Diffusion of Point Defects in a Stressed Simple Cubic Lattice
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DIFFUSION OF POINT DEFECTS IN A STRESSED SIMPLE CUBIC LATTICE
DIMITRIOS MAROUDAS AND ROBERT A. BROWN Massachusetts Institute of Technology, Department of Chemical Engineering, Cambridge, MA 02139
ABSTRACT A systematic theoretical and computational study is presented for the diffusion of point defects in a stressed cubic lattice. The study combines an atomistic description of point defect migration with a continuum model for point defect transport. Moment analysis of the macroscale transport equation gives expressions for the drift velocity and the diffusivity tensor of the point defects, which are calculated by a dynamic Monte Carlo simulation of the defect migration process. Results are presented for the case of diffusion under a constant force of interaction between the applied stress on the crystal and the defects. The continuum model gives the general constitutive model for stress-assisted point defect diffusion. 1. INTRODUCTION Quantitative analysis of the influence of processing conditions on the number and type of defects in crystalline semiconductors requires the development of macroscale constitutive equations that relate the number, creation and motion of defects in the lattice to the mechanical stress and temperature history of the material. The first step in such a development is an analysis of the diffusion of point defects, either vacancies or interstitial atoms, in the presence of applied or internal mechanical stresses. Migration of point defects in a crystal takes place in a stress field. The distribution of point defects is characterized by its number density n(r, t). A force F(r) is exerted on each of the point defects in the distribution through the interaction potential U(r) between the stress field and the point defect; F(r) = -VU(r). Quantitative modelling of the point defect dynamics in the lattice is based on the solution of the macroscopic transport equation of point defect migration On -5T = _-.
(udn) + V.- (D. Vn),
(1)
where D is the diffusivity tensor and Ud is the drift velocity of the point defect distribution due to the potential U that is superimposed on the perfect lattice potential. A constitutive model for the dependence of D and Ud on the force F and the temperature field T(r, t) in the crystal is necessary to solve eq. (1). Such a constitutive model is obtained by analysis of the diffusion process in the crystalline lattice. 2. ANALYSIS OF THE POINT DEFECT DIFFUSION PROCESS Point defect migration is modelled on the lattice scale as a thermally activated jumpj) process between nearest neighbor equilibrium lattice sites. If a particle occupies an eqlui-
Mat. Res. Soc. Symp. Proc. Vol. 163. 'c,1990 Materials Research Society
616
librium position i, at either a substitutional or an interstitial site, it can move from this site to Z nearest neighbor sites labelled j, where Z is the coordination number of the lattice. Each i - j jump is characterized by a jump frequency wij, that is proportional to the Boltzmann factor exp(-Qij/kT), where Qij is the potential energy barrier
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