Layer Growth Model Research Articles

Nonequilibrium lattice models of epitaxial growth

We illustrate the use of the Chapman-Kolmogorov and Master equations as a frame-work for solving models of epitaxial growth. Within this probabilistic approach, we specify rules governing the incorporation and migration of individual atoms, which then determine the transition rates between surface configurations. The Chapman-Kolmogorov and Master equations are then solved for either the full probability distribution of surface configurations, or for a reduced distribution function. Several idealized models for epitaxial growth are formulated and solved based upon rules for adatom incorporation into the growth front. Atoms are immobile once attached to the surface. The models studied are: (i) the random deposition model, wherein incident atoms attach to the point of initial contact with the surface; (ii) the perfect layer growth model, in which incident atoms are incorporated into the highest unfilled layer; (iii) the local perfect layer growth model, where incident atoms are incorporated into the highest unfilled layer within subsections of an adjustable size; and (iv) the local island model, which takes into account the tendency of an adsorbing atom to select sites with nearest-neighbor atoms. Inclusion of this modification results in the growth of square or rectangular islands whose size increases with the size of a subsection. These models are exactly solvable using the Chapman-Kolmogorov equation, yet some exhibit complex behavior, including decaying oscillations in the diffracted intensity and non-trivial spatial correlations within the surface. The mechanism for the damping of the oscillations is shown to be phase incoherence between growth occurring in different subsections, which is also manifested by an increasing surface roughness as measured by the root-mean-square distribution of occupied layers. Examining the relationship between the evolutions of the step density and kinematic reflection high-energy electron-diffraction (RHEED) intensities in these models underscores the role of surface correlations in determining the profile of measured RHEED intensities.

Boriding of Fe and Fe–C, Fe–Cr, and Fe–Ni alloys; Boride-layer growth kinetics

Specimens of pure Fe and of Fe-0.8 mass % C, Fe-0.5 mass % Cr, Fe-4.0 mass % Cr, Fe-4.0 mass% Ni, and Fe-10.0 mass% Ni alloys were borided in boriding powder. A boron-compound layer developed consisting of a surface-adjacent “FeB” sublayer on top of an “Fe2B” sublayer. Layer-growth kinetics were analyzed by measuring the extent of penetration of the “FeB” and “Fe2B” sublayers as a function of boriding time and temperature in the range 1025–1275 K. Layer growth is dominated by B diffusion through “FeB/Fe2B”. This diffusion process is of strongly anisotropic nature. Consequently, ragged interfaces occur between the substrate and the boride layers. The depths of the tips of the most deeply penetrated “FeB” and “Fe2B” needles have been taken as measures for diffusion in the easy [001] diffusion directions. Assuming unidirectional B diffusion and parabolic growth, a simple model of layer growth has been given. It accounts for the specific volume difference between “FeB” and “Fe2B”. In contrast with earlier work, the model provides values for the kinetic parameters for growth of each of the phases in the boron-compound layer.

Low-angle X-ray diffraction study of solid state amorphization in an [formula omitted] multilayer

Successive stages in the annealing of an Ni Ti multilayer at 523 K were studied by X-ray diffraction at small scattering angles. For a correct interpretation the Darwin-Prins dynamic scattering theory was generalized by removing the discretization errors. The problem of extracting a composition profile from a diffraction pattern and the influence of diffusion on a diffraction pattern are discussed. Although a simple layer growth model did not explain the experimental diffraction patterns in detail, the results indicate that planar layer growth is the dominant mechanism for amorphization: sharp interfaces and a regular periodicity are maintained throughout all stages of annealing. The first internal Ni Ti interface remained inactive.

Differential scanning calorimetry studies of solid state amorphization in multilayer [formula omitted

The reduction in free energy which accompanies solid state amorphization arises primarily from the negative enthalpy of mixing the elements which constitute the diffusion couple. This evolution of enthalpy enables the amorphization reaction to be analysed using differential scanning calorimetry (DSC) and quantitative thermodynamic and kinetic data to be obtained. We present a diffusion-controlled layer growth model for computing DSC traces given by sputtered thin film Ni Zr diffusion couples. We also examine mechanisms which can affect the layer thickness of amorphous alloy formed at interfaces in d.c. magnetron sputtered Ni Zr multilayers during deposition. We discuss phenomena which may cause decreases in the rate of reaction during annealing of Ni Zr multilayers with large modulation wavelengths.

A diffusion kinetic model for epitaxial growth of gallium arsenide layers from the gas phase

The growth model of gallium arsenide epitaxial layers has been further extended in two ways. First of all, the location of the substrate in experiments in the reactor has been taken into account, and a general relationship has been derived for dependence of the mean effective thickness of the boundary layer on the distance of the substrate from the leading edge and on the substrate size. Secondly, the effect of rate of the chemical heterogeneous reaction has been investigated in addition to the diffusion rate on the resulting growth velocity of epitaxial gallium arsenide layers. A quantitative model comprising the rates of the both partial processes has been formulated. The theoretical relationships obtained in this way have been confronted with experimental results.

A dual equilibrium diffusion model for epitaxial growth of gallium arsenide layers from the gas phase and an a priori computation of growth rates

A model is proposed and quantitatively treated of epitaxial growth of gallium arsenide layers, where the rate controlling step consists in the diffusion of reactants through a stagnant gas film adhering to the substrate, and where chemical equilibria are established between the reactants in the main gas stream and at the surface of substrate. The boundary layer theory is applied to the hydrodynamic part of the model which is simplified by introducing a mean effective film thickness, and the system of Ga-As-Cl-H is reduced to six molecular species and to three chemical reactions. With this basis and using estimated values of diffusion coefficients, the growth rates of epitaxial gallium arsenide layers have been a priori computed in dependence of the feed rate, its composition and on temperature. The predicted three dependences are discussed from the view-point of their courses and of the significance of computed results.

