A Model for Curvature in Film-Substrate System
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A Model for Curvature in Film-Substrate System G. Vanamu1, T. A. Khraishi2 and A. K. Datye1 Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, NM-87131, U.S.A. 2 Mechanical Engineering Department, University of New Mexico, Albuquerque, NM87131, U.S.A. 1
ABSTRACT Growth of lattice mismatched films creates bending in the whole structure. There has been great interest in the study of these curvatures in epitaxially-grown materials. An analytical solution for the radius of curvature produced by stresses developed in growing lattice mismatched materials has been obtained. The analyses were based on beam bending theory and strain partitioning theory introduced by our group earlier. The expressions for radius of curvature were obtained for a two-layer heterostructure. The variation of the radius of curvature with the relative thicknesses, relative lattice constants, and relative elastic constants of the layers was determined. The model was verified by applying it to a symmetric tri-laminate structure. The above model can also be extended to determine the curvature for multi-layered heterostructures. INTRODUCTION Growth of lattice mismatched films; combining very dissimilar materials have generated much interest because a variety of novel properties are realized. Heteroepitaxial structures (such as Si-Ge alloys) are gaining prominence for advanced electronic [1-2] and optoelectronic devices [3-5]. In order to produce relaxed SixGe1-x material on a Si substrate, the most promising scheme is the linearly graded buffer layer (SixGe1-x) [6-10]. Growth of lattice mismatched structures causes wafer bending or curvature. Curvature in an epitaxially grown heterostructure is a very critical quantity. First, stress and strain fields can be determined using curvature measurements of the multilayered system. Second, there is a need to decrease the curvature in a heterostructure as it could pose a problem for device fabrication [11]. In recent years there has been great interest in the study of stresses and strains in epitaxially grown multilayers. These elastic fields can be determined using curvature measurements of the multilayer system. It is necessary to find the variables that affect the curvature. The calculations for the curvature were done in two ways. In first case authors [12-15] used force and moment balance equations and in second case authors [16-17] used minimization of the total elastic energy of the system. This paper calculates the curvature using the strain formulations obtained from the model developed in our previous work [18]. In this work we have developed a model to calculate curvature for a two-layer structure. We have verified our model by applying it to a sandwich beam, which consists of two equally-thick layers of the same material on either side of a middle core. THEORETICAL MODEL Consider a planar heterostructure made up of two layers with cubic unit cells as shown in Figure 1. This figure describes the heterostructure after growth, where the
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Figure 1.
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