The Critical Thickness of the Dislocation-Free Stranski-Krastanow Pattern of Growth
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THE CROTOCAL THOCO(NESS OF THE DOSLOCATOON-FREE STRANSC(DK(RASTANOW PATTERN OF GROWTH. MICHAEL A. GRINFELD Department of Mathematics, Rutgers University, New Brunswick, NJ 08903 A•STRACT , It was demonstrated earlier (1,2] in the framework of equilibrium thermodynamics that the morphological stability of the free boundaries and interfaces in crystals is extremely sensitive to the presence of shear stresses. Relying on that idea we have established the formula H = pAC/'2 of a critical thickness of solidifying He 4 films and of the dislocation-free Stranski-Krastanow growth of epitaxial films (where a - the coefficient of surface tension, g - the shear module of the crystal, "t - the external or misfit stress). In this report we present certain facts pertaining to possible patterns of the growing corrugations and introduce the second critical thickness at which a symmetry change in the patterns has to occur. Introduction It was demonstrated theoretically in [1,2] that in the absence of surface tension a flat boundary of non-hydrostatically stressed solid of any symmetry is always unstable with respect to "mass rearrangement". The physical mechanisms of the rearrangement can be different, as, for instance, a)melting-freezing or vaporizationsublimation processes at liquid-solid or vapor-solid phase boundaries, b)surface diffusion of particles along free or interfacial boundaries, c)adsorption-desorbtion of the atoms in epitaxial crystal growth, etc... This universal instability delivers new insights and provides new opportunities in different branches of materials and other science a part of which is discussed in [3]. In particular, it has already allowed to predict stress driven corrugations of thin solid films of'He 4 [4] and to explain the phenomenon of dislocation-free Stranski-Krastanov pattern of growth of epitaxial films of GaAs on Si substrates (see Ref. [5]-[7]). The key formula for the critical thickness of such films was announced in [8] and published in [3]. [9]. 2D theory of elasticity was the basis of the above mentioned studies, and the 3D approach is becoming urgent necessity in view of (i)typical misfit stresses in the problems of epitaxy and (ii)the wide opportunities appeared in the experiments with solid He4 films. Below we present some recent results pertaining to 3D theory [10]. In particular, we examine evolution of the corrugations appearing at the phase boundary of a destabilized prestressed isotropic elastic film attached to a rigid substrate, and establish the dispersion relation of amplification factor of different Fourier components. This relation allows one to study morphological patterns of the unstable corrugations (islands) in prestressed solid films and, in particular, possible changes of symmetry of the surface morphology which accompany the process of thickening of the film.
Mat. Res. Soc. Symp. Proc. Vol. 290. @1993 Materials Research Society
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The Static Method Let us consider a thin solid film of a thickness H attached to a rigid substrate which is uniformly stressed (Fig. 1
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