Dynamic Scaling Theory Research Articles

Roughening and smoothing behavior of single crystal Si by low energy Ar + ion bombardment

We examined the dependence of smoothing or roughening behavior on the initial surface roughness of single crystal Si. An atomically flat cleaved Si surface of roughness 0.038 nm rms and a Si wafer substrate roughness of 0.12 nm rms were sputtered by 0.2–3 keV Ar + ion beam at normal ion incidence. Surface morphology was observed by atomic force microscope (AFM) under DFM mode. Result shows that the rough surface ( R = 0.12 nm rms) becomes smooth ( R = 0.068 nm rms) and the smooth surface ( R = 0.038 nm rms) becomes rough ( R = 0.068 nm rms) due to low energy Ar + ion beam sputtering process and both finally saturates at 0.068 nm rms. The saturated roughness R sat depends on ion beam energy and increases with increasing beam energy. To understand the roughening mechanism, we studied the dynamic scaling theory and measured the roughness exponent α and growth exponent β. The values of α = 1.03 and β = 0.28 are in agreement with the Cuerno’s one dimensional simulation of the Linear diffusion equation with noise, that corresponds to the initial stage of dynamic scaling of ion bombardment.

Application of dynamic scaling to the surface properties of organic thin films: Energetic materials

The surface evolution of pentaerythritol tetranitrate (PETN) organic films, formed by room-temperature thermal evaporation on 16-mercaptohexadecanoic acid self-assembly monolayer, has been investigated using atomic force microscopy (AFM). Results are analyzed using dynamic scaling theory. High values are obtained for dynamic scaling coefficients β = 0.56 ± 0.02 and 1/ z = 0.35 ± 0.01, where β is the growth exponent and 1/ z is the dynamic exponent. Using mound height and in-plane correlation length the dependence of mound slope m on PETN thickness is obtained. Two thickness regimes are identified. For thickness between 40 and 300 nm, m grows rapidly due to high bulk surface diffusion with smooth PETN morphology. For thickness from 300 to 800 nm, m continues to grow but at a reduced rate. Facets are evident and crevices form in this thickness range. The surface evolution behavior is attributed to bulk surface diffusion and the presence of an Ehrlich–Schwoebel barrier, which preferentially enhances step-up diffusion.

Evolution of resist roughness during development: stochastic simulation and dynamic scaling analysis

The study of surface-roughness evolution of resist films during development may elucidate the material origins of line-edge roughness. We use a stochastic simulator of resist development and analyze the resulting surface roughness evolution with dynamic scaling theory in polymers with homogeneous and inhomogeneous local dissolution behavior. In all cases, a power-law increase of root mean square (rms) roughness and correlation length is found. In homogenous polymers, a slow rms roughness increase is reported and the scaling exponents are shown to obey the dynamic scaling hypothesis of Family-Viscek. Dissolution inhomogeneity is inserted on a monomer scale (using copolymers), on a chain scale (using mixture of copolymers), or on a film scale by the activation of photo acid generator (PAG) molecules and the concomitant acid diffusion. Our simulator predicts that any kind of inhomogeneity may cause much larger rms roughness than homogeneous solubility. Furthermore, PAG-induced inhomogeneity results in a faster increase and larger values of roughness compared to chain-level inhomogeneity and this, in turn, leads to larger values compared to polymers with monomer-scale inhomogeneity. The differences in roughness are abruptly magnified when one reduces the resist solubility to levels in the range of the clearing dose. A comparison with experimental results shows good agreement with the simulation predictions.

