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We present design equations for error function (or “S-graded”) graded buffers for use in accommodating lattice mismatch of heteroepitaxial semiconductor devices. In an S-graded metamorphic buffer layer the composition and lattice mismatch profiles follow a normal cumulative distribution function. Minimum-energy calculations suggest that the S-graded profile may be beneficial for control of defect densities in lattice-mismatched devices because they have several characteristics which enhance the mobility and glide velocities of dislocations, thereby promoting long misfit segments with relatively few threading arms. First, there is a misfit-dislocation-free zone (MDFZ) adjacent to the interface, which avoids dislocation pinning defects associated with substrate defects. Second, there is another MDFZ near the surface, which reduces pinning interactions near the device layer which will be grown on top. Third, there is a large built-in strain in the top MDFZ, which enhances the glide of dislocations to sweep out threading arms. In this paper we present approximate design equations for the widths of the MDFZs, the built-in strain, and the peak misfit dislocation density for a general S-graded semiconductor with diamond or zincblende crystal structure and (001) orientation, and show that these design equations are in fair agreement with detailed numerical energy-minimization calculations for ZnSySe1−y/GaAs (001) heterostructures.

We have conducted a theoretical study of the equilibrium strain and misfit dislocation density profiles for “S-graded” buffer layers of InxGa1−xAs on GaAs (001) substrates in which the compositional profile follows a normal cumulative distribution function. On the basis of this modeling work we show that the S-graded layer exhibits misfit dislocation-free regions near the substrate interface and the free surface (or device interface). The equilibrium peak misfit dislocation density as well as the thicknesses of the dislocation-free regions may be tailored by design of the compositional profile; this in turn should enable minimization of the density of electronically active threading dislocations at the top surface. S-graded buffer layers may therefore facilitate the achievement of metamorphic device structures with improved performance compared with similar structures having uniform or linearly graded buffers.

Exponentially graded semiconductor layers are of interest for use as buffers in heteroepitaxial devices because of their tapered dislocation density and strain profiles. Here we have calculated the critical layer thickness for the onset of lattice relaxation in exponentially graded InxGa1−xAs/GaAs (001) heteroepitaxial layers. Upwardly convex grading with \( x = x_{\infty } \left( {1 - {\rm e}^{ - \gamma /y} } \right) \) was considered, where y is the distance from the GaAs interface, γ is a grading length constant, and x∞ is the limiting mole fraction of In. For these structures the critical layer thickness was determined by an energy-minimization approach and also by consideration of force balance on grown-in dislocations. The force balance calculations underestimate the critical layer thickness unless one accounts for the fact that the first misfit dislocations are introduced at a finite distance above the interface. The critical layer thickness determined by energy minimization, or by a detailed force balance model, is approximately \( h_{\rm{c}} \approx 0.243\;\mu {\hbox{m}}\left( {\gamma /1\;\mu {\hbox{m}}} \right)^{0.5} \left( {x_{\infty } /0.1} \right)^{ -0.54} . \) Although these results were developed for exponentially graded InxGa1−xAs/GaAs (001), they may be generalized to other material systems for application to the design of exponentially graded buffer layers in metamorphic device structures such as modulation-doped field-effect transistors and light-emitting diodes.

We report an experimental and modeling study of ZnS y Se1−y /GaAs (001) structures, all of which comprised a uniform top layer of ZnS0.014Se0.986 grown on a compositionally graded buffer layer or directly on the GaAs substrate. High-resolution x-ray diffraction was used to estimate dislocation densities on type A slip systems, with misfit dislocation (MD) line segments oriented along the $$ [1\bar{1}0] $$ direction, and type B slip systems, with MD line segments oriented along a [110] direction. A control sample having no graded buffer exhibits equal dislocation densities on the two types of slip systems (D A ≈ D B ≈ 1.5 × 108 cm−2), but a forward-graded (FG) structure (grading coefficient of 27 cm−1) exhibits 20% more dislocations on the type B slip systems (D A ≈ 1.6 × 108 cm−2 and D B ≈ 1.9 × 108 cm−2) and a steep forward-graded structure (grading coefficient of 54 cm−1) exhibits 50% more type B dislocations (D A ≈ 2 × 108 cm−2 and D B ≈ 3 × 108 cm−2). The insertion of an overshoot interface reduced the dislocation densities in the uniform top layer by promoting annihilation and coalescence reactions, but type B dislocations were removed more effectively. Based on equilibrium calculations the overshoot graded layer in the steep graded overshoot structure is expected to exhibit large compressive and tensile strains, with a reversal in the sign of the strain near its middle, which may promote annihilation and coalescence reactions between threading dislocations.

We show that for a mismatched heteroepitaxial layer with linear compositional grading, the first misfit dislocations will be introduced at a finite distance yC from the substrate interface. This is of practical as well as fundamental importance; it alters the value of the critical layer thickness for lattice relaxation and it moves the misfit dislocations away from the interface, where contaminants and defects may cause dislocation pinning or mobility reduction. We have calculated the position of the initial misfit dislocations yC for linearly graded Si1−xGex/Si(001) heteroepitaxial layers with lattice mismatch given by f=Cfy, where Cf is the grading coefficient and y is the distance from the interface. The distance of the first misfit dislocations from the interface yC decreases with increasing grading coefficient but can exceed 40 nm in layers with shallow grading (|Cf

We have calculated the equilibrium misfit dislocation density and strain profiles in reverse-graded heteroepitaxial layers, using Si1?xGex/Si (0?0?1) as a model material system. In these structures there is a finite lattice mismatch at the substrate interface and the composition is graded linearly in such a way that the lattice mismatch decreases with distance from this interface. Reverse-graded layers exhibit two distinct behaviors which may be referred to as unkinked and kinked. In the unkinked reverse-graded layer the misfit dislocations are confined to a thin region near the substrate interface, the residual strain is a linear function of distance from the interface and the strain exhibits a sign reversal within the layer. Kinked reverse-graded layers exhibit a kink in the strain versus distance characteristic, and this kinked region contains a nearly constant density of misfit dislocations. We have developed models for the behavior of both types of reverse-graded layers, and we show that these models provide accurate predictions for Si1?xGex/Si (0?0?1) reverse-graded layers.

We present a computational approach for the determination of the equilibrium misfit dislocation density and strain in a semiconductor heterostructure with an arbitrary compositional profile. We demonstrate that there is good agreement between our computed results and known analytical solutions for heterostructures containing a single linearly graded layer or a single uniform composition layer. We have calculated the dislocation density and strain profiles in Si1−xGex/Si(001), InxGa1−xAs/GaAs(001), and ZnSySe1−y/GaAs(001) heterostructures, each containing a uniform composition layer (uniform layer) on a linearly graded buffer layer (graded layer). The density of misfit dislocations in the graded layer is inversely proportional to its grading coefficient and is unchanged by the presence of the uniform layer, but the dislocated thickness increases with the uniform layer thickness. If the uniform layer is sufficiently thick, misfit dislocations will exist throughout the graded layer, but additional misfit dis...