Submonolayer Growth Research Articles

Island density scaling in reversible submonolayer growth with attachment barriers.

Island nucleation and growth play an important role in thin-film growth. One quantity of particular interest is the exponent χ, which describes the dependence of the saturation island density N_{sat}∼(D_{h}/F)^{-χ} on the ratio D_{h}/F of the monomer hopping rate D_{h} to the deposition rate F. While standard rate equation(RE) theory predicts that χ=i/(i+2) (where i is the critical island size), more recently it has been predicted that in the presence of a strong barrier to the attachment of monomers to islands, a significantly larger value χ=2i/(i+3) may be observed. While this prediction has recently been tested using kinetic Monte Carlo simulations for the case of irreversible growth corresponding to i=1, it has not been tested for the case of reversible island growth corresponding to i>1. Here we present a mean-field self-consistent RE method which we have used to study the dependence of the effective value of χ on D_{h}/F and barrier-strength for i=1,2,3, and 6. Both the no-nucleation-barrier case in which there exists a barrier for monomers to attach to islands larger than the critical island size (but not to smaller islands) and the nucleation-barrier case in which there is a barrier for monomers to attach to islands of all sizes are studied. In all cases, we find that the existence of attachment barriers significantly increases the effective value of χ for a given barrier strength. In addition, for i=1 we find good agreement between our extrapolated asymptotic value of χ and the theoretical strong-barrier prediction both with and without a nucleation barrier. In contrast, for i>1 the value of χ is significantly larger in the presence of a nucleation barrier than in its absence. In particular, while an asymptotic analysis of our results for i>1 also leads to excellent agreement with the strong barrier prediction in the presence of a nucleation barrier, in the absence of a nucleation barrier the asymptotic values are significantly lower.

Anomalous intralayer growth of epitaxial Si on Ag(111)

The epitaxial growth of silicene has been the subject of many investigations, controversies and non-classical results. In particular, the initially promising deposition of Si on a metallic substrate such as Ag(111) has revealed unexpected growth modes where Si is inserted at the beginning of the growth in the first atomic plane of the substrate. In order to rationalize this anomalous growth mode, we develop an out-of-equilibrium description of a lattice-based epitaxial growth model, which growth dynamics are analyzed via kinetic Monte-Carlo simulations. This model incorporates several effects revealed by the experiments such as the intermixing between Si and Ag, and surface effects. It is parametrized thanks to an approach in which we show that relatively precise estimates of energy barriers can be deduced by meticulous analysis of atomic microscopy images. This analysis enables us to reproduce both qualitatively and quantitatively the anomalous growth patterns of Si on Ag(111). We show that the dynamics results in two modes, a classical sub-monolayer growth mode at low temperature, and an inserted growth mode at higher temperatures, where the deposited Si atoms insert in the first layer of the substrate by replacing Ag atoms. Furthermore, we reproduce the non-standard Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} shape of the experimental plot of the island density as a function of temperature, with a shift in island density variation during the transition between the submonoloyer and inserted growth modes.

Spatially modulated interface states in a two-dimensional potential: Single-layer RbI on Ag(111)

Alkali halides are known to exhibit interface electronic states (IES) when deposited on metal surfaces with ultra-thin coverage. Here, we examine the IES formed by sub-monolayer RbI growth on Ag(111), which exhibits spatial variations in electronic structure in surprising contrast to the results previously obtained for other alkali halides. We find that this spatially dependent behavior can be qualitatively modeled by using a two-dimensional cosine potential commensurate with the moiré superstructure, where the IES is constructed from the well-known analytical solutions to the Mathieu equation. Our results indicate this potential is more corrugated than for similar potentials reported for other alkali halides, a result of substrate–adlayer charge transfer interactions that are stronger for RbI. This two-dimensional effective potential leads to anisotropy in the effective electron mass, in surprising contrast to previous results for other alkali halides, which report a single isotropic mass.

An ab initio approach to anisotropic alloying into the Si(001) surface.

