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Metamorphic semiconductor devices such as high electron mobility transistors (HEMTs), light-emitting diodes (LEDs), laser diodes, and solar cells are grown on mismatched substrates and typically exhibit a high degree of lattice relaxation. In order to minimize the incorporation of threading defects it is common to use a linearly-graded buffer layer to accommodate the mismatch between the substrate and device layers. However, some work has suggested that buffer layers with non-linear grading could offer superior performance in terms of limiting the surface density of threading defects. In this work, we have compared S-graded buffer layers with different orders and thicknesses. To do so we calculated the expected surface threading dislocation density for each buffer design assuming a GaAs (001) substrate. The threading dislocation densities were calculated using the LMD model, in which the coefficient for second-order annihilation and coalescence reactions between threading dislocations is considered to be equal to the length of misfit dislocations.

Metamorphic realization of semiconductor devices has become increasingly important due to the great freedom it affords in layer compositions and thicknesses. However, metamorphic growth is often accompanied by the introduction of high densities of threading dislocation defects. This behavior may be understood by using an annihilation and coalescence model for the threading dislocation behavior which is based on the dislocation interaction length Lint. For its application we considered only glide of dislocations, so the interaction length was assumed to be equal to the length of misfit dislocation segments LMD. The length of misfit segments was determined approximately by the Matthews, Mader, and Light model [J. Appl. Phys., 41, 3800 (1970)] for lattice relaxation, and was assumed to be independent of the distance from the interface. Within this framework we have applied the annihilation and coalescence model to chirped semiconductor superlattices to evaluate these superlattices as strainrelaxed buffers for metamorphic devices. In this work we have studied two basic types of InGaAs/GaAs chirped superlattice buffers: type I superlattices are compositionally modulated while type II superlattices are thickness modulated.

Several simple models have been developed for the threading dislocation behavior in heteroepitaxial semiconductor materials. Tachikawa and Yamaguchi [Appl. Phys. Lett., 56, 484 (1990)] and Romanov et al. [Appl. Phys. Lett., 69, 3342 (1996)] described models for the annihilation and coalescence of threading dislocations in uniform-composition layers, and Kujofsa et al. [J. Electron. Mater., 41, 2993 (2013)] extended the annihilation and coalescence model to compositionally-graded and multilayered structures by including the misfit dislocation-threading dislocation interactions. However, an important limitation of these previous models is that they involve empirical parameters. The goal of this work is to develop a predictive model for annihilation and coalescence of threading dislocations which is based on the dislocation interaction length Lint. In the first case if only in-plane glide is considered the interaction length is equal to the length of misfit dislocation segments while in the second case glide and climb are considered and the interaction length is a function of the distance from the interface, the length of misfit dislocations, and the density of the misfit dislocations. In either case the interaction length may be calculated using a model for dislocation flow. Knowledge of the dislocation interaction length allows predictive calculations of the threading dislocation densities in metamorphic device structures and is of great practical importance. Here we demonstrate the latter model based on glide and climb. Future work should compare the two models to determine which is more relevant to typical device heterostructures.

Chirped superlattices are of interest as buffer layers in metamorphic semiconductor device structures, because they can combine the mismatch accommodating properties of compositionally-graded layers with the dislocation filtering properties of superlattices. Important practical aspects of the chirped superlattice as a buffer layer are the surface strain and surface in-plane lattice constant. In this work two basic types of InGaAs/GaAs chirped superlattice buffers have been studied. In design I (composition modulated), the average composition is varied by modulating the composition of one of the two layers in the superlattice period, but the individual layer thicknesses were fixed. In design II (thickness modulated), the individual layer thicknesses were modulated, but the compositions were fixed. In this paper the surface strain and surface in-plane lattice constant for these chirped superlattices are presented as functions of the top composition and period for each of these basic designs.

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 present an approximate model for the threading dislocations in III–V heterostructures and have applied this model to study the defect behavior in metamorphic triple-junction solar cells. This model represents a new approach in which the coefficient for second-order threading dislocation annihilation and coalescence reactions is considered to be determined by the length of misfit dislocations, LMD, in the structure, and we therefore refer to it as the LMD model. On the basis of this model we have compared the average threading dislocation densities in the active layers of triple junction solar cells using linearly-graded buffers of varying thicknesses as well as S-graded (complementary error function) buffers with varying thicknesses and standard deviation parameters. We have shown that the threading dislocation densities in the active regions of metamorphic tandem solar cells depend not only on the thicknesses of the buffer layers but on their compositional grading profiles. The use of S-graded buffer layers instead of linear buffers resulted in lower threading dislocation densities. Moreover, the threading dislocation densities depended strongly on the standard deviation parameters used in the S-graded buffers, with smaller values providing lower threading dislocation densities.

Chirped superlattices are of interest as buffer layers in metamorphic semiconductor device structures, because they can combine the mismatch accommodating properties of compositionally-graded layers with the dislocation filtering properties of superlattices. Important practical aspects of the chirped superlattice as a buffer layer are the surface strain and surface in-plane lattice constant. In this work we study two basic types of InGaAs/GaAs chirped superlattice buffers; type I is compositionally modulated while type II is thickness modulated. For both types, we have studied the equilibrium surface strain and surface in-plane lattice constant, and we have compared the surface strain behavior to linearly-graded buffers having the same top target composition of indium.

