The critical epilayer thickness for the formation of misfit dislocations at the interface between an epilayer and a substrate with a finite thickness is derived in the present study. The analysis is based on the energy approach, in which the self-energy of dislocation, the interaction energy between the dislocation and free surfaces, and the lattice mismatch energy of substrate and epilayer are calculated. To satisfy the free surface condition, the methodology of superposition principle and Fourier transformation are used in analyzing the stress field due to the interface dislocation. The critical epilayer thickness is compared with those reported in the literature.

Criteria for formation of interface dislocations in a finite thickness epilayer deposited on a substrate

Strain evolution during In0.2Ga0.8As/GaAs (001) growth by molecular beam epitaxy has been monitored in real time. We have detected that three main relaxation stages, related to different mechanisms, take place during growth, and we have obtained the thickness range where those mechanisms are active. The in situ measured relaxation behavior in the plastic stages has been described by means of a simple equilibrium model that takes into account dislocations generation and interaction between them. The excellent agreement between the experimental data and the model allows us to determine the value of the formation energy per unit length of a misfit dislocation and the extent of the interaction between dislocations in this material system.

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Study of the relaxation process during InGaAs/GaAs (001) growth from in situ real-time stress measurements

This paper describes the modeling of the dislocation distribution and of the strain relief in linearly and step-graded structures, based on work-hardening considerations. The model considers the energy variation in an InGaAs/GaAs system upon introducing a new dislocation into the interfacial fixed array of misfit dislocations. An analytical expression for the strain relaxation in graded-buffer structures is proposed. Transmission electron microscopy observation confirms the model predictions and reveals that the saturation state of relaxation is reached in the buried layers.

A work-hardening based model of the strain relief in multilayer graded-buffer structures

Using previously published relaxation models [D. J. Dunstan, P. Kidd, L. K. Howard and R. H. Dixon, Appl. Phys. Lett. 59, 3390 (1991) and D. Gonzalez, D. Araujo, G. Aragon, and R. Garcia, Appl. Phys. Lett. 71, 2475 (1997)] that predict the strain relaxation in the InGaAs/GaAs system, before and during the stage of relaxation saturation, the critical thickness where dislocation interactions begin to limit the plastic relaxation is estimated. The approximations used to deduce an analytical expression are shown to be appropriate for describing the regime of relaxation considered. A good agreement with experimental data previously published by other authors permits a physical explanation for the different observed regimes of relaxation to be given.

Critical thickness for the saturation state of strain relaxation in the ingaas/gaas systems

The mismatch stress field in a graded epilayer and its substrate is analyzed by analogy to thermal stresses. In particular, a straight dislocation line and a half-rectangular dislocation loop in the graded epilayer are investigated. For the straight dislocation line, there are two positions in the epilayer at which the resultant force acting on the dislocation is null. If the dislocation is generated from the surface and passes the first zero force position, it will continue moving toward the second zero force position, where the dislocation is at a stable equilibrium. It is the stable equilibrium position that determines the dislocation formation energy and then the critical thickness of the graded epilayer. For a half-rectangular dislocation loop, its optimum shape occurs at a saddle point of the total energy profile as a function of the dislocation width and length. The activation energy for emitting a perfect or a partial dislocation loop, the dislocation width, and the aspect ratio of dislocation length to width are all related to the graded layer thickness. All three, the activation energy, the dislocation width, and the aspect ratio, decrease quickly when the thickness increases from its critical value and then gradually approach extreme value as the thickness approaches infinity. The extreme values of the activation energy, the dislocation width, and the aspect ratio correspond to those for a dislocation loop emitted from the surface of an epilayer with a uniform composition. The results indicate that 1. dislocation emission is more difficult in a compositionally graded layer than in a compositionally uniform layer and that 2. the aspect ratio of threading dislocation density to mismatch dislocation density is lower in a compositionally graded layer than in a compositionally uniform layer. The theoretical results agree with experimental observations.

A dislocation in a compositionally graded epilayer

The total energy of an array of dislocations in a strained epitaxial layer is composed of the self energy of the dislocations, the strain energy which arises from the lattice mismatch between the layer and its substrate and the interaction energy between the dislocations and the mismatch strains. The sum of the self energy and the interaction energy represents the formation energy of the dislocations. In this study, the self energy is formulated using complex potentials. Two limiting conditions are used to check the solution. The first is that the self energy of the array reduces to that for an isolated single dislocation as the dislocation spacing in the array approaches infinity. Secondly, as the layer thickness approaches infinity, the self energy reduces to that for a dislocation wall. A negative formation energy promotes dislocation generation while a positive formation energy implies a suppression of dislocation generation. A critical thickness required for the generation of an isolated dislocation ...

An array of dislocations in a strained epitaxial layer. I. Elastic energy

The sequential generation of dislocations in a strained epitaxial layer is considered. It is found that an additional work component results from dislocation‐dislocation interactions after some dislocations are generated and located in the interface between the epitaxial layer and the substrate. The interaction energies induced by a single dislocation and a dislocation array are derived. It is found that, in general, when the distance between a fresh dislocation and the nearest pre‐existing dislocation is comparable to the layer thickness, the additional work component achieves the level of the self energy of an isolated dislocation. The additional work increases sharply with decreasing distance between the fresh and pre‐existing dislocations. If the spacing between the dislocations exceeds approximately 20 times the layer thickness, the additional work becomes insignificant. These results are consistent with experimental observations.

An array of dislocations in a strained epitaxial layer. II. Work hardening

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An array of dislocations in a strained epitaxial layer. I. Elastic energy

An array of dislocations in a strained epitaxial layer. II. Work hardening