By employing density functional theory calculations, we explore the initial stage of competitive alloying of co-deposited silver and indium atoms into a silicon surface. In particular, we identify respective adsorption positions and activation barriers governing their diffusion on a dimer-reconstructed silicon surface. Furthermore, we develop a growth model that appropriately describes diffusion mechanisms and silicon morphology with the account of silicon dimerization and the presence of C-type defects. Based on the surface kinetic Monte Carlo simulations, we examine the dynamics of bimetallic adsorption and elaborate on the temperature effects on the submonolayer growth of an Ag-In alloy. A close inspection of adatom migration clearly indicates effective nucleation of Ag and In atoms, followed by the formation of orthogonal atomic chains. We show that the epitaxial bimetallic growth might potentially lead to exotic ordering of adatoms in the form of anisotropic two-dimensional lattices via orthogonally oriented single-metal rows. We argue that this scenario becomes favorable provided above room temperature, while our numerical results are shown to be in agreement with the experimental findings.

The Modeling of growth process on the surface of crystal

The article is devoted to modelling the growth of thin films on the surfaces of crystals having a similar crystal structure with a small parameter of mismatch of the lattice of substances from which the film and the crystal substrate are formed. A review of modelling methods based on both analytical expressions and computational methods is made. A number of methods for modelling the most typical processes: surface formation in the form of pyramidal formations (so-called needle crystals), two-dimensional with initial islands of growth and three-dimensional uneven growth processes. To model the process of growth of needle crystals, it is proposed to use a method based on Gaussian statistics of surface height increments. The model of three-dimensional growth of the crystal surface, which uses the iterative algorithm of Foss, and which makes it possible to investigate the processes of stepped, uneven growth of crystals, is also considered. In contrast to stepwise growth, a model of submonolayer growth of a film based on the Monte Carlo method is considered. For submonolayer growth of the film, pseudo-random sequences are used, which simulate the initial arrangement of the nuclei of the nucleus of the next layer on the crystal surface. The computational characteristics of this method are determined, namely the dependence of the number of iterations on the initial surface filling coefficient.

Effects of short-range attachment barriers on submonolayer growth

A variety of effects can lead to short-range attachment barriers in thin-film growth. While it has been predicted that the exponent χ which describes the dependence of the island density N on deposition rate F in the submonolayer regime (where N∼Fχ) crosses over from the diffusion-limited value i/(i+2) (where i is the critical island size) in the absence of an attachment barrier to the attachment-limited value 2i/(i+3) for a strong attachment barrier, this prediction has not been confirmed. Furthermore, the dependence of the effective value of χ on the barrier strength and ratio R=D/F (where D is the monomer hopping rate) has not been studied. Here we consider the effects of attachment barriers in irreversible growth (i=1) for both the case of a barrier to island nucleation and attachment as well as that of an island attachment barrier but no nucleation barrier. Our results indicate that in both cases the effective value of χ increases with increasing R to a maximum value χmax(Rmax) which depends on barrier strength before decreasing very slowly toward the diffusion-limited value. In addition, both χmax and Rmax increase as the barrier strength increases. The results of self-consistent rate-equation calculations are also presented and good agreement is found with our simulations. We also present a scaling analysis for the dependence of Rmax on the barrier strength for arbitrary critical island-size i and good agreement is found with our simulation results for the case in which there is both a nucleation barrier and a barrier to island attachment.

Machine learning predictions of surface migration barriers in nucleation and non-equilibrium growth

Machine learning is playing an increasing role in the discovery of new materials and may also facilitate the search for optimum growth conditions for crystals and thin films. Here, we perform kinetic Monte-Carlo simulations of sub-monolayer growth. We consider a generic homoepitaxial growth scenario that covers a wide range of conditions with different diffusion barriers (0.4–0.55 eV) and lateral binding energies (0.1–0.4 eV). These simulations are used as a training data set for a convolutional neural network that can predict diffusion barriers and binding energies. Specifically, a single Monte-Carlo image of the morphology is sufficient to determine the energy barriers with an accuracy of approximately 10 meV and the neural network is tolerant to images with noise and lower than atomic-scale resolution. We believe this new machine learning method will be useful for fundamental studies of growth kinetics and growth optimization through better knowledge of microscopic parameters.