Metamorphic semiconductor devices such as high electron mobility transistors (HEMTs), light-emitting diodes (LEDs), laser diodes, and solar cells are grown on mismatched substrates and typically exhibit a high degree of lattice relaxation. In order to minimize the incorporation of threading defects it is common to use a linearly-graded buffer layer to accommodate the mismatch between the substrate and device layers. However, some work has suggested that buffer layers with non-linear grading could offer higher performance in terms of limiting the surface density of threading defects. In this work, we compare bi-parabolic graded buffer layers to linear and S-graded buffer layers in terms of the surface threading dislocation density by considering a uniform InGaAs layer grown on a GaAs substrate with an intermediate graded InGaAs buffer layer. The threading dislocation densities were calculated using the LMD model, in which the coefficient for second-order annihilation and coalescence reactions between threading dislocations is considered to be equal to the length of misfit dislocations.

The critical layer thickness (CLT) determines the criteria for dislocation formation and the onset of lattice relaxation. Although several theoretical models have been developed for the critical layer thickness, experimentally-measured CLTs in ZnSe/GaAs (001) heterostructures are often at variance with one another as well as with established theories. In a previous work [T. Kujofsa et al., J. Vac. Sci. Technol. B, 34, 051201 (2016)], we showed that the experimentally measured CLT may be much larger than the equilibrium value when using finite experimental resolution. In this work, we apply a general dislocation flow model to determine the apparent critical layer thickness as a function of the experimental resolution for ZnSe/GaAs (001) heterostructures. More importantly, we compare the results utilizing different equilibrium theories and therefore varying driving forces for the lattice relaxation in order to determine which established models are consistent with several measured values of CLT for ZnSe/GaAs (001) once kinetically-limited relaxation and finite experimental strain resolution are taken into account.

The extent of strain relaxation in semiconducting device heterostructures has important implications in the design of high electron mobility transistors, light-emitting diodes, and laser diodes, in which the residual strain affects the device characteristics. In this work, we develop the theoretical framework for understanding strain compensation in a semiconductor device layer using a uniform buffer layer which can be intentionally mismatched to the material above. Specifically, we determined the critical condition for complete strain compensation in the device layer by intentionally introducing a compositional mismatch at the device-buffer interface. We present minimum energy calculations and show that for a given device layer with fixed mismatch and layer thickness, the buffer layer may be designed with the appropriate combination of thickness and mismatch such that the device layer will have zero residual strain in equilibrium. Such a structure can be referred to as a completely strain-compensated design. In the more general case, there may be partial strain compensation, and we give a simple physics-based Gaussian-type function describing the residual strain in the device layer. We have applied this general framework to In x Ga1−x As/GaAs (001) heterostructures for the purpose of illustration, but the work is applicable to any diamond or zinc blende (001) heteroepitaxial material system.

For the design and analysis of a strained-layer semiconductor device structure, the equilibrium strain profile may be determined numerically by energy minimization but this method is computationally intense and non-intuitive. Here we present an electric circuit model approach for the equilibrium analysis of an epitaxial stack, in which each sublayer may be represented by an analogous configuration involving a current source, a resistor, a voltage source, and an ideal diode. The resulting node voltages in the analogous electric circuit correspond to the equilibrium strains in the original epitaxial structure. This new approach enables analysis using widely accessible circuit simulators, and an intuitive understanding of electric circuits may be translated to the relaxation of strained-layer structures. In this paper, we describe the mathematical foundation of the electrical circuit model and demonstrate its application to epitaxial layers of Si1−x Ge x grown on a Si (001) substrate.

The critical layer thickness hc for the onset of lattice relaxation has important implications for the design of pseudomorphic and metamorphic II–VI device structures on lattice-mismatched substrates. Several theoretical models have been developed for the critical layer thickness, including the well-known force-balance model of Matthews and Blakeslee [J. Cryst. Growth 27, 188 (1974)]. Experimentally measured critical layer thicknesses in ZnSe/GaAs (001) heterostructures are often at variance with one another as well as the Matthews and Blakeslee model. By assuming that the lattice relaxation is a fixed fraction of the equilibrium relaxation (constant γ/γeq), Fritz [Appl. Phys. Lett. 51, 1080 (1987)] has shown that the measured hc may be much larger than the equilibrium value when using a finite experimental resolution. However, the assumption of constant fractional relaxation is not applicable to any heterostructure exhibiting kinetically limited lattice relaxation. In order to reconcile the conflicting r...

We have analyzed the strain resolution of x-ray rocking curve profiles from measurements of the peak position and peak width made with finite counting statistics. In this work, we have considered x-ray rocking curves which may be Gaussian or Lorentzian in character and have analyzed the influence of the effective number of counts, full-width-at-half-maximum (FWHM) and the Bragg angle on the resolution. Often experimental resolution values are estimated on the order of 10−5 whereas this work predicts more sensitive values (10−9) with smaller FWHM and larger effective counts and Bragg angles.

