only recently as a potential giant-gap spin Hall insulator. has been measured by combining XRD and HRTEM.30 This In the first study by MD a buckling effect of 0.071 nm has been experimental value agrees well with the results of independent simulated.26 However, also a larger buckling height of 0.093 nm first-principles calculations of 0.340 nm,19 and 0.338 nm.31 The has been suggested that is comparable to the stanene value.27 By bilayer interaction energy of 156 meV per atom (trilayer: adding this value to the vdW diameter of 0.404 nm a thickness of 224 meV per atom) for AA stacking (similar values for AB 0.497 nm is obtained for the corrugated plumbene layer. stacking) indicates the formation of 2D allotropes with an The geometrical slab volume applied by adding the buckling effective charge on the outer bilayer.31 Note that these interlayer effect to the vdW diameter delivers a larger volume than the real interaction energies are surprisingly large as will be pointed out electron density-based volume of the corrugated monolayer. There- in the discussion below. The space group of the planar SiC fore, the corresponding thickness presents an upper limit of this monolayer is P6%m2.15 quantity. The corrugation effect increases the effective volume, For GeC a similar interlayer spacing of 0.369 nm was found however, not so much as assumed by the simple geometrical for the energetically preferred bilayer structure.32 A somewhat slab model. To gain further information on the error involved in smaller interlayer distance of 0.350 nm for AB stacking of the suggested procedure a quantitative estimate of the effective GeC has been reported in a previous publication.19 By using thickness of the real volume of corrugated monolayers is crucial. the first-principles pseudopotential method, for buckled GeSi

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an interlayer spacing of 0.323 nm has been reported for AB monolayers phosphorene, arsenene, antimonene, and bismuthene stacking.19 For this configuration DFT calculations with CA and their potential applications in electronics, optoelectronics, parametrization yield a covalently bonded bilayer structure with thermoelectrics, and topological spintronics have been discussed an interlayer separation of only 0.249 nm and a binding energy of in a comprehensive review.35 Phosphorene has already been 240 meV per atom.33 manufactured by exfoliating black phosphorus with a first measure- As shown in Fig. 1, for planar SiC and GeC the spacing is ment of the thickness by AFM of B0.85 nm (theory 0.616 nm),36 slightly smaller than the vdW diameter owing to electrostatic while monolayers of antimony, arsenic, and bismuth could be interaction. In the case of siligene (GeSi), buckling causes a prepared by bottom-up growth on various surfaces under ultrahigh larger difference. This finding of a smaller spacing can be vacuum (UHV) conditions. Free-standing few-layer films of arsenene rationalized by the relatively large interlayer binding energy, couldbeisolatedbytheadhesive tape method and those of which is about a factor of three larger than the value proposed antimonene by ultrasonication.37 Table 2 presents the geometry or for vdW interaction in graphite.33 For the other buckled com- spacegroup,vdWdiameter,corrugation,thicknessvalues,theore- pounds, no information on the interlayer separation and bind- tical methods, and information on bi- and few-layer systems. Fig. 2 ing energy could be found. shows the results for the monolayer thickness and interlayer spacing, The vdW diameter agrees reasonable-well with both the including the corrugation effects. experimental and theoretical interlayer separations of SiC and The pnictogen monolayers exist in two stable phases, which GeC. The assumption of a modest enforcement of vdW inter- may have nearly the same stability, the puckered a-phase (wash- action is consistent with the interlayer interaction energies of board or w-phase) realized in orthorhombic black phosphorus SiC bilayers, which vary between 34 and 55 meV per atom, and the buckled b-phase (b-phase) realized in blue phosphorus, depending on stacking.34 Contrary to this situation stronger usually holding the minimum energy configuration. The term covalent-type interlayer bonding and therefore larger deviations buckling is used here for a small zigzag-like corrugation that are found for buckled GeSi.33 Obviously, the tabulated vdW radii can be characterized by one lattice constant and one bond allow a reasonable estimate of the monolayer thickness only for length, whereas the pucker effect (symmetric or antisymmetric) the weakly interacting plane binary group-IV compounds. designates a washboard-like corrugation that must be described by two lattice constants and two bond lengths. 2.3 Thickness of elemental group-V monolayers Contradicting results on the most stable phase, sometimes The enormous progress achieved recently in experimental and found in the literature, are due to the small difference in the theoretical investigations of the hexagonal elemental group-V cohesive energy. For example, while puckered antimonene has

Table 2 Geometry or space group, vdW diameter, buckling, monolayer thickness, theoretical method, interlayer spacing, and interlayer bonding of elemental and binary group-V compounds

Geometry vdW diam. Buck., pucker Thickness Method Inter. spacing Inter. binding space group d (nm) Dz (nm) d + Dz (nm) monolayer d (nm) (meV per atom) Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. w w sp Phosphorene hex., buckl.41 0.36014 0.12441 (b) 0.484 (b) DFT-PBE41 0.40941 (b) 642 Pmna59 0.25139 (a) 0.611 (a) DFT-PBE39 0.52440 (a) 2036 Arsenene P3%m145 0.37014 0.14043 (b) 0.510 (b) DFT-PBE43 0.46743 (b) 30,44 89.648 Pmna45,59 0.23943 (a) 0.609 (a) DFT-PBE43 0.55444 (a) 5344 Antimonene hex., buckl.41 0.41214 0.16541 (b) 0.577 (b) First princip.41 0.50237 (b) 124,38 86.048 Pmna59 0.28239 (a) 0.694 (a) DFT-PBE39 0.61638 (a) 6838 Bismuthene hex., buckl.41 0.41414 0.17141 (b) 0.585 (b) First princip.41 0.48046 (b) 136,46 12347 Pmna59 0.27239 (a) 0.686 (a) DFT-PBE39 0.61246 (a) 12046