Kinetically driven island morphology in growth on strained Cu(100).

We study the effects of strain on the monomer and dimer diffusion mechanisms and island morphology during the growth of Cu on a biaxially strained Cu(100) substrate. We find an approximately linear dependence of the activation barriers on strain. In particular, while hopping is favored for compressive and/or small (<2%) tensile strain, for greater than 2% tensile strain, the exchange mechanism is favored. We then present the results of temperature-accelerated dynamics simulations of submonolayer growth at 200K. For the case of 2% compressive strain we find that, as in previous kinetic Monte Carlo simulations of Cu/Ni(100) growth, the competition between island growth and multi-atom relaxation ("pop-out") events leads to an island morphology with a mixture of open and closed steps. At slightly higher coverage, island coalescence then leads to elongated islands. However, annealing leads to a significant decrease in the number of open steps. In contrast, for the case of 8% tensile strain, only one large strongly anisotropic island is formed. Surprisingly, we find that despite the large strain, the island anisotropy is not due to energetics but is instead due to anisotropic attachment barriers that favor the exchange-mediated attachment of monomers to corners over close-packed step-edges. An explanation for the asymmetry in attachment barriers is provided. Our results provide a new general kinetic mechanism for the formation of anisotropic islands in the presence of isotropic diffusion and tensile strain.

Change in Topography of InAs Submonolayer Nanostructures at the 2D to 3D Transition

A strong topographical evidence of the 2D to 3D transition in InAs/GaAs nanostructures assembled by the submonolayer (SML) growth mode using molecular beam epitaxy (MBE) is reported. Whereas the 2D to 3D transition in InAs nanostructures assembled by the Stranski–Krastanov (SK) growth mode is well established with a critical thickness of around 1.7 monolayer (ML), the analogous phenomenon in SML‐grown nanostructures is not yet well understood. Herein, atomic force microscopy (AFM) is utilized to investigate the topographical changes associated with the 2D to 3D transition in SML nanostructures, and it is unambiguously shown that for a given number of stacks and GaAs matrix (spacer) layer thickness, there exists a critical thickness in the amount of deposited InAs per cycle, beyond which the nucleation of 3D nanostructures abruptly begins. It is also shown that the critical thickness decreases with increasing number of stacks as well as with decreasing GaAs matrix layer thickness. Moreover, the present work shows that InAs/GaAs SML nanostructures exist in two distinct forms: 2D wetting layer (WL)‐like islands and 3D quantum dot (QD)‐like structures.

Mechanism of stacking fault formation in metal(100) heteroepitaxial growth

Strain in metal(100) heteroepitaxial thin-film growth can lead to a variety of effects such as the formation of stacking faults (SFs). To gain a better understanding we have carried out temperature-accelerated dynamics (TAD) simulations of the submonolayer growth of Cu islands on a biaxially strained Cu substrate at 200 K. In the case of 4% compressive strain we find that SF formation occurs with a morphology very similar to the structures found experimentally in Cu/Ni(100) growth. We also find that islands play a key role by lowering the barrier for vacancy formation. In particular, once two substrate vacancies are formed and diffuse to form the appropriate configuration this leads to the formation of a SF in both the substrate and island. While the activation barrier for SF formation is very high, due to the presence of a large number of low-frequency vibrational modes, the saddle-point entropy is large while the corresponding Vineyard prefactor is more than 10 orders of magnitude larger than is typical of atomic processes in fcc metals. Similarly, an analysis of the entropy of the SF state indicates that the reverse prefactor is much smaller, while the free energy of this state is lower than that of the initial state.

