Let's break down the solution step by step: --- ### **Given:** - **Diffusion Time (\(t\))** = 5 hr = 18,000 s - **Temperature (\(T\))** = 1100°C = 1373 K --- ### **Step 1: Find the required oxide thickness for boron** #### **Explanation:** To mask the diffusion, the oxide thickness should be such that the diffusion through the oxide is negligible compared to that through silicon. The **masking factor** is the ratio of diffusivity in silicon to that in silicon dioxide: \[ \text{Masking Factor} = \frac{D_{Si}}{D_{SiO_2}} \] Typically, the oxide is considered an effective mask if it reduces the dopant flux by a factor of 1000. So, we find the oxide thickness (\(x_{ox}\)) that makes the flux through it negligible. ##### **For Boron:** - **Diffusivity of Boron in Si at 1100°C:** \(D_{Si}^{B} \approx 6 \times 10^{-15} \text{ cm}^2/\text{s}\) - **Diffusivity of Boron in SiO₂ at 1100°C:** \(D_{SiO_2}^{B} \approx 3 \times 10^{-17} \text{ cm}^2/\text{s}\) \[ \frac{D_{Si}^{B}}{D_{SiO_2}^{B}} = \frac{6 \times 10^{-15}}{3 \times 10^{-17}} = 200 \] We use the "**1% rule**" for masking: The oxide thickness should be at least \(\sqrt{D_{Si} \cdot t}\), but to be effective, we often use 10 times that value. \[ x_{ox}^{B} = 10 \times \sqrt{D_{SiO_2}^{B} \cdot t} \] Calculate: \[ x_{ox}^{B} = 10 \times \sqrt{3 \times 10^{-17} \cdot 1.8 \times 10^4} \] \[ = 10 \times \sqrt{5.4 \times 10^{-13}} \] \[ = 10 \times 7.35 \times 10^{-7} \text{ cm} \] \[ = 7.35 \times 10^{-6} \text{ cm} = 73.5 \text{ nm} \] --- ### **Step 2: Repeat for phosphorus** #### **Explanation:** ##### **For Phosphorus:** - **Diffusivity of P in Si at 1100°C:** \(D_{Si}^{P} \approx 3 \times 10^{-13} \text{ cm}^2/\text{s}\) - **Diffusivity of P in SiO₂ at 1100°C:** \(D_{SiO_2}^{P} \approx 2 \times 10^{-15} \text{ cm}^2/\text{s}\) \[ x_{ox}^{P} = 10 \times \sqrt{D_{SiO_2}^{P} \cdot t} \] \[ = 10 \times \sqrt{2 \times 10^{-15} \cdot 1.8 \times 10^4} \] \[ = 10 \times \sqrt{3.6 \times 10^{-11}} \] \[ = 10 \times 6 \times 10^{-6} \text{ cm} \] \[ = 6 \times 10^{-5} \text{ cm} = 600 \text{ nm} \] --- ### **Final Answers:** #### **(a) For boron:** - **Required oxide thickness = 73.5 nm (or about 74 nm)** #### **(b) For phosphorus:** - **Required oxide thickness = 600 nm** --- **Summary Table:** | Dopant | Required SiO₂ Thickness | |-------------|------------------------| | Boron | 74 nm | | Phosphorus | 600 nm | Let me know if you need the step-by-step calculations shown more explicitly or if you want the answer in a different format!

