Height Correlation Function Research Articles

Fluctuations of Permeable Interfaces in Water-in-Water Emulsions

The fluctuations of highly permeable interfaces, encountered in phase-separated biopolymer solutions, liposomes, polymersomes, or colloidosomes, are investigated. An expression for the power spectrum of the height correlation function is derived for a multicomponent system, incorporating the effects of mass transfer across the interface, using nonequilibrium thermodynamics. We also derive an expression for the relaxation time of the height correlation function, and calculate the relaxation time for a phase-separated gelatin-dextran-water system. Comparing our expression with the expression for an impermeable interface shows that mass transfer has a significant impact on the relaxation time of the interface.

Scratch test induced shear banding in high power laser remelted metallic glass layers

Laser remelted surface layers of a Cu-based metallic glass forming alloy have been produced with fully amorphous depths up to 350 μm for single track widths of around 1.3 mm and have been checked by transmission of synchrotron radiation. They have been subjected to indentation hardness and scratch testing, and the development of shear bands in both situations has been addressed. During the cross-sectional hardness indentation tests, Vickers values of over 735 HV2 have been found through the depth of the treated layer, and the scratch testing has revealed extremely low friction coefficient values (<0.02 at 10 N in single-pass and 0.02 at 18 N multi-pass regimes against a diamond stylus). The shear band formation has been related to both scratch test speed (strain rate) and load (contact stress) by methods such as atomic force microscopy measurements and subsequent surface roughness characterization by a height–height correlation function.

Nonequilibrium Thermodynamic Model of Water-in-Water Emulsions

The dynamic behavior of highly permeable interfaces in phase-separated biopolymer solutions, liposomes, polymersomes, and colloidosomes is investigated. Using nonequilibrium thermodynamics, an expression for the correlation function of the height of a flat interface is derived for a multicomponent system, incorporating the effects of mass transfer across the interface. In addition, an expression is derived for the relaxation time of the height correlation function. This relaxation time is calculated for a phase-separated gelatin−dextran−water system. Comparing our expression with the expression for an impermeable interface shows that mass transfer has a significant impact on the fluctuations of the interface. At small values for the amplitude, the relaxation of fluctuations is completely dominated by the permeability of the interface. At high values for the amplitude, the relaxation is dominated by the viscosities and densities of the bulk phases. In this regime, the long-time limit of the height correlat...

A unified study of crack propagation in amorphous silica: Using experiments and simulations

Atomistic aspects of dynamic fracture in amorphous silica are investigated with molecular dynamics (MD) simulations. Simulations on amorphous silica were performed for two system sizes, 15 million and 113 million atoms. Crack propagation in these systems is accompanied by nucleation and growth of nanometer scale cavities up to 20 nm ahead of the crack tip. Cavities coalesce and merge with the advancing crack to cause mechanical failure. This scenario was also observed experimentally during stress corrosion ultra-slow fracture of glass using atomic force microscopy (F. Célarié et al., Phys. Rev. Lett. 90 (2003) 075504; S. Prades, D. Bonamy, D. Dalmas, E. Bouchaud, C. Guillot, Int. J. Sol. Struct. 42 (2004) 637). This mechanism has macroscopic consequences in terms of sample life-time and deformation field. The morphology of the fracture surfaces has also been studied by calculating the height–height correlation function. In general experiments reveal two universal roughness exponents, 0.5 for small length scales and 0.8 for large length scales. The MD simulations of the 15 million and 113 million atoms system find the first roughness exponent (0.5), but the second exponent (0.8) occurs over length scales inaccessible to MD simulations. Finally, the 113 million atoms simulation was used to map out the morphology and dynamics of the whole crack front.

AFM analysis of MgB 2 films and nanostructures

For the fabrication of superconducting devices based on nanostructured thin films, the quality of the starting surface is often of crucial importance. For example, the transport properties of superconducting nanobridges are strongly affected by the geometry and the edge definition of the nanostructures. In this work, we report about AFM characterization of magnesium diboride films and nanobridges fabricated in view of application in superconducting electronics. MgB 2 films, obtained by co-deposition method followed by annealing in situ on silicon nitride substrate, have been nanostructured by electron beam lithography and ion milling. The analysis of the AFM images by the height–height correlation function shows that the films have a self-affine smooth textured surface with a RMS roughness of 20 nm. Furthermore, the nanobridges are continuous, with a well-defined geometry and a rounded profile, and the nanostructuration process does not significantly affect the film morphology.