Dominant contributions to the apparent activation energy in two-dimensional submonolayer growth: comparison between Cu/Ni(111) and Ni/Cu(111)

For surface-mediated processes in general, such as epitaxial growth and heterogeneous catalysis, a constant slope in the Arrhenius diagram of the rate of interest, R, against inverse temperature, log R vs 1/T, is traditionally interpreted as the existence of a bottleneck elementary reaction (or rate-determining step), whereby the constant slope (or apparent activation energy, ) reflects the value of the energy barrier for that elementary reaction. In this study, we express as a weighted average, where every term contains the traditional energy barrier for the corresponding elementary reaction plus an additional configurational term, while identifying each weight as the probability of executing the corresponding elementary reaction. Accordingly, the change in the leading (most probable) elementary reaction with the experimental conditions (e.g. temperature) is automatically captured and it is shown that a constant value of is possible even if control shifts from one elementary reaction to another. To aid the presentation, we consider kinetic Monte Carlo simulations of submonolayer growth of Cu on Ni(111) and Ni on Cu(111) at constant deposition flux, including a large variety of single-atom, multi-atom and complete-island diffusion events. In addition to analysing the dominant contributions to the diffusion constant of the complete adparticle system (or tracer diffusivity) and its apparent activation energy as a function of both coverage and temperature for the two heteroepitaxial systems, their surface morphologies and island densities are also compared.

Statistics of adatom diffusion in a model of thin film growth.

We study the statistics of the number of executed hops of adatoms at the surface of films grown with the Clarke-Vvedensky (CV) model in simple cubic lattices. The distributions of this number N are determined in films with average thicknesses close to 50 and 100 monolayers for a broad range of values of the diffusion-to-deposition ratio R and of the probability ε that lowers the diffusion coefficient for each lateral neighbor. The mobility of subsurface atoms and the energy barriers for crossing step edges are neglected. Simulations show that the adatoms execute uncorrelated diffusion during the time in which they move on the film surface. In a low temperature regime, typically with Rε≲1, the attachment to lateral neighbors is almost irreversible, the average number of hops scales as 〈N〉∼R^{0.38±0.01}, and the distribution of that number decays approximately as exp[-(N/〈N〉)^{0.80±0.07}]. Similar decay is observed in simulations of random walks in a plane with randomly distributed absorbing traps and the estimated relation between 〈N〉 and the density of terrace steps is similar to that observed in the trapping problem, which provides a conceptual explanation of that regime. As the temperature increases, 〈N〉 crosses over to another regime when Rε^{3.0±0.3}∼1, which indicates high mobility of all adatoms at terrace borders. The distributions P(N) change to simple exponential decays, due to the constant probability for an adatom to become immobile after being covered by a new deposited layer. At higher temperatures, the surfaces become very smooth and 〈N〉∼Rε^{1.85±0.15}, which is explained by an analogy with submonolayer growth. Thus, the statistics of adatom hops on growing film surfaces is related to universal and nonuniversal features of the growth model and with properties of trapping models if the hopping time is limited by the landscape and not by the deposition of other layers.

Tracing the two- to three-dimensional transition in InAs/GaAs sub-monolayer quantum dot growth

A two- to three-dimensional (2D to 3D) transition in InAs/GaAs sub-monolayer quantum dot (SML-QD) growth was investigated, and its evolution was traced. From the AFM results, the SML-QDs disappeared gradually and plateaus around each island formed with increasing GaAs capping layer thickness (tCL) from 1.5 ML to 4.5 ML. The SML-QDs become fully covered with a GaAs capping layer, leading to a planarization process caused by strain relaxation. In addition, when the individual InAs coverage was is insufficient to cause a 3D transition, similar plateaus formed when stacking with an increasing number of stacking layers (S) from 2 to 6. On the other hand, as the InAs coverage was increased from 0.3 to 0.5 ML, 3D dots with a comparable size to typical SK-QDs were observed beyond S = 4. This means that as the total InAs deposition thickness increases above a critical thickness (~2.0 ML), 2D-small islands transform spontaneously into 3D-pyramid QDs. Such structures then show similar optical characteristics to SK-QDs. Overall, the transition from 2D to 3D in InAs/GaAs SML-QD growth could be controlled by careful choice of the parameters tcl, θ, and S.