PN P3m156 0.33514 0.08649 (b) 0.421 (b) DFT-PBE49 0.57653 (b)— 56 49 49 53 Pmn21 0.190 (a) 0.525 (a) DFT-PBE 0.635 (a) AsN P3m156 0.34014 0.09756 (b) 0.436 (b) DFT-PBE56 —— 56 56 56 Pmn21 0.221 (a) 0.561 (a) DFT-PBE SbN P3m156 0.36114 0.10256 (b) 0.462 (b) DFT-PBE56 0.33255 (b)— 56 56 56 54 Pmn21 0.243 (a) 0.603 (a) DFT-PBE 0.590 (a) BiN P3m156 0.36214 0.10556 (b) 0.467 (b) DFT-PBE56 —— 56 56 56 Pmn21 0.259 (a) 0.621 (a) DFT-PBE AsP P3m150 0.36514 0.13249 (b) 0.497 (b) DFT-PBE49 —— 59 49 49 Pmn21 0.235 (a) 0.600 (a) DFT-PBE SbP P3m150 0.38614 0.14149 (b) 0.527 (b) DFT-PBE49 0.33455 (b)— 59 49 49 Pmn21 0.285 (a) 0.671 (a) DFT-PBE BiP hex., buckl.58 0.38714 0.14858 (b) 0.535 (b) First princip.58 0.34255 (b)— 59 50 50 54 Pmn21 0.293 (a) 0.680 (a) DFT-PBE 0.606 (a) SbAs P3m150 0.39114 0.15249 (b) 0.543 (b) DFT-PBE49 —— 59 49 49 Pmn21 0.280 (a) 0.671 (a) DFT-PBE BiAs P3m150 0.39214 0.15550 (b) 0.548 (b) DFT-PBE50 —— 59 50 50 Pmn21 0.295 (a) 0.687 (a) DFT-PBE BiSb P3m150 0.41314 0.16950 (b) 0.582 (b) DFT-PBE50 —— 59 50 50 Pmn21 0.335 (a) 0.748 (a) DFT-PBE

