Sensitivity Analysis

Sensitivity analysis in semiconductor simulation determines which input parameters have the greatest influence on output performance — identifying the critical "knobs" that drive process variability and guiding where to focus engineering effort for maximum impact.

Why Sensitivity Analysis Matters

• Semiconductor processes involve dozens of parameters (temperatures, pressures, times, doses, thicknesses, etc.).
• Not all parameters matter equally — typically a few parameters dominate while most have negligible impact.
• Sensitivity analysis identifies the vital few, enabling engineers to:
• Focus process control on the most impactful parameters.
• Prioritize DOE factors — study the important ones first.
• Set specification limits — tighter specs for sensitive parameters, relaxed specs for insensitive ones.
• Allocate metrology resources — measure the critical parameters more frequently.

Methods of Sensitivity Analysis

• One-at-a-Time (OAT): Vary each parameter individually by ±Δ while holding others constant. Simple but misses interactions and can be misleading in nonlinear systems.
• Sensitivity coefficient: $S_i = \frac{\partial y}{\partial x_i} \cdot \frac{x_i}{y}$ (normalized).

• Variance-Based (Sobol Indices): Decompose the total output variance into contributions from each input parameter (and their interactions).
• First-Order Index ($S_i$): Fraction of output variance due to parameter $x_i$ alone.
• Total-Effect Index ($S_{Ti}$): Fraction of output variance due to $x_i$ and all its interactions.
• If $S_i \approx S_{Ti}$, the parameter acts mainly independently. If $S_{Ti} \gg S_i$, the parameter interacts strongly with others.

• Regression-Based: Fit a regression model (linear, quadratic) to simulation or experimental data and examine the coefficients.
• Standardized Regression Coefficients (SRC): Coefficients normalized by input and output standard deviations — directly comparable across parameters.

• Morris Method (Elementary Effects): A screening method that efficiently ranks parameters by importance using a small number of simulations — useful as a first pass before more expensive analysis.

Semiconductor Applications

• Gate Length Sensitivity: How much does a 1 nm change in gate length affect Vth, Idsat, and Ioff? (Typically high sensitivity.)
• Oxide Thickness: Impact of ±0.1 nm variation on gate capacitance and Vth.
• Implant Dose/Energy: Sensitivity of junction depth and doping concentration to implanter settings.
• Etch Process: Which etch parameter (power, pressure, gas ratio) most affects CD, profile angle, and selectivity?

Practical Workflow

1. Screen with Morris method or OAT — quickly identify the top 5–8 parameters.
2. Quantify with Sobol indices or regression — determine exact variance contributions.
3. Optimize with DOE/RSM — focus on the sensitive parameters identified.
4. Control with SPC — monitor the sensitive parameters with tight control limits.

Sensitivity analysis is the essential first step in process optimization — it tells you where to invest your limited engineering time and resources for maximum yield and performance improvement.