Wetting Properties of Polycrystalline TiO2 Surfaces: A Scaling Approach to the Roughness Factors

This work presents a thorough study on the wettability of polycrystalline anatase TiO(2) thin films prepared at 250 °C in a microwave plasma enhanced chemical vapor deposition (MW-PECVD) reactor with Ar/O(2) plasmas. Anatase polycrystalline thin films with different microstructures, textures, and surface roughness were obtained as a function of their thickness. The water contact angle of the samples was analyzed within the assumptions of the Wenzel, Cassie, and Miwa models to ascertain the effect of roughness and other surface heterogeneities on their characteristic parameters. The roughness factors defined in the different models were calculated from the atomic force microscopy (AFM) images of the films for two different observation scales within the premises of the dynamic scaling theories. The obtained results indicate that the wetting angle of an equivalent flat anatase surface with a value of 82° can only be properly estimated for observation scales of 5 × 5 μm(2) and using the Miwa model. The analysis of the UV induced hydrophilization of the surface state of the anatase films and the posterior recovery of the partially hydrophobic character of these surfaces in the absence of UV photons suggest a clear dependence of the light induced wettability on their texture and size of crystalline domains.

Space-resolved dynamics of a tracer in a disordered solid

The dynamics of a tracer particle in a glassy matrix of obstacles displays slow complex transport as the free volume approaches a critical value and the void space falls apart. We investigate the emerging subdiffusive motion of the test particle by extensive molecular dynamics simulations and characterize the spatio-temporal transport in terms of two-time correlation functions, including the time-dependent diffusion coefficient, the frequency-dependent conductivity, as well as the wavenumber-dependent intermediate scattering function. We rationalize our findings within the framework of critical phenomena and compare our data to a dynamic scaling theory.

Self-organization of bimetallic PdAu nanoparticles on SiO2 surface

Bimetallic PdAu nanoparticles on SiO2 substrate were produced by a sequential room-temperature sputtering deposition method. By the atomic force microscopy technique we studied the nanoparticles self-organization mechanisms in various conditions. First, Pd nucleation and growth proceeds at the substrate defects and the Pd nanoparticles density increase rapidly. During the second sputtering deposition, Au atoms adsorb on the SiO2 and diffuse toward Pd nanoparticles without forming new nuclei. The Au atoms are trapped by the preformed Pd nanoparticles, forming PdAu bimetallic nanoparticles which size increases. Furthermore, fixing the amount of deposited Pd and increasing the amount of deposited Au, we analyzed the evolution of the PdAu film surface morphology: we observe that the PdAu grows initially as three-dimensional islands; then the PdAu film morphology evolves from compact three-dimensional islands to partially coalesced worm-like structures, followed by a percolation morphology and finally to a continuous and rough film. The application of the interrupted coalescence model allowed us to evaluate the critical mean island diameter Rc ≈ 2.8 nm for the partial coalescence process. The application of the dynamic scaling theory of growing interfaces allowed us to evaluate the dynamic growth exponent β = 0.21 ± 0.01 from the evolution of the film surface roughness. Finally, fixing the amount of deposited Pd and Au we studied the self-organization mechanism of the PdAu nanoparticles induced by thermal processes performed in the 973–1173 K temperature range. The observed kinetic growth mechanism is consistent with a surface diffusion-limited ripening of the nanoparticles with a temperature-dependent growth exponent. The dependence of the growth exponent on the temperature is supposed to be linked to the variation with the temperature of the characteristics of the PdAu alloy. The activation energy for the surface diffusion process was evaluated in 0.54 ± 0.03 eV.

Kinetic growth mechanisms of sputter-deposited Au films on mica: from nanoclusters to nanostructured microclusters

Au nanostructured film was deposited on mica by room temperature RF sputtering. The growth mechanism of the film was studied analyzing the evolution of the film morphology as a function of its thickness by the atomic force microscopy. In the early stages of a growth the film evolution proceeds by the nucleation and growth of nanoclusters. After a critical thickness the growth of microclusters formed by the joining of nanoclusters in preferential nucleation sites, onto a quasicontinuous film, is observed. We quantified the evolution of the mean nanoclusters height and surface density and of the film roughness. This data were analyzed by the dynamic scaling theory of growing interfaces obtaining the scaling and roughness exponents z and β whose values suggest a conservative growth process. We also quantified the growth of the microclusters showing that it is consistent with a coalescence/impingement dynamic. About the formation of the microclusters, furthermore, we speculate that their origin is strongly correlated to the features of the sputtering technique in connection with the deposition on a high-diffusivity substrate.

