Chapter 14, Problem 14.47P

If an amount of energy $Q_0^{"}\left({\text{J/m}}^2\right)$ is released instantaneously, as, for example, from a pulsed laser, and it is absorbed by the surface of a semi-infinite medium, with no attendant losses to the surroundings,the subsequent temperature distribution in the medium is
$T\left(x,t\right)-T_i=\frac{Q_0^{"}}{pc\left(r\alpha t\right)}\text{exp}\left(\frac{-x^2}{4\alpha t}\right)$
where ${\text{T}}_{\text{i}}$ is the initial, uniform temperature of the medium. Consider an analogous mass transfer process involving deposition of a thin layer of phosphorous (P) on a silicon (Si) wafer at room temperature. If the wafer is placed in a furnace, the diffusion of P into Si is significantly enhanced by the high-temperature environment.

A Si wafer with 1 -tm-thick P film is suddenly placed in a furnace at 1000°C, and the resulting distribution of P is characterized by an expression of the form
$C_P\left(x,t\right)=\frac{M_{p,0}^{"}}{{\left(\pi D_{P-Si}\right)}^{1/2}}\text{exp}\left(\frac{-x^2}{4D_{Psi}t}\right)$
where $M_{p,0}^{"}$ is the molar area density $\left(kmol/m^2\right)$ of P associated with the film of concentration $C_p$ and thickness $d_0$ .
(a) Explain the correspondence between variables in the analogous temperature and concentration distributions.
(b) Determine the mole traction of P at a depth of $\text{0}\text{.1μm}$ in the Si after 30 s. The diffusion coefficient ${\text{D}}_{\text{p-Si}}\text{= 1}{\text{.2 × 10}}^{\text{-19}}{\text{m}}^{\text{2}}\text{/s}$ . The mass densities of P and Si are 2000 and $2300kg/m^2$ , respectively, and their molecular weights are 30.97 and $\text{28}\text{.09 kg/kmol}$ .