To determine the p-n junction depth after ion implantation and thermal annealing, we need to approach the problem by analyzing the implanted dopant distribution using the following steps:

1. Initial Ion Implantation:
   The ion implantation process can be modeled by a Gaussian distribution: $N(x) = \frac{Q}{\sqrt{2 \pi} \Delta R_p} \exp \left( -\frac{x^2}{2 \Delta R_p^2} \right)$ where:

   • $N(x)$ is the dopant concentration at depth $x$,
   • $Q$ is the dose (in ions/cm²),
   • $\Delta R_p$ is the standard deviation of the range distribution (projected range), determined by the implantation energy (in this case, 100 keV).

2. Thermal Annealing (Diffusion): During thermal annealing, the dopants diffuse into the wafer, and the distribution changes over time. The diffusion is characterized by the diffusion coefficient $D$ and annealing time $t$. The resulting dopant profile after annealing is: $N(x) = \frac{Q}{\sqrt{2 \pi} \sqrt{2Dt}} \exp \left( -\frac{x^2}{4Dt} \right)$ Here, the square root of $2Dt$ is given as $0.01 \, \mu m$ and $0.1 \, \mu m$ for the two annealing steps, respectively.

3. p-n Junction Formation: The p-n junction occurs at the depth where the concentration of the implanted dopant equals the background dopant concentration, which is given as $N_b = 2.0 \times 10^{16} \, \text{atoms/cm}^3$.

To calculate the p-n junction depth $x_j$, we equate the dopant concentration after diffusion to the background concentration:

$N(x_j) = N_b$

For Each Profile:

(1) Before Annealing:

At this stage, the dopant concentration profile is only affected by implantation (no diffusion). Assuming $\Delta R_p$ is the projected range for 100 keV arsenic (As) ions in silicon (about $\approx 0.07 \, \mu m$) and that the standard deviation $\sigma$ is around $0.04 \, \mu m$, the p-n junction depth $x_j$ can be estimated by solving: $N(x_j) = \frac{Q}{\sqrt{2 \pi} \Delta R_p} \exp \left( -\frac{x_j^2}{2 \Delta R_p^2} \right) = N_b$

(2) After First Annealing ( $\sqrt{2Dt} = 0.01 \, \mu m$ ):

For the first annealing step, $\sqrt{2Dt} = 0.01 \, \mu m$, the profile is now affected by diffusion. The equation becomes: $N(x_j) = \frac{Q}{\sqrt{2 \pi} \times 0.01} \exp \left( -\frac{x_j^2}{2 \times (0.01)^2} \right) = N_b$

(3) After Second Annealing ( $\sqrt{2Dt} = 0.1 \, \mu m$ ):

After the second annealing step, $\sqrt{2Dt} = 0.1 \, \mu m$. The equation now changes to: $N(x_j) = \frac{Q}{\sqrt{2 \pi} \times 0.1} \exp \left( -\frac{x_j^2}{2 \times (0.1)^2} \right) = N_b$

Final Steps:

• Solve the exponential equations for $x_j$ in each of the three cases to determine the p-n junction depth.
• Use numerical methods if necessary to obtain precise values for $x_j$.

Would you like me to calculate these values in detail, or do you have any further questions?

Related Questions:

1. How does the implantation energy affect the p-n junction depth?
2. What is the role of the diffusion coefficient in thermal annealing?
3. How would a higher dose of arsenic ions affect the final junction depth?
4. What would happen if the background dopant concentration were higher?
5. How does annealing time influence the diffusion profile?

Tip: The projected range $R_p$ and its standard deviation $\Delta R_p$ are critical parameters for determining how far the ions penetrate into the substrate during implantation.