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Monte Carlo Analysis
Local and Global Variations
We also plot the regression line Y = aX + b which relates the dependent variable Y=IR2 to
X=IR1. It is a well known result that the slope of this line is given by the formula:
and the intercept is .
Local and Global Variations
In statistical models for transistor level simulation, there are two types of variability in device
properties:
• Inter-die (between chips) variations or global variations.
Typical examples of statistical parameters with global tolerances are oxide thickness or
channel length reduction.
• Intra-die (within chip) variations are local variations that affect transistors individually.
A typical example of statistical parameters with local tolerances are the transistor
threshold voltages.
Inter-die variability consists of the accumulated fluctuations of material characteristics between
lot-to-lot, wafer-to-wafer and die-to-die (chip-to-chip). However, once the wafers have been cut
into individual chips there is no longer a traceable correlation among them. Therefore, lot-to-lot,
wafer-to-wafer and die-to-die variations are pooled together into one term: the inter-die
variability. This variability equally affects all devices on a given chip. In simulation terms, this
means that all devices of a certain type use the same statistical model.
Local variations lead to an increase in the number of statistical parameters relative to the
number of transistors in a circuit. In a circuit with 100 transistors, there will be one global
threshold voltage V th, glob that affects all transistors equally and 100 local threshold voltages
V ith, loc for i = 1, ..., 100 that affect each transistor individually and independently from others.
Therefore, you can consider that the effective threshold voltage for the i-th device is the sum:
According to these definitions, global statistical parameters are not correlated with local
statistical parameters, and local statistical parameters are not correlated with each other.
Example
In this example, LOT and DEV variations are defined on the length and width of the resistor
model. By definition, there are two independent random variables associated to the inter-die
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Eldo® User's Manual, 15.3