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Asymptotic estimates of diffusion times for rapid thermal annealing
1.Because of the form of Eq. (3), Dt is often used instead of θ to denote this parameter. Since it is experimentally measurable, it is used to compute values for and E. Strictly speaking, the integral in Eq. (2) should be carried out over the time interval of the anneal. However, since is essentially zero for values of t outside this interval, the limits of integration may be extended without changing the value of the integral. This parameter is also used to solve the diffusion equation when is independent of concentration and position;
1.see R. Ghez, G. S. Oehrlein, T. O. Sedgwick, F. F. Morehead, and Y. H. Lee, Appl. Phys. Lett. 45, 881 (1984).
2.P. A. Redhead, Vacuum 12, 203 (1962).
3.Thermally Stimulated Relaxation in Solids, P. Bräunlich, ed. (Springer, Berlin, 1979).
4.J. L. Ham, R. M. Parke, and A. J. Herzig, Trans. Amer. Soc. Metals 31, 877 (1943).
5.S. J. Rothman, in Diffusion in Crystalline Solids, edited by G. E. Murch and A. S. Nowick (Academic, NY, 1984), pp. 1–61.
6.See, for example, C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw‐Hill, NY, 1978), pp. 261–263.
7.In the laboratory the order of contact is determined by how fast the heat source brings the sample to the plateau temperature: the longer and slower the approach, the higher the order of contact (cf. Fig. 2).
8.Only the sections of the ramps within roughly 20 ° of the plateau cause a significant amount of diffusion. Physically, this is the key in the use of Laplace’s method.
9.Experimental details of the measurement procedure have been described by T. O. Sedgwick, S. A. Cohen, G. S. Oehrlein, V. R. Deline, R. Kalish, and S. Shatas, in VLSI Science and Technology/1984. edited by K. E. Bean and G. A. Rozgonyi (The Electrochemical Society, Pennington, NJ, 1984), p. 192.
10.Equation (10) is also interesting since it may be possible to calculate an activation energy from a single plateau temperature using this equation and several profiles with sufficiently small δ values.
11.Such effects have been reported by G. S. Oehrlein, R. Ghez, J. D. Fehribach, E. F. Gorey, T. O. Sedgwick, S. A. Cohen, and V. R. Deline, in Proceedings of the Thirteenth International Conference on Defects in Semiconductors, edited by J. M. Parsey (The Metallurgical Society of AIME, Cleveland, 1984).
12.These calculations may have to be iterative since the correction terms contain β which in tum depends on the activation energy.
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