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of 0.455 nm and 0.554 nm, respectively.44 Asmallerinterlayer interaction energy of -30 meV per atom (AA) has been calculated for b-As than for a-As at 53 meV per atom (AB) by the DMC method.44 For buckled arsenene the space group is P3%m1 and for the puckered configuration it is Pmma.45 Like arsenene, antimonene is of interest due to its semiconducting properties and its wide band gap. For the buckled phase (b-Sb) an interlayer spacing of multilayers of only 0.365 nm (AB), with a bilayer interaction energy of 124 meV per atom, and for the puckered phase (a-Sb) a much larger spacing of 0.616 nm (AB), with a smaller bilayer binding energy of 68 meV per atom, have been reported.38 For comparison, an interlayer spacing of 0.448 nm (AB) has been calculated for the bulk rhombohedral honeycomb lattice of b-Sb.41 Furthermore, an Fig. 2 Monolayer thickness values approximated by the vdW diameter, even larger interlayer spacing of 0.502 nm has been reported for the sum of vdW diameter and buckling or puckering effect, as well as the bulk b-phase of antimonene by an independent study.37 With calculated interlayer spacing of elemental and binary group-V compounds. the buckling effect of 0.165 nm41 and the vdW diameter of 0.412 nm a monolayer thickness of 0.577 nm can be extracted for the b-phase. Accordingly, the wide pucker height of 0.282 nm39 the minimum energy configuration, the cohesive energy of causes a larger thickness of 0.694 nm for the a-phase. buckled antimonene is close and b-Sb is more stable than The honeycomb monolayer of the heaviest group-V element a-Sb in multilayers with more than 3 layers.38 bismuthene has just recently attracted interest as a room- The interest in anisotropic puckered phosphorene arises temperature TI, showing great promise for spintronic applications. from its direct band gap and the high carrier mobility. By adding Bismuthene is a narrow band gap semiconductor. First-principles the large puckering height of 0.251 nm to the vdW diameter of DFT-PBE calculations for multilayer b-bismuthene delivered a bulk 0.360 nm we find a layer thickness of 0.611 nm for the most interlayer spacing of 0.442 nm (AB), including a buckling effect of stable a-phase.39 This value can be compared with an indepen- 0.171 nm.41 The interlayer distances between nearest atoms in dent estimate of 0.524 nm, based on the nearest layer distance of bilayer b-Bi (AA) at 0.309 nm and a-Bi (AA) at 0.340 nm result in adjoined layers in a multilayer of 0.307 nm and a smaller pucker interlayer spacings of 0.480 nm and 0.612 nm, using the buckling effect of 0.217 nm.40 Interestingly, the interlayer interaction effect of 0.171 nm and a pucker height of 0.272 nm,39 energy of the a-phase with AB stacking is only 20 meV per respectively.46 For b-Bi (AA) stacking an interlayer binding atom.36 A much smaller interlayer spacing of 0.412 nm has been energy of 123 meV per atom has been obtained, by employing determined for the bulk b-phase using 0.288 nm for the nearest the DFT-PBE approximation without SOC and 170 meV per 47 Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. interlayer distance of adjoined layers and 0.124 nm for atom with SOC. With the buckling effect of 0.171 nm and the buckling.41 However, based on ab initio calculations, also a vdW diameter of 0.414 nm a monolayer thickness of 0.585 nm larger interlayer spacing of 0.563 nm has been reported for can be extracted for the b-phase and with the pucker effect of the buckled b-phase of blue phosphorus.42 Similar to the results 0.272 nm a thickness of 0.686 nm is obtained. for the actual stacking configuration (AB) of the a-phase an The interlayer separations of a-phase and b-phase bilayers extremely low interlayer interaction energy of 6 meV per atom and multilayers of pnictogens have been determined by a series has been reported for the b-phase.42 of first-principles methods. As can extracted from the discussion The wide band gap, which changes from indirect to direct by presented above, the corresponding interlayer spacing of phos- applying strain or an external field, stimulated the first-principles phorene, arsenene, antimonene, and bismuthene scatter around calculations performed recently for arsenene and its multilayers. thickness values which are lower than those defined by the vdW For the most stable buckled lattice of b-As the following bulk and diameter and the combination of vdW diameter and corrugation tri-layer interlayer spacings of 0.455 nm,37 0.536 nm (AB),41 and effect. While the effect of an increasing number of layers on the 0.450 nm (AAA)43 have been evaluated by adding the buckling interlayer spacing is not negligible going from a bilayer to effect of 0.140 nm to the reported nearest interlayer distances multilayers, the stacking order also can change the interlayer between atoms of adjacent layers.43 Note that for the bilayer of spacing substantially and therefore is shown together with the b-As an interlayer spacing of 0.467 nm has been reported.43 With spacing. the vdW diameter of 0.370 nm and the buckling effect of To select the best stacking configuration for an eventual 0.140 nm a formal monolayer thickness of 0.510 nm for b-As evaluation of the isolated monolayer thickness, it is useful to and with the pucker height of 0.239 nm43 a thickness of 0.609 nm consider the invers behavior between interlayer spacing and for a-As is derived. A systematic investigation of the nearest interlayer binding energy. For both a-P and b-P AB stacking leads interlayer distances of bilayers by diffusion quantum Monte to small interlayer binding energies of 20 meV per atom36 and Carlo (DMC) and DFT calculations resulted in 0.352 nm for 6meVperatom.42 Therefore, a reasonable agreement between b-As (AA) and 0.315 nm for a-As (AB), resulting in layer spacings the calculated spacing and the vdW-diameter-based thickness