Let's carefully work through each part of the problem with detailed explanations. --- ### **Given Data:** - **Diffusion time, \( t \) = 5 hours = 5 × 3600 = 18,000 seconds** - **Temperature, \( T \) = 1100°C = 1373 K** --- ### **Part (a): Find the required silicon dioxide (SiO₂) thickness to mask boron diffusion** #### **Step 1: Understand the masking principle** **Explanation:** To prevent the boron from diffusing into silicon, a silicon dioxide layer is used as a mask. The effectiveness depends on the oxide's ability to slow or block the dopant flux. The key is to determine the oxide thickness such that the dopant flux through the oxide is negligible during the diffusion period. The diffusivities of boron in silicon and silicon dioxide at 1100°C are known: - **Diffusivity in silicon \( D_{Si}^{B} \) ≈ \( 6 \times 10^{-15} \) cm²/sec** - **Diffusivity in SiO₂ \( D_{SiO_2}^{B} \) ≈ \( 3 \times 10^{-17} \) cm²/sec** **Explanation:** Since the diffusivity in SiO₂ is much lower, the oxide acts as a barrier. The ratio of diffusivities indicates how much more resistant the oxide is. --- #### **Step 2: Calculate the masking factor** \[ \text{Masking factor} = \frac{D_{Si}^{B}}{D_{SiO_2}^{B}} = \frac{6 \times 10^{-15}}{3 \times 10^{-17}} = 200 \] **Interpretation:** The diffusivity in silicon is 200 times higher than in SiO₂, indicating the oxide is a good mask. --- #### **Step 3: Determine the oxide thickness needed** **Explanation:** A common approximation for masking is to ensure the oxide layer provides enough resistance such that the dopant flux is reduced by about 1000 times (or more). The oxide thickness \( x_{ox} \) should satisfy: \[ x_{ox} \approx 10 \times \sqrt{D_{SiO_2} \times t} \] This is based on Fick's laws, assuming a steady-state or quasi-steady diffusion. Calculate: \[ x_{ox}^{B} = 10 \times \sqrt{3 \times 10^{-17} \times 1.8 \times 10^{4}} \] \[ = 10 \times \sqrt{5.4 \times 10^{-13}} \] \[ = 10 \times 7.35 \times 10^{-7} \, \text{cm} \] \[ = 7.35 \times 10^{-6} \, \text{cm} \] Convert to nanometers: \[ 1\, \text{cm} = 10^{7}\, \text{nm} \] \[ x_{ox}^{B} = 7.35 \times 10^{-6} \, \text{cm} \times 10^{7} \, \text{nm/cm} = 73.5\, \text{nm} \] **Final answer for part (a):** **Approximately 74 nm of SiO₂ is required to mask boron diffusion.** --- ### **Part (b): Repeat for phosphorus** #### **Step 1: Diffusivity values** - **Diffusivity in silicon \( D_{Si}^{P} \) ≈ \( 3 \times 10^{-13} \) cm²/sec** - **Diffusivity in SiO₂ \( D_{SiO_2}^{P} \) ≈ \( 2 \times 10^{-15} \) cm²/sec** #### **Step 2: Calculate the masking factor** \[ \frac{D_{Si}^{P}}{D_{SiO_2}^{P}} = \frac{3 \times 10^{-13}}{2 \times 10^{-15}} = 150 \] Again, the oxide must be thick enough to reduce diffusion effectively. #### **Step 3: Calculate the oxide thickness** \[ x_{ox}^{P} = 10 \times \sqrt{D_{SiO_2}^{P} \times t} \] \[ = 10 \times \sqrt{2 \times 10^{-15} \times 1.8 \times 10^{4}} \] \[ = 10 \times \sqrt{3.6 \times 10^{-11}} \] \[ = 10 \times 6 \times 10^{-6} \, \text{cm} \] \[ = 6 \times 10^{-5} \, \text{cm} \] Convert to nanometers: \[ 6 \times 10^{-5} \, \text{cm} \times 10^{7} = 600\, \text{nm} \] **Final answer for part (b):** **Approximately 600 nm of SiO₂ is required to mask phosphorus diffusion.** --- ### **Summary:** | **Dopant** | **Required SiO₂ Thickness** | |------------|------------------------------| | **Boron** | **74 nm** | | **Phosphorus** | **600 nm** | --- **In conclusion,** - To effectively mask boron diffusion during a 5-hour process at 1100°C, approximately **74 nm** of SiO₂ is needed. - For phosphorus, about **600 nm** of SiO₂ is necessary. Feel free to ask if you'd like further clarification!