From bell shapes to pyramids: A reduced continuum model for self-assembled quantum dot growth

A continuum model for the growth of self-assembled quantum dots that incorporates surface diffusion, an elastically deformable substrate, wetting interactions and anisotropic surface energy is presented. Using a small slope approximation a thin-film equation for the surface profile that describes faceted growth is derived. A linear stability analysis shows that anisotropy acts to destabilize the surface. It lowers the critical height of flat films and there exists an anisotropy strength above which all thicknesses are unstable. A numerical algorithm based on spectral differentiation is presented and simulations are carried out. These clearly show faceting of the growing islands and a power law coarsening behavior.     

Renormalization group study of Mullins' equation for molecular beam epitaxy with conserved noise

The dynamics of driven interfaces under conserved noise in a continuum model of growth by a molecular beam has been studied by means of the Noziéres-Gallet dynamic renormalization group technique, using the results of Sun and Plischke for the case of non-conserved noise. Relaxation of the growing film is due to both surface tension and surface diffusion. In (1 + 1) dimensions, four growth regimes have been found. None of these are purely diffusive. One of these fixed points has negative surface tension and is stable with respect to renormalization group flow. This is an unstable growth state in which the creation of large slopes in the interface configuration is expected. In (2 + 1) dimensions, seven growth regimes have been found, in which three are purely diffusive. There is also one fixed point with a negative surface tension. However, this fixed point is unstable with respect to renormalization group flow, and is therefore expected to crossover into the other growth regimes at large system size and long times.     

Revisiting surface diffusion in random deposition

An investigation of the effect of surface diffusion in random deposition model is made by analytical methods and reasoning. For any given site, the extent to which a particle can diffuse is decided by the morphology in the immediate neighbourhood of the site. An analytical expression is derived to calculate the probability of a particle at any chosen site to diffuse to a given length, from first principles. This method may become particularly important in cases where obtaining the continuum limit and solving the corresponding differential equation may not be feasible. Numerical simulation of surface diffusion in random deposition model with varying extents of diffusion are performed and their results are interpreted in the light of the analytical calculations. Systems with surface diffusion show an initial random deposition-like growth upto monolayer deposition, then a deviation due to correlation effects and eventual saturation. An explanation for this behaviour is discussed and the point of departure from the linear form is estimated analytically.     

含关联噪声的空间分数阶随机生长方程的动力学标度行为研究

为探讨含关联噪声的空间分数阶随机生长方程的动力学标度行为,本文利用Riesz分数阶导数和Grümwald-Letnikov分数阶导数定义方法研究了关联噪声驱动下的空间分数阶Edwards-Wilkinson (SFEW)方程在1+1维情况下的数值解,得到了不同噪声关联因子和分数阶数时的生长指数、粗糙度指数、动力学指数等,所求出的临界指数均与标度分析方法的结果相符合.研究表明噪声关联因子和分数阶数均影响到SFEW方程的动力学标度行为,且表现为连续变化的普适类.     

Effect of symmetry on volume conserving surface models

We study the effect of symmetry on volume conserving models without deposition and evaporation. By using the master equation approach, we identify two types of stochastic continuum equation with a conservative noise, depending on the symmetry of hopping rate in diffusion rules. In the model with symmetric hopping rate, a Laplacian term is essentially absent from the continuum equation. The dynamic scaling of this model is thus determined by the nonlinear fourth order equation with a conservative noise. When the symmetry is broken, a Laplacian term may be present, so the asymptotic scaling behavior is governed by the Laplacian term with nonzero coefficient. We verify this result by investigating a simple discrete model analytically.     

Jeans type analysis of chemotactic collapse

We perform a linear dynamical stability analysis of a general hydrodynamic model of chemotactic aggregation [P.H. Chavanis, C. Sire, Physica A 384 (2007) 199]. Specifically, we study the stability of an infinite and homogeneous distribution of cells against “chemotactic collapse”. We discuss the analogy between the chemotactic collapse of biological populations and the gravitational collapse (Jeans instability) of self-gravitating systems. Our hydrodynamic model involves a pressure force which can take into account several effects like anomalous diffusion or the fact that the organisms cannot interpenetrate. We also take into account the degradation of the chemical which leads to a shielding of the interaction like for a Yukawa potential. Finally, our hydrodynamic model involves a friction force which quantifies the importance of inertial effects. In the strong friction limit, we obtain a generalized Keller–Segel model similar to the generalized Smoluchowski–Poisson system describing self-gravitating Langevin particles. For small frictions, we obtain a hydrodynamic model of chemotaxis similar to the Euler–Poisson system describing a self-gravitating barotropic gas. We show that an infinite and homogeneous distribution of cells is unstable against chemotactic collapse when the “velocity of sound” in the medium is smaller than a critical value. We study in detail the linear development of the instability and determine the range of unstable wavelengths, the growth rate of unstable modes and the damping rate, or the pulsation frequency, of the stable modes as a function of the friction parameter and shielding length. For specific equations of state, we express the stability criterion in terms of cell density.     

