Critical Layer Thickness in Exponentially Graded Heteroepitaxial Layers

Abstract

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 In _x Ga_1−x As/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 In _x Ga_1−x As/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.