Question:

Please convert the below content to rendered equations without any errors.Alright, let’s walk through this cleanly and step-by-step — this is a **classic lithography thermal overlay** question. --- ## Given * **Wafer field size** = (36 \text{ mm} \times 25 \text{ mm}) * **Overlay requirement** = better than **10 nm** * **Mask material** = Quartz * Coefficient of thermal expansion (CTE): [ \alpha_{\text{quartz}} \approx 0.5 \times 10^{-6} ; /^\circ\text{C} ] * **Stepper reduction**: * Case 1: **4× mask** * Case 2: **5× mask** We always consider the **largest dimension** because it gives the worst overlay error. --- ## Key idea (very important ?) Thermal expansion of the **mask** causes overlay error on the **wafer**, reduced by the stepper magnification: [ \text{Overlay error on wafer} = \frac{\alpha , L_{\text{mask}} , \Delta T}{M} ] Where * (L_{\text{mask}} = M \times L_{\text{wafer}}) * (M) = magnification (4 or 5) Notice something nice happens ? The magnification cancels out: [ \text{Overlay} = \alpha , L_{\text{wafer}} , \Delta T ] So **overlay depends only on wafer field size**, not directly on mask size. --- ## Use the worst-case dimension [ L_{\text{wafer}} = 36 \text{ mm} = 36 \times 10^{-3} \text{ m} ] --- ## Overlay condition [ \alpha , L , \Delta T \le 10 \text{ nm} ] Substitute values: [ (0.5 \times 10^{-6}) \times (36 \times 10^{-3}) \times \Delta T \le 10 \times 10^{-9} ] --- ## Solve for temperature control [ \Delta T \le \frac{10 \times 10^{-9}} {0.5 \times 10^{-6} \times 36 \times 10^{-3}} ] [ \Delta T \le 0.56^\circ \text{C (approximately)} ] --- ## ✅ Final Answers ### ? For **4× quartz mask** [ \boxed{\Delta T \le \pm 0.56^\circ \text{C}} ] ### ? For **5× quartz mask** [ \boxed{\Delta T \le \pm 0.56^\circ \text{C}} ] --- ## ⭐ Key takeaway (exam gold) * **Overlay error from thermal expansion is independent of magnification** * It depends on: * Wafer field size * Mask material CTE * Temperature stability This is why **thermal control at the sub-degree level** is absolutely critical in photolithography. If you want, I can also: * Compare with **MoSi masks** * Explain how **real steppers reach ±0.1 °C** * Convert this into a **GATE / VLSI exam-style short answer** Just say the word ?