Kinetic Monte-Carlo simulation of the homoepitaxial growth of MgO{001} thin films by molecular deposition

A lattice-based kinetic Monte-Carlo (KMC) code has been developed to investigate the MgO{001} crystal growth from deposition of MgO molecules, as a prototypical case of the growth of oxide thin films. The KMC approach has been designed on the basis of an extensive database including all possible diffusion mechanisms. The corresponding activation energies have been computed through first-principles calculations at zero temperature or from Arrhenius plots of the frequencies obtained by molecular dynamics simulations with empirical potentials. Crystal growth occurs layer by layer, as experimentally observed, and the diffusion of admolecules leads to a high capacity of nucleation, which is enhanced by vacancy diffusion. We have characterized the growth through surface roughness, size distribution and density of the islands, and filling ratios of the growing layers. Moreover, we have analysed the influence of each elementary mechanism on the growth. The best quality of the deposited layers is reached for temperatures larger than 700 K and for pressures smaller than 0.1 Torr. For these conditions, the simulated surface roughness is fully consistent with available experimental results.     

Dirac–Kähler Equation

Tensor, matrix, and quaternion formulations of Dirac–Kähler equation for massive and massless fields are considered. The equation matrices obtained are simple linear combinations of matrix elements in the 16-dimensional space. The projection matrix-dyads defining all the 16 independent equation solutions are found. A method of computing the traces of 16-dimensional Petiau–Duffin–Kemmer matrix product is considered. We show that the symmetry group of the Dirac–Kähler tensor fields for charged particles is SO(4, 2). The conservation currents corresponding this symmetry are constructed. We analyze transformations of the Lorentz group and quaternion fields. Supersymmetry of the Dirac–Kähler fields with tensor and spinor parameters is investigated. We show the possibility of constructing a gauge model of interacting Dirac–Kähler fields where the gauge group is the noncompact group under consideration.     

Roughness measurement of metallic surfaces based on the laser speckle contrast method

A method of measuring surface roughness of flat lapped, ground and polished metallic surfaces, by the far-field speckle contrast method is presented in this paper. The laser speckle contrast technique depends on the existence of an approximately linear relationship between the speckle contrast and the roughness of the illuminated surface. Initially it was shown that the linear relationship existed up to 0.1 μm Ra (centre-line average) roughness using Helium–Neon light, after which a saturation effect was observed. The effect of varying the incident angle of illumination was investigated with a view to extending the measurement range. The use of high incident angles of illumination has been found to increase the surface roughness range up to 0.4 μm measurement Ra.     

Atomic force microscopy study of growth kinetics: Scaling in TiN-TiB2 nanocomposite films on Si(1 0 0)

We used the reactive unbalanced close-field dc-magnetron sputtering growth of TiN-TiB₂ on Si(1 0 0) at room temperature to determine if scaling theory provides insight into the kinetic mechanisms of two-phase nanocomposite thin films. Scaling analyses along with height-difference correlation functions of measured atomic force microscopy (AFM) images have shown that the TiN-TiB₂ nanocomposite films with thickness ranging from 70 to 950 nm exhibit a kinetic surface roughening with the roughness increasing with thickness exponentially. The roughness exponent α and growth exponent β are determined to be ∼0.93 and ∼0.25, respectively. The value of dynamic exponent z, calculated by measurement of the lateral correlation length ξ, is ∼3.70, agreeing well with the ratio of α to β. These results indicate that the surface growth behavior of sputter-deposited TiN-TiB₂ thin films follows the classical Family-Vicseck scaling and can be reasonably described by the noisy Mullins diffusion model, at which surface diffusion serves as the smoothing effect and shot noise as the roughening mechanism.     

Kinetic Monte Carlo simulation of thin film growth

A three-dimensional kinetic Monte Carlo technique has been developed for simulating growth of thin Cu films. The model involves incident atom attachment, diffusion of the atoms on the growing surface, and detachment of the atoms from the growing surface. The related effect by surface atom diffusion was taken into account. A great improvement was made on calculation of the activation energy for atom diffusion based on a reasonable assumtion of interaction potential between atoms. The surface roughness and the relative density of the films were simulated as the functions of growth substrate temperature and film thickness. The results showed that there exists an optimum growth temperatureT _opt at a given deposition rate. When the substrate temperature approaches toT _opt, the growing surface becomes smoothing and the relative density of the films increases. The surface roughness minimizes and the relative density saturates atT _opt. The surface roughness increases with an increment of substrate, temperature when the temperature is higher thanT _opt.T _opt is a function of the deposition rate and the influence of the deposition rate on the surface roughness depends on the substrate temperatures. The simulation results also showed that the relative density decreases with the increasing of the deposition rate and the average thickness of the film.     