In situ laser reflectivity to monitor and control the nucleation and growth of atomically thin 2D materials*

The growth of atomically thin two-dimensional (2D) layered and other quantum materials is typically performed without in situ monitoring or control. Here, a simple laser reflectivity approach is demonstrated to provide in situ control over sub-monolayer thickness and growth kinetics during pulsed laser deposition (PLD) of MoSe2 layers. First, the general technique is presented with emphasis on designing the maximum sensitivity of the optical contrast through consideration of Fresnel’s equations with proper choice of layer thickness, substrate, and laser monitoring wavelength, incidence angle, and laser polarization. Then the 632.8 nm optical reflectivity of MoSe2 layers on SiO2/Si substrates was predicted and compared with in situ monitoring of MoSe2 growth by PLD under actual growth conditions using a probe HeNe laser beam. The measurements showed high sensitivity and excellent agreement with MoSe2 surface coverages calculated from atomic resolution STEM analysis of 2D layers deposited in arrested growth experiments. Growth kinetics revealed by these measurements showed sigmoidal nucleation and growth stages in the formation of the 2D MoSe2 layers that are described by a simple model, indicating the promise of the laser reflectivity technique for in situ monitoring and control of 2D materials deposition.

Modeling of nonequilibrium surface growth by a limited-mobility model with distributed diffusion length.

Kinetic Monte Carlo (KMC) simulations are a well-established numerical tool to investigate the time-dependent surface morphology in molecular beam epitaxy experiments. In parallel, simplified approaches such as limited mobility (LM) models characterized by a fixed diffusion length have been studied. Here we investigate an extended LM model to gain deeper insight into the role of diffusional processes concerning the growth morphology. Our model is based on the stochastic transition rules of the Das Sarma-Tamborena model but differs from the latter via a variable diffusion length. A first guess for this length can be extracted from the saturation value of the mean-squared displacement calculated from short KMC simulations. Comparing the resulting surface morphologies in the sub- and multilayer growth regime to those obtained from KMC simulations, we find deviations which can be cured by adding fluctuations to the diffusion length. This mimics the stochastic nature of particle diffusion on a substrate, an aspect which is usually neglected in LM models. We propose to add fluctuations to the diffusion length by choosing this quantity for each adsorbed particle from a Gaussian distribution, where the variance of the distribution serves as a fitting parameter. We show that the diffusional fluctuations have a huge impact on cluster properties during submonolayer growth as well as on the surface profile in the high coverage regime. The analysis of the surface morphologies on one- and two-dimensional substrates during sub- and multilayer growth shows that the LM model can produce structures that are indistinguishable to the ones from KMC simulations at arbitrary growth conditions.

Kinetic Monte Carlo simulations for the submonolayer growth of homoepitaxial SrTiO 3 thin films

Perovskite ferroelectric thin films have attracted much attention for the potential applications in modern microelectronics, micro-electro-mechanical systems, and information storage. Computer simulation is helpful for optimising the process parameters and manufacturing high-performance ferroelectric thin films. Herein the work presents a kinetic Monte Carlo method for simulating the submonolayer growth of homoepitaxial SrTiO3 (STO) thin films, and investigate the effects of substrate temperature, incident kinetic energy of adatoms, and deposition rate on the growth of STO thin films. The results show that with the increase of the substrate temperature, and the incident kinetic energy, or the decrease of the deposition rate, the density of islands decreases, and the bonding ratio increases. This study provides an understanding of submonolayer STO thin films growth at atomic scale.

