Statistics behind Variation Models

Keywords

OCV (On Chip Variation), AOCV (Advanced OCV), POCV (Parametric OCV), SOCV (Statistical OCV), LVF (Liberty Variation Format), CCS (Composite Current Source), NLDM (Non-Linear Delay Model), STA (Static Timing Analysis), VLSI characterization, timing variation model

To test robustness of fabricated integrated circuits (ICs) a semiconductor manufactures (like TSMC, GlobalFoundries etc.) will manufacture a group of wafers with process parameters set to extreme operating values. ICs, thus manufactured, are tested in varying environmental conditions (such as voltage, temperature, clock frequency etc.) to ensure normal operation. This is called corner-lot analysis. Corner lot analysis is used to determine combinations of process corners and environmental conditions essential for sign-off analysis including timing sign-off.

To enable sign-off analysis, VLSI cell libraries are needed at the same combination of process and environmental conditions including voltage and temperature. This condition is represented as ‘operating_condition’ group in liberty file.

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Facets of Variation Models

Variation of local process parameters affect circuit behavior (i.e., currents and voltages). This effect shows up in circuit measurements of timing, power, noise etc. Magnitude of variation effect (caused by local process and environmental conditions) on timing has been pronounced on timing models since 130nm and below, whereas it fails to make significant mark in power and noise models. For that reason, power variation and noise variation models have been ignored. In the following section we’ll review characterization and modeling of variation with respect to timing only.

Local Corner Timing Variation Models

There are 3 main types of models

1. On-Chip Variation Model (aka OCV)

2. Advanced On-Chip Variation Model (aka AOCV)

3. Liberty Variation Format (aka LVF)

These three models are similar in principle and mainly differ in the level of granularity to meet the demand for higher levels of accuracy by accounting for higher number of dominant variable in the modeling. These models require basic understanding of statistics. So let’s establish some notations before getting ahead of ourselves.

Back to Basic Statistics

Since variation is a statistical topic, it’s good time to review our basics before we proceed further. Consider letter x represents the statistical variable, x_i denotes i^th observation of that variable in the group. With that, here are four basic definitions that we’ll use in the blog later.

Statistical parameter mean μ and sigma σ are depicted in the next picture as part of distribution graph of cell delay for a fixed slope, load and stimulus with varying PVT conditions. Delay distribution shown here is shown as Gaussian distribution for the sake of simplicity Actual distribution may not be symmetric or Gaussian. It is clear from the picture that 3σ distance from the mean μ covers approximately 99.77% of the delay spread and is sufficient for modeling.

As we shall see later in other blogs that mean μ and sigma σ of timing groups (like cell delay, cell transition time etc.) characteristics are the basis behind all variation models including OCV, AOCV, POCV, LOCV, SBOCV, SOCV and LVF.

For example, OCV models are created by dividing cells into 8 different categories given below:

1. Rise transition, Data path, Early analysis

2. Rise transition, Data path, Late analysis

3. Rise transition, Clock path, Early analysis

4. Rise transition, Clock path, Late analysis

5. Fall transition, Data path, Early analysis

6. Fall transition, Data path, Late analysis

7. Fall transition, Clock path, Early analysis

8. Fall transition, Clock path, Late analysis

For each one of these categories, Monte-Carlo transient simulations is performed to provide group of delay measurement for each of measurements. Mean (μ) and Sigma (σ) is computed for these group and derate is computed as 3-sigma distance from the mean value of delays.

Summary

In this article, we connected PVT variation to basic statistics to get a better understanding of overall variation modeling in the historical context.

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