desirability function

**The desirability function** is a mathematical technique for **combining multiple response variables into a single optimization metric**, enabling simultaneous optimization of competing objectives — a common requirement in semiconductor process development where multiple outputs must be balanced. **Why Desirability?** - Real semiconductor processes have multiple responses that must all be acceptable: - **Etch**: Maximize etch rate, minimize roughness, target specific CD, maximize selectivity. - **Deposition**: Target film thickness, minimize stress, maximize uniformity. - **CMP**: Target removal rate, minimize dishing, minimize defects. - These responses often conflict — settings that improve one may worsen another. - The desirability function transforms each response into a **0–1 scale** and combines them into a single overall metric. **Individual Desirability Functions** For each response $y_i$, a desirability $d_i$ is defined: - **Target-is-Best** (e.g., CD = 30 nm): - $d = 1$ when $y$ equals the target. - $d = 0$ when $y$ reaches the lower or upper acceptable limit. - Decreases smoothly from 1 to 0 as $y$ deviates from target. - **Larger-is-Better** (e.g., maximize selectivity): - $d = 0$ when $y$ is at or below the minimum acceptable value. - $d = 1$ when $y$ reaches the maximum desired value. - **Smaller-is-Better** (e.g., minimize roughness): - $d = 1$ when $y$ is at or below the minimum desired value. - $d = 0$ when $y$ reaches the maximum acceptable level. **Shape Parameter (s)** - The exponent $s$ controls the shape of the desirability curve: - $s = 1$: Linear — equal penalty for any deviation from target. - $s > 1$: Convex — emphasis on getting very close to target (stringent). - $s < 1$: Concave — acceptable performance over a wider range (lenient). **Overall Desirability** $$D = \left(d_1^{w_1} \cdot d_2^{w_2} \cdot ... \cdot d_k^{w_k}\right)^{1/\sum w_i}$$ - The **geometric mean** of individual desirabilities, with **weights** $w_i$ reflecting the relative importance of each response. - If **any** individual desirability is zero, the overall desirability is zero — ensuring no response is completely sacrificed. **Optimization Workflow** - **Fit Response Models**: Use RSM (CCD or Box-Behnken DOE) to model each response as a function of the process factors. - **Define Desirability**: Set targets, limits, and weights for each response. - **Optimize**: Search the factor space for the settings that maximize overall desirability $D$. - **Verify**: Run confirmation experiments at the optimal settings. The desirability function is the **standard method** for multi-response optimization in semiconductor DOE — it provides a principled, transparent way to balance competing process requirements.