Critical thickness of a strained epitaxial film

Consider a single crystal film of thickness hf grown epitaxially on a substrate of thickness 3 h. For this case, the evolution of substrate curvature for arbitrary combinations of substrate and film moduli Ms and Mf, respectively, was analyzed in Section 2.2.1. If the modulus ratio Mf/Mg is of order unity then, to lowest order in the thickness ratio hf/hg, the curvature of the substrate as given in (2.19) is zero, the elastic strain in the substrate is zero, and the mismatch Cm between the two materials is completely accommodated by elastic strain in the film. It follows that the uniform state of strain in the film is 

For many film-substrate systems, the elastic properties of the two materials are similar and it is assumed for the present that Mg Mf = 2fif l + Uf)/ — i f). If the difference in elastic constants is taken into 

The spatially uniform state of stress in the perfectly coherent film is 

This is the stress field that is available to do work as a dislocation is formed in the film. The corresponding elastic energy in the film per unit area of the interface is For a film of thickness /if = 50 nm, a biaxial 

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Fig. 7.9. Illustration of the results established as critical conditions for formation of two arrays of periodic interface misfit dislocations, one array orthogonal to the other, at the interface between a strained epitaxial film and its substrate. The lower curve is a plot of the result (7.21) for insertion of the last dislocation necessary to complete one of the arrays, the other being already complete, and the upper curve is the equivalent result (7.22) based only on mean strain measures and the critical thickness condition for insertion of an isolated misfit dislocation. The graphs are based on the screw dislocation model with a mismatch strain 7m = 0.01.

To examine the relationship between mismatch strain and critical thickness implied by the condition (6.16), consider a thin film of a cubic material for which the free surface is the (001) plane and for which the slip plane is the (111) plane illustrated in Figure 6.6. If a dislocation were to form spontaneously on the slip plane in this strained epitaxial film, its line would be oriented along the [110] direction, and the slip direction would be parallel to either the [101] direction or the [Oil] direction see Figure 6.6. The x, y and z coordinate directions are then [110], [001] and [110], respectively, in crystallographic directions. 

Early observations of elastic strain relaxation during growth of epitaxial layers led to paradoxical results. An attempt to interpret the observations on the basis of the critical thickness theory in its most elementary form suggested that, once the thickness of a film exceeded the critical thickness, the final elastic strain of the film should be determined by the thickness of the film alone, independent of the original, or fuUy coherent, mismatch strain. This is implied by the result in (6.27), which states that that the mean elastic strain predicted by the equilibrium condition G(/if) = 0 is completely determined by hf beyond critical thickness, no matter what the value of Cni- However, it was found that the post-growth elastic strain as measured by x-ray diffraction methods did indeed vary with the initial elastic mismatch strain, and it did so in different ways for different film thicknesses (Bean et al. 1984). As a consequence, the critical thickness theory came under question, and various alternate models were proposed to replace it. However, further study of the problem has revealed the relaxation process to be much richer in physical phenomena than anticipated, with the critical thickness theory revealing only part of the story. 

The increase in the TD density in the films grown on relatively thick (6-8 pm) PSC is most probably caused by a specific plastic relaxation process, occurring as a reaction to a particular state of strain that appears in these epitaxial films. This can be stated on the basis of strain inversion in the films grown on PSC, as well as on the increase in compressive stress with the thickness of the PSC layer increasing. These effects show that apart from the stress caused by the GaN/SiC lattice mismatches, an additional built-in stress arises in the films. Obviously, the additional stress is caused by the presence of (0001) PDs, because one can expect that a part of GaN film within the faulted region may have altered its mechanical properties as compared with unfaulted material [72]. Then the increase in dislocation density in GaN grown on relatively thick PSC can be explained by a plastic relaxation process, which relieves the built-in stress and occurs because this internal stress/(0001) PD density reaches a certain critical value. 

Suppose that a strained layer with uniform mismatch strained Cm is deposited on a substrate to a thickness h i that exceeds the critical thickness her for the layer itself. Then, an unstrained layer of uniform thickness is deposited on the surface of the strained layer. For example, suppose a SiGe alloy film is deposited epitaxially on a Si substrate to a thickness beyond its critical thickness, and then a Si capping layer is deposited on the surface of the alloy layer. The total thickness of the composite film is /if = /igi + /i i. The case of a Si capping layer is the simplest case that can be considered because, in the absence of a dislocation within a strained layer, the epitaxial capping layer is free of mismatch stress this particular system was studied experimentally by Nix et al. (4990). The case of a capping layer that is not matched to the substrate can be handled in essentially the same way, but with slightly greater complexity in the details an example is included as an exercise. 

The physical system studied is depicted in Figure 7.1. A thin film of thickness h is epitaxially bonded to relatively thick substrate, and the lateral extent of the interface is assumed to be very large compared to h. The lattice mismatch between the film and substrate materials is represented by the mismatch shear strain 7m. Prior to formation of any dislocations in the film, the elastic strain in the film is uniform and is given by xz = lyz = 0. Dislocation formation occurs at the expense of the energy stored in this strain field. The critical thickness condition for dislocation formation is given in (6.46) for general cut off radius ro, and it is restated here for the particular value Vo= as... 

A thin film of SiGe, a cubic material, is epitaxially grown on a Si( 100) substrate. Show how the substrate thickness influences the film mismatch strain at which dislocation formation first becomes energetically possible in the film-substrate bilayer as a function of the film thickness. Assume that the system relaxes by the formation of 60° dislocations as described in Section 6.2.2, i/ = 0.25 and = b. For this material combination, the mismatch Cm < 0 is negative. Compare the critical conditions for dislocation formation with those that prevail for a thin film on a relatively thick substrate. 

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