Quasi-steady-state heat transfer/thermal stress model for dislocation generation in the vertical gradient freeze growth of GaAs

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We have formulated a tractable model of the vertical gradient freeze (VGF) process for GaAs, providing dislocation density contour lines in terms of geometrical and physical parameters. First, the temperature distribution in a cylindrical boule has been determined in closed form involving modified Bessel functions of the first kind, order zero (I0) by solving the quasi-steady-state partial differential equation for heat conduction. Subsequently, the principal thermoelastic stress components have been evaluated and then resolved in the {111}, <11̄0≳ slip system which in excess of the critical resolved shear stress (CRSS) introduce dislocations by slip. We present dislocation density contour maps for 2- and 3-in.-diam undoped (100) GaAs grown by VGF under a variety of linear thermal gradients (v) imposed on the periphery of the boule. We show that for large v the dislocation distribution is similar to that observed in liquid-encapsulated Czochralski (LEC) material but lowering v effectively suppresses dislocation generation even in boules larger than 2 in. in diameter. A comparison of dislocation generation in VGF and standard LEC growth using very recent CRSS data is also given.