Misfit dislocations in epitaxy
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NTRODUCTION AND BACKGROUND
EPITAXY, as we all know, is the phenomenon in which a single crystalline substrate A imposes the same orientation, e.g., a parallel one, on a (mostly different) single-crystal B; each time B is grown under appropriate conditions by deposition of B atoms onto the surface of crystal A. The phenomenon of epitaxial growth was first observed by Frankenheim in 1839, first subjected to quantitative experimental study by Royer in 1928 and first studied theoretically by Frank and van der Merwe (FvdM)[1] in 1949, using the so-called Frenkel–Kontorowa (FK) model[2] in introducing the concept of misfit (interfacial) dislocations. The FvdM theory revealed most of the interesting features of epitaxy, e.g., the attainment of perfection in crystallinity and miniaturization, for which the phenomenon became almost indispensible in the fabrication of devices. The FvdM theory was subsequently refined and generallized and supplemented, particularly with the application of the Volterra model by Matthews and others. Because of their groundlaying features concerning the theme of this symposium, those features, as presented by the FvdM theory, will be briefly highlighted. It needs to be emphasized that the Frenkel–Kontorowa (FK) model[2] applied in the FvdM theory[1] only qualitatively models the real system—an epilayer on a crystalline substrate. This model comprises the following: (1) a onedimensional chain of particles to model a two-dimensional (2-D) monolayer (ML) or a three-dimensional multilayer of atoms; (2) elastic particle-particle interaction, instead of electronic atom-atom interaction; and (3) a one-dimensional JAN H. VAN DER MERWE, Professor-Extraordinarius, is with the Department of Physics, University of South Africa, Pretoria, 0003 Unisa, South Africa. This article is based on a presentation in the symposium “Interfacial Dislocations: Symposium in Honor of J.H. van der Merwe on the 50th Anniversary of His Discovery,” as part of the 2000 TMS Fall Meeting, October 11-12, 2000, in St. Louis, Missouri, sponsored under the auspices of ASM International, Materials Science Critical Technology Sector, Structures. METALLURGICAL AND MATERIALS TRANSACTIONS A
(1-D) Fourier series, truncated at first-order harmonic, to represent the complicated 2-D epilayer-substrate interaction. The relevant refinements to more closely represent the real system will be briefly dealt with though. We take the freedom in what follows to refer to the chain of identical particles often as “epilayer.” II. FvdM THEORY A. Governing Equation and its Solution Frenkel and Kontorowa[2] introduced their model, named after them, to present their concept of caterpillar motion in plastic deformation of crystalline solids. The model illustrated by the diagram in Figure 1 consists of a straight chain of particles (“atoms”) connected by identical elastic springs of stiffness and natural length b, to model the intralayer interaction, whereas the crystalline substrate acted as a source of a periodic potential V(x) of periodicity a. Frenkel and
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