When impurities are diffused into a sample from an unlimited source such that the surface concentration $N_{0}$ is held constant, the impurity distribution (profile) is given by
$$
N(x, t)=N_{0} \operatorname{erfc}\left(\frac{x}{2 \sqrt{D t}}\right)
$$
where $D$ is the diffusion coefficient for the impurity, $t$ is the diffusion time, and erfc is the complementary error function.
If a certain number of impurities are placed in a thin layer on the surface before diffusion, and if no impurities are added and none escape during diffusion, a gaussian distribution is obtained:
$$
N(x, t)=\frac{N_{s}}{\sqrt{\pi D t}} e^{-\left(x / 2 \sqrt{D_{t}}\right)^{2}}
$$
where $N_{s}$ is the quantity of impurity placed on the surface $\left(\right.$ atoms $\left./ \mathrm{cm}^{2}\right)$ prior to $t=0$. Notice that this expression differs from Eq. (4-44) by a factor of two. Why?
Figure P5-2 gives curves of the complementary error function and gaussian factors for the variable $u$, which in our case is $x / 2 \sqrt{D t}$. Assume that boron is diffused into $\mathrm{n}$ -type $\mathrm{Si}$ (uniform $\left.N_{d}=5 \times 10^{16} \mathrm{~cm}^{-3}\right)$ at $1000^{\circ} \mathrm{C}$ for 30 minutes. The diffusion coefficient for $\mathrm{B}$ in $\mathrm{Si}$ at this temperature is $D=3 \times 10^{-14} \mathrm{~cm}^{2} / \mathrm{s}$
(a) Plot $N_{a}(x)$ after the diffusion, assuming that the surface concentration is held constant at $N_{0}=5 \times 10^{20} \mathrm{~cm}^{-3} .$ Locate the position of the junction below the surface.
(b) Plot $N_{a}(x)$ after the diffusion, assuming that $\mathrm{B}$ is deposited in a thin layer on the surface prior to diffusion $\left(N_{s}=5 \times 10^{13} \mathrm{~cm}^{-2}\right)$, and no additional B atoms are available during the diffusion. Locate the junction for this case.
Hint: Plot the curves on five-cycle semilog paper, with an abscissa varying from zero to $\frac{1}{2} \mu \mathrm{m}$. In plotting $N_{a}(x)$, choose values of $x$ that are simple multiples of $2 \sqrt{D t}$.