Dislocation Arrays in Epitaxial Interfaces
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DISLOCATION ARRAYS IN EPITAXIAL INTERFACES. R. BEANLAND AND R. C. POND Department of Materials Science and Engineering, The University of Liverpool, P.O. Box 147, Liverpool, L69 3BX, England. ABSTRACT A method for analysing the form of dislocation arrays which accommodate misfit at epitaxial interfaces based on the Frank-Bilby expression of the dislocation content of an interface is presented. Emphasis is placed on the deformation of a crystal in the presence of a dislocation array, and ease of formulation through use of an orthogonal reference frame. Several aspects of epitaxial growth are subsequently addressed, including the uniqueness of a dislocation array accommodating a given misfit. In the case of growth on vicinal cubic surfaces, it is shown that dislocations generated to relieve misfit may also lead to misorientation of the overlayer. From experimental measurements of dislocation densities using transmission electron microscopy, it is possible to calculate the state of strain in a metastable misfitting epitaxial layer. When misfit is not isotropic in the interface (such as for silicon or niobium grown on sapphire) it is shown that dislocation line directions may not lie along low index directions in either crystal, a point which is particularly important with regard to studies of interfacial structure by high resolution electron microscopy. THEORY The dislocation content of interfaces has been studied both theoretically and experimentally fox many years. The most successful description of interfacial dislocation content has been the FrankBilby equation, which describes the Burgers vector content of an interface between two crystal lattices [1,2]. We give a brief derivation of the equation, placing emphasis on a simple visual description of the deformation produced by an array of interfacial dislocations. This is followed by a discussion of the relevance of the equation to epitaxial systems, illustrated by application of the theory to some epitaxial systems which are usually difficult to deal with. Firstly, consider an array of edge dislocations with line direction t, spacing d and Burgers vector b lying in the xy plane of an orthogonal coordinate system (Figure 1a). The extra half planes lie above the xy plane when b x t is parallel to -z according to the FS/RH convention. We define a vector p which lies parallel to and at some distance -z below the xy plane. This vector is moved in the direction of increasing z; it will change length abruptly at z=O when it crosses the dislocation lines to become a vector p', where g'= p"± nb =12 + nb sgnp
(1)
where (p is the angle (less than ir) between t and p., taken to be positive if t to U is clockwise looking along +z, and n is the number of dislocations crossed. The value of n is given by IRI/d sin (p,where d is the dislocation spacing. Making 2 a unit vector gives 2' = p + pb sin (p
(2)
where p is the linear interfacial dislocation density, i.e. l/d. Expressing equation (2) in terms of the reference frame xyz gives p'= y + pb sin (0-4) = x cos 0 +y sin 0 + pb
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