The Critical Thickness of Layers Subject to Anisotropic Misfit
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THE CRITICAL THICKNESS OF LAYERS SUBJECT TO ANISOTROPIC MISFIT.
RICHARD BEANLAND Department of Materials Science and Engineering, The University of Liverpool, P.O. Box 147, Liverpool L69, 3BX, England.
ABSTRACT It is well known that it becomes energetically favourable for misfit dislocations to be introduced into strained epitaxial layers above a certain 'critical' layer thickness, hc. To date, theoretical calculations of hc have only been made for cases of isotropic misfit - i.e. cases where the misfit is the same for every direction in the interface. Using a new formulation of the Frank-Bilby equation and the concept of coherency dislocations, it is now possible to treat cases of anisotropic misfit, such as silicon on sapphire (SOS). The method used to obtain the critical thickness is described, and values of hc and equilibrium dislocation density are given for various materials systems. INTRODUCTION The critical thickness theory of Matthews and co-workerslll-[4] is a simple and robust model of equilibrium dislocation density and critical thickness of strained epitaxial layers. Here the principles of the model are applied to systems with anisotropic misfit - i.e. systems in which the misfit is not the same in all directions in the interface. This generalisation is made possible by a recent application of the Frank-Bilby equation[ 5,6l which allows the deformation tensor (and hence strain and stress tensors) to be derived for a strained epitaxial layer. All the assumptions of the Matthews model apply to the present theory. Two of these assumptions may be questionable in some systems. These are: a) the elastic boundary conditions will only be valid for a layer-by-layer growth mode, and b) the materials are allowed to be elastically similar for calculation of dislocation self energies. It is also assumed that the materials behave in a linear, isotropic elastic manner, which may not be true for the large strains encountered in many epitaxial systems. THEORY The model relies upon the calculation of the total energy per unit area of a strained layer and finding the minimum (equilibrium) as a function of dislocation density. The total energy per unit area of a strained layer is
(1)
Etot = Estrain + Earray
The strain energy term has contributions from the coherency strain and the strain relief due to the misfit dislocation array. Calculations are greatly simplified when performed in an orthogonal reference frame fl, where [001]n is a unit vector pointing from the interface towards the overlayer surface. In this reference frame, a dislocation array is related to the deformation of the layer by the Frank-Bilby equations (Beanland 1991) pbýx = D[010]0,
pb~y = - DI1100]l
(2)
where p is the interfacial dislocation density, b=[bx by bjz] is the Burgers vector, •=[x O]r is the line direction and D is the deformation tensor written in the Q frame. Applying the boundary condition that no stress acts on the surface gives the engineering strain produced by the array (Beanland 1991) ,y
Mat. Res. Soc. Symp. Proc. Vol. 239. 0
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