Dislocation Emission at Surfaces
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The appearance of misfit dislocations during epitaxial growth has been observed experimentally to coincide with the attainment of a critical thickness, which in turn depends on the misfit strain, elastic constants, and the orientation of the various slip systems involved (e.g., Matthews [1], Hull et al. [2], and Houghton et al. [3]). Theoretical studies of this problem have established the validity of the critical thickness concept, starting with Frank and van der Merwe [4] and leading up to recent work by Freund et al. [5-7]. Most analyses to date, however, have concentrated on the stability of a single pre-existing threading dislocation, or a threading dislocation in the presence of an array of other dislocations. Another issue that must be addressed more carefully is the actual nucleation of dislocations. The prevention of dislocation formation at the source, rather that the reduction of already-formed dislocations, constitutes an alternative approach for defect minimization. A possible location at which dislocations are actually generated during film growth is the film surface. The primary purpose of this paper is to re-examine the process of dislocation nucleation at a crystal surface, in light of recent developments with the theory of dislocation formation. First, an expression for the shear stress due to a general shear dislocation loop perpendicular to a free surface is presented, and immediately utilized to derive an accurate expression for the elastic self energy of a semicircular loop. Next, conditions for nucleation of the semicircular loop are worked out in the continuum framework described by Hirth [8] and subsequently by Fitzgerald et al. [9], but with the modified expression for the self energy of the loop. Finally, the potential utility of accurate expressions for stresses in connection with atomistic-type calculations is pointed out. 93
Mat. Res. Soc. Symp. Proc. Vol. 356 01995 Materials Research Society
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Figure 1. Schematic of an epitaxial substrate-layer system, with a slip plane (shaded) containing a threading dislocation segment joining an interfacial misfit with the surface, as well as an incipient loop.
THE ENERGY OF A SEMICIRCULAR DISLOCATION LOOP AT A FREE SURFACE We present here a calculation of the self energy of a dislocation loop emerging perpendicularly from a free surface in an isotropic, elastic half-space. To understand the nature of the principal result of this section, recall that the elastic self energy for a full circular dislocation loop of radius r in an infinite elastic solid is given by [10]
Ufull =gb~r( 2 -v)lIn
(8r(1
where yu is the shear modulus, b is the Burgers vector, v is Poisson's ratio, ro is the core cutoff radius. In an analogous fashion to what Gao and Rice [11] have done for a general dislocation loop ahead of a crack, we will show here that the energy of the semicircular loop of radius r emerging from the free surface has the form Uhalf_=b2r (2-v) In 8mr 2
8 (1-v)
e ro
(2)
where m is a geometry-dependent correction factor. Early analyses of dislo
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