Diffusion, lattice

The relevance of crystal faces to the subject of electrociystalhzation comes up as follows Each of the crystal faces just described contains all the microfeatures that have been described in previous sections, steps, kinks, etc. Further, the same phenomena of deposition—the ions crossing the electrified interface to form adions, the surface diffusion, lattice incorporation of adions, screw dislocation, growth spirals, etc.—occur on all the facets. 

Stochastic approximations such as random walk or molecular chaos, which treat the motion as a succession of simple one- or two-body events, neglecting the correlations between these events implied by the overall deterministic dynamics. The analytical theory of gases, for example, is based on the molecular chaos assumption, i.e. the neglect of correlations betweeen consecutive collision partners of the same molecule. Another example is the random walk theory of diffusion in solids, which neglects the dynamical correlations between consecutive jumps of a diffusing lattice vacancy or interstitial. 

Pore migration mechanisms include surface diffusion, lattice diffusion, gas diffusion and evaporation/condensation, as shown schematically in Figure 11.1. Under a capillary driving force of grain boundary migration, the atoms in front of the moving boundary around the pore are under compression while those behind the boundary are under tension. This pressure distribution results in a... 

To make possible efficient explorations of the dependence of interfacial dynamics on microscopic interactions, we have recently developed a novel, dynamic MF approach, which we call the convective-diffusive lattice-gas (CDLG) model. Lattice-gas representations of microscopic interactions and kinetic energies are used as the starting point. As in the old theory of transport in hquids developed by Eyring [12] and Frenkel [13] (hence referred to as EF theory), we assume that in a fluid, molecules are surrounded by cages of neighbors, but some cages are vacant. 

For simplicity, we present here a two-dimensional, isothermal version of the convective-diffusive lattice-gas model, appropriate for liquid and vapor phases of a single-species system in a microcapillary. Consider a rectangular slab A oi N = Lx X Ly sites r = (x, y) on a square lattice, with lattice constant a and unit vectors ei,2 = + 1,0)a, and 63,4 = (0) 1) - The boundary layers and B2 ait y = 0 and y = (Ly — l)a are adjacent to solid walls Wi and W2 ait y = — a and y = Ly a, respectively. We will assume periodic boundary conditions in the y direction. Sites are assigned spin variables S, = + 1, representing occupancy by a single particle species of mass n, or a vacancy at site r, respectively. Furthermore, assume that this closed microcapillary system between the two walls contains a fixed number of particles, and that the system is isothermal, as if each site v/ere in contact with a heat reservoir at temperature T. Let us define a fundamental timestep At = z. At any given time, we will assume a velocity cUf, defined at each site, where c = a/i is a unit velocity, and u, is a dimensionless velocity field measured in fractions of the unit velocity. For the remainder of this paper, length and time will be expressed in units of a and x, respectively. 

However, real solids are more complex. In the common case where the concentration of the diffusant is much higher than that of vacant sites, one must consider vacancies as the diffusing entities, in a similar fashion to the hole formalism used in semiconductors. More generally, the concentration to be used is that of the diffusing lattice defects, e.g. vacancies, interstitial ions, etc. The result is still of the Arrhenius form, however, and plots of In(crr) vs. l/T are linear with a slope of — EJkT. Consequently, the requirement for high mobility is simply a low activation energy, as expected for an open structure. 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

The characteristic time of the tliree-pulse echo decay as a fimction of the waiting time T is much longer than the phase memory time T- (which governs the decay of a two-pulse echo as a function of x), since tlie phase infomiation is stored along the z-axis where it can only decay via spin-lattice relaxation processes or via spin diffusion. 

Theoretical studies of diffusion aim to predict the distribution profile of an exposed substrate given the known process parameters of concentration, temperature, crystal orientation, dopant properties, etc. On an atomic level, diffusion of a dopant in a siUcon crystal is caused by the movement of the introduced element that is allowed by the available vacancies or defects in the crystal. Both host atoms and impurity atoms can enter vacancies. Movement of a host atom from one lattice site to a vacancy is called self-diffusion. The same movement by a dopant is called impurity diffusion. If an atom does not form a covalent bond with siUcon, the atom can occupy in interstitial site and then subsequently displace a lattice-site atom. This latter movement is beheved to be the dominant mechanism for diffusion of the common dopant atoms, P, B, As, and Sb (26). 

Charge carriers in a semiconductor are always in random thermal motion with an average thermal speed, given by the equipartion relation of classical thermodynamics as m v /2 = 3KT/2. As a result of this random thermal motion, carriers diffuse from regions of higher concentration. Applying an electric field superposes a drift of carriers on this random thermal motion. Carriers are accelerated by the electric field but lose momentum to collisions with impurities or phonons, ie, quantized lattice vibrations. This results in a drift speed, which is proportional to the electric field = p E where E is the electric field in volts per cm and is the electron s mobility in units of cm /Vs. 

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