Dynamics and kinetic roughening of interfaces in two-dimensional forced wetting

We consider the dynamics and kinetic roughening of wetting fronts in the case of forced wetting driven by a constant mass flux into a 2D disordered medium. We employ a coarse-grained phase field model with local conservation of density, which has been developed earlier for spontaneous imbibition driven by capillary forces. The forced flow creates interfaces that propagate at a constant average velocity. We first derive a linearized equation of motion for the interface fluctuations using projection methods. From this we extract a time-independent crossover length ξ_×, which separates two regimes of dissipative behavior and governs the kinetic roughening of the interfaces by giving an upper cutoff for the extent of the fluctuations. By numerically integrating the phase field model, we find that the interfaces are superrough with a roughness exponent of χ= 1.35 ±0.05, a growth exponent of β= 0.50 ± 0.02, and ξ_×∼v^-1/2 as a function of the velocity. These results are in good agreement with recent experiments on Hele-Shaw cells. We also make a direct numerical comparison between the solutions of the full phase field model and the corresponding linearized interface equation. Good agreement is found in spatial correlations, while the temporal correlations in the two models are somewhat different.     

Kinetic Models for Granular Flow

The generalization of the Boltzmann and Enskog kinetic equations to allow inelastic collisions provides a basis for studies of granular media at a fundamental level. For elastic collisions the significant technical challenges presented in solving these equations have been circumvented by the use of corresponding model kinetic equations. The objective here is to discuss the formulation of model kinetic equations for the case of inelastic collisions. To illustrate the qualitative changes resulting from inelastic collisions the dynamics of a heavy particle in a gas of much lighter particles is considered first. The Boltzmann–Lorentz equation is reduced to a Fokker–Planck equation and its exact solution is obtained. Qualitative differences from the elastic case arise primarily from the cooling of the surrounding gas. The excitations, or physical spectrum, are no longer determined simply from the Fokker–Planck operator, but rather from a related operator incorporating the cooling effects. Nevertheless, it is shown that a diffusion mode dominates for long times just as in the elastic case. From the spectral analysis of the Fokker–Planck equation an associated kinetic model is obtained. In appropriate dimensionless variables it has the same form as the BGK kinetic model for elastic collisions, known to be an accurate representation of the Fokker–Planck equation. On the basis of these considerations, a kinetic model for the Boltzmann equation is derived. The exact solution for states near the homogeneous cooling state is obtained and the transport properties are discussed, including the relaxation toward hydrodynamics. As a second application of this model, it is shown that the exact solution for uniform shear flow arbitrarily far from equilibrium can be obtained from the corresponding known solution for elastic collisions. Finally, the kinetic model for the dense fluid Enskog equation is described.     

Coarsening process in one-dimensional surface growth models

Surface growth models may give rise to instabilities with mound formation whose typical linear size L increases with time (coarsening process). In one dimensional systems coarsening is generally driven by an attractive interaction between domain walls or kinks. This picture applies to growth models for which the largest surface slope remains constant in time (corresponding to model B of dynamics): coarsening is known to be logarithmic in the absence of noise ( L(t) ∼ ln t) and to follow a power law ( L(t) ∼t ^1/3) when noise is present. If the surface slope increases indefinitely, the deterministic equation looks like a modified Cahn-Hilliard equation: here we study the late stages of coarsening through a linear stability analysis of the stationary periodic configurations and through a direct numerical integration. Analytical and numerical results agree with regard to the conclusion that steepening of mounds makes deterministic coarsening faster : if α is the exponent describing the steepening of the maximal slope M of mounds ( M ^α∼L) we find that L(t) ∼t ⁿ: n is equal to for 1≤α≤2 and it decreases from to for α≥2, according to n = α/(5α - 2). On the other side, the numerical solution of the corresponding stochastic equation clearly shows that in the presence of shot noise steepening of mounds makes coarsening slower than in model B: L(t) ∼t ^1/4, irrespectively of α. Finally, the presence of a symmetry breaking term is shown not to modify the coarsening law of model α = 1, both in the absence and in the presence of noise. Received 28 September 2001 and Received in final form 21 November 2001     

Effect of the amorphous thin layer on the surface growth of amorphous/crystalline binary multilayer films

The effect of the amorphous thin layer on the surface growth of amorphous/crystalline binary multilayer films has been studied by using a continuum model. It is shown that both the surface roughness and the growth exponent of amorphous/crystalline binary multilayer films decrease with increasing thickness ratio between amorphous and crystalline layers. Our simulations have also revealed, in contrast to the monotonous rise in surface roughness observed in single-layer films grown on flat substrates, the surface growth of a multilayer film consists of two processes: interface smoothing and roughening, namely the film roughness decreases during the growth of amorphous thin layers but increases monotonously during the growth of crystalline thin layers. The observed interface smoothing and roughening can be obviously influenced by the change in the thickness ratio between amorphous and crystalline layers. The rise in thickness ratio between amorphous and crystalline layers enhances the interface smoothing effect but lowers the interface roughening effect and consequently shows a marked smoothing effect on the surface roughness.     