Equilibrium studies of metamorphic (partially relaxed) buffer layers are important in understanding the strain and misfit dislocation density configurations. The authors present a theoretical study of the equilibrium strain and misfit dislocation density profiles as well as appropriate design equations for nonlinearly graded (logarithmic) buffers for use in accommodating the lattice mismatch of heteroepitaxial InxGa1−xAs/GaAs (001) semiconductor devices. Minimum energy calculations show that the nonlinearly graded profile may be beneficial for the control of defect densities in lattice-mismatched devices because they have several characteristics which enhance the mobility and glide velocities of dislocations, thereby promoting longer misfit segments with relatively few threading arms. This study suggest that the use of nonlinear metamorphic buffer layers are beneficial because they contain (1) a misfit dislocation free zone (MDFZ) adjacent to the interface, which avoids dislocation pinning defects associated with substrate defects, (2) a misfit dislocation free zone near the surface, which reduces pinning interactions near the device layer which will be grown on top, and (3) a large built-in strain in the top MDFZ, which enhances the glide of dislocations to sweep out threading arms. In addition, the authors show that the use of nonlinear compositionally grading may be superior to linearly graded layers depending on the specific application of the heterostructure. Moreover, the use of a nonlinearity coefficient (deviation of the average lattice mismatch) enables comparison of nonlinearly graded metamorphic buffer layers to traditionally grown linearly graded heterostructures. The authors also present approximate design equations for the widths of the misfit dislocation free zones, the built-in strain, and peak misfit dislocation density for the general logarithmically graded semiconductor with diamond or zinc blende crystal structure and (001) orientation, and show that these design equations are in fair agreement with detailed numerical energy minimization calculations.

The inclusion of metamorphic buffer layers (MBL) in the design of lattice-mismatched semiconductor heterostructures is important in enhancing reliability and performance of optical and electronic devices. These metamorphic buffer layers usually employ linear grading of composition, and materials including InxGa1-xAs and GaAs1-yPy have been used. Non-uniform and continuously graded profiles are beneficial for the design of partially-relaxed buffer layers because they reduce the threading dislocation density by allowing the distribution of the misfit dislocations throughout the metamorphic buffer layer, rather than concentrating them at the interface where substrate defects and tangling can pin dislocations or otherwise reduce their mobility as in the case of uniform compositional growth. In this work we considered heterostructures involving a linearly-graded (type A) or step-graded (type B) buffer layer grown on a GaAs (001) substrate. For each structure type we present minimum energy calculations and compare the cases of cation (Group III) and anion (Group V) grading. In addition, we studied the (i) average and surface in-plane strain and (ii) average misfit dislocation density for heterostructures with various thickness and compositional profile. Moreover, we show that differences in the elastic stiffness constants give rise to significantly different behavior in these two commonly-used buffer layer systems.

Recent results have shown that nonlinearly graded buffer layers may be beneficial for the reduction of threading dislocation densities in metamorphic semiconductor devices. In this work, the authors have studied the equilibrium strain relaxation and misfit dislocation densities in nonlinearly graded heterostructures with logarithmic grading, and compare the cases of InxGa1−xAs/GaAs and GaAs1−yPy/GaAs buffer layers. The authors show that differences in the elastic stiffness constants give rise to significantly different behavior in these two commonly used buffer layer systems. Moreover, the width of the dislocated region, the average misfit dislocation density, and surface in-plane strain may be related to the nonlinearity coefficient of the grading profile.

Metamorphic semiconductor devices are commonly fabricated with linearly-graded buffer layers, but equilibrium modeling studies suggest that S-graded buffers, following a normal cumulative distribution function, may enable lower threading defect densities. The present work involves a study of threading dislocation density behavior in S-graded ZnS x Se 1-x buffer layers for metamorphic devices on mismatched GaAs (001) substrates using a kinetic model for lattice relaxation and misfit-threading dislocation interactions. The results indicate that optimization of an S-graded buffer layer to minimize the surface threading dislocation density requires adjustment of the standard deviation parameter and cannot be achieved by varying the buffer thickness alone. Furthermore, it is possible to tailor the design of the S-graded buffer layer in such a way that the density of mobile threading dislocations at the surface vanishes. Nonetheless, the threading dislocation behavior in these heterostructures is quite complex, and a full understanding of their behavior will require further experimental and modeling studies.

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 strain and misfit dislocation density profiles for heteroepitaxial Si1−xGex/Si (001) with convex exponential grading of composition. A graded layer of this type exhibits two regions free from misfit dislocations, one near the interface of thickness y1 and another near the free surface of thickness h−yd, where h is the layer thickness. The intermediate region contains an exponentially tapered density of misfit dislocations. We report approximate analytical models for the strain and dislocation density profile in exponentially graded Si1−xGex/Si (001) which may be used to calculate the effective stress and rate of lattice relaxation. The results of this work are readily extended to other semiconductor material systems and may be applied to the design of exponentially graded buffer layers for metamorphic device structures including transistors and light emitting diodes.