A Generalized Model of the Yield Drop for Impure Semiconductors
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A Generalized Model of the Yield Drop for Impure Semiconductors Boris V. Petukhov Institute of Crystallography, Russian Academy of Sciences, Leninskii pr. 59, Moscow, 117333 Russia ABSTRACT The Alexander-Haasen theory describing the deformation behavior of low-dislocated semiconductor crystals is generalized taking into account the dynamic ageing of dislocations due to the impurities dragging. The constitutive equations describing kinetics of the plastic deformation are modified for the case of a spectrum of age-dependent internal stresses. Besides the solution hardening, the theory developed explains a number of qualitative distinctions of the elastic-plastic transition in silicon crystals grown by the Czochralski and the float zone methods. Particularly, this concerns a dependence of the yield drop on the initial dislocation density, and a weakening of the strain rate sensitivity of the yield stress. INTRODUCTION The yield drop in semiconductors is the most pronounced transient deformation instability. However, for silicon, the classic Alexander-Haasen model of the avalanche dislocation multiplication [1,2] (see also [3]), qualitatively good, does not describe some features of the phenomenon, for example, a great difference in behavior of Czochralski- and float zonegrown crystals. To account for such difference, it is natural to ascribe it to a dynamic interaction between solute atoms and moving dislocations, and to incorporate this interaction into the model of the yield drop. This contribution presents a combined approach, which unites the AlexanderHaasen-Johnston-Gilman (AHJG) dislocation multiplication model and impurities dragging model. An accumulation of solutes in a cloud surrounding the dislocation changes its dynamical properties with time. A dependence of the dragging force on the dislocation prehistory makes it necessary to modify the ordinary concept of a common effective stress to a more detailed description using a spectrum of particular dislocation age-dependent effective stresses. As a consequence, a modification of the constitutive equations describing the plastic deformation of considered materials is required. In the following, a generalized model of the yield drop phenomenon for impure semiconductors is suggested and solved. Results are qualitatively compared with experimental data. DYNAMIC AGEING OF DISLOCATIONS Due to the interaction with dislocations, solute atoms diffuse to dislocation cores and segregate at them. In result, the dislocation mobility decreases with time, in other words, dislocations are “ageing”. This process takes place for static dislocations as well as for ones moving with a sufficiently small velocity, when solutes have time to follow them. The solutes dragging will be described by equation dc rV = c0 2 . dt a
(1)
BB3.3.1
Here c is the solute concentration at dislocations, c0 is the solute concentration in the volume, V is the dislocation velocity, r is the capture range, a is the lattice period. Equation (1) implies that in unite of time all solutes are captured in
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