Non-monotonous dependence of surface roughness on factors influencing energy of adatoms during thin island film growth

Many experimental results show that surface roughness of thin films can increase, decrease, stay constant or pass through the minimum with the change in substrate temperature, energy of arriving atoms or assisted beam (electrons, photons, ions), depending on material and interval of variation of those parameters. The aim of this paper is to explain and analyze this non-monotonous behavior of surface roughness by proposed kinetic model. The model is based on rate equations and includes processes of surface diffusion of adatoms, nucleation, growth and coalescence of islands in the case of thin films growth in Volmer-Weber mode. It is shown by modeling that non-monotonous dependence of surface roughness on the factors influencing energy of adatoms (e.g. temperature, assisted beam irradiation, accelerating voltage) occurs as a result of interplay between diffusion length of adatoms and size of islands, because both parameters depend on energy of adatoms. Variation of island size and diffusion length results in atomic jumps from islands forming rougher or smoother surface. The functions of surface roughness, island size, island density on diffusion length of adatoms and on other parameters are calculated and analyzed in this work.     

各向异性表面张力对定向凝固中共晶生长形态稳定性的影响

研究各向异性表面张力对定向凝固中共晶生长形态稳定性的影响.应用多重变量展开法导出了共晶界面表达式和扰动振幅的变化率满足的色散关系.结果表明,共晶生长系统有两种整体不稳定性机理:由非震荡导致的"交换稳定性"机理和由震荡导致的"整体波动不稳定性"机理.震荡有四种典型模式,即:反对称-反对称(AA-),对称-反对称(SA-)、反对称-对称(AS-)和对称-对称(SS-)模式.稳定性分析表明:共晶界面形态稳定性取决于Peclet数ε的某一个临界值ε_*,当ε大于临界值ε_*时,共晶界面形态不稳定;当ε小于临界值ε_*时,共晶界面形态稳定.随着各向异性表面张力增大,对应于AA-,SA-和SS-模式的临界值ε_(aa*),ε_(sa*)和ε_(ss*)随之减小,表明各向异性表面张力减小这三种模式的稳定性区域;然而,随着各向异性表面张力增大,对应于AS-模式的临界值ε_(as*)随之增大,表明各向异性表面张力增大AS-模式的稳定性区域.     

Model and theory for the determination of diffusion coefficients by Auger electron spectroscopy measurements and an application to oxygen diffusion along the [0001] and [101¯0] axes in single crystal zirconium

A simple hopping model of the diffusion of adsorbed species from a surface into the bulk of a material has been formulated and solved mathematically. The difference in the energy barriers for an atom moving between the atomic layers at the surface and in the bulk are explicitly considered. This model is also capable of describing the initial stages of diffusion, something that conventional solutions of the continuum diffusion equation cannot handle. Auger electron spectroscopy has been used to measure the dissolution rate of oxygen from Zr(0001) and Zr(101¯0) surface into the bulk. Satisfactory results were obtained by applying our model to the diffusion data for these two zirconium surfaces for two different heating schedules: (i) rapid temperature ramp-and-hold and (ii) continuous linear heating with respect to time. The resulting Arrhenius expressions for diffusion are: D = (0.115 ± 0.031)exp[(−44.45 ± 4.82)kcal/RT]cm²/s along Zr[0001] and D = (1.07 ± 0.26)exp[(−46.18 ± 4.22)kcal/RT]cm²/s along Zr[101¯0].     

The effect of Ehrlich-Schwoebel step-edge barrier on the formation of self-organized Si nanodots by ion-sputter erosion

The ion flux dependence of the self-organized Si nanodots induced by 1.5 keV Ar⁺ ion sputter erosion has been studied. It shows that for the regime with ion flux >∼280 μA/cm², the currently adopted Bradley-Harper (BH) model, which is incorporated in a dynamic continuum equation holds valid. However, for ion flux <∼280 μA/cm², the measured dot size and surface roughness deviate drastically from the BH model. To interpret the data for this lower ion flux regime, the effect of the Ehrlich-Schwoebel (ES) step-edge barrier was introduced into the continuum equation. A consistency between the calculated and the experimental results was reached, furthermore, a reasonable trend was found, that is, the effective ES diffusion decreases steadily with the increasing ion flux, and at ∼280 μA/cm², it became negligibly small.