Stress Calculations in Thin Films and Multilayers Using Distributions of Infinitesimal Dislocations.
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STRESS CALCULATIONS IN THIN FILMS AND MULTILAYERS USING DISTRIBUTIONS OF INFINITESIMAL DISLOCATIONS.
JEAN PRILH]f and NICOLE JUNQUA, Laboratoire de M~tallurgie Physique, Universiti de Poitiers, 40 avenue du Recteur Pineau, 86022 POITIERS, France.
ABSTRACT Distributions of infinitesimal dislocations have been used to determine stress components near lateral surfaces of strained multilayers.The nucleation of misfit dislocations at these surfaces are discussed. INTRODUCTION. When a thin film is deposited epitaxially onto a monocrystalline substrate, the difference between the two lattice parameters induces coherency stresses in the film. These coherency stresses can be described by distributions of infinitesimal dislocations located at the interface [1,2] and are called "coherency dislocations". A family of infinitesimal edge dislocations of Burgers vector I5bl = ba introduced at each interatomic distance as transforms a film deposited without coherency and without internal stress into an epitaxial film under stress. In the general case, epitaxy occurs in two directions and two families of coherency dislocations are needed. In the present paper, epitaxy in one direction only is considered. The more general case can be deduced by a simple extension of the derivation. This formalism can be useful in studying the behavior of dislocations in films in substrates or in multilayers. The theory of image dislocations is then applicable to take free surfaces and interfaces into account. Coherency dislocations have already been used [3,4 ] to calculate equilibrium position of misfit dislocations when the elastic modulus of film and substrate are very different. The results show that the misfit dislocations are not located at the interface but at four to six interatomic distances in the softer material. The repulsive force due to the interface is then balanced by the attractive force of the coherency dislocations. In this paper, coherency dislocations are used to calculate the components of the stress tensor near the lateral surfaces of a multilayer. The possibility of introducing misfit dislocations by gliding from lateral surfaces is discussed. METHOD OF CALCULATION In a multilayer composed of epitaxial monolayers of two materials (1) and (2) isotropic in the plane of the film, the internal stresses oxx(i) and oyy(i)are given by:
hi with:
hi)
(2)
=
h2 x
EBIEB2 hI h 2 ba E 11 hI + E12 h 2 aeq
6a = a2 - at, a~i = ai(l + ex(i)) , exx(') = Oxx(')/EBi and Oxx(i) = Yyy(i).
EBi is the biaxial modulus of the material (i), ai the lattice parameter, hi the thickness of the monolayers, a. the equilibrium lattice parameter of the multilayer. It is well known that there is no shear stress parallel to the film but is no longer true near lateral surfaces. To calculate the shear stresses it is supposed that hl >> h 2 so that OXX(2) = oyy(2) = 0, that the epitaxy occurs only in one direction and that the elastic constants pi and v of the two materials are the same. The more general case can be deduced by a simple extension of the deri
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