Unusual ultra-low-frequency fluctuations in freestanding graphene

Defects induce phase transition from dynamic to static rippling in graphene

Two-dimensional (2D) materials display nanoscale dynamic ripples that significantly impact their properties. Defects within the crystal lattice are the elementary building blocks to tailor the material's morphology. While some studies have explored the link between defective structures and rippling dynamics in 2D materials, a comprehensive understanding of this relationship has yet to be achieved. Here, we address this using machine learning-driven molecular dynamics simulations. Specifically, we find that above a critical concentration of defects, free-standing graphene sheets undergo a dynamic transition from freely propagating to static ripples. Our computational approach captures the dynamics with atomic resolution, and reveals that the transition is driven by elastic interactions between defects. The strength of these interactions is found to vary across defect types and we identify a unifying set of principles driving the dynamic-to-static transition in 2D materials. Our work not only rationalizes puzzling experimental results for defective 2D materials, but also paves the way to design two-dimensional devices with tailored rippling dynamics. These insights could lay the foundations for a class of disorder-based catalytic and interfacial materials.

Enhancing magnetic coupling in MN4-graphene via strain engineering

MN4-embedded graphene (MN4-G) layers, with transition metal elements M, are experimentally accessible two-dimensional (2D) materials and show great potential for stable nanoscale magnetization. In these materials, the exchange couplings between magnetic atoms are predominantly governed by Ruderman-Kittel-Kasuya-Yosida (RKKY) coupling, exhibiting an unusual prolonged decay of r-n, where r is the M-M separation distance, and 0.5 ≤ n ≤ 2. In this paper, we explore the effects of induced strain on the electronic and magnetic properties of MN4-G layers through ab initio density functional theory. We employ a specific method to apply strain by positioning atoms from one layer within the equilibrium structure of another layer, thereby inducing strain in the form of either tension or compression. The induced strain results in an approximate ±0.4% variation in the unit-cell area of the MN4-G lattice. Our findings reveal that while the exchange coupling mechanism remains unaffected, the strength, amplitude, and decay rate of the RKKY coupling are significantly influenced by the induced strain. Notably, the CoN4-G layer exhibits a remarkable increase in the strength and oscillation amplitude of the RKKY coupling, along with a reduced decay rate. Additionally, the electronic and magnetic properties of the CuN4-G layers remain unchanged under induced strain.

Understanding the Curvature Effect on the Structure and Bonding of MoCy Nanoparticles on Carbon Supports

The interaction between molybdenum carbide (MoCy) nanoparticles and both flat and curved graphene surfaces, serving as models for carbon nanotubes, was investigated by means of density functional theory. A variety of MoCy nanoparticles with different sizes and stoichiometries have been used to explore different adsorption sites and modes across models with different curvature degrees. On flat graphene, off-stoichiometric MoCy featuring more low-coordinated Mo atoms exhibits stronger interaction and increased electron transfers from the carbide to the carbon substrate. This preferentially occurs through support C and Mo atoms leading to the formation of additional Mo-C bonds. Notably, the MoCy adsorption strength increases on concave surfaces and decreases on convex surfaces, showing a strong linear correlation with the surface curvature. This curvature-dependent behavior alters the charge state of the nanoparticles, making them more/less positively charged in concave/convex regions. The present results demonstrate that the interaction strength can be effectively tuned by manipulating the carbide stoichiometry, the substrate curvature, and the local concave/convex environments, providing valuable guidelines for the rational design of MoCy/C-based catalysts.

Catalytic selectivity of nanorippled graphene

Experiments have shown that nanoscale ripples in a graphene membrane exhibit unexpectedly high catalytic activity with respect to hydrogen dissociation. Nonetheless, the catalytic selectivity of nanorippled graphene remains unknown, which is an equally important property for assessing a catalyst's potential and its fit-for-purpose applications. Herein, we examine the catalytic selectivity of nanorippled graphene using a model reaction of molecular hydrogen with another simple but double-bonded molecule, oxygen, and comparing the measurement results with those from splitting of hydrogen molecules. We show that although nanorippled graphene exhibits a high catalytic activity toward hydrogen dissociation, the activity for catalyzing the hydrogen-oxygen reaction is quite low, translating into a strong catalytic selectivity. The latter reaction involves the reduction of oxygen molecules by the dissociated hydrogen adatoms, which requires additional energy cost and practically determines the selectivity. In this sense, the well-established information about reactions in general of atomic hydrogen with many other species in the literature could potentially predict the selectivity of nanorippled graphene as a catalyst. Our work provides implications for the catalytic properties of nanorippled graphene, especially its selectivity. The results would be important for its extension to a wider range of reactions and for designer technologies involving hydrogen.