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could be envisaged for phosphorene. For arsenene and antimo- second layer directly stacks on the first layer with no rotation or nene DFT calculations delivered comparable interlayer inter- translation.54 Note that significantly lower interlayer spacings have action energies of 89.6 meV per atom and 86.0 meV per atom, been evaluated for ABC stacking of the bulk rhombohedral crystals respectively.48 However, also substantially smaller values of of b-SbN (0.332 nm), b-SbP (0.334 nm), and b-BiP (0.342 nm) in 30 meV per atom (b-As) and 53 meV per atom (a-As) have been their ground state structure.55 The sum of the vdW radii of these suggested, indicating the difficulties encountered in the deter- and the other binary compounds, as well as the inclusion of their mination of accurate interaction energies. Since these values are buckling and pucker effects, are displayed in Table 2. on the upper limit of vdW interaction still useful thicknesses Additional data on buckling and puckering has been taken may result from the interlayer separations. According to the from a study on nitride compounds with auxiticity.56 Moreover, larger bilayer binding energies of 124 meV per atom (b-Sb) from studies on strain induced QSH insulator behavior of and 68 meV per atom (a-Sb)38 and of 123 meV per atom b-BiSb and strain induced topological phase transitions in binary (DFT-PBE) and 170 meV per atom (DFT-PBE + SOC) for b-Bi bismuth compounds.57,58 Finally, important information on the larger deviations are expected.47 relevance of the layer thickness on thermal transport properties Altogether, the presented results are consistent with the has been published.59 The present status of both theory and assumption that primarily the interlayer interaction energy experiments of 2D V–V binary materials has been reviewed.60 controls the suitability of calculated and measured interlayer Inspection of Fig. 2 shows that the sum of the vdW radii distances and spacings for a realistic prediction of free mono- increases continuously from 0.335 nm for PN to 0.413 nm for layer thicknesses. Whenever sp3-like hybrids contribute to BiSb. This slight increase of the vdW diameters with atomic interlayer bonding or interaction with a surface, covalent forces number arises from the increase of the orbital size within the diminish the thickness of a free monolayer with a conforming chemically related group members of the periodic table. The change of the multilayer properties. Interestingly, quasi-free- buckling effects enlarge the vdW thicknesses, especially in the standing monolayers of heavier elements could be realized by case of the heavier compounds. Since both the vdW diameters adding a suitable buffer layer between surface and monolayer. and buckling effects are well-established by independent studies, the interlayer spacings of b-SbN, b-SbP, and b-BiP are much 2.4 Thickness of binary group-V monolayers smaller than the monolayer thicknesses. Owing to the increased First-principles investigations of binary group-V monolayers are contribution of sp3 hybrids strong bonding takes place between finding increasing attention, driven by the search for ferro- the layers of nitrides and phosphates.55 It is not clear yet, why the electric materials, TIs, and dissipationless transport devices. interlayer spacing of b-PN is much larger than expected. The few While theoretical studies predict stable binary honeycomb interlayer spacings available for the puckered compounds a-PN, semiconductors with direct and indirect band gaps the experi- a-SbN, and a-BiP are in qualitative agreement with the layer mental verification of these predictions is still widely lacking. thicknesses resulting from the sum of vdW diameter and pucker For the stable phases of binary group-V compounds both the effect, despite large scatter. a-phase and b-phase configurations will be considered (see 49–53 2.5 Thickness of group III–V monolayers Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. Table 2). This includes the compounds PN, AsN, SbN, BiN, AsP, SbP, BiP, SbAs, BiAs, and BiSb. Unfortunately, detailed The semiconducting members of group III–V compounds are of information on the corresponding structural properties of their interest owing to their large band gaps and piezoelectric properties. multilayers is mostly absent. Fig. 2 shows the thickness values The following compounds with an atom of the second row of the with and without corrugation and available interlayer spacings periodic table have a planar honeycomb structure (BN, BP, BAs, BSb, of a-phase and b-phase compounds. AlN, GaN, InN). The other compounds achieve their stable hexago- Using first-principles calculations, a stable puckered mono- nal structure by buckling in the vertical direction (AlSb, GaP, GaAs, layer of a-PN and a stable buckled monolayers of b-PN have GaSb, InP, InAs, InSb).61 The buckling effects presented in Table 3, been predicted (note that the nomenclature of the authors has taken from the most comprehensive work,61 are in good agreement been changed).53 By adding the buckling effect of 0.086 nm49 to with those of other publications.62,63 The data available for this the nearest layer distances of 0.490 nm (AA) and 0.541 nm (AB) group, including the space groups,64 is collected in Table 3. Fig. 3 of layered b-PN, interlayer spacings of 0.576 nm and 0.627 nm shows the interlayer spacings and monolayer thicknesses derived are found.53 Even larger interlayer spacings are obtained by with the vdW diameter. adding the large pucker effect of 0.190 nm49 to the nearest Boronitrene currently belongs to the best studied mono- interlayer distances of a-PN of 0.446 nm (AA) and 0.445 nm layers. The monolayer thickness of 0.333 nm, derived from (AB), yielding interlayer spacings of 0.636 nm (AA) and 0.635 nm interlayer spacing, agrees surprisingly well with that of graphite, (AB), respectively.53 By adding the buckling effect of 0.086 nm despite the expected contribution of electrostatic interlayer and pucker height of 0.190 nm to the sum of the vdW radii of forces.65 Already early X-ray studies of boron nitride attained the two constituents a vdW diameter of 0.335 nm and layer an interlayer spacing of 0.333 nm.66 This has been confirmed by thicknesses of 0.421 nm (b-PN) and 0.525 nm (a-PN) are found. the layer spacing of 0.33 nm obtained by recent XRD measure- Ab initio calculations of the bulk interlayer spacing of a-SbN ments on 50 nm thick h-BN films, synthesized by ambient at 0.590 nm and a-BiP at 0.606 nm were based on the a-structure of pressure CVD.67 In excellent agreement with these results are phosphorene and the most stable AA stacking order, where the detailed calculations of the binding energy, also predicting an

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Table 3 Geometry or space group, vdW diameter, buckling, monolayer thickness, theoretical methods, interlayer spacing, and interlayer bonding of elemental and binary group III–V compounds

Geometry vdW diam. Buckling Thickness Method Int. spacing Inter. binding space group dw (nm) Dz (nm) dw + Dz (nm) monolayer dsp (nm) (meV per atom) h-BN P6%m264 0.34714 061 0.347 DFT-PBE61 0.33365 (26–86)9,10 h-AlN P4/nmm64 0.33914 061 0.339 DFT-PBE61 0.21369 125,69 29270 h-GaN P6%m264 0.34214 061 0.342 DFT-PBE61 0.24771 140,72 29071 h-InN P6%m264 0.34814 061 0.348 DFT-PBE61 0.26773 — h-TlN P3%m175 0.35114 0.02974 0.380 DFT-PBE74 0.25175 18575 BP P6%m264 0.37214 061 0.372 DFT-PBE61 0.33976 112,76 13476 BAs P6%m264 0.37714 061 0.377 DFT-PBE61 0.33577 — BSb P6%m264 0.39814 061 0.398 DFT-PBE61 —— AlP Pmn2164 0.36414 061 0.364 DFT-PBE61 —— AlAs P4/nmm64 0.37914 061 0.369 DFT-PBE61 —— GaP Pmn2164 0.36714 0.04861 0.415 DFT-PBE61 —— GaAs Pmn2164 0.37214 0.06161 0.433 DFT-PBE61 0.28377 — InP Pmn2164 0.37314 0.05661 0.429 DFT-PBE61 —— InAs Pmn2164 0.37814 0.07161 0.449 DFT-PBE61 0.31577 — AlSb P4/nmm64 0.39014 0.06261 0.452 DFT-PBE61 —— GaSb Pmn2164 0.39314 0.06861 0.461 DFT-PBE61 —— InSb Pmn2164 0.39914 0.07761 0.476 DFT-PBE61 —— TlP P3%m175 0.37614 0.06674 0.442 DFT-PBE74 0.28075 9175 TlAs P3%m175 0.38114 0.07474 0.455 DFT-PBE74 0.28775 7975 TlSb P3%m175 0.40214 0.08274 0.484 DFT-PBE74 0.303 7675