Effect of substrate on surface morphology evolution of Cu thin films deposited by magnetron sputtering

Copper thin films were deposited onto rough Al 2O 3 and Si (100) substrates using R.F. magnetron sputtering technique at room temperature. Surface morphology of thin films was detected by atomic force microscopy (AFM) and was characterized by dynamic scaling theory. The effect of substrate on the surface evolution of thin films was investigated. Roughness exponent α and growth exponent β were obtained by height–height correlation function and root–mean–square (RMS) roughness measurement. The results show that surface roughness of Cu thin films on Al 2O 3 substrates first decrease and then increase with increasing thickness, which can be explained by the contribution of rough substrate surface profile. During the initial deposition, the rough substrate morphology leads to a decrease of the measured growth exponent value, while it has no effect on the measurement of roughness exponent. With regards to the effect of the rough substrate surface profile, α and β of thin films deposited on Al 2O 3 are 0.83 ± 0.05 and 0.32 ± 0.03, respectively, while those of thin films deposited on Si substrates are 0.78 ± 0.02 and 0.37 ± 0.04, which indicates that substrate type has an important influence on surface dynamic evolution.

Statistical characterization of surface morphologies

We implement and assess several statistical methods for rough surface morphology characterization, which can largely be divided into two classes. The first class is based on estimation of two-point, quadratic statistical functions and includes: estimates of the sample autocovariance function, sample height–height correlation (also, structure) function, and periodogram estimate of the surface power spectrum. The second class incorporates estimation of up to the fourth statistical moments of the local curvature on a fixed scale. We apply these methods first on computer-simulated epitaxial surfaces, which permits the characterization using large sets of “data” and rich statistics. Then we deal with real surfaces, whose roughness profiles are measured using atomic force microscopy (AFM). In both cases we infer and discuss scaling properties, degree of anisotropy and deviation from Gaussian distribution of surface heights.

The Investigation of Ferroelectric Domain Behavior Affected by Thin Metallic Electrode

The physical and chemical properties of metallic electrodes on Pb(Zr,Ti)O3 (PZT) are investigated by atomic force microscopy. Pt, a catalyst, tends to change itself into a dielectric oxide. Such a passive layer should cause a depolarization field driving the domain structure into the polystate. Therefore, the effective thickness of the Pt electrode becomes smaller than that of Ir electrode. Thus, we observed an opposite domain contrast around a region covered by thin Pt on PZT, which implied the failure of the screening of the depolarizing field. This multidomain configuration can cause a suppression of the remnant polarization of PZT capacitors. Also, Ir is better than Pt in regard to film smoothness, indicating that the adhesion property of Ir on PZT is superior to that of Pt on PZT. We prove it using height–height correlation function. This study is significant on the scaling issues of ferroelectric capacitors, such as the thickness limit of the electrode and the requirement for high surface smoothness of ferroelectric films.

Anisotropic self-affine properties of experimental fracture surfaces

The scaling properties of post-mortem fracture surfaces of brittle (silica glass), ductile (aluminum alloy) and quasi-brittle (mortar and wood) materials have been investigated. These surfaces, studied far from the initiation, were shown to be self-affine. However, the Hurst exponent measured along the crack direction is found to be different from the one measured along the propagation direction. More generally, a complete description of the scaling properties of these surfaces call for the use of the two-dimensional (2D) height–height correlation function that involves three exponents ζ ≃ 0.75, β ≃ 0.6 and z ≃1.25 independent of the material considered as well as of the crack growth velocity. These exponents are shown to correspond to the roughness, growth and dynamic exponents respectively, as introduced in interface growth models. They are conjectured to be universal.

Multifractal behavior of the Korean stock-market index KOSPI

We investigate multifractality in the Korean stock-market index KOSPI. The generalized qth order height–height correlation function shows multiscaling properties. There are two scaling regimes with a crossover time around t c = 40 min . We consider the original data sets and the modified data sets obtained by removing the daily jumps, which occur due to the difference between the closing index and the opening index. To clarify the origin of the multifractality, we also smooth the data through convolution with a Gaussian function. After convolution we observe that the multifractality disappears in the short-time scaling regime t < t c , but remains in the long-time scaling regime t > t c , regardless of whether or not the daily jumps are removed. We suggest that multifractality in the short-time scaling regime is caused by the local fluctuations of the stock index. But the multifractality in the long-time scaling regime appears to be due to the intrinsic trading properties, such as herding behavior, information outside the market, the long memory of the volatility, and the nonlinear dynamics of the stock market.