Atomic force microscopy study of the growth mechanisms of nanostructured sputtered Au film on Si(111): Evolution with film thickness and annealing time

Nanostructured Au films were deposited on Si(111) by room-temperature sputtering. By the atomic force microscopy technique we studied the evolution of the Au film morphology as a function of the film thickness h and annealing time t at 873 K. By the study of the evolution of the mean vertical and horizontal sizes of the islands forming the film and of their fraction of covered area as a function of h from 1.7×1017 to 1.0×1018 Au/cm2 we identified four different growth stages such as: (1) 1.7×1017≤h≤3.0×1017 Au/cm2, nucleation of nanometric three-dimensional (3D) hemispherical Au clusters; (2) 3.0×1017<h≤5.2×1017 Au/cm2, lateral growth of the Au clusters; (3) 5.2×1017<h≤7.7×1017 Au/cm2, coalescence of the Au clusters; (4) 7.7×1017<h≤1.0×1018 Au/cm2, vertical growth of the coalesced Au clusters. The application of the dynamic scaling theory of growing interfaces allowed us to calculate the dynamic scaling exponent z=3.8±0.3, the dynamic growth exponent β=0.38±0.03, the roughness exponent α=1.4±0.1 and the Avrami exponent m=0.79±0.02. Finally, the study of the evolution of the mean Au clusters size as a function of annealing time at 873 K allowed us to identify the thermal-induced self-organization mechanism in a surface diffusion limited ripening of 3D structures and also the surface diffusion coefficient of Au on Si(111) at 873 K was estimated in (8.2×10−16)±(3×10−17) m2/s.

Island Nucleation and Growth Dynamics during Submonolayer Vapor Deposition Polymerization

The evolution of island formation during submonolayer growth of poly(chloro-p-xylylene) films on naturally grown silicon oxide substrates is investigated using atomic force microscopy (AFM), and analyzed within the framework of dynamic scaling theory. A transition of film growth mechanism from diffusion-limited aggregation to reaction-limited aggregation is dramatically displayed in an unprecedented progression of the island density distribution functions. As surface coverage increases, a conventional bell-shaped monomodal distribution in early growth regime (θ = 0.07) gradually changes to a monotonically decreasing function, approximately following a power-law. The marginal power-law scaling behavior of the island distribution functions, reminiscent of the behavior of colloid aggregation, strongly suggests that film growth by vapor deposition polymerization is a new type of surface growth governed by reaction-limited aggregation process in which the competition between diffusion and deposition does not define the typical time scale of film growth.

Cluster-resolved dynamic scaling theory and universal corrections for transport on percolating systems

Cluster-resolved dynamic scaling theory and universal corrections for transport on percolating systems

Homologous temperature dependence of global surface scaling behaviors of polycrystalline copper films

A strategy integrating structure zone model with dynamic scaling theory was proposed to study the global surface dynamics of polycrystalline Cu films deposited at different homologous temperature Ts/Tm. The evolution of roughness exponent α and growth exponent β reveals a transition from random deposition to surface diffusion dominated smoothening in the lower Ts/Tm regime and then to rapid surface roughening in the higher Ts/Tm regime. In contrast to that of amorphous films, the distinct scaling behavior in higher Ts/Tm regime arises from the change of anisotropic mass transport mechanisms, which could be related to the texture evolution during growth.

Scaling of surface roughness in sputter-deposited ZnO:Al thin films

We have studied surface roughness scaling of ZnO:Al thin films grown by rf magnetron sputtering of a compound target within framework of the dynamic scaling theory using atomic force microscopy. We have observed a crossover in scaling behavior of surface roughness at a deposition time of 25 min. Both the regimes are characterized by power-law dependence of local surface width w(r,t) on deposition time for small r, typical of anomalous scaling. The scaling exponents for the first regime indicate the existence of a new dynamics. For t≥25 min, the films follow super-rough scaling behavior with global exponents α=1.5±0.2 and β=1.03±0.01, and local exponents αlocal=1 and βlocal=0.67±0.05. The anomaly in the scaling behavior of the films is discussed in terms of the shadowing instability and bombardment of energetic particles during growth of the films.

