Defect dynamics

T. Sinno, R. A. Brown, W. Van Ammon, E. Dornberger. Point defect dynamics and the oxidation-induced stacking-fault ring in Czochralski-grown silicon crystals. J Electrochem Soc 145 302, 1998. 

Chuang. 1. el al. Cosmology in the Laboratory Defect Dynamics in Liquid Crystals," Science, 1336 (March 5, 1991). 

As was noted earlier in the present chapter, one of the key ways in which an effective description of material response can be constructed is through the construction of defect dynamics approaches in which the fundamental degrees of... 

In chap. 8 we developed many of the fundamental tools needed to examine the behavior of one or several dislocations. However, an equally challenging and important problem is the statistical problem posed by a collection of large numbers of dislocations. We made a certain level of progress in confronting the statistical questions that attend the presence of multiple interacting dislocations in the previous chapter, and now revisit these questions from the standpoint of the hierarchical approaches being described here and in particular in terms of the variational approaches to defect dynamics introduced in section 12.3.2. 

Reaction-diffusion systems encounter difficulties even for the seemingly simple question of mere existence of rigidly rotating spiral waves. With the technically demanding tool of spatial dynamics in the (logarithmic) radial direction, this difficulty has been overcome for small amplitude waves in a celebrated paper by Scheel [66]. Subsequently, interesting consequences have been derived most notably a first classification of possible instabilities of defect dynamics, reiating to core and far-field break-up [71]. 

Chuang, I., Durrer, R., Turok, N.. and Yurke, B., Cosmology in the laboratory defect dynamics in liquid crystals. Science, 251, 1336-1342 (1991). 

The mathematical model of point defect dynamics can be adequately used on the basis of the physical model in which the impurity precipitation process occurs before the formation of microvoids or dislocation loops (V.I. Talanin I.E. Talanin, 2010b). The model of point defect dynamics can be considered as component of the diffusion model for formation grown-in microdefects. 

As mentioned earlier the defect formation processes in a semiconductor crystal, in general, and in silicon, in particular, have been described using the model of point defect dynamics in this case, the crystal has been considered a dynamic system and real boundary conditions have been specified. However, the model of point defect dynamics has not been used for calculating the formation of interstitial dislocation loops and microvoids under the... 

The experimentally determined temperature range of the formation of microvoids in crystals with a large diameter is 1403... 1343 K (Kato et al, 1996 Itsumi, 2002). In this respect, the approximate calculations for the solution in terms of the model of point defect dynamics were performed at temperatures in the range 1403...1073 K. The computational model uses the classical theory of nucleation and formation of stable clusters and, in strict sense, represents the size distribution of clusters (microvoids) reasoning from the time process of their formation and previous history. 

The calculations were carried out in the framework of the model of point defect dynamics, i.e., for the same crystals with the same parameters as in already the classical work on the simulation of microvoids and interstitial dislocation loops (A-microdefects) (Kulkarni et al., 2004). According to the analysis of the modern temperature fields used when growing crystals by the Czochralski method, the temperature gradient was taken to be G = 2.5 K/ mm (Kulkarni et al., 2004). The simulation was performed for crystals 150 mm in diameter, which were grown at the rates Vg = 0.6 and 0.7 mm/ min. These growth conditions correspond to the growth parameter Vg/ G >... 

Detailed calculations are presented in the articles (V.I. Talanin I.E. Talanin, 2010b).Our results somewhat differ from those obtained in (Kulkarni et al., 2004). These differences are as follows (i) the nucleation rate of microvoids at the initial stage of their formation is low and weakly increases with a decrease in the temperature and (ii) a sharp increase in the nucleation rate, which determines the nucleation temperature, occurs at a temperature T 1333 K. These differences result from the fact that the recombination factor in our calculations was taken to be kjy = 0. For kjy 0, consideration of the interaction between impurities and intrinsic point defects in the high-temperature range becomes impossible, which is accepted by the authors of the model of point defect dynamics (Kulkami et al., 2004). In this case, in terms of the model, there arises a contradiction between the calculations using the mathematical model and the real physical system, which manifests itself in the ignoring of the precipatation process (Kulkami et al., 2004). 

The calculations of the formation of microvoids and dislocation loops (A-microdefects) demonstrated that the above assumptions do not lead to substantial differences from the results of the previous calculations in terms of the model of point defect dynamics. This circumstance indicates that the mathematical model of point defect dynamics can be adequately used on the basis of the physical model in which the impurity precipitation process occurs before the formation of microvoids or interstitial dislocation loops. Moreover, the significant result of the calculations is the confirmation of the coagulation mechanism of the formation of microvoids and the deformation mechanism of the formation of interstitial dislocation loops. Therefore the model of the dynamics of point defects can be considered as component part of the diffusion model for formation grown-in microdefects. 

Dornberger, E. Ammon, von W. Virbulis, J. Hanna, B. Sinno T. (2001). Modeling of transient point defect dynamics in Czochralski silicon crystal. Journal Crystal Growth, Vol. 230, No. 1-2, pp. 291-299, ISSN 0022-0248. 

Kulkami, M.S. Voronkov, V.V. Falster, R. (2004). Quantification of defect dynamics in unsteady-state and steady-state Czochralski growth of monocrystalline silicon. Journal Electrochemical Society, Vol. 151. - No. 5, pp. G663-G669, ISSN 0013-4651. 

Kulkami, M.S. (2007). Defect dynamics in the presence of oxygen in growing Czochralski silicon crystals. Journal Crystal Growtit, Vol. 303, No. 2, pp. 438-448, ISSN 0022-0248. 

Figure 10 Schematic representation of HMM applications. In (a) simulation of a macroscopic process for which the constitutive relations have to be obtained from modeling at the microscale. The macroscopic system is solved using a grid xj ) and only a small region around each macroscopic-solver grid point is used for the atomistic calculation (the shaded area represents the atomic cell at grid point The time step (TS) used for the macroscopic calculations is much larger than the microscopic one (ts), and times, TS are necessary to equilibrate the atomistic calculations. In (b) and (c) isolated defect calculations, i.e., problems where the coupling with the microscale model is needed only in a limited part of the system (near the defect itself). If the time scale for the defect dynamics is much larger than the time scale for the relaxation of the defect structure (case b), then only a short time At TS is simulated using the atomistic model for each macroscopic time step, otherwise (case c) the whole time history of the defect should be computed atomistically.

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