(Al over N and N over Al).69 Independently, a similar interlayer distance of 0.218 nm and a bilayer interaction energy of 292 meV per atom have been calculated for the ground-state configuration.70 For AA stacking (Al over Al and N over N) the interlayer spacing increases to 0.376 nm and the binding energy decreases to 32 meV per atom (see Table 3).70 Optimum stacking AA0AA0... (i.e. hexagons on top of each other with Ga above N) of bilayer, trilayer, and multilayer h-GaN aggregates also exhibits large interlayer interaction energies of 142 meV per atom, 200 meV per atom, and 327 meV per atom, respectively.5 Surprisingly, the authors argue that interlayer interaction is dominated by van der Waals forces despite the Fig. 3 Monolayer thickness values approximated by the vdW diameter, Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. the sum of vdW diameter and buckling, as well as measured and calculated small interlayer separations of 0.252 nm and 0.249 nm for bi- and interlayer spacing of group III–V compounds. tri-layers, respectively. Only in multilayers with 0.244 nm separation a slight chemical contribution from anion–cation interaction is proposed.5 In a study based on vdW-DFT calculations a similar interlayer distance of 0.333 nm and interlayer interaction energies interlayer distance of 0.247 nm but 290 meV per atom for the of 26 to 86 meV per atom from bilayer to multilayer.9,10,68 An interaction energy of AA0 stacked bilayers has been obtained.71 estimation of the thickness by the vdW diameter using the sum of DFT-PBE calculations confirmed both the small interlayer van der Waals radii of the boron atom (0.192 nm) and nitrogen separation of 0.247 nm and the large interlayer binding energy atom (0.155 nm) yields 0.347 nm.14 The small deviation suggests of 140 meV per atom.72 Conversely, these authors consider that the sum of vdW radii provides a useful description of the size the electrostatic interaction of Ga (N) and N (Ga) between the of planar binary compounds. layers as main contribution. Besides h-BN few-layer systems of the group-III nitrides Likewise, small interlayer separations of 0.267 nm, 0.266 nm, h-AlN and h-GaN have received most attention due to their and 0.256 nm for the bi-, tri-, and four-layer systems of h-InN semiconducting properties with a wide band gap. Contrary to have been found, owing to the substantial diﬀerence in electro-

the space group given in Table 3, the space group P63/mmc has negativity. This again confirms the strong effect of ionicity on been assumed for h-AlN.69 The stacking orders with the smallest the effective interlayer spacings of nitrides and consequently on interlayer interaction energy offer the most realistic estimate of their electronic properties.73 As described before, the sum of the the thickness of an isolated monolayer. Clearly, in the considered tabulated vdW radii was employed to estimate the quite similar compounds interlayer anion–cation interaction enlarges the vdW diameters of h-AlN (0.339 nm), h-GaN (0.340 nm), and inherent bonding by vdW forces. For AA0 stacking with the h-InN (0.348 nm), which are comparable with that of h-BN.14 largest interlayer coupling energy of highly polar h-AlN the bilayer Thus, the real monolayer thickness is substantially larger than distance is only 0.213 nm. This is due to the interlayer the interlayer spacing derived for few-layer systems of nitrides interaction energy of 125 meV per atom for AA0 stacking with strong ionic layer-by-layer interaction.

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On the search for 2D TIs a new family of thallium com- interlayer spacing and monolayer thickness found for the pounds h-TlA (A = N, P, As, and Sb) has been discovered.74 The group-III arsenides also reflects the strength of interlayer

geometric structures were calculated by first principles and bonding. The realization of p–p bonding by overlapping pz simulated by MD, yielding slightly buckled hexagonal structures, orbitals in planar sp2 geometry of BAs changes to buckled GaAs including the nitride h-TlN, as shown in Table 3. The resulting and InAs with dominant sp3-type bonding due to increasing

thickness values of dw + Dz = 0.38 nm (TlN), dw + Dz = 0.442 nm mixing of pz and s orbitals.