Binary liquid concentration fluctuations linked to elementary reactions

Acoustical relaxation spectra of solutions and mixtures exhibiting a micro-heterogeneous liquid structure are discussed in view of non-critical and critical fluctuations in the local concentration. A unifying model of non-critical concentration fluctuations that includes all previous theories is shown to favourably apply to the relevant experimental spectra. This model assumes a parallel development of fluctuations by diffusion as well as by a rate process. Allowance is also made for spatial correlations in the fluctuations. For various aqueous solutions a correlation between the fluctuation correlation length and the hydrophobic properties of the solute becomes obvious. In addition, experimental results are reported which enable the scaling function of critically demixing binary systems to be determined without adjustable parameter. It is found that the Bhattacharjee-Ferrell dynamic scaling theory applies nicely to the experimental findings. Spectra for more complicated systems, such as revealing critical micelle formation are also discussed briefly, in order to indicate couplings between the fluctuations and elementary reactions.

Aggregation of magnetic holes in a rotating magnetic field

We experimentally investigated field-induced aggregation of nonmagnetic particles confined in a magnetic fluid layer when rotating magnetic fields were applied. After application of a magnetic field rotating in the plane of the fluid layer, the single particles start to form two-dimensional clusters, like dimers, trimers, and more complex structures. These clusters aggregated again and again to form bigger clusters. During this nonequilibrium process, a broad range of cluster sizes was formed, and the scaling exponents z and z;{'} of the number of clusters N(t) approximately t;{-z;{'}} and average cluster size S(t) approximately t;{z} were calculated. The process could be characterized as diffusion-limited cluster-cluster aggregation. We found that all sizes of clusters that occurred during an experiment fall on a single curve, as the dynamic scaling theory predicts. However, the characteristic scaling exponents z;{'},z and crossover exponents Delta were not universal. A particle tracking method was used to find the dependence of the diffusion coefficients D_{s} on cluster size s . The cluster motions show features of Brownian motion. The average diffusion coefficients D_{s} depend on the cluster size s as a power law D_{s} proportional, variants;{gamma} where values of gamma as different as gamma=-0.62+/-0.19 and gamma=-2.08+/-0.51 were found in two of the experiments.

Random walk on topologically biased anisotropic percolation clusters and transport properties

Unbiased random walks are performed on topologically biased anisotropic percolation clusters (APC). Topologically biased APCs are generated using suitable anisotropic percolation models. New walk dimensions are found to characterize the anisotropic behaviour of the unbiased random walk on the biased topology. Critical properties of electro and magneto conductivities are characterized estimating respective dynamical critical exponents. A dynamical scaling theory relating dynamical and static critical exponents has been developed. The dynamical critical exponents satisfy the scaling relations within error bar.

Scaling behavior of internal stress in electrodeposited nickel thin films

We have investigated the effects of the current density, film thickness, temperature, and additive agent on the internal stress in electrodeposited nickel thin films using a bent strip measurement. The internal stress is found to obey a scaling law expressed in terms of the current density and film thickness. The additive agent is shown to behave as a noise leading to different exponents in the dynamic scaling theory. In addition, the Arrhenius temperature dependence of the internal stress indicates the presence of thermal activation related to grain growth.

Growth mechanisms of vapor-born polymer films

The surface morphologies of poly(chloro-$p$-xylylene) films were measured using atomic force microscopy and analyzed within the framework of the dynamic scaling theory. The evolution of polymer films grown with fixed experimental parameters showed drastic changes in dynamic roughening behavior, which involve unusually high growth exponent $(\ensuremath{\beta}=0.65\ifmmode\pm\else\textpm\fi{}0.03)$ in the initial growth regime, followed by a regime characterized by $\ensuremath{\beta}\ensuremath{\sim}0$, and finally a crossover to $\ensuremath{\beta}=0.18\ifmmode\pm\else\textpm\fi{}0.02$ in a steady growth regime. Detailed scaling analysis of the surface fluctuation in Fourier space in terms of power spectral density revealed a gradual crossover in the global roughness exponent, analogous to a phase transition between two equilibrium states, from a morphology defined by $\ensuremath{\alpha}=1.36\ifmmode\pm\else\textpm\fi{}0.13$ to the other morphology characterized by $\ensuremath{\alpha}=0.93\ifmmode\pm\else\textpm\fi{}0.04$ as the film thickness increases. Our experimental results which significantly deviate from the well established descriptions of film growth clearly exhibit that the dynamic roughening of polymer film is deeply affected by strong molecular interactions and relaxations of polymer chains.

