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INTRODUCTION

Thin films deposited on an elastic substrate often develop residual stresses associated with thermal expansion mismatch, epitaxy or phase transitions. These stresses which can be as large as several tenths of the shear modulus can be partially relieved by mechanisms such as dislocation motion, surface roughening and cracking. Another possible relief mechanism by

which misfit (transformation) strains can be accommodated is through twinning, as suggested by Roitburd [1]. Extensive twinning is observed in the case of ferroelectric thin films of BaTiO 3 as well as superconducting thin films of YBa 2Cu 30 7 . The presence of twins in these high Tc superconductors can modify the critical temperature as well as the flux pinning capabilities. In ferroelectric thin films, these twins can alter the aging characteristics and optical properties. The energetics of twin domain formation in epitaxial films is controlled by an interplay between two competing effects - the decreasing stored elastic strain energy that accompanies the formation of twinned domains and the increased interface energy contribution due to the twin domain boundaries. These two competing effects conspire to determine the equilibrium width of the twinned domains as a function of film thickness and physical and crystallographic properties. The elastic displacements and the elastic strain energy associated with misfitting epitaxial ferroelectric films have been solved approximately [1,21 and, more recently, by numerical integration of the Green's function [3]. In this paper, we will show that the general elasticity problem for a thin film with a periodic distribution of twinned domains can be solved analytically and that by further employing an asymptotic analysis of the resultant stress fields, the elastic solution for a film containing a single embedded twinned domain can be obtained. We then apply these results to determine the equilibrium twinned domain width as well as the equilibrium periodic spacing of the twins when a tetragonal film is deposited on a cubic substrate. More importantly, we determine the equilibrium twin width and spacing as a function of the film thickness and the ratio of the twin boundary interface energy to strain energy. 149

Mat. Res. Soc. Symp. Proc. Vol. 389 © 1995 Materials Research Society

2. MATHEMATICAL FORMULATION AND METHOD OF SOLUTION We consider the case of a misfitting isotropic thin film on an isotropic, linear elastic as shown in Fig.l. In ,Ysubstrate order to make the analyses tractable, we assume that the elastic constant mismatch between the film and the substrate is negligible and hence the film-twin domain-substrate _.system is treated as elastically x homogeneous. The twinned /Z domains within the film are indicated by the shaded regions in figure 1. The film thickness SUBSTRATE h can be taken to be the characteristic length and hence all other length scales can be normalized with respect to this Fig.1 Schematic of film-twin-substrate system parameter. Therefore, in terms of dimensionless variables