Materials Science and Engineering A309–310 (2001) 473–477 Critical epitaxial film thickness for forming interface dislocations Sanboh Lee a,∗ , Shing-Dar Wang b , Chun-Hway Hsueh c a Department of Materials Science, National Tsing Hua University, Hsinchu, Taiwan, ROC b Department of Mechanical Engineering, Chung Chou Junior College of Technology and Commerce, Chang-Hua, Taiwan, ROC c Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6068, USA Abstract The system of an epitaxial film on a semi-infinite substrate of a different material is considered and the critical thickness of the film to form misfit interface dislocations is derived in the present study. The energy approach is used to predict the critical thickness and both the self-energy of the dislocation and the interaction energy between the dislocation and the mismatch strain are analyzed. The elastic stress field due to the interface dislocation is required in analyzing the energies and both the superposition principle and Fourier integral are adopted to derive this elastic stress field. The predicted stress fields in the system satisfy both the free surface condition at the film surface and the continuity condition at the interface. The predicted critical film thickness for forming interface dislocation decreases with the increase in the shear modulus ratio of the film to the substrate. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Epitaxial film; Dislocation; Critical thickness; Modeling 1. Introduction Many semiconductor devices and high temperature su- perconducting films require high quality of crystalline films grown epitaxially on substrates of different crystals. How- ever, the film and the substrate generally have different lat- tice parameters. As a result, the lattice mismatch exists at the film/substrate interface and internal stresses are induced in the system. These internal stresses provide a driving force for the formation of interface dislocations [1–7] which, in turn, degrades the device performance. Hence, the study of the condition for the interface dislocation to form in the epi- taxial film/substrate system is imperative in the materials design. The existence of a critical epitaxial film thickness for the interface dislocation to form was first proposed by Frank and van der Merwe [4]. When the film is sufficiently thin, the mismatch at the interface can be accommodated by the distortion of the lattice spacing of both the film and the substrate. As the film becomes thicker, there exists a criti- cal thickness at which alignment between the film and the substrate can no longer be maintained and interface disloca- tions are formed. There have been many studies to analyze this critical film thickness [1–9]. However, due to the com- plexity of the problem, various simplifications in modeling have been adopted in order to obtain the solutions. ∗ Corresponding author. Tel.: +886-3-571-9677; fax: +886-3-572-2366. E-mail address: sblee@mse.nthu.edu.tw (S. Lee). The purpose of the present study is to develop a better analytical model with a closed-form solution for the critical film thickness in forming the interface dislocation. First, the existing models are reviewed. Then, a new analytical model is presented in the present study. Finally, the critical film thickness for the interface dislocation to form is predicted and compared with existing solutions. 2. Summary of existing models The major difficulty in analyzing the problem is the derivation of the elastic stress field due to the interface dislocation. This difficulty results from the following two factors. First, the dislocation is located at the interface of two different materials. Second, the stress field needs to sat- isfy the free surface condition at both the film surface and the substrate surface when both the film and the substrate have finite thickness. In order to simplify the problem, the early analyses made two assumptions: (1) the film and the substrate are isotropic elastic materials and have the same elastic constants and (2) the substrate is infinitely thick. With these assumptions, two approaches have been adopted to predict the critical film thickness for the interface dislo- cation to form. The first one is the force approach developed by Matthews and Blakeslee in which the force exerted by the misfit strain and the approximate tension in the disloca- tion line were considered [1]. The second one is the energy approach, in which the work of forming the dislocation due to the presence of internal stresses and the self-energy of 0921-5093/01/$ – see front matter © 2001 Elsevier Science B.V. All rights reserved. PII:S0921-5093(00)01717-2