The theoretical solution (2.7) of the diffusion equation can be used to model the diffusion of atoms in a thin film mediator. An appropriate measurement technique is used to obtain a reasonable set of experimental data. This data is then compared numerically with the theoretical relation (2.7) with various additional terms and considerations to account for mechanisms in the real system that either enhance or stunt the diffusion process.

The functional $x$-dependence of concentration (2.7) can be rewritten in terms of the average root mean square of $x$, the diffusion length,

Equation 2.7 may be rewritten,

An experimental result gives[1] $\bar{x} = 1.3\times$10$^4$ Å at $400^\circ \textrm{C}$. Figure 2.5 shows the effective concentration at $1.3\times 10^4$Å is $C_s(1/e) \approx 0.33 C_s$ as expected after $t = 1 \textrm{s}$ of diffusing time. In general, the thickness of the Al thin film should be on the order of $\sim10^4$Å for effective transport of Si diffusors to the buried Si/Al interface.

In striking comparison, the diffusion of Al in Si[2] would require $\sim10^8$ years to reach the length of $1.3\times 10^4$ Å. Dynamically, a Si diffusor in an Al mediator is a suitable combination for this technology. Moreover, the diffusion length of Si in Al is appropriate for an industrial fabrication process. The Si growth rate can be as high as 20 Å/s from an electron beam evaporation source, fully compatible with mass production and processing of Si devices ($\sim 2$ Å/s).

In the investigated case of an Al mediator and Si diffusor, the Al-Al bonds in the bulk lattice are much stronger than Al-Si bonds. Thus, atomic Si diffusion in Al may be considered in terms of hard atomic spheres. The volume of a Si atom is about half the volume of an Al atom (Table 1.1).

References