A Thermoelastic Analysis of Dislocation Generation in Pulled GaAs Crystals

Abstract

The optical and electronic characteristics of devices based on GaAs (LEDs, laser diodes, etc.) are adversely affected by the dislocations originating in the substrates. We demonstrate by means of thermoelastic analysis that the primary cause for the observed dislocation density patterns in Czochralski-pulled GaAs single crystals, which serve as a source for substrates, is crystallographic glide, induced by the excessive thermal stresses arising during the growth process. First, we formulate a tractable model for crystal growth. We obtain the temperature distribution in the crystal by solving the quasi-steady-state partial differential equation for heat conduction subject to appropriate boundary conditions. The closed-form solution includes time, pull rate, axial location, radius, convective and radiative heat transfer coefficients (h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> + h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> ), and a fixed ambient temperature (T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</inf> ) among the variables. Next, from the temperature profiles we determine the radial, tangential, and axial stress components acting on the GaAs boule. These stresses permit the evaluation of the 12 resolved shear stress components for the {111}, 〈1&bar;10〉 slip system. We postulate the sum of the absolute values of the 12 components (σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tot</inf> ) to be proportional to the dislocation density within an additive constant. Employing σ <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">tot</inf> as a parameter, we have constructed dislocation distribution contour maps for {100} GaAs wafers which are in good accord with the dislocation patterns observed on KOH-etched wafers cut from near the top end of Cr and Te-doped GaAs boules. A detailed examination of the effect of the numerous parameters on the dislocation density of Czochralski-pulled GaAs is also given. Only by a drastic increase of T <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</inf> and a substantial decrease of h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</inf> + h <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> would one be able to overcome the natural limitations imposed by the thermal and mechanical properties on dislocation density. Finally, we pay attention to the effects of elastic anisotropy and interfacial heat flux, discuss the philosophical and mathematical difficulties associated with finding a true transient solution, provide some practical suggestions