Coverage dependent molecular assembly of anthraquinone on Au(111).

A scanning tunneling microscopy study of anthraquinone (AQ) on the Au(111) surface shows that the molecules self-assemble into several structures depending on the local surface coverage. At high coverages, a close-packed saturated monolayer is observed, while at low coverages, mobile surface molecules coexist with stable chiral hexamer clusters. At intermediate coverages, a disordered 2D porous network interlinking close-packed islands is observed in contrast to the giant honeycomb networks observed for the same molecule on Cu(111). This difference verifies the predicted extreme sensitivity [J. Wyrick et al., Nano Lett. 11, 2944 (2011)] of the pore network to small changes in the surface electronic structure. Quantitative analysis of the 2D pore network reveals that the areas of the vacancy islands are distributed log-normally. Log-normal distributions are typically associated with the product of random variables (multiplicative noise), and we propose that the distribution of pore sizes for AQ on Au(111) originates from random linear rate constants for molecules to either desorb from the surface or detach from the region of a nucleated pore.

Chemical insight from density functional modeling of molecular adsorption: Tracking the bonding and diffusion of anthracene derivatives on Cu(111) with molecular orbitals.

We present a method of analyzing the results of density functional modeling of molecular adsorption in terms of an analogue of molecular orbitals. This approach permits intuitive chemical insight into the adsorption process. Applied to a set of anthracene derivates (anthracene, 9,10-anthraquinone, 9,10-dithioanthracene, and 9,10-diselenonanthracene), we follow the electronic states of the molecules that are involved in the bonding process and correlate them to both the molecular adsorption geometry and the species' diffusive behavior. We additionally provide computational code to easily repeat this analysis on any system.

How "hot precursors" modify island nucleation: a rate-equation model.

We propose a novel island nucleation and growth model explicitly including transient (ballistic) mobility of the monomers deposited at rate F, assumed to be in a hot precursor state before thermalizing. In limiting regimes, corresponding to fast (diffusive) and slow (ballistic) thermalization, the island density N obeys scaling N∝F(α). In between is found a rich, complex behavior, with various distinctive scaling regimes, characterized by effective exponents α(eff) and activation energies that we compute exactly. Application to N(F,T) of recent organic-molecule deposition experiments yields an excellent fit.

Do two-dimensional "noble gas atoms" produce molecular honeycombs at a metal surface?

Anthraquinone self-assembles on Cu(111) into a giant honeycomb network with exactly three molecules on each side. Here we propose that the exceptional degree of order achieved in this system can be explained as a consequence of the confinement of substrate electrons in the pores, with the pore size tailored so that the confined electrons can adopt a noble-gas-like two-dimensional quasi-atom configuration with two filled shells. Formation of identical pores in a related adsorption system (at different overall periodicity due to the different molecule size) corroborates this concept. A combination of photoemission spectroscopy with density functional theory computations (including van der Waals interactions) of adsorbate-substrate interactions allows quantum mechanical modeling of the spectra of the resultant quasi-atoms and their energetics.

Coalescence of 3-phenyl-propynenitrile on Cu(111) into interlocking pinwheel chains.

3-phenyl-propynenitrile (PPN) adsorbs on Cu(111) in a hexagonal network of molecular trimers formed through intermolecular interaction of the cyano group of one molecule with the aromatic ring of its neighbor. Heptamers of trimers coalesce into interlocking pinwheel-shaped structures that, by percolating across islands of the original trimer coverage, create the appearance of gear chains. Density functional theory aids in identifying substrate stress associated with the chemisorption of PPN's acetylene group as the cause of this transition.

Power of confinement: adsorbate dynamics on nanometer-scale exposed facets.

The diffusion and arrangements of CO adsorbates within nanometer-scale pores on a copper surface are investigated by low-temperature scanning tunneling microscopy. In contrast to extended terraces, confinement stabilizes dislocation lines that expose more than one-fourth of the adsorbate population to potentially more reactive adsorption configurations. Confinement allows correlation between adsorbate diffusivity and the number of adsorbates in the pore. A marked increase is found that coincides with the absence of dense films on the exposed facets. In combination, we find that in confinement CO molecules are much more likely to be at adsorption sites that allow lateral access, in contrast to the dense and uniform films on extended terraces.

Strong quantum size effects in Pb(111) thin films mediated by anomalous Friedel oscillations.

Using first-principles calculations within density functional theory, we study Friedel oscillations (FOs) in the electron density at different metal surfaces and their influence on the lattice relaxation and stability of ultrathin metal films. We show that the FOs at the Pb(111) surface decay as 1/x with the distance x from the surface, different from the conventional 1/x(2) power law at other metal surfaces. The underlying physical reason for this striking difference is tied to the strong nesting of the two different Fermi sheets along the Pb(111) direction. The interference of the strong FOs emanating from the two surfaces of a Pb(111) film, in turn, not only results in superoscillatory interlayer relaxations around the center of the film, but also determines its stability in the quantum regime. As a simple and generic picture, the present findings also explain why quantum size effects are exceptionally robust in Pb(111) films.

Adsorbates in a box: titration of substrate electronic states.

