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

• 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.