K.A. Jackson

On the nature of crystal growth from the melt

Abstract The theory of interface motion as applied to crystal growth by Cahn and his coworkers is examined in detail. This theory, as derived, applied to systems which can have a second order phase transformation but not to liquid-solid or vapor-solid phase transformations which are first order. In this paper, the formalism of the theory is applied to these first order phase transformations. Reasonable agreement with experiment still cannot be obtained. This is because the molecular configuration of the interface is averaged out in the theory by considering a diffuse interface, rather than taking it into account properly in calculating the growth rate. Experimental data on crystal growth have been accumulated and analyzed. It is concluded from this analysis that the theory of Jackson on interface roughness qualitatively predicts crystal growth morphology. Most of the quantitative data on crystal growth are, however, of questionable reliability or are not sufficiently complete for detailed comparison with theory. None of the existent theories of crystal growth can account for these data.

The present state of the theory of crystal growth from the melt

Abstract In this paper, recent advances in our understanding of crystal growth from the melt are reviewed. These include statistical mechanical treatments and computer simulation results which can be compared directly with experiment. The cooperative processes by which crystal growth takes place can be treated properly by these methods, which give insight both into the details of the crystal growth process and into the properties of interfaces.

Experimental observation of the surface roughening transition in vapor phase growth

Abstract The surface roughening transition has been observed in growth from the vapor phase in both C 2 Cl 6 and NH 4 Cl. Both these materials are plastic crystals which have cubic structures and sublime at atmospheric pressure rather than melting. The surface roughening transition was detected by observing changes in the growth morphology in these two materials. In C 2 Cl 6 the roughening transition is approximately at 100°C where the equilibrium vapor pressure is 40 mm, and in NH 4 Cl it is at approximately 365°C and 3 atm pressure.

Instability in radiatively melted silicon films

Abstract Bosch and Lemons [Phys. Rev. Letters 47 (1981) 1151] were first to report that on heating of silicon with a laser, the heated area can break up into small regions of solid and liquid. Thus phenomenon produces undesirable surface roughness on silicon which has been melted using irradiation from a laser or heat lamps. It is due to the higher reflectivity of liquid silicon so that radiative heating produces small regions of superheated solid in contact with small regions of supercooled liquid. In this paper, the instabilities resulting from this unusual thermal situation have been analyzed. It is shown that a stable pattern can develop provided that the spacing between the solid and liquid is small enough. For a 1/2 μm thick layer of polysilicon on silica, the calculated stable spacing is less than about 10 μm, in accord with experiment.

On the theory of crystal growth: Growth of small crystals using periodic boundary conditions

Abstract A theory of crystal growth is presented which gives explicit expressions for the growth rate of crystals. The structure of the interface is not assumed but rather is determined by considering the various possible configurations of the interface and the transitions between them. The growth rate of the crystal is then calculated on the basis of the probability of finding the various configurations. The method used requires the enumeration and analysis of all the possible configurations of the interface and cannot therefore be applied to large crystal interfaces. It has been applied to small segments of crystal interfaces assuming cyclic boundary conditions. In particular, the analysis has been carried through in detail for a two-dimensional square crystal growing on the {10} and {11} faces with a repeat distance of 2, 3, and 4 unit cell spacings. The calculated growth rate does not depend strongly on the repeat distance for these three cases. In addition, calculations have been made for the {100}, {110}, and {111} faces of a simple cubic material. The face in each case contained 2×2 unit cells. In all cases the predicted growth rates are relatively independent of direction for a small entropy change and the growth rate depends strongly on direction for high entropy change, in accord with experiment. In all cases the growth rate is predicted to be linear with undercooling, which subsequent work has shown to be a consequence of the small repeat distance which is assumed.

