1063-7397/02/3104- $27.00 © 2002 MAIK “Nauka /Interperiodica”
Russian Microelectronics, Vol. 31, No. 4, 2002, pp. 260–264. Translated from Mikroelektronika, Vol. 31, No. 4, 2002, pp. 307–313.
Original Russian Text Copyright © 2002 by Bodyagin, Vikhrov, Mursalov, Tarasov.
Structural irreproducibility is a serious problem
commonly encountered in the fabrication of monocrys-
talline semiconductors and noncrystalline materials
alike. An example of the latter is hydrogenated amor-
phous silicon (a-Si:H) . For crystals, irreproducibil-
ity manifests itself in lattice defects, thus reducing cir-
cuit yield. The irreproducibility of structure and prop-
erties is often found in nature as well (recall
snowﬂakes). It is impossible to accurately predict how
the degree of reproducibility will vary over time.
In the context of materials technology, reproducibil-
ity is usually deﬁned as the accuracy to which a value
that characterizes the material can be repeated in suc-
cessive runs if all the process variables are kept con-
stant to a maximum possible accuracy.
Reproducibility is normally evaluated only for eco-
nomically important properties of the material. It is
quantiﬁed as the probability with which a speciﬁed
variable falls within a speciﬁed range, whose size is set
from practical considerations. However, this approach
cannot yield a deﬁnition of reproducibility that would
be applicable to all fabrication technologies.
It is common practice to associate the structural irre-
producibility of materials with the variability of pro-
cess conditions. Accordingly, great care is given to
maintaining critical process variables at speciﬁed val-
ues. This strategy is to a large extent responsible for the
complexity of existing process technologies. However,
it often proves to be ineffective and cannot satisfy up-
to-date requirements. If it does work, the gains are
often too small compared with the cost.
We think that the above difﬁculties in microelec-
tronics processing stem from lack of the theoretical
understanding of irreproducibility. In fact, this phe-
nomenon has been viewed from an engineering stand-
point, not as a fundamental problem of science. By con-
trast, the present study traces materials irreproducibil-
ity to the physics of atomic interactions, proposes some
numerical measures of irreproducibility that can be
applied to any process technology, and deﬁnes strate-
gies to combat the irreproducibility.
2. THEORETICAL BACKGROUND
We address the irreproducibility in the fabrication of
solid materials by treating the development of three-
dimensional structural regularity as self-organization in
a nonlinear system. The dynamics of the system is
regarded as a key factor responsible for materials irre-
producibility, with the limited degree of precision in
process variables considered less important.
It has been shown that the concepts and techniques
from the theory of self-organizing systems can in prin-
ciple be applied to solidiﬁcation, which displays bifur-
cations, symmetry breaking, nonequilibrium behavior,
and dissipation [1, 2]. The formation of solid-state reg-
ularity is often investigated by statistical methods.
However, they have been successful only with systems
near thermal equilibrium, a condition that rarely mate-
rializes in the real world. Far more detailed knowledge
of a much wider variety of structures can be gained
with the tools of chaotic dynamics .
The role of chaotic dynamics in solidiﬁcation is
illustrated by the ﬁgure, which represents in mathemat-
ical form the evolution of a material to the solid state.
During stage 1, the material behaves in a random man-
ner. Stage 2 is a cascade of bifurcations. The material is
now characterized by having solidiﬁcation nuclei cor-
related with each other. These correlations develop to
extend over the whole system, and the number of pos-
sible bifurcations increases. In stage 3, the system
exhibits deterministic chaos, i.e.,
behavior. Mathematically, stage 3 is associated with a
strange attractor, which evolves according to external
conditions. With increasing correlations between dif-
ferent regions, the dimension of the attractor decreases.
In stage 4, the strange attractor turns into simpler attrac-
tors: limit cycles and stable equilibrium points. Finally,
Structural and Behavioral Irreproducibility of Solid Materials:
A New Insight into the Problem
N. V. Bodyagin*, S. P. Vikhrov, S. M. Mursalov, and I. V. Tarasov
Ryazan State Academy of Radio Engineering, Ryazan, Russia
Received October 26, 2001
—Reasons for the structural irreproducibility of solid materials are discussed. Some numerical mea-
sures of irreproducibility are deﬁned within the theory of dynamical systems. The concept of reproducibility
limit is framed. A link between the degree of reproducibility and the mass of the specimen is established.