Analysis of InAsSb/GaAs submonolayer stacks

InAsSb submonolayer (SML) islands separated by GaAs spacer layers show three-dimensional charge carrier localization centers of very high density. We advance the understanding of the submonolayer growth technique by combination of structural and optical data. Our analysis of the Sb incorporation by Langmuir-type adsorption model reveals involvement of a slow reaction component. In search for the governing mechanism for electronic coupling we monitored structural parameters such as interface roughness by X-ray reflection measurements. Interestingly, no significant structural changes with respect to the spacer thickness are found. Still, a critical upper thickness limit to observe electronic changes of 1.8–1.9 MLs GaAs is found. Below this critical thickness, three-dimensional charge carrier localization is present in the SML stack and the average localization depth can be controlled through spacer thickness.

Effective diffusion length and elementary surface processes in the concurrent growth of nanowires and 2D layers

In this work, we revisit models of the diffusion-induced growth of nanowires (NWs) accompanied by the growth of crystalline layers on the nonactivated areas of the substrate surface. In the conventional approach, the presence of the surface steps is taken into account implicitly by introducing effective diffusion length of adatoms before desorption or incorporation. In contrast, here we treat adatom incorporation as an outcome of a sequence of elementary surface processes on the terraces and step edges. The surface diffusion problems are formulated and solved for two modes of the lateral growth at the substrate surface: (i) submonolayer growth of 2D islands and (ii) permanent presence of the steps forming a closed loop around the NW base. In the limiting cases where the substrate adatoms with high probability either reach the sidewall of a NW or leave the substrate surface by desorption or attachment to 2D islands, our analysis demonstrates the correspondence between this approach and the conventional one under the relevant and physically transparent choice of the effective diffusion length. It has been shown that at relatively large NW heights the steps act as the sinks not only for the atoms impinging the substrate surface but also for the atoms impinging the NW sidewalls. This explains the failure of the simple material balance arguments to predict the NW height evolution observed in the experiments on the Au-assisted growth of InAs and GaAs NWs.

A case study of thin film stress evolution at a dissimilar material interface via molecular dynamics simulations

Evolution of deformation and stress in growing thin films has been studied in this work using computational simulations that resolve matter at atomic length and time scales. For the surface layers of films laying on the substrate of a dissimilar material, the stress distribution analysis around defects becomes more challenging. Herein, spatial and temporal distribution of deformation and associated stress evolution are presented for different thin film formation events including (1) sub-monolayer growth during an early film nucleation stage and (2) coalescence of adjacent monolayer “islands.” Validity of the stress computed via local computations of the virial expression for stress in a system of interacting particles was checked by comparing to results obtained from considerations of local atomic deformation in conjunction with existing expressions for epitaxial thin film growth stress. For the geometries studied here, where a monolayer of film with a highly characterized linear defect, as in the case of a stacking fault, was simulated for coalescence, fairly good agreement was found. This result demonstrates that, for similar defects at the surface layer, with sufficient sub-ensemble averaging of the standard virial expression for stress, semiquantitative spatial stress distribution information can be obtained from atomic scale simulations. Using our validated stress computation method, we reveal significant stress localization during thin film growth processes, leading to pronounced differences in maximum and minimum stress observed over very small spatial extent (of order multiple GPa over 3–6 nm distances). One prominent mechanism of stress localization revealed here is coalescence between adjacent growing islands. For geometries explored here, stress manifesting during coalescence is highly localized.

Atomic structure governed diversity of exchange-driven spin helices in Fe nanoislands: Experiment and theory

We combine spin-polarized scanning tunneling microscopy (SP-STM) and first-principles calculations to demonstrate the control of the wavelength of helical spin textures in Fe nanoislands by varying their atomic structure. We make use of the complexity of submonolayer growth of Fe on Cu(111) to prepare nanoislands characterized by different thickness and in-plane atomic structure. SP-STM results reveal that the magnetic states of different nanoislands are spin helices. The wavelength of the spin helices varies strongly. Calculations performed for Fe films with different thickness and in-plane atomic structure explain the strong variation of the wavelength by a subtle balance in the competition between ferromagnetic and antiferromagnetic exchange interactions. We identify the crucial role of the effectively enhanced weak antiferromagnetic exchange interactions between distant atoms.