Water flow in graphene nanochannels driven by imposed thermal gradients: the role of flexural phonons

Accurate control of fluid transport in nanoscale structures is key to enable the design of foreseeable nanofluidic devices with applications in many fields such as chip cooling, energy conversion, drug delivery and medical diagnosis. Here, inspired by the experimental observation of intrinsic thermal ripples in graphene and by recent advances in the manipulation of 2D nanomaterials, we introduce a graphene-based thermal nanopump which produces controlled and continuous liquid flow in nanoslit channels. We investigate the performance of this thermal nanopump employing large scale molecular dynamics simulations. Upon systematically imposing thermal gradients, a net water flow towards the low-temperature zone is observed, achieving flow velocities up to 4 m s^-1. We observe that water flow rates increase monotonically due to larger ripple fluctuations on the graphene layers as higher thermal gradients are applied. Moreover, we find that the out-of-plane flexural phonons in graphene are responsible for flow generation wherein lower frequency phonon branches are activated with higher imposed thermal gradients. Furthermore, by modifying the wettability of the channel walls, an increase of 50% in the water flow rates is observed, showing that the efficiency of the proposed thermal pump can be enhanced by tuning the channel wall hydrophobicity. Our results indicate that thermal gradients can be employed to drive continuous water flow in graphene nanoslit channels with potential applications in nanofluidic devices.

Construction and physical properties of low-dimensional structures for nanoscale electronic devices

Over the past decades, construction of nanoscale electronic devices with novel functionalities based on low-dimensional structures, such as single molecules and two-dimensional (2D) materials, has been rapidly developed. To investigate their intrinsic properties for versatile functionalities of nanoscale electronic devices, it is crucial to precisely control the structures and understand the physical properties of low-dimensional structures at the single atomic level. In this review, we provide a comprehensive overview of the construction of nanoelectronic devices based on single molecules and 2D materials and the investigation of their physical properties. For single molecules, we focus on the construction of single-molecule devices, such as molecular motors and molecular switches, by precisely controlling their self-assembled structures on metal substrates and charge transport properties. For 2D materials, we emphasize their spin-related electrical transport properties for spintronic device applications and the role that interfaces among 2D semiconductors, contact electrodes, and dielectric substrates play in the electrical performance of electronic, optoelectronic, and memory devices. Finally, we discuss the future research direction in this field, where we can expect a scientific breakthrough.

Thermalized buckling of isotropically compressed thin sheets

The buckling of thin elastic sheets is a classic mechanical instability that occurs over a wide range of scales. In the extreme limit of atomically thin membranes like graphene, thermal fluctuations can dramatically modify such mechanical instabilities. We investigate here the delicate interplay of boundary conditions, nonlinear mechanics, and thermal fluctuations in controlling buckling of confined thin sheets under isotropic compression. We identify two inequivalent mechanical ensembles based on the boundaries at constant strain (isometric) or at constant stress (isotensional) conditions. Remarkably, in the isometric ensemble, boundary conditions induce a novel long-ranged nonlinear interaction between the local tilt of the surface at distant points. This interaction combined with a spontaneously generated thermal tension leads to a renormalization group description of two distinct universality classes for thermalized buckling, realizing a mechanical variant of Fisher-renormalized critical exponents. We formulate a complete scaling theory of buckling as an unusual phase transition with a size-dependent critical point, and we discuss experimental ramifications for the mechanical manipulation of ultrathin nanomaterials.

Oscillations of the Spacing between van Hove Singularities Induced by sub-Ångstrom Fluctuations of Interlayer Spacing in Graphene Superlattices

Physical properties of two-dimensional van der Waals (vdWs) structures depend sensitively on both stacking orders and interlayer interactions. Yet, in most cases studied to date, the interlayer interaction is considered to be a "static" property of the vdWs structures. Here we demonstrate that applying a scanning tunneling microscopy (STM) tip pulse on twisted bilayer graphene (TBG) can induce sub-Ångstrom fluctuations of the interlayer separation in the TBG, which are equivalent to dynamic vertical external pressure of about 10 GPa on the TBG. The sub-Ångstrom fluctuations of the interlayer separation result in large oscillations of the energy separations between two van Hove singularities (VHSs) in the TBG. The period of the oscillations of the VHSs spacing is extremely long, about 500-1000 sec, attributing to tip-induced local stress in the atomic-thick TBG. Our result provides an efficient method to tune and measure the physical properties of the vdWs structures dynamically.

