In the case of interstitial diffusion the impurity diffuses by squeezing between the lattice atoms and taking residence in the interstitial space between lattice sites. Since this mechanism does not require the presence of a vacancy, it proceeds much faster than substitutional diffusion Conventional dopants such as B, P, As, and Sb diffuse by the substitutional method. This is beneficial in that the diffusion process is much slower and can therefore be controlled more easily in the manufacturing process. Many of the undesired impurities such as Fe, Cu, and other heavy metals diffuse by the interstitial mechanism and therefore the process is extremely fast. This again is beneficial in that at the temperatures used, and in the duration of fabri unwanted metals can diffuse completely through the Si wafer Gettering creates trapping sites on the back surface of the wafer for these impurities that would otherwise remain in the silicon and cause adverse device effects Regardless of the diffusion mechanism, it can be formalized mathematically in the same way by introducing a diffusion coefficient, D(cm2/sec), that accounts for the diffusion rate. The diffusion constants follow an behavi ng to the E where Do is the prefactor, EA the activation energy, k Boltzmanns constant, and T the absolute temperature. Conventional silicon dopants(substitutional diffusers) have diffusion coefficients on the order of 10-to 10-12 at 1100@C, whereas heavy metal interstitial diffusers(Fe, Au, and Cu) have diffusion coefficients of 10- to 10 at this temperature The diffusion process can be described using Ficks Laws. Ficks first law says that the flux of impurity, F crossing any plane is related to the impurity distribution, N(x, t)per cm, by (23.8) dx in the one-dimensional case. Ficks second law states that the time rate of change of the particle density in turn is related to the divergence of the particle flux: an dF 239) Combining these two equations gives an aaN a-N (23.10) in the case of a constant diffusion coefficient as is often assumed This partial differential equation can be solved by separation of variables or by Laplace transform techniques for specified boundary conditions For a constant source diffusion the impurity concentration at the surface of the wafer is throughout the diffusion process. Solution of Eq (23. 10)under these boundary conditions, asst infinite wafer, results in a complementary error function diffusion profile: c2000 by CRC Press LLC© 2000 by CRC Press LLC In the case of interstitial diffusion the impurity diffuses by squeezing between the lattice atoms and taking residence in the interstitial space between lattice sites. Since this mechanism does not require the presence of a vacancy, it proceeds much faster than substitutional diffusion. Conventional dopants such as B, P, As, and Sb diffuse by the substitutional method. This is beneficial in that the diffusion process is much slower and can therefore be controlled more easily in the manufacturing process. Many of the undesired impurities such as Fe, Cu, and other heavy metals diffuse by the interstitial mechanism and therefore the process is extremely fast. This again is beneficial in that at the temperatures used, and in the duration of fabrication processes, the unwanted metals can diffuse completely through the Si wafer. Gettering creates trapping sites on the back surface of the wafer for these impurities that would otherwise remain in the silicon and cause adverse device effects. Regardless of the diffusion mechanism, it can be formalized mathematically in the same way by introducing a diffusion coefficient, D (cm2 /sec), that accounts for the diffusion rate. The diffusion constants follow an Arrhenius behavior according to the equation: (23.7) where D0 is the prefactor, EA the activation energy, k Boltzmann’s constant, and T the absolute temperature. Conventional silicon dopants (substitutional diffusers) have diffusion coefficients on the order of 10–14 to 10–12 at 1100°C, whereas heavy metal interstitial diffusers (Fe, Au, and Cu) have diffusion coefficients of 10–6 to 10–5 at this temperature. The diffusion process can be described using Fick’s Laws. Fick’s first law says that the flux of impurity, F, crossing any plane is related to the impurity distribution, N(x,t) per cm3 , by: (23.8) in the one-dimensional case. Fick’s second law states that the time rate of change of the particle density in turn is related to the divergence of the particle flux: (23.9) Combining these two equations gives: (23.10) in the case of a constant diffusion coefficient as is often assumed. This partial differential equation can be solved by separation of variables or by Laplace transform techniques for specified boundary conditions. For a constant source diffusion the impurity concentration at the surface of the wafer is held constant throughout the diffusion process. Solution of Eq. (23.10) under these boundary conditions, assuming a semi￾infinite wafer, results in a complementary error function diffusion profile: (23.11) D D E kT A = -È Î Í Í ˘ ˚ ˙ ˙ 0 exp F D N x = ¶ ¶ ¶ ¶ ¶ ¶ N t F x = ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ N t x D N x D N x = Ê Ë Á ˆ ¯ ˜ = 2 2 N N x Dt (,) x t = Ê Ë Á ˆ ¯ 0 ˜ 2 erfc