Silicon self-diffusion

For silicon self-diffusion, the total diffusion coefficient could be expressed as... 

It is clear Ifom Fig. 7 that the silicon self-diffusion coefficient in undoped crystals is lower than in doped crystals. An opposite dependence was observed for the carbon self-diffusion coefficient. According to the law of acting masses, the silicon vacancy (acceptor) concentration increases with increasing donor dopant concentration (nitrogen). AU this leads to an increase in the diffusion coefficient in the presence of a vacancy mechanism of self-diffusion. Carbon vacancies are donors, and their concentration decreases when the donor concentration is increased. This leads to a decrease in the carbon self-diffusion coefficient. 

The technology of silicon and germanium production has developed rapidly, and knowledge of die self-diffusion properties of diese elements, and of impurity atoms has become reasonably accurate despite die experimental difficulties associated widi die measurements. These arise from die chemical affinity of diese elements for oxygen, and from die low values of die diffusion coefficients. 

Because of die rigidity and directionality of die covalent bonds die energies of self-diffusion have been found to be higher diaii diose of metals. In die case of silicon, it appears drat a furdier complication is drat an intersti-tialcy mechanism predominates above 1000°C. Below diis teiiiperamre, bodi elements appear to self-diffuse by atom-vacancy exchange as for die metals. 

It will be noted that because of the low self-diffusion coefficients the numerical values for representations of self-diffusion in silicon and germanium by Anhenius expressions are subject to considerable uncertainty. It does appear, however, that if this representation is used to average most of the experimental data the equations are for silicon... 

Fig.3.1.9 (a) The adsorption-desorption isotherm (circles, right axis) and the self-diffusion coefficients D (triangles, left axis) for cyclohexane in porous silicon with 3.6-nm pore diameter as a function of the relative vapor pressure z = P/PS1 where Ps is the saturated vapor pressure, (b) The self-diffusion coefficients D for acetone (squares) and cyclohexane (triangles) as a function of the concentration 0 of molecules in pores measured on the adsorption (open symbols) and the desorption (filled symbols) branches. 

Lesher C.E., Hervig R.L., and Tinker D. (1996) Self diffusion of network formers (silicon and oxygen) in naturally occurring basaltic liquid. Geochim. Cosmochim. Acta 60, 405-413. 

W. Frank. The interplay of solute and self-diffusion—A key for revealing diffusion mechanisms in silicon and germanium. In D. Gupta, H. Jain, and R.W. Siegel, editors, Defect and Diffusion Forum, volume 75, pages 121-148, Brookfield, VT, 1991. Sci-Tech Publications. 

Theoretical studies of diffusion aim to predict the distribution profile of an exposed substrate given the known process parameters of concentration, temperature, crystal orientation, dopant properties, etc. On an atomic level, diffusion of a dopant in a silicon crystal is caused by the movement of the introduced element that is allowed by the available vacancies or defects in the crystal. Both host atoms and impurity atoms can enter vacancies. Movement of a host atom from one lattice site to a vacancy is called self-diffusion. The same movement by a dopant is called impurity diffusion. If an atom does not form a covalent bond with silicon, the atom can occupy in interstitial site and then subsequently displace a lattice-site atom. This latter movement is believed to be the dominant mechanism for diffusion of the common dopant atoms, P, B, As, and Sb (26). 

Point Defects. Point defects are defined as atomic defects. Atomic defects such as metal ions can diffuse through the lattice without involving themselves with lattice atoms or vacancies (Figure 9), in contrast to atomic defects such as self-interstitials. The silicon self-interstitial is a silicon atom that is bonded in a tetrahedral interstitial site. Examples of point defects are shown in Figure 9. 

Point Defect Models of Diffusion in Silicon. Under conditions of thermal equilibrium, a Si crystal contains a certain equilibrium concentration of vacancies, C v°, and a certain equilibrium concentration of Si self-interstitials, Cz°. For diffusion models based on the vacancy, Cv° Cf and the coefficients of dopant diffusion and self-diffusion can be described by equation 27 (15)... 

Point Defect Generation During Phosphorus Diffusion. At Concentrations above the Solid Solubility Limit. The mechanism for the diffusion of phosphorus in silicon is still a subject of interest. Hu et al. (46) reviewed the models of phosphorus diffusion in silicon and proposed a dual va-cancy-interstitialcy mechanism. This mechanism was previously applied by Hu (38) to explain oxidation-enhanced diffusion. Harris and Antoniadis (47) studied silicon self-interstitial supersaturation during phosphorus diffusion and observed an enhanced diffusion of the arsenic buried layer under the phosphorus diffusion layer and a retarded diffusion of the antimony buried layer. From these results they concluded that during the diffusion of predeposited phosphorus, the concentration of silicon self-interstitials was enhanced and the vacancy concentration was reduced. They ruled out the possibility that the increase in the concentration of silicon self-interstitials was due to the oxidation of silicon, which was concurrent with the phosphorus predeposition process. 

Silicon nitride is a highly covalent bonded compound with self-diffusion coefficient of the nitrogen atoms of 6.3 x 10 2(1 cm2/s at 1400°C.22 Therefore, densification without any sintering additives is nearly impossible. In 1961, Deeley and Herbert.23 was the first to report that Si3N4 ceramics could be... 

Kijama, K., Shirasaki, S. (1976), Nitrogen self diffusion in silicon nitride , J. Chem. Phys., 65, 2668. 

In the past, attempts to prepare such ternary nitrides by reaction of the respective binary nitrides always have failed because the binary nitrides do not melt congruently and also because of the low self-diffusion coefficients of these materials. However, for the synthesis of SiPNj a molecular precursor Cl3SiNPCl3 has been proven to be specifically useful [5]. In this compound the required structural element of two vertex sharing tetrahedra centered by phosphorus and silicon and connected via a common nitrogen atom is pre-organized on a molecular level. The precursor compound is obtained (Scheme 3) in a three-step synthesis starting from ((CH3)3Si)2NH which is commercially available. 

The randomizaton takes place by a a diffusion of atoms that is implicit in our earlier description of the initial randomization process as being akin to melting [1]. Later it was shown that the root-mean-square displacement of each atom must be of the order of the nearest-neighbor distance in order that the network lose all memory of the original crystal structure as measured by the structure factor S q) [21]. In this context, the melting point can be defined as that temperature for which the mean square displacement increases linearly with time. It appears, though, that a sequence of bond switches as illustrated in Fig. 1 is not the primary mechanism for self-diffusion in silicon [31,32]... 

In general, covalently bonded materials are difficult to sinter because of their inherently low value of self-diffusivity. However, using microwave heating, silicon nitride with 20 wt.% yttria-doped zirconia has been sintered at 1400°C in a nitrogen atmosphere at a pressure 0.1 MPa. In contrast, via a conventionally heated hot isostatic press, a sintering temperature of 1850°C and a nitrogen pressure of about 180 MPa were required to densify the same silicon nitride/zirconia composition as completely as was achieved by microwave heating at 1400°C. ... 

Blochl P. E., Smargiassi E., Car R., Laks D. B., Andreoni W. and Pantelides S. T, First-Principles Calculations of Self-Diffusion Constants in Silicon, Phys. Rev. Lett. 70, 2435 (1993). 

Tang M., Colombo L., Zhu J. and Diaz de la Rubia T., Intrinsic Point Defects in Crystalline Silicon Tight-Binding Molecular Dynamics Studies of Self-Diffusion, Interstitial-vacancy Recombination and Formation Volumes, Phys. Rev. B55, 14279 (1997). 

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