Anomalous Flexural Elasticities of Graphene Membranes Unveiled by Manipulating Topology

Mechanical behavior of atomically thin membranes is governed by bending rigidity and the Gaussian modulus. However, owing to methodological drawbacks, these two parameters have not been investigated sufficiently. We employed atomic force microscopy to demonstrate that the bending rigidity can be extracted from a quadratic relationship of adhesion energy with monolayer curvatures of rolled and unrolled graphene. The tip-induced topological defects revealed the Gaussian modulus; to the best of our knowledge, this is the first study on these parameters. Our study may hold great significance because existing investigations have been performed only on flat graphene. The configurational (strain) energy was evaluated via changes in the surface geometry, with subatomic resolution, by three-dimensional analyses of attractive interatomic forces. The mechanical parameters, evaluated at the hollow sites of the honeycomb lattice, were consistent with the isotropic elastic attributes. The remarkably large negative Gaussian modulus, observed when a single carbon atom was located at the center of the tip-induced bump, revealed attractive interactions between the topological defects and geometric potentials of the Gaussian curvature. Our approach will aid in developing two-dimensional materials and understanding cell biology.

Locomotion of the C60-based nanomachines on graphene surfaces

We provide a comprehensive computational characterization of surface motion of two types of nanomachines with four C₆₀ "wheels": a flexible chassis Nanocar and a rigid chassis Nanotruck. We study the nanocars' lateral and rotational diffusion as well as the wheels' rolling motion on two kinds of graphene substrates-flexible single-layer graphene which may form surface ripples and an ideally flat graphene monolayer. We find that the graphene surface ripples facilitate the translational diffusion of Nanocar and Nanotruck, but have little effect on their surface rotation or the rolling of their wheels. The latter two types of motion are strongly affected by the structure of the nanomachines instead. Surface diffusion of both nanomachines occurs preferentially via a sliding mechanism whereas the rolling of the "wheels" contributes little. The axial rotation of all "wheels" is uncorrelated.

Fluctuation-induced current from freestanding graphene

At room temperature, micron-sized sheets of freestanding graphene are in constant motion, even in the presence of an applied bias voltage. We quantify the out-of-plane movement by collecting the displacement current using a nearby small-area metal electrode and present an Ito-Langevin model for the motion coupled to a circuit containing diodes. Numerical simulations show that the system reaches thermal equilibrium and the average rates of heat and work provided by stochastic thermodynamics tend quickly to zero. However, there is power dissipated by the load resistor, and its time average is exactly equal to the power supplied by the thermal bath. The exact power formula is similar to Nyquist's noise power formula, except that the rate of change of diode resistance significantly boosts the output power, and the movement of the graphene shifts the power spectrum to lower frequencies. We have calculated the equilibrium average of the power by asymptotic and numerical methods. Excellent agreement is found between experiment and theory.

Wrinkle-induced highly conductive channels in graphene on SiO2/Si substrates

A graphene wrinkle is a quasi-one-dimensional structure and can alter the intrinsic physical and chemical activity, modify the band structure and introduce transport anisotropy in graphene thin films. However, the quasi-one-dimensional electrical transport contribution of wrinkles to the whole graphene films compared to that of the two-dimensional flat graphene nearby has still been elusive. Here, we report measurements of relatively high conductivity in micrometer-wide graphene wrinkles on SiO2/Si substrates using an ultrahigh vacuum (UHV) four-probe scanning tunneling microscope. Combining the experimental results with resistor network simulations, the wrinkle conductivity at the charge neutrality point shows a much higher conductivity up to ∼33.6 times compared to that of the flat monolayer region. The high conductivity can be attributed not only to the wrinkled multilayer structure but also to the large strain gradients located mainly in the boundary area. This method can also be extended to evaluate the electrical-transport properties of wrinkled structures in other two-dimensional materials.