Pair approximation for interface kinetics

Abstract A pair approximation method is described for the interface kinetics of a simple cubic (100) surface with no surface migration. Condensation and evaporation are considered explicitly in a system of master equations applied to a pair of sites; and the effect of the neighboring sites is treated with a set of parameters which are determined self-consistently. Runge Kutta numerical integration of the equations yields the time evolution of the surface. The results for a high entropy of transformation indicate a region of metastable states (zero growth rate) for small driving force, and the instantaneous growth rate is a periodic function of time for larger driving force. The period of the variations is the time required to grow a monolayer, and the amplitude is less than that obtained by Temkin using the mean field approximation. The variations appear as artifacts of most approcimation methods which treat a limited number of sites. The average growth rate is less than the limiting Wilson-Frenkel law for all growth conditions studied. This results since the surface becomes quite rough in the presence of a large condensation rate, and therefore its evaporation rate is greater than the evaporation rate of an equilibrium surface assumed in the Frenkel model. The metastable region is absent at low entropies of transformation and here the growth rate is nearly linear in the driving force. Agreement with Monte Carlo data is excellent for a low entropy of transformation and for all materials at high growth rates.

The use of an electric analogue to solve the lamellar eutectic diffusion problem

Abstract An electrical analogue technique has been developed to obtain numerical solutions to the diffusion equation in the liquid ahead of a growing lamellar eutectic. Using the solutions it has been possible to examine certain of the approximations made in earlier treatments of the problem. A modification has been made to the Jackson and Hunt model of eutectic growth which becomes important when composition differences in the interfacial liquid composition become large (i.e. at very high growth velocities). The modified treatment no longer predicts υλ 2 = constant although the variation of υλ 2 with growth velocity is not likely to be detected easily.

On the theory of crystal growth: The fundamental rate equation

Abstract The series of molecular configurations through which a crystal surface must pass in order to grow is considered. Analysis of the steady state distribution of these configurations leads to an expression for the growth rate of a crystal in terms of the probabilities of molecules joining and leaving each of the configurations through which the interface passes. For cases where the rate of attachment of molecules is independent of the surface configuration, the growth rate can be expressed in terms of the free energies of the sequence of configurations. The growth rate equation reduces to previously derived expressions when appropriate approximations are made.

Computer simulation of vapor deposition

Abstract The initial nucleation which occurs during vapor growth on a crystalline substrate has been simulated in a computer by the use of Monte Carlo methods. The crystalline surface is represented by a 30 × 30 array and the deposition site is selected randomly as is the alternative between adsorbed atom migration or evaporation. The output was programmed so as to be displayed graphically or tabulated. During each simulation run atoms were given the opportunity of moving between two and three thousand times. The result of simulations at three different temperatures where surface activation energies and binding energies were chosen so as to be on the range reported for silver on silver chloride showed that while good agreement between the simulation and the two-dimensional nucleation theory of Zinsmeister occured at the start of nucleation, agreement could only be obtained at later stages if direct impingement of depositing atoms is considered. In summary, the computer simulations were found to be in general agreement with two-dimensional nucleation theory. The most important points of disagreement are: (1) the temperature dependence of the total cluster concentration which is found is to be temperature-dependent at all temperatures investigated; and (2) the cluster size distribution, which is much broader than predicted. The simulation results indicate that the assumption of no direct impingement of depositing atoms is more critical than the assumption of the size independence of the collision cross section. Finally, it has been shown that the nucleation theory can be applied to the co-deposition of two different atomic species.

Stability of a melting interface

Abstract Comparison between experimental observations of a melting interface and a theoretical analysis of interface stability has been made. The theoretical analysis follows the method of Mullins and Sekerka but includes diffusion in the growing phase. The analysis, which is appropriate for melting where diffusion in the growing phase is rapid, predicts that a melting interface should be more stable than a freezing interface in the same material. Comparison of this analysis has been made with observation of the stability of a melting planar interface of transparent organic binary mixtures which have low entropies of fusion. Experimentally, ripples were observed at the interface under some conditions of melting. It was found that the wavelength of the ripples decreased with increasing melting rate and increasing solute concentration. The experimental data are in good agreement with the theoretical analysis both for the onset of instability, and the wavelength of the ripples. These experiments confirm that the onset of melting instability is predicted correctly by stability theory, and that it does not coincide with the onset of constitutional superheating. Calculations of the wavelength of the instability have also been made for the case of a freezing interface. The wavelength was found to be independent of concentration and to depend primarily on the diffusion layer width, Dl/V, for normal growth conditions, in accord with observation.