Nanoscale confinement of adsorbed CO molecules in an anthraquinone network on Cu(111) with a pore size of ≈4 nm arranges the CO molecules in a shell structure that coincides with the distribution of substrate confined electronic states. Molecules occupy the states approximately in the sequence of rising electron energy. Despite the sixfold symmetry of the pore boundary itself, the adsorbate distribution adopts the threefold symmetry of the network-substrate system, highlighting the importance of the substrate even for such quasi-free-electron systems.

Effects of impurities on surface morphology: some examples.

Small amounts of impurities are known to have remarkably great influence on surface morphology. We discuss three examples that arise in our research. First, we consider impurities codeposited during epitaxial growth, paying particular attention to Cu(100). After many layers of growth, the impurities can dramatically alter the surface morphology, changing the wavelength of the meandering instability and producing small square-base pyramids. Second, we consider the decoration of both island and vacancy island edges on Ag(111) with C(60). We use this system as a prototype to model how edge decoration with impurities can have a striking effect on the shape of the island as well as the dynamics of edge fluctuations. Finally, we show that about one per cent of pentacene quinone impurities alters the size of the critical nucleus in submonolayer pentacene deposition. This provides a platform on which to discuss our recent work characterizing the capture-zone distribution associated with the islands in terms of the generalized Wigner distribution, a simple one-parameter expression in which the characteristic exponent is the size of the smallest stable island.

Two-step unconventional protocol for epitaxial growth in one dimension with hindered reactions.

We study the effect of hindered aggregation and/or nucleation on the island formation process in a two-step growth protocol. In the proposed model, the attachment of monomers to islands and/or other monomers is hindered by additional energy barriers which decrease the hopping rate of the monomers to the occupied sites of the lattice. For zero and weak barriers, the attachment is limited by diffusion while for strong barriers it is limited by reaction. We describe the time evolution of the system in terms of the monomer and island densities, N_{1} and N. We also calculate the gap length, the capture zone and the island distributions. For all the sets of barriers considered, the results given by the proposed analytical model are compared with those from kinetic Monte Carlo simulations. We found that the behavior of the system depends on the ratio of the nucleation barrier to the aggregation barrier. The two-step growth protocol allows more control and understanding on the island formation mechanism because it intrinsically separates the nucleation and aggregation processes in different time regimes.

Fragmentation approach to the point-island model with hindered aggregation: Accessing the barrier energy.

We study the effect of hindered aggregation on the island formation process in a one- (1D) and two-dimensional (2D) point-island model for epitaxial growth with arbitrary critical nucleus size i. In our model, the attachment of monomers to preexisting islands is hindered by an additional attachment barrier, characterized by length l_{a}. For l_{a}=0 the islands behave as perfect sinks while for l_{a}→∞ they behave as reflecting boundaries. For intermediate values of l_{a}, the system exhibits a crossover between two different kinds of processes, diffusion-limited aggregation and attachment-limited aggregation. We calculate the growth exponents of the density of islands and monomers for the low coverage and aggregation regimes. The capture-zone (CZ) distributions are also calculated for different values of i and l_{a}. In order to obtain a good spatial description of the nucleation process, we propose a fragmentation model, which is based on an approximate description of nucleation inside of the gaps for 1D and the CZs for 2D. In both cases, the nucleation is described by using two different physically rooted probabilities, which are related with the microscopic parameters of the model (i and l_{a}). We test our analytical model with extensive numerical simulations and previously established results. The proposed model describes excellently the statistical behavior of the system for arbitrary values of l_{a} and i=1, 2, and 3.

Refined evaluation of the level-spacing distribution of symplectic ensembles: moments and implications.

To obtain a more precise value for the variance sigma2 of the joint probability distribution of a symplectic ensemble, we extend previous numerical evaluations of a power series. Our result sigma2 approximately 0.1041 shows that the excellent approximation using the analytically simple Wigner surmise fractionally overestimates this value. This behavior is important in establishing the trend of a generalization of the surmise to describe the terrace-width distribution on vicinal surfaces. We also obtain precise estimates of the skewness and the kurtosis of the exact distribution, as well as the related moments.

Beyond the Wigner distribution: Schrödinger equations and terrace width distributions.

The so-called generalized Wigner distribution has earlier been shown to be an excellent approximation for the terrace width distribution (TWD) of vicinal surfaces characterized by step-step interactions that are perpendicular to the average step direction and fall off as the inverse square of the step spacing. In this paper, we show that the generalized Wigner distribution can be derived from a plausible, phenomenological model in which two steps interact with each other directly and with other steps through a position-dependent pressure. We also discuss generalizations to more general step-step interactions and show that the predictions are in good agreement with TWDs derived from numerical transfer-matrix calculations and Monte Carlo simulations. This phenomenological approach allows the step-step interaction to be extracted from experimental TWDs.

Interacting steps with finite-range interactions: analytical approximation and numerical results.