Phonon-Induced Ratchet Motion of a Water Nanodroplet on a Supported Black Phosphorene

Phonons are not supposed to carry any physical momentum as lattice vibrational modes; thus, it is believed no mass transport could be induced by phonons. In this Letter, we show that a ratchet motion of a water nanodroplet could be induced on a two-dimensional puckered lattice like black phosphorene (BP) by exciting its flexural phonons through a moving substrate. The water nanodroplet exhibits a forward motion along the armchair or a backward motion along the zigzag directions on a BP lattice that is supported on a substrate possessing a relative armchair or zigzag forward motion with BP. Through the analysis of the structure and vibrational density states of BP, it is found that in-plane lattice displacement asymmetry and the in-plane vibration asymmetry are induced by the excited flexural phonons, which determine the water nanodroplet motion as an anisotropic Brownian motor. Simulations of the nanodroplet motion as functions of the substrate relative motion speed and direction and also the substrate coupling strength with BP are performed. Results of the nanodroplet ratchet motion exhibit good agreement with the theoretical predications from calculating the Brownian motor asymmetry. Our findings reveal a promising mass transport strategy and a further understanding of phonon-related interactions in crystalline solids.

Critical Stable Length in Wrinkles of Two-Dimensional Materials

The emergent two-dimensional (2D) materials are atomically thin and ultraflexible, promising for a variety of miniaturized, high-performance, and flexible devices in applications. On one hand, the ultrahigh flexibility causes problems: the prevalent wrinkles in 2D materials may undermine the ideal properties and create barriers in fabrication, processing, and quality control of materials. On the other hand, in some cases the wrinkles are used for the architecturing of surface texture and the modulation of physical/chemical properties. Therefore, a thorough understanding of the mechanism and stability of wrinkles is highly needed. Herein, we report a critical length for stabilizing the wrinkles in 2D materials, observed in the wrinkling and wrinkle elimination processes upon thermal annealing as well as by our in situ TEM manipulations on individual wrinkles, which directly capture the evolving wrinkles with variable lengths. The experiments, mechanical modeling, and self-consistent charge density functional tight binding (SCC-DFTB) simulations reveal that a minimum critical length is required for stabilizing the wrinkles in 2D materials. Wrinkles with lengths below a critical value are unstable and removable by thermal annealing, while wrinkles with lengths above a critical value are self-stabilized by van der Waals interactions. It additionally confirms the pronounced frictional effects in wrinkles with lengths above critical value during dynamical movement or sliding.

Wrinkling of two-dimensional materials: methods, properties and applications

Recently, two-dimensional (2D) materials, including graphene, its derivatives, metal films, MXenes and transition metal dichalcogenides (TMDs), have been widely studied because of their tunable electronic structures and special electrical and optical properties. However, during the fabrication of these 2D materials with atomic thickness, formation of wrinkles or folds is unavoidable to enable their stable existence. Meaningfully, it is found that wrinkled structures simultaneously impose positive changes on the 2D materials. Specifically, the architecture of wrinkled structures in 2D materials additionally induces excellent properties, which are of great importance for their practical applications. In this review, we provide an overview of categories of 2D materials, which contains formation and fabrication methods of wrinkled patterns and relevant mechanisms, as well as the induced mechanical, electrical, thermal and optical properties. Furthermore, these properties are modifiable by controlling the surface topography or even by dynamically stretching the 2D materials. Wrinkling offers a platform for 2D materials to be applied in some promising fields such as field emitters, energy containers and suppliers, field effect transistors, hydrophobic surfaces, sensors for flexible electronics and artificial intelligence. Finally, the opportunities and challenges of wrinkled 2D materials in the near future are discussed.

Postprocessing Algorithm for Driving Conventional Scanning Tunneling Microscope at Fast Scan Rates

We present an image postprocessing framework for Scanning Tunneling Microscope (STM) to reduce the strong spurious oscillations and scan line noise at fast scan rates and preserve the features, allowing an order of magnitude increase in the scan rate without upgrading the hardware. The proposed method consists of two steps for large scale images and four steps for atomic scale images. For large scale images, we first apply for each line an image registration method to align the forward and backward scans of the same line. In the second step we apply a "rubber band" model which is solved by a novel Constrained Adaptive and Iterative Filtering Algorithm (CIAFA). The numerical results on measurement from copper(111) surface indicate the processed images are comparable in accuracy to data obtained with a slow scan rate, but are free of the scan drift error commonly seen in slow scan data. For atomic scale images, an additional first step to remove line-by-line strong background fluctuations and a fourth step of replacing the postprocessed image by its ranking map as the final atomic resolution image are required. The resulting image restores the lattice image that is nearly undetectable in the original fast scan data.