Fe(001) on vicinal Au(001): correlation of topography and magnetic surface anisotropy

For an epitaxial, vicinal surface of a magnetic film, the twofold step-induced symmetry about the film normal in the atomic coordination implies a corresponding twofold magnetic surface anisotropy. We report a strong correlation between the in-plane structural anisotropy in the height–height correlation function and the step-induced magnetic anisotropy of epitaxial Fe(001) films deposited on Au surfaces vicinal to the (001) plane, tilted about the in-plane [100]Au or [110]Au axis. Surface topography and crystallography are analysed using scanning tunnelling microscopy and low energy electron diffraction. The 5 × 20 reconstruction of Au is found to be affected by the tilt axis orientation and the vicinal angle. The orientation of the magnetic easy axis is found to be closely related to the anisotropic surface topography.

Evaluation of a rough soil surface description with ASAR-ENVISAT radar data

The input roughness parameters for electromagnetic backscattering modelling need to be accurate to estimate radar measurements correctly over bare soils, particularly in agricultural environments. This paper proposes to evaluate the roughness description in terms of several characterisations through a correlation function using a numerical backscattering model. The experimental database used in this study is based on ASAR-ENVISAT experimental campaigns in the Beauce region (France). Two presentations of the surface height correlation function are proposed in this study. The first one, referred to as the “ α function” fits the experimental correlation functions up to the correlation length, while the second one, the “( a, b) function”, fits the correlation function for scales corresponding to positive values. A relationship is proposed between the rms height of soil surface and the shape of the correlation function. Using the α function, comparisons between radar measurements for high incidence angles and simulations based on the numerical backscattering model (moment method) show a good agreement for soil surfaces with an rms height smaller than 2 cm with medium and high soil moisture.

Nano-Structured Thin Films: A Lorentz Transmission Electron Microscopy and Electron Holography Study

This paper aims at applying advanced transmission electron microscopy (TEM) to functional materials, such as ultra-soft magnetic films for high-frequency inductors, to reveal the structure-property relationship. The ultimate goal is to delineate a more quantitative way to obtain information of the magnetic induction and local magnetization. Nano-crystalline Fe-Zr-N films have been prepared by DC magnetron reactive sputtering with a thickness between 50 and 500 nm. Conventional TEM and selected area diffraction (SAD), reveal crystallites of sizes ranging between 2 and 30 nm. The films showed a granular or hillock type of roughness with an rms amplitude of 5 nm. In particular this paper concentrates on an analysis of phase maps in electron holography and intensity maps in Lorentz transmission electron microscopy including the thickness variation over the sample. For a particular statistical description of the roughness and values for the roughness it is shown that analytical expressions can be obtained. We demonstrate that starting from the concept of the vector potential in classical electrodynamics these results can be achieved assuming independent stationary Gaussian distributions for the height correlation functions.

Simulation of growth process of thin film on non-planar substrate

In (2 + 1) dimension, growth process of thin film on non-planar substrate in Kuramoto–Sivashinsky model is studied with numerical simulation approach. 15 × 15 semi-ellipsoids arranged orderly on the surface of substrate are used to represent initial rough surface. The results show that at the initial stage of growth process, the surface morphology of thin film appears to be grid-structure, and the interface width constantly decreases with the growth time, then reaches minimum. However, the grid-structure becomes ambiguous, and granules of different sizes distribute evenly on the surface of thin film with the increase of growth time. Thereafter, the average size of granules and the interface width gradually increase, and the surface morphology of thin film presents fractal properties. The numerical results of height–height correlation functions of thin film verify the surface morphology of thin film to be fractal for a longer growth time. By fitting of the height–height correlation functions of thin film with different growth times, the growth process is described quantitatively.

Line edge roughness and critical dimension variation: Fractal characterization and comparison using model functions

Line edge (or width) roughness (LER or LWR) of photoresists lines constitutes a serious issue in shrinking the critical dimensions (CD) of the gates to dimensions of a few tens of nanometers. In this article, we address the problem of the reliable LER characterization as well as the association of LWR with the CD variations. The complete LER characterization requires more parameters than the rms value σ since the latter neglects the spatial aspects of LER and does not predict the dependence on the length of the measured line. The further spatial LER descriptors may be the correlation length ξ and the roughness exponent α, which can be estimated through various methods. One aim of the present work is to perform a systematic comparative study of these methods using model edges generated by a roughness algorithm, in order to show their advantages and disadvantages for a reliable and accurate determination of the spatial LER parameters. In particular, we compare the results from (a) the study of the height–height correlation function (HHCF), (b) the Fourier [or power spectrum (PS)] analysis, and (c) the variation of rms value σ with measured line edge L [σ(L) curve]. It is found that the HHCF can be considered approximately a rescaled version of σ(L) and that the value of σ becomes almost independent of the measured edge length for lengths larger than ten times the correlation length. As regards the PS, it is shown that the finite length of the edge may harmfully affect the reliable estimation of α and ξ. Finally, we confirm theoretically and generalize an experimental observation [Leunissen et al., Microelectron. Eng. (to be published)] regarding the relationship between LWR and the σ of the CD variations within a die of a wafer. It is shown that they behave in a complimentary way as line length increases so that the sum of their squares remains constant and equal to the square of the LWR σ of the infinite line.

