1063-7397/03/3201- $25.00 © 2003 MAIK “Nauka /Interperiodica”
Russian Microelectronics, Vol. 32, No. 1, 2003, pp. 51–62. Translated from Mikroelektronika, Vol. 32, No. 1, 2003, pp. 62–76.
Original Russian Text Copyright © 2003 by Bogdanov, Bogdanova, Dshkhunyan.
As early as in 1964, Murphy  introduced the com-
pound Poisson distribution into microelectronics, hav-
ing found that the traditional Poisson distribution is not
always adequate to predict yield in integrated-circuit
(IC) manufacture. The point is that process-induced
faults tend to occur unevenly over the wafer, appearing
as clusters. Murphy’s approach was developed by
Seeds , Okabe
, Stapper [4, 5], and other
The compound Poisson distribution differs from the
traditional one in that the parameter
denoting the fault
density is regarded as a random variable. Experience
indicates that the gamma distribution is probably the
most accurate model for
[6–10]. Also note that the
compound Poisson distribution is a limiting case of the
compound binomial distribution, the latter arising
within Pólya’s urn model .
Statistical yield models based on compound distri-
butions have proven to be useful in design, manufac-
ture, and product evaluation alike. Yield enhancement
aims to make the product fault-tolerant (i.e., less sensi-
tive to process-induced faults) by adding a degree of
redundancy to the IC (error-correcting codes are an
example) and by optimizing its ﬂoorplan and layout
[12–18]. Concerning the manufacture phase, we cite
the Bayesian approach. Applied to in-process product
control, the method allows one to reﬁne yield predic-
tion and make decisions as the batches progress along
the processing line . Compound-distribution mod-
els also help one to calculate yield distribution over
wafers, to estimate costs, to evaluate manufacturing
efﬁciency, to predict yield losses, etc. [20, 21].
In this study, we propose and develop a hierarchical
approach to the construction of compound distributions
for process-induced faults in IC manufacture.
Section 2 describes the origin and main properties of
a yield model that is built around the compound Pois-
son distribution and has been accepted by the electron-
ics industry .
Section 3 deﬁnes the compound binomial distribu-
tion and describes some of its properties. This model
includes the compound Poisson distribution as a limit-
ing case in much the same way as the traditional bino-
mial distribution leads to the Poisson distribution. The
reader is also introduced to Pólya’s urn model, within
which the compound binomial distribution arises.
Section 4 develops a theory of compound-distribu-
tion hierarchies. Within this framework, the above-
mentioned distributions belong to level 0 or 1. It is
shown that main formulae can be written in compact
analytical form by means of generating functions.
Section 5 deals with applied aspects. Compared
with previous results, a more general formalism for
mean yields is presented. Also included are equations
for yield probability densities. The yield is thus treated
as a hierarchical random variable.
Appendices 1 and 2 are concerned with a generali-
zation of Pólya’s urn model and with the Bayesian
approach to the estimation of fault density, respectively.
2. COMPOUND POISSON DISTRIBUTION
Primitive yield models employ the binomial distri-
bution. Consider a chip with
components. If each of
them has the same probability
of being faulty and
faults arise independently of one another, then the prob-
Statistical Yield Modeling for IC Manufacture:
Hierarchical Fault Distributions
Yu. I. Bogdanov*, N. A. Bogdanova**, and V. L. Dshkhunyan*
* OAO Angstrem, Moscow, Russia
** Moscow Institute of Electronic Engineering (Technical University), Moscow, Russia
Received February 12, 2002
—A hierarchical approach to the construction of compound distributions for process-induced faults
in IC manufacture is proposed. Within this framework, the negative binomial distribution and the compound
binomial distribution are treated as level-1 models. The hierarchical approach to fault distribution offers an inte-
grated picture of how fault density varies from region to region within a wafer, from wafer to wafer within a
batch, and so on. A theory of compound-distribution hierarchies is developed by means of generating functions.
With respect to applications, hierarchies of yield means and yield probability-density functions are considered
and an in-process measure of yield loss is introduced. It is shown that the hierarchical approach naturally
embraces the Bayesian approach.