Questions and Answers on the Activation Strain
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In contrast, the growth rate decreases with uniaxialcompression in the plane of the interface, and is enhanced by uniaxial tension, as shown in Fig. 1 [2]. The bar-shaped samples in Fig. 1 (a) were deformed elastically at temperatures high enough for solid phase epitaxy to proceed but low enough to prevent plastic deformation of the crystal. The stress state so induced is uniaxially compressive in the x direction (left-right on the page) on the top side of the wafer and uniaxially tensile on the bottom side; the magnitude of the stress varies linearly with position from the center out to the left and right edges. These apparently contradictory effects of hydrostatic vs. nonhydrostatic compression have been explained by an extension of the theory of thermally activated growth to nonhydrostatic stress states. Guide
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THEORY OF THERMALLY ACTIVATED GROWTH The rate of a kinetic process is generally written as the product of a mobility for that process with some measure of the driving force for the process. In this work we are addressing the dependence of the mobility, rather than of the driving force, on nonhydrostatic stress. In the theory of thermally activated growth, applied to any defect-mediated, unimolecular growth process occurring under hydrostatic pressure, the rate is given by Rate = (constant) C . m. [I -exp(AG/kT)]
(1)
where C is the defect concentration, m the defect mobility, kT has the usual meaning, and AG in the thermodynamic factor is the Gibbs free energy of the growing phase minus that of the parent phase (AG < 0). The defect mobility is proportional to a Boltzmann factor in the barrier height, exp(-AGm/kT), where AGm is the standard Gibbs free energy of the system in the transition state (saddle point in configuration space) minus that in the initial state [3]. If the defect concentration equilibrates rapidly, C is proportional to a Boltzmann factor in the standard Gibbs free energy of defect formation, exp(-AGo/kT). It is also possible that the time scale for equilibration of the defect concentration is much longer than the duration of the experiment, in which case C is a constant. The formation and migration terms are often combined into a free energy of activation, resulting in Rate thermodynamic factor
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where M is the mobility for the reaction and AG* = AGm + AGO or AG* = AGm as the case may be. In SPEG, the very slow variation of the thermodynamic factor (compared to that of the Boltzmann factor) with
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