Three-dimensional surface topography of graphene by divergent beam electron diffraction

There are only a handful of scanning techniques that can provide surface topography at nanometre resolution. At the same time, there are no methods that are capable of non-invasive imaging of the three-dimensional surface topography of a thin free-standing crystalline material. Here we propose a new technique—the divergent beam electron diffraction (DBED) and show that it can directly image the inhomogeneity in the atomic positions in a crystal. Such inhomogeneities are directly transformed into the intensity contrast in the first-order diffraction spots of DBED patterns and the intensity contrast linearly depends on the wavelength of the employed probing electrons. Three-dimensional displacement of atoms as small as 1 angstrom can be detected when imaged with low-energy electrons (50–250 eV). The main advantage of DBED is that it allows visualization of the three-dimensional surface topography and strain distribution at the nanometre scale in non-scanning mode, from a single shot diffraction experiment.

Graphene, and other 2D materials, do not exist as strictly planar sheets but instead have topographic ripples on the sub-nanometre scale. Here, Latychevskaia et al. present a method to non-invasively image ripples in 2D materials with a single-shot, wide-area, electron diffraction measurement.

Anomalous Dynamical Behavior of Freestanding Graphene Membranes

We report subnanometer, high-bandwidth measurements of the out-of-plane (vertical) motion of atoms in freestanding graphene using scanning tunneling microscopy. By tracking the vertical position over a long time period, a 1000-fold increase in the ability to measure space-time dynamics of atomically thin membranes is achieved over the current state-of-the-art imaging technologies. We observe that the vertical motion of a graphene membrane exhibits rare long-scale excursions characterized by both anomalous mean-squared displacements and Cauchy-Lorentz power law jump distributions.

Visualising the strain distribution in suspended two-dimensional materials under local deformation

We demonstrate the use of combined simultaneous atomic force microscopy (AFM) and laterally resolved Raman spectroscopy to study the strain distribution around highly localised deformations in suspended two-dimensional materials. Using the AFM tip as a nanoindentation probe, we induce localised strain in suspended few-layer graphene, which we adopt as a two-dimensional membrane model system. Concurrently, we visualise the strain distribution under and around the AFM tip in situ using hyperspectral Raman mapping via the strain-dependent frequency shifts of the few-layer graphene's G and 2D Raman bands. Thereby we show how the contact of the nm-sized scanning probe tip results in a two-dimensional strain field with μm dimensions in the suspended membrane. Our combined AFM/Raman approach thus adds to the critically required instrumental toolbox towards nanoscale strain engineering of two-dimensional materials.

The shallow shell approach to Pogorelov's problem and the breakdown of 'mirror buckling'

We present a detailed asymptotic analysis of the point indentation of an unpressurized, spherical elastic shell. Previous analyses of this classic problem have assumed that for sufficiently large indentation depths, such a shell deforms by 'mirror buckling'-a portion of the shell inverts to become a spherical cap with equal but opposite curvature to the undeformed shell. The energy of deformation is then localized in a ridge in which the deformed and undeformed portions of the shell join together, commonly referred to as Pogorelov's ridge. Rather than using an energy formulation, we revisit this problem from the point of view of the shallow shell equations and perform an asymptotic analysis that exploits the largeness of the indentation depth. This reveals first that the stress profile associated with mirror buckling is singular as the indenter is approached. This consequence of point indentation means that mirror buckling must be modified to incorporate the shell's bending stiffness close to the indenter and gives rise to an intricate asymptotic structure with seven different spatial regions. This is in contrast with the three regions (mirror-buckled, ridge and undeformed) that are usually assumed and yields new insight into the large compressive hoop stress that ultimately causes the secondary buckling of the shell.

Fast diffusion of water nanodroplets on graphene

Diffusion across surfaces generally involves motion on a vibrating but otherwise stationary substrate. Here, using molecular dynamics, we show that a layered material such as graphene opens up a new mechanism for surface diffusion whereby adsorbates are carried by propagating ripples in a motion similar to surfing. For water nanodroplets, we demonstrate that the mechanism leads to exceedingly fast diffusion that is 2-3 orders of magnitude faster than the self-diffusion of water molecules in liquid water. We also reveal the underlying principles that regulate this new mechanism for diffusion and show how it also applies to adsorbates other than water, thus opening up the prospect of achieving fast and controllable motion of adsorbates across material surfaces more generally.