(TlP), dw + Dz = 0.455 nm (TlAs), and dw + Dz = 0.484 nm (TlSb) are The results collected for this group of III–V compounds consistent with those of the related compounds presented in undoubtedly reveal the reduction of interlayer spacing caused Table 3, with the exception that TlN is no longer planar. The by strong electrostatic interactions in planar few-layer III–V general picture, however, noticed for bilayers of this group agrees sheets. In the buckled systems, covalent interlayer interaction with the behavior of the other nitrides. The bilayer of TlN has the takes place by mixed hybridization and efficient stacking smallest interlayer spacing of 0.251 nm and strongest interlayer orders. It is important to note that these types of stronger interaction of 185 meV per atom followed by TlP (0.280 nm, interlayer interaction of monolayers likewise occur on surfaces. 91 meV per atom), TlAs (0.287 nm, 79 meV per atom), and TlSb (0.303 nm, 76 meV per atom).75 2.6 Thickness of group IV–VI monolayers Boron-phosphide (BP) has attracted recent interest owing to In analogy to the already discussed III–V compounds binary its flat honeycomb monolayer with direct band gap. For the group IV–VI compounds or group-IV monochalcogenides are the most stable configuration of bilayers with AB stacking and isoelectronic counterparts of elemental group-V semiconductors direct band gap the interlayer interaction energy is 134 meV such as phosphorene. Different to phosphorene the monolayers per atom and for the most unfavorable AA stacking it is exhibit SOC since inversion symmetry is missing. Some members 112 meV per atom.76 Accordingly, a larger interlayer spacing of this group of binary monolayers MX (M = C, Si, Ge, Sn and of 0.393 nm was found for AA stacking and only 0.339 nm for AB X = S, Se, Te) have found increasing attention, owing to their stacking approaching the vdW diameter (see Table 3). For tri- and high carrier mobility, low thermal conductivity, enhanced four-layer systems the interlayer interaction energy increases slightly piezoelectricity, chemical stability, and strong absorbance of to 152 (ABC) and 161 meV per atom (ABCA), respectively.76 visible light. Therefore, they are candidates for future applica- First-principles calculations have also been performed for tions in electronics, optoelectronics, and photovoltaics. The the most stable few-layer systems of planar BAs (AC), buckled MTe compounds with the vdW diameters of 0.376 nm (CTe), GaAs (AB), and buckled InAs (AB).77 These calculations delivered 0.416 nm (SiTe), 0.417 nm (GeTe), and 0.423 nm (SnTe) are not the following interlayer spacings of bilayers of 0.335 nm (BAs), presented in Table 4, however, they are included in Fig. 4.14 0.283 nm (GaAs), and 0.315 nm (InAs), which are partly smaller From the three free-standing geometrical configurations, than the trilayer spacings of 0.339 nm (BAs), 0.326 nm (GaAs), namely planar, buckled, and puckered the slightly deformed and 0.315 nm (InAs). The buckling eﬀects of 0.086 nm (GaAs) puckered orthorhombic configuration (antisymmetric a phase) and 0.088 nm (InAs), presented by these authors, are slightly normally is the most stable structure of these compounds with

Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. larger than those given in Table 3. For bilayer GaN the authors the exceptions of SiS and SiSe, which have nearly the same present an interlayer spacing of 0.252 nm consistent with the cohesive energy for the buckled configuration. For some puckered value listed in Table 3. monolayers the indirect band gap can be transformed into direct Inspection of Fig. 3 demonstrates agreement between the band gap because of the small diﬀerence between direct and experimental interlayer spacing, theoretical interlayer spacing, indirect band gap. Table 4 covers the space group, vdW diameter, and vdW diameter only for h-BN, whereas the few available pucker, thickness, theoretical method, interlayer spacings, and first-principles calculations of bilayer and few-layer spacings of interlayer binding for puckered compounds. Fig. 4 displays the the other nitrides deviate considerably from the mean vdW vdW diameter, corrugated monolayer thickness, and interlayer diameter. One reason for the short interlayer distances is the spacing of selected group IV–VI compounds. strong ionic interlayer interaction, as discussed for h-AlN and First-principles calculations of the bulk interlayer spacing of h-GaN.5,69,72 In such cases of strong electrostatic interaction of orthorhombic structures have been performed for GeS (0.541 nm), bilayers their separation is no longer suitable to assess the free- GeSe (0.566 nm), SnS (0.569 nm), and SnSe (0.591 nm).78 These standing monolayer thickness. In h-GaN bilayers, for example, values agree well with the thicknesses of GeS (0.56 nm), GeSe the N atoms are positioned exactly above Ga atoms in the most (0.54 nm), SnS (0.56 nm), and SnSe (0.57 nm) presented in a stable stacking configuration with 0.244 nm spacing.5 review.79 For multiple SnS layers the theoretical findings have been Only in very few cases, stacking configurations exist, where confirmed by HRTEM measurements, which acquired 0.56 nm, the interlayer spacing approaches the vdW diameter. This matching the DFT-based values.80 The theoretical interlayer situation is encountered for BP, where the spacing of the weak separations, estimated from the size of the orthorhombic primitive bonding configuration approaches the vdW-based thickness unit cell of the bulk material, are collected in Table 4, together with (see Fig. 3).75 The absence of charge transfer between the AA- the early XRD data on the orthorhombic crystals of GeS (0.524 nm), stacked layers produces a relatively weak interlayer interaction, GeSe (0.541 nm), SnS (0.560), and SnSe (0.525 nm).81 leading to the good agreement between interlayer distance and The computation of the nearest interlayer distance of the vdW-based monolayer thickness. The varying agreement between atoms of adjacent layers oﬀers an alternative approach to estimate

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Table 4 Geometry or space group, vdW diameter, pucker eﬀect, monolayer thickness, theoretical methods, interlayer spacing, and interlayer bonding of elemental and binary group IV–VI compounds

