# SPAYN

Recent Developments

Introduction

An important aspect of statistical process control
in IC production is the ability to predict circuit performance variation
in the manufacturing process. Two new features in **SPAYN** allow
the user, for a particular circuit performance parameter, to rapidly
calculate an estimate of the standard deviation and also generate
a yield distribution utilising Monte Carlo simulations, thus allowing
a full statistical analysis of the circuit performance parameter
distribution.

For any given data base containing SPICE model
parameters, an initial exploratory data analysis usually consists
of trying to identify relationships and interdependencies between
the specified variables. This is usually achieved by looking at
the correlation structure, and examining histogram and scattergram
plots of the various circuit parameters. However due to the large
number of SPICE model parameters that are generally extracted, it
is of great benefit to be able to generate these plots automatically.
The new **SPAYN** "Macro Command" feature allows the
user to do just that.

Statistical Analysis of Circuit Performance Parameters

One of the primary objectives of statistical ``worst-case'' yield modelling is to predict best and worst circuit performance due to fluctuations in the IC manufacturing process. The aim is to isolate those parameters causing the largest variation in circuit performance, so that they can be strictly monitored and controlled. An understanding of the sources of variation provides an insight, at the design stage, into where any problems are likely to occur during production.

There are two new features in **SPAYN** that
allow the user to statistically analyse a particular circuit performance
parameter. The circuit simulations are performed by connecting **SPAYN**
to an external simulator such as **SmartSpice**, **SPAYN**
is then used to analyse the results. The first new feature, known
as the ``Gradient Analysis'' method [1], computes an accurate estimate
of the standard deviation for a given circuit performance parameter.
The second new feature allows the statistical investigation of a
performance parameter generated through Monte Carlo simulations.

The ``Gradient Analysis'' approach [1] permits designers to rapidly calculate an estimate of the circuit performance standard deviation utilising independent or quasi-independent reference design parameters such as principal component factors or dominant parameters. The Gradient Analysis standard deviation estimate is based on a linear approximation by considering the circuit performance parameter as a function of independent reference design parameters.

Monte Carlo simulations of a given circuit can
be performed using SmartSpice, and the performance parameter stored
as a new **SPAYN** variable for further analysis. Thus allowing
an engineer to examine the distribution of a particular circuit
performance parameter, calculate various descriptive statistics
or utilise any of **SPAYN**'s other statistical capabilities.

As an example of the above new features (Figure
1), consider a circuit consisting of a single MOS transistor biased
with -2.5 volts on the gate, and device dimensions W/L = 20/2.5
microns. The **SmartSpice** control file then specifies the drain
source voltage to be ramped from 0 -> -5 volts in steps of 0.2
of a volt. For this analysis let the circuit performance parameter
be the maximum value of the drain current (ID_MAX) for a given set
of parameter values. In this case the independent reference design
parameters are taken as the first four principal components, which
account for 89% of the variation in the data.

Figure 1. SPAYN simulation
interface window, with gradient analysis

standard deviation estimate and Monte Carlo simulation dialog (inset).

Utilising the Gradient Analysis method, the estimated standard deviation of ID_MAX is 1.10e-41 compared to the sample standard deviation of 1.52e-41 generated from the Monte Carlo simulations. Figure 2 shows a histogram plot of the Monte Carlo simulations for the circuit performance parameter ID_MAX. The vertical lines on the plot indicate the +/- 3 limits of the Gradient Analysis standard deviation estimate. Using the distribution fit option in the histogram window, a Gamma distribution best describes this particular set of circuit performance parameter data.

Figure 2. Histogram of the circuit performance parameter ID.MAX.

Generating multiple Histogram and Scattergram plots

With any statistical data analysis an applications engineer is interested in examining possible relationships between the various parameters in a particular circuit design. This is especially important with the extraction of a large number of interdependent circuit parameters. The identification of such relationships is crucial to the understanding of how individual variables affect circuit performance. In particular, if a user it attempting to locate sources of variation, isolate outliers, or investigate marginal distributions in a given data base.

The histogram and scattergram are two of the key
exploratory data analysis tools used to investigate the graphical
relationships between variables in a given data base. It is now
possible, within **SPAYN**, to rapidly produce multiple histogram
and scattergram plots in a postscript file format. The postscript
files can then be directly incorporated into technical reports,
presentations or product documentation. This is accomplished utilising
the ``Macro Commands'' facility, which is accessed from the ``Analysis''
menu of the main **SPAYN** window (Figure 3).

Figure 3. Macro command window for generating multiple histograms and scattergrams.

Consider, for example, a data base where the NMOS
threshold voltage parameter VTO_N is of particular interest. The
user would like to investigate the relationship between VTO_N and
the other variables in the data base. An examination of the correlation
matrix will reveal which parameters are correlated with VTO_N, this
can then be confirmed graphically by generating multiple scattergram
plots of those sets of parameters using the ``Macro Commands'' option.
Alternatively, suppose a user needs a graphical verification that
all of the parameters in a given data base can be modelled with
a Gaussian distribution. Rather than producing plots individually,
using the **SPAYN** histogram window, the ``Macro Commands''
option can be employed to automatically generate any number of plots
in a postscript file format, ready for comparison. In both of these
examples the user has the flexibility to select the destination
directory. All of the plots are uniquely identified with a plot
type label, the system PID (**SPAYN** process identification
number) and a plot number tag.

**References**

[1] "Realistic Statistical Worst-Case Simulations of VLSI Circuits" M. Bolt, M. Rocchi and J. Engel. IEEE Transactions on Semiconductor Manufacturing Vol. 4. August 1991.