Control of surface roughness in amorphous thin-film growth

The growth of optical layers of SiO2 and Nb2O5 on amorphous substrates is investigated. We develop a kinetic Monte Carlo model which mimics the amorphous structure by randomly fluctuating binding energies. The resulting surface profiles are characterized by their root-mean-square roughness, height–height correlation functions, and growth exponents. For strong random fluctuations, the growth exponents exceed the value of 0.5, in good agreement with experiment.

Low energy Ar + ion beam induced kinetic roughening of thin Pt films on a Si substrate

A 30 nm Pt thin film evaporated onto a Si wafer was sputtered by 9.7 keV oblique Ar + bombardment at various ion fluences. The evolution of the modified sputtered films was monitored by atomic force microscopy (AFM) and high resolution scanning electron microscopy (HRSEM). The most interesting observation was the formation of mound-like structures on the metal surface. Morphological data were quantitatively analysed within the framework of the dynamic scaling theory. Analyses of the height–height correlation function for different ion fluences yield roughness exponents α in the range 0.65–0.87, while the root-mean-square roughness amplitude w evolves with the ion fluence φ as a power law w∝ φ β , with the growth exponent, β=0.36. The results are discussed in terms of the existing continuum models developed for the ion sputtering process.

Microstructure, surface morphology, and mechanical properties of nanocrystalline TiN/amorphous Si3N4 composite films synthesized by ion beam assisted deposition

Two-phase nanocrystalline TiN/amorphous Si3N4 composite films were synthesized at room temperature by ion beam assisted deposition. Film bonding states and microstructure were investigated by x-ray photoelectron spectroscopy, x-ray diffraction, and transmission electron microscopy. Surface morphology was measured by atomic force microscopy and quantitatively analyzed using height–height correlation function. The mechanical properties were evaluated from nanoindentation measurements. The films consisted of nanometer-sized TiN crystallites embedded in an amorphous Si3N4 matrix. A maximum hardness of about 42 GPa was observed in a film with silicon content of 11.4 at. %. The improved mechanical properties of Ti–Si–N films with the addition of Si into TiN were attributed to their densified microstructure with development of fine grain size and reduced surface roughness. The combination of hardness and grain size was correlated with film phase composition and structural analyses. The results have been interpreted with the help of a Monte Carlo Potts model simulation.

Interfacial correlation function for adsorption on a disc

Within the Landau–Ginzburg–Wilson approach we derive the effective Hamiltonian governing fluctuations of an interface between two phases, one of them adsorbed on a disc-like substrate. For large disc radii and for temperatures close to the wetting temperature of the corresponding planar substrate the expressions for the height–height correlation function and the accompanying correlation angle are derived. Their dependence on the disc radius and the reduced temperature is discussed and the relevant scaling regimes are identified.

Quantitative description of surface morphology evolution in a growing film

A quantitative description of the surface morphology evolution in thin film growing via 3D island mechanism is presented. The treatment is based on the calculation of the survival probability for any point on (or above) a substrate during growth process. The paraboloid growth model is considered. In this model, the lateral growth of an island occurs in a circle form via capture of migrating adatoms, while its vertical growth proceeds at constant velocity owing to direct impingement of deposited atoms, so that a steady (in time) and random (in space) surface relief consisting of paraboloid growth hillocks is developed eventually. For this surface relief, forming at paraboloid growth, the height–height autocorrelation function is derived for different hillock shape and different nucleation mode generating different hillock space distribution. It is found that the hillock shape strongly affects on the overall surface roughness and the hillock space distribution influences on both the autocorrelation function form and the rms roughness. The surface height distribution is asymmetric, non-Gaussian with negative skewness. The random rough surface of thin film with surely non-Gaussian height distribution is characterized by close-to-Gaussian height–height correlation function.