We calculate an analytical expression for the terrace-width distribution P(s) for an interacting step system with nearest- and next-nearest-neighbor interactions. Our model is derived by mapping the step system onto a statistically equivalent one-dimensional system of classical particles. The validity of the model is tested with several numerical simulations and experimental results. We explore the effect of the range of interactions q on the functional form of the terrace-width distribution and pair correlation functions. For physically plausible interactions, we find modest changes when next-nearest neighbor interactions are included and generally negligible changes when more distant interactions are allowed. We discuss methods for extracting from simulated experimental data the characteristic scale-setting terms in assumed potential forms.

Mean-field approximation for spacing distribution functions in classical systems.

We propose a mean-field method to calculate approximately the spacing distribution functions p((n))(s) in one-dimensional classical many-particle systems. We compare our method with two other commonly used methods, the independent interval approximation and the extended Wigner surmise. In our mean-field approach, p((n))(s) is calculated from a set of Langevin equations, which are decoupled by using a mean-field approximation. We find that in spite of its simplicity, the mean-field approximation provides good results in several systems. We offer many examples illustrating that the three previously mentioned methods give a reasonable description of the statistical behavior of the system. The physical interpretation of each method is also discussed.

Voronoi cell patterns: theoretical model and applications.

We use a simple fragmentation model to describe the statistical behavior of the Voronoi cell patterns generated by a homogeneous and isotropic set of points in 1D and in 2D. In particular, we are interested in the distribution of sizes of these Voronoi cells. Our model is completely defined by two probability distributions in 1D and again in 2D, the probability to add a new point inside an existing cell and the probability that this new point is at a particular position relative to the preexisting point inside this cell. In 1D the first distribution depends on a single parameter while the second distribution is defined through a fragmentation kernel; in 2D both distributions depend on a single parameter. The fragmentation kernel and the control parameters are closely related to the physical properties of the specific system under study. We use our model to describe the Voronoi cell patterns of several systems. Specifically, we study the island nucleation with irreversible attachment, the 1D car-parking problem, the formation of second-level administrative divisions, and the pattern formed by the Paris Métro stations.

Spacing distribution functions for the one-dimensional point-island model with irreversible attachment.

We study the configurational structure of the point-island model for epitaxial growth in one dimension. In particular, we calculate the island gap and capture zone distributions. Our model is based on an approximate description of nucleation inside the gaps. Nucleation is described by the joint probability density p(n)(XY)(x,y), which represents the probability density to have nucleation at position x within a gap of size y. Our proposed functional form for p(n)(XY)(x,y) describes excellently the statistical behavior of the system. We compare our analytical model with extensive numerical simulations. Our model retains the most relevant physical properties of the system.

One-dimensional model of interacting-step fluctuations on vicinal surfaces: analytical formulas and kinetic Monte Carlo simulations.

We study analytically and numerically a one-dimensional model of interacting line defects (steps) fluctuating on a vicinal crystal. Our goal is to formulate and validate analytical techniques for approximately solving systems of coupled nonlinear stochastic differential equations (SDEs) governing fluctuations in surface motion. In our analytical approach, the starting point is the Burton-Cabrera-Frank (BCF) model by which step motion is driven by diffusion of adsorbed atoms on terraces and atom attachment-detachment at steps. The step energy accounts for entropic and nearest-neighbor elastic-dipole interactions. By including Gaussian white noise to the equations of motion for terrace widths, we formulate large systems of SDEs under different choices of diffusion coefficients for the noise. We simplify this description via (i) perturbation theory and linearization of the step interactions and, alternatively, (ii) a mean-field (MF) approximation whereby widths of adjacent terraces are replaced by a self-consistent field but nonlinearities in step interactions are retained. We derive simplified formulas for the time-dependent terrace-width distribution (TWD) and its steady-state limit. Our MF analytical predictions for the TWD compare favorably with kinetic Monte Carlo simulations under the addition of a suitably conservative white noise in the BCF equations.

Impurity decoration for crystal shape control: C60 on Ag(111).

The decoration of hexagonal Ag/Ag(111) monolayer islands by chains of C60, observed via STM at 300 K, dramatically changes the nanocrystalline shape and fluctuations of the islands. We tune coverage so that a single chain of C60 fully decorates each Ag island boundary, forming a closed circular "necklace." We model the C60-induced rounding in terms of competing energetic and entropic effects. We thereby characterize the decorated-step fluctuations and estimate the C60-Ag and C60-C60 attractions to be approximately 0.13 and approximately 0.03 eV, respectively. Generalizations of our model show that decorating molecules of both circular and rectangular surface-projected symmetry will similarly lower the energy of fully kinked boundaries, leading to corner rounding and reorientations by 30 degrees on (111) surfaces and 45 degrees on (100) surfaces.

Capture-zone scaling in island nucleation: universal fluctuation behavior.

In island nucleation and growth, the distribution of capture zones (in essence proximity cells) can be described by a simple expression generalizing the Wigner surmise (power-law rise, Gaussian decay) from random matrix theory that accounts for spacing distributions in a host of fluctuation phenomena. Its single adjustable parameter, the power-law exponent, can be simply related to the critical nucleus of growth models and the substrate dimensionality. We compare with extensive published kinetic Monte Carlo data and limited experimental data. A phenomenological theory elucidates the result.