Liquid-assisted tip manipulation: fabrication of twisted bilayer graphene superlattices on HOPG

We use the tip of a scanning tunneling microscope (STM) to manipulate single weakly bound nanometer-sized sheets on a highly oriented pyrolytic graphite (HOPG) surface through artificially increasing the tip and sample interaction by pretreatment of the surface using a liquid thiol molecule. By this means it is possible to tear apart a graphite sheet against a step and fold this part onto the HOPG surface and thus generate graphene superlattices with hexagonal symmetry. The tip and sample surface interactions, including the van der Waals force, electrostatic force and capillary attraction force originating from the Laplace pressure due to the formation of a highly curved fluid meniscus connecting the tip and sample, are discussed quantitatively to understand the formation mechanism of a graphene superlattice induced by the STM tip. The capillary force plays a key role in manipulating the graphite surface sheet under humid conditions. Our approach provides a simple and feasible route to prepare controllable superlattices and graphene nanoribbons and also to better understand the process of generation of a graphene superlattice on the surface of HOPG with the tip.

Nematic Liquid Crystal on a Two Dimensional Hexagonal Lattice and its Application

We have studied the alignment of liquid crystal adsorbed onto graphene and hexagonal boron nitride by using a polarized optical microscope. From the experimental data, it was found that there were 6 different alignment orientations of the liquid crystal molecules on a single crystal substrate. This result has never been reported and is quite different from other previous results. As the hexagonal lattice has a threefold rotational symmetry, three different alignment orientations were expected, but our result seems counter-intuitive. We explain this result considering the bending of the tail of the liquid crystal molecules. Using this anchoring effect with six accurate discrete angles, a novel non-volatile display can be developed with micron-scale pixel size, due to the molecular level accuracy of the alignment.

The rare two-dimensional materials with Dirac cones

Inspired by the great development of graphene, more and more research has been conducted to seek new two-dimensional (2D) materials with Dirac cones. Although 2D Dirac materials possess many novel properties and physics, they are rare compared with the numerous 2D materials. To provide explanation for the rarity of 2D Dirac materials as well as clues in searching for new Dirac systems, here we review the recent theoretical aspects of various 2D Dirac materials, including graphene, silicene, germanene, graphynes, several boron and carbon sheets, transition-metal oxides (VO2)n/(TiO2)m and (CrO2)n/(TiO2)m, organic and organometallic crystals, so-MoS2, and artificial lattices (electron gases and ultracold atoms). Their structural and electronic properties are summarized. We also investigate how Dirac points emerge, move, and merge in these systems. The von Neumann–Wigner theorem is used to explain the scarcity of Dirac cones in 2D systems, which leads to rigorous requirements on the symmetry, parameters, Fermi level, and band overlap of materials to achieve Dirac cones. Connections between existence of Dirac cones and the structural features are also discussed.

Effects of thermal rippling on the frictional properties of free-standing graphene

A unique case of dynamically corrugated surfaces is presented for the first time, attributed to the effect of significant thermally excited flexural waves in atomically thin layers on kinetic friction.

Molecular dynamics simulation of thermal ripples in graphene with bond-order-informed harmonic constraints

We describe the results of atomistic molecular dynamics simulations of thermal rippling in graphene with the use of a generic harmonic constraint model. The distance and angular constraint constants are calculated directly from the second-generation bond-order interatomic potential that describes carbon binding in graphene. We quantify the thermal rippling process in detail by calculating the overall rippling averages, the normal-normal correlation distributions and the height distributions. In addition, we consider the effect of a dihedral angular constraint, as well as the effect of sample size on the simulated rippling averages. The dynamic corrugation morphologies of simulated graphene samples obtained with the harmonic constraint model at various temperatures are, overall, consistent with those obtained with the bond-order potential and are in qualitative accord with previously reported findings. Given the wide availability of the harmonic constraint model in various molecular mechanics implementations, along with its high computational efficiency, our results indicate a possible use for the presented model in multicomponent dynamic simulations, including atomically thin layers.