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Thickness of Elemental and Binary Single Atomic Monolayers Cite This: Nanoscale Horiz., 2020, 5,385 Peter Hess Nanoscale Horizons View Article Online FOCUS View Journal | View Issue 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 mÀ2 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 mÀ1 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 This journal is © The Royal Society of Chemistry 2020 Nanoscale Horiz., 2020, 5,385--399 | 385 View Article Online Focus Nanoscale Horizons 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
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