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.