Geometry vdW diam. Near. dist. Pucker Thickness Method Spacing Inter. binding space group dw (nm) dn (nm) Dz (nm) dw or dn + Dz monolayer dsp (nm) (meV per atom) 85 14 CS Pmn21 0.350 ——— ——— 85 14 SiS Pmn21 0.390 ——— ——— 85 14 84 14,84 84 78 GeS Pmn21 0.391 — 0.256 0.647 DFT-PBE 0.541 — — 0.34682 0.25684 0.60282,84 DFT-PBE82 0.52481 — 85 14 84 14,84 84 78 SnS Pmn21 0.397 — 0.285 0.682 DFT-PBE 0.569 — — 0.36082 0.28584 0.64582,84 DFT-PBE82 0.56081 — 85 14 CSe Pmn21 0.360 ——— ——— 85 14 SiSe Pmn21 0.400 ——— ——— 85 14 84 14,84 84 78 83 GeSe Pmn21 0.401 — 0.262 0.663 DFT-PBE 0.566 37.5 — 0.35282 0.26284 0.61482,84 DFT-PBE82 0.54181 52.383 85 14 84 14,84 84 78 87 SnSe Pmn21 0.407 — 0.275 0.682 DFT-PBE 0.591 110 — 0.36382 0.27584 0.63882,84 DFT-PBE82 0.52581 14686

and SnSe (0.638 nm) are compared with the ideal interlayer spacings derived from the unit cell and measured interlayer spacings in Fig. 4. Owing to the extreme pucker effect, the layer thicknesses derived from the mean vdW diameters are far too small (see Fig. 4). As discussed before, the upper limit of the thickness of puckered monolayers can be predicted, by using the calculated pucker distances given above.84 A measurement of the monolayer thickness of SnS by cross-sectional TEM yielded B0.7 nm.85 Inspection of Fig. 4 shows that the pucker-vdW-diameter-based thickness is in reasonable agreement with the TEM experiment including the one estimated by the closest-layer distances. The bulk layer spacings measured by XRD are in general lower and in better agreement with the DFT calculations of the interlayer spacings, pointing to an enlargement of interlayer interaction with the number of layers. The interlayer binding energies presented above for GeSe Fig. 4 Monolayer thickness values approximated by the vdW diameter, suggest that vdW forces are responsible for the interaction the sum of vdW diameter or nearest layer distance and puckering effect, 83 Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. between puckered layers. However, a much stronger interlayer as well as measured and calculated interlayer spacing of group IV–VI compounds. interaction of 146 meV per atom has been calculated for layered bulk SnSe by taking into account different exchange– correlation functionals and vdW interaction in a DFT-based the layer thickness. Such calculations have been performed for study.85,86 These authors assume strong charge transfer between different stacking orders.82 For AD stacking, where the upper layers by lone pair electrons and expect difficulties for preparation and under layers are the mirror image of each other (mutual by exfoliation methods. Different from this conclusion other location of the same atoms), the minimal interlayer interaction authors using the DFT-PBE functionals concluded that SnSe leads to the following nearest layer distances GeS (0.346 nm), GeSe exhibits with 110meVperatomthelimitoftheinterlayer (0.352 nm), SnS (0.360 nm), and SnSe (0.363 nm).82 With the interaction energy of group IV–VI compounds and considered this exception of GeSe, showing slightly more stable AA stacking, the as still favorable for exfoliation.87 corresponding distances of the most stable AB stacking are roughly 20% lower. Independent calculations of the closest interlayer distance of adjacent atoms reported for bilayer GeSe provide a similar 3. Discussion of thickness values result of 0.33 nm for AD stacking and an interlayer binding energy of 37.5 meV per atom.83 For the most stable AA stacking of GeSe For completion, a brief discussion of recent progress made in the binding energy is 52.3 meV per atom. studies of group-III monolayers is necessary. Besides boron, the To find the effective layer thickness of corrugated mono- information on the group III elements is at an early stage. layers the pucker distance is needed. This has been provided by Boron is the lightest element that can form stable covalent first-principles calculations for the four compounds with the monolayers. Remarkably, the stability of borophene has been following values: GeS (0.256 nm), GeSe (0.262 nm), SnS predicted first by theory and then has been confirmed by (0.285 nm), and SnSe (0.275 nm).84 The resulting interlayer experiments. With its three valence electrons boron atoms form distances of GeS (0.602 nm), GeSe (0.614 nm), SnS (0.645 nm), delocalized three-center, two-electron bonds involving three