Anomalous scaling and super-roughness in the growth of CdTe polycrystalline films

CdTe films grown on glass substrates covered by fluorine doped tin oxide by Hot Wall Epitaxy (HWE) were studied through the interface dynamical scaling theory. Direct measures of the dynamical exponent revealed an intrinsically anomalous scaling characterized by a global roughness exponent $\alpha$ distinct from the local one (the Hurst exponent $H$), previously reported [Ferreira \textit{et al}., Appl. Phys. Lett. \textbf{88}, 244103 (2006)]. A variety of scaling behaviors was obtained with varying substrate temperature. In particular, a transition from a intrinsically anomalous scaling regime with $H\ne\alpha<1$ at low temperatures to a super-rough regime with $H\ne\alpha>1$ at high temperatures was observed. The temperature is a growth parameter that controls both the interface roughness and dynamical scaling exponents. Nonlocal effects are pointed as the factors ruling the anomalous scaling behavior.

Relaxation rate and scaling function of the critical system 3-methylpentane-nitroethane-cyclohexane

The critical system 3-methylpentane-nitroethane-cyclohexane (3-MP-NE-CH) has been investigated and compared to the limiting binary systems 3-MP-NE as well as NE-CH in order to study the degree of renormalization in the critical exponents of the ternary system. The solubility curves of the 3-MP-NE-CH system have been determined at various molar ratios of the nonpolar constituents in order to obtain the plait points as a function of mixture composition. At the col point (the mixture with the lowest transition temperature) and two further plait point compositions shear viscosity, dynamic light scattering, and frequency-dependent ultrasonic attenuation coefficient measurements have been performed as a function of temperature near the critical temperatures. The fluctuation correlation length and the relaxation rate of fluctuations display power law behavior as a function of reduced temperature, with universal critical exponents nu = 0.63 and nuZ(0) = 1.928, respectively, as characteristic for binary critical mixtures. In conformity with the 3-MP-NE and NE-CH critical mixtures the scaling function in the ultrasonic spectra nicely agrees with the empirical scaling function of the Bhattacharjee-Ferrell dynamic scaling theory. Hence with respect to power laws and scaling the 3-MP-NE-CH system behaves like a quasibinary mixture. The individual amplitudes of the relaxation rate show a minimum at the col point composition, corresponding with a maximum in the background viscosity of the liquids. The amount of the adiabatic coupling constant g, derived from the amplitudes in the ultrasonic spectra, increases monotonously when going from NE-CH (/g/ = 0.1) to 3-MP-NE (/g/ = 0.26).

Kinetic mechanism of the thermal-induced self-organization of Au/Si nanodroplets on Si(100): Size and roughness evolution

Very thin Au layer was deposited on Si(100) using the sputtering technique. By annealing at 873 K Au/Si nanodroplets were formed and their self-organization was induced changing the annealing time. The evolution of droplet size distribution, center-to-center distance distribution, and droplet density as a function of the annealing time at 873 K was investigated by Rutherford backscattering spectrometry, atomic force microscopy (AFM), and scanning electron microscopy. As a consequence of such study, the droplet clustering is shown to be a ripening process of hemispherical three-dimensional structures limited by the Au surface diffusion. The application of the ripening theory allowed us to calculate the surface diffusion coefficient and all other parameters needed to describe the entire process. Furthermore, the AFM measurements allowed us to study the roughness evolution of the sputtered Au thin film and compare the experimental data with the dynamic scaling theories of growing interfaces.