Effect of Internal Stresses on Thermodynamics of Heterophase Structures in Epitaxial Layers
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EFFECT OF INTERNAL STRESSES ON THERMODYNAMICS OF HETEROPHASE STRUCTURES IN EPITAXIAL LAYERS ALEXANDER L. ROYTBURD Department of Materials and Nuclear Engineering University of Maryland, College Park, MD20742-2115 ABSTRACT The effect of the elastic energy of internal stresses in a system of coherent phases within an epitaxial layer is considered. The equilibrium two-phase layer has a transversely modulated structure with a modulation period dependent on the layer thickness. The phase diagrams for the phases in an epitaxial layer can differ qualitatively from standard phase diagrams, and this difference also depends on the layer thickness. INTRODUCTION The thermodynamic theory of heterophase transversely modulated structures in epitaxial layers is elaborated in this paper. As shown before [1,2] the elastic interaction of an epitaxial layer with a substrate (or with other layers in a multilayer composite) may result in the transformation of the layer to a system of periodically alternating lamellas. The lamellas may be either of differently oriented domains of one phase (twins) or of the different phases (Fig. 1). The equilibrium period of the modulation depends on the layer thickness. It will be shown in this paper that elastic interactions between individual phases within an epitaxial layer and their interaction with a substrate result in considerable changes of the phase diagrams in comparison with standard phase diagrams for incoherent phases. Polymorphic and isomorphic transformation of solid solutions axe discussed as examples. FREE ENERGY OF A COHERENT HETEROPHASE EPITAXIAL SYSTEM The equilibrium heterophase structure in an epitaxial layer corresponds to the minimum of the free energy of an epitaxial system. This free energy consists of the free energies of the undistorted phases (ff , f•) and the elastic energy due to the crystalline coherency. The elastic energy is determined by the misfits between the phases and the substrate, and by a self-strain accompanying the mutual transformation of the phases in an epilayer. The different components of the elastic energy of a coherent two-phase epitaxial layer are shown in Fig. 2. The two single phase states of the epitaxial layer are shown in Fig. 2a. The strains, el and 2, characterize the geometric differences of the phases in the epitaxial state and in the free state. These "3D-misfits" determine the elastic energies, el and e2 , arising in the phases due to the epitaxy:
e=11)62 2 2=
,
e2
=
j.1
(2)o2 2
(1)
where G(') and G(,2) axe the planar elastic modulii of phase 1 and phase 2, respectively. The modulii axe dependent on the crystallographic orientation, n-,between the layer and the substrate, and their anisotropic elastic properties [1,3]. One of the components of the total elastic energy of the two-phase system is connected with the coherency of the phases and is determined by the self-strain between the phases. Mat. Res. Soc. Symp. Proc. Vol. 311. p1993 Materials Research Society
138
D
(a)
(b)
Figure 1: Formation of two-phase modulated structure
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