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boronatoms.Anexampleistheregular triangular borophene by the increasing influence of sp3 hybrids. Adding the buckling monolayer with buckled structure.88 This so-called striped boro- eﬀect to the vdW diameter yields a first estimate of thickness. phene has been observed on a stabilizing surface by scanning As can be seen in Fig. 5, the elemental group-IV and group-V tunneling microscopy (STM).89 The thickness of 0.478 nm, esti- monolayers with mixed sp2–sp3 hybridization obviously define mated by the vdW diameter of 0.384 nm and the buckling height an upper limit of thickness for buckled single-atom mono- of 0.094 nm, agrees well with the general behavior of other buckled layers. Only the thickness of plumbene is much smaller than compounds, as delineated in Fig. 5. Furthermore, by suitable expected. The heaviest compounds often deviate from normal combination of electron donors, provided by the three-center flat group behavior, which may be due to the influence of d and f triangular regions, and two-center hexagonal regions acting as electrons. The thicknesses of the corresponding binary buckled electron acceptors in the right proportion boron can form stable compounds are usually smaller. They appear between the line planar polymorphs consisting of hexagonal sites with holes and defined by the elemental buckled monolayers and the vdW triangular sites (variation of boron hexagons with and without a diameter line (see PN, AsN, SbN, and BiN). According to Fig. 5, boron atom at the center of the hexagon).90 The description of the thickness of buckled compounds varies in the wider range thickness by the vdW diameter is in reasonable agreement with between about 0.4 nm and 0.6 nm. that of other planar monolayers. With their extra bonding electron group-V elemental and Fig. 5 shows the development of the thickness of group-III their binary compounds form besides the buckled phase a up to group-VI elements and compounds from period number 2 puckered phase that often has a similar cohesive energy and to 6 of the periodic table. We distinguish between three groups, stability. Due to the stronger corrugation the thickest mono- planar monolayers with a thickness defined by the vdW dia- layers belong to the puckered compounds with a thickness of meter, buckled monolayers obtained by adding vdW diameter about 0.5–0.7 nm. Here the strongest contribution of sp3-type and buckling eﬀect, and puckered monolayers expressed by bonding occurs, controlling the thickness of both the a-phase vdW diameter plus pucker height. Note that for clarity this and b-phase of phosphorene, arsenene, antimonene, and bis- overview scheme encloses not all thickness values gathered in muthene. Note that the formal increase of the layer thickness Tables 1–4. from period to period due to the pucker effect is significantly Planar sp2 hybridization realized by atoms with four binding larger than the slowly growing vdW diameter, owing to an electrons (octet rule) and isoelectronic III–V compounds form increasing influence of sp3 hybrids in interlayer bonding and three plane s bonds and delocalized p–p bonds if the atoms are corrugation.

small enough to allow overlapping of pz orbitals. For group IV elements this is only carbon owing to the enormous increase of atom size already for silicon. Obviously, there is only one 4. Conclusions exception (TlN) from the rule that in binary III–V compounds the planar configuration is stable if one element belongs to the The comprehensive database of thickness values presented in second period. Note that the increase of thickness from the second this study includes single atomic monolayers of elemental and

Published on 03 December 2019. Downloaded 10/1/2021 11:12:59 PM. to the sixth period matches that of the vdW diameter, which grows binary group-IV and group-V compounds, as well as isoelectronic only slightly from 0.34 nm to about 0.40 nm (see Fig. 5). III–V and IV–VI mixed compounds. Available measurements of

If the atom size is too large for an eﬃcient overlap of the pz the spacing of multilayers are in good agreement with theory, orbitals of group-IV elements or in the case of five valence however, may deviate from the thickness of free-standing mono- electrons of group-V elements, the monolayer becomes buckled layers. Thickness measurements on free-standing monolayers are notoriously too large. A breakthrough has been achieved just recently by controlling the thickness of growing high quality a-Sb on a suitable substrate in a layer-by-layer fashion, by using STM.91 Here it is demonstrated for the first time, how the thickness of the quasi-monolayer of B0.78 nm decreases to the theoretical valueof0.61nmduringgrowthoffewlayers(B0.63 nm for the 6th layer). Since systematic experiments on mono- and few-layer systems are still missing, the discussion of related group elements of the periodic table is mainly based on DFT-PBE calculations. In the case of planar monolayers, we apply the simple model of a rectangular slab with constant thickness, taking tabulated vdW diameters as the basic thickness. For graphene and boronitrene all methods besides nanoindentation measurements are in good agreement, including the thickness obtained by electron density-based volume calculations. The values derived Fig. 5 The vdW diameter-based thicknesses of planar graphene-like monolayers, buckled monolayers described by adding vdW diameter and from interlayer spacing of bilayers and multilayers are often buckling height, and puckered monolayers defined by vdW diameter and smaller than the vdW diameters, owing to stronger inter- pucker eﬀect. layer interaction in comparison to graphene and boronitrene.

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This occurs for bulk interlayer interaction energies around 6 J. D. Bernal, The Structure of Graphite, Proc. R. Soc. London, 80 meV per atom. The DFT study of the electron density-based Ser. A, 1924, 106, 749–773. volume clearly supports the vdW-diameter-based geometrical slab 7 M. Patel, H.-S. Kim and J. Kim, Wafer-Scale Production of model for uniform nano-objects with quasi planar geometry. Vertical SnS Multilayers for High-Performing Photoelectric To estimate the thickness of corrugated monolayers the Devices, Nanoscale, 2017, 9, 15804–15812. buckling or pucker eﬀect has been added to the vdW diameter. 8 K. S. Novoselov, A. Mishchenko, A. Carvalho and This is a first approximation to this more complex layer geo- A. H. Castro Neto, 2D Materials and van der Waals Hetero- metry, because the resulting volume of the geometric slab model structures, Science, 2016, 353, aac9439. may be larger than the electron density-based volume of the 9 G. Graziano, J. Klimesˇ, F. 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