Simulation of slider-induced convection in horizontal LPE slider system

The control of thickness uniformity and compositional homogeneity is extremely important in the growth of high quality heterostructure lasers. Slider-induced convection within the growth solution in the horizontal LPE system can affect both of these factors and has been largely neglected in the modeling of layer growth. A plexiglas boat with water and dyes was used to simulate the fluid motions in the melts of the LPE horizontal slider system. The simulation shows that there are substantial fluid motions induced by the movement (push) of the slider. The duration of the convective cell thus generated can be comparable to the growth time of thin layers. Therefore, a substantial portion of the growth time for thin layers is under non-steady-state conditions. Furthermore, convection dominates in the initial period of epitaxial growth. The dependence of the convective cell on melt geometry is discussed. A video tape was made showing the solution dynamics of several boat designs used in the growth of (GaAl)As heterostructure lasers.

Mathematical modeling of diffusion during multiphase layer growth

Finite difference techniques have been applied to the modeling of multiphase layer growth for the reaction between Ni wires embedded in an Al matrix. The mathematical model considered the formation of Ni2Al3 and NiAl3 phase layers and the sequence of growth and dissolution processes involved in the attainment of an equilibrium state of NiAl3 rods in a Ni-saturated Al matrix. A means for establishing effective diffusion coefficients and solubility limits for intermetallic phases where this information is missing or incomplete has been developed. The approach is predicated upon employing readily obtained phase-layer growth data in lieu of the unknown information.

LPE growth of InP thin layers from supercooled solutions by the two-phase technique

A model for LPE growth of thin layers of InP from supercooled solutions by the two-phase technique is presented. This model relative to the case where the melt is entirely covered by a crystal source is based on the diffusion limited theory for growth and takes into account the temperature dependence of the solute diffusion coefficient. Calculations are performed in two steps: (1) determination of the exact solute concentration profile in the melt before the growth. (2) calculation of the deposited film thickness on the substrate versus time. Theoretical results are given for various sets of experimental parameters: growth temperature, cooling rate, melt thickness, time duration needed for controlling both supercooling and growth. It is demonstrated that this technique enables the control of thin layers grown under moderate supercooled conditions, by means of a proper choice of experimental parameters. Experimental results for InP and quaternary alloys show a fairly good agreement with this model.

The turbulent heat flux from arctic leads

The turbulent transfer of heat from Arctic leads in winter is one of the largest terms in the Arctic heat budget. Results from the AIDJEX Lead Experiment (ALEX) suggest that the sensible component of this turbulent heat flux can be predicted from bulk quantities. Both the exponential relation N = 0.14R x 0.72 and the linear relation N = 1.6 × 10−3 R x+ 1400 fit our data well. In these, N is the Nusselt number formed with the integrated surface heat flux, and R x is the Reynolds number based on fetch across the lead. Because of the similarity between heat and moisture transfer, these equations also predict the latent heat flux. Over leads in winter, the sensible heat flux is two to four times larger than the latent heat flux. The internal boundary layer (IBL) that develops when cold air encounters the relatively warm lead is most evident in the modified downwind temperature profiles. The height of this boundary layer, δ, depends on the fetch, x, on the surface roughness of the lead, z 0 and on both downwind and upwind stability. A tentative, empirical model for boundary layer growth is % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWcaaqaaiabes% 7aKbqaaiaadQhadaWgaaWcbaGaaGimaaqabaaaaOGaeyypa0JaeqOS% di2aaeWaaeaacqGHsisldaWcaaqaaiaadQhadaWgaaWcbaGaaGimaa% qabaaakeaacaWGmbaaaaGaayjkaiaawMcaamaaCaaaleqabaGaaGim% aiaac6cacaaI4aaaaOWaaeWaaeaadaWcaaqaaiaadIhaaeaacaWG6b% WaaSbaaSqaaiaaicdaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqa% baGaaGimaiaac6cacaaI0aaaaaaa!472D!\[\frac{\delta }{{z_0 }} = \beta \left( { - \frac{{z_0 }}{L}} \right)^{0.8} \left( {\frac{x}{{z_0 }}} \right)^{0.4} \] where L is the Obukhov length based on the values of the momentum and sensible heat fluxes at the surface of the lead, and Β is a constant reflecting upwind stability. Velocity profiles over leads are also affected by the surface nonhomogeneity. Besides being warmer than the upwind ice, the surface of the lead is usually somewhat rougher. The velocity profiles therefore tend to decelerate near the surface, accelerate in the mid-region of the IBL because of the intense mixing driven by the upward heat flux, and rejoin the upwind profiles above the boundary layer. The profiles thus have distinctly different shapes for stable and unstable upwind conditions.

The anodic oxidation of thiocyanate ion dissolved as KSCN in dimethylsulphoxide

The anodic oxidation of SCN − ion dissolved as KSCN in dimethylsulphoxide, on platinum electrodes, was investigated at ca 25, 60 and 160°C by means of various non-stationary electrochemical techniques. At low temperatures one anodic and two main cathodic processes were found. The anodic oxidation of SCN − ion yields as primary product the SCN radical, which readily produces (SCN) 2. The latter can be in part reduced back to SCN − ion, because it reacts in part yielding solvated hydrogen ions which cause the second cathodic reaction. In the region of 60°C, no (SCN) 2 cathodic current is observed. In the region of 160°C the only reaction is SCN − ion oxidation and the primary product polymerizes to (SCN) x , which forms a film on the electrode, causing passivation. On the basis of kinetic data obtained for the different reactions, a mechanism for the anodic oxidation of the SCN − ion and for the passivating film formation is suggested. The second process only partly fits the Müller model for the electrochemical growth of insoluble layers.