J. Phys. C: Solid State Phys.,
16
(1983)
21452156.
Printed in Great Britain
Theoretical basis for the statistics
of
dielectric
breakdown
R M Hill and L A Dissado
The Dielectrics Group, Chelsea College, University of London, Pulton Place.
London SW6 5PR, UK
Received 30 September 1982
Abstract.
The statistics of dielectric breakdown processes are presented and compared with
those derived from consideration of the fluctuation components of the dielectric response.
It is shown that the Weibull statistic parameters have a direct physical meaning and that
breakdown processes can be considered as cooperative events.
1. Introduction
Weibull statistics have generally been applied to the failure of materials under the action
of external forces such as mechanical stress (Trustrum and Jayatilaka 1979) and electrical
stress (Metra
et a1
1975, Stone and Lawless 1979). It is the intention here to examine the
basis of the Weibull approach, to show its applicability to dielectric breakdown and to
derive the Weibull form in terms of manybody cooperative fluctuations within the
dielectric medium.
The Weibull statistic (Gumbel 1958) is based on a stability postulate. If Nsamples,
each of size
n,
are taken from the same population then in each sample there is a smallest
value, and the smallest value of the total
Nn
observations must also be the smallest value
obtained from the samples of size
n.
Asymptotic expansions for the probability function
lead to three similar solutions of the form
F(Z)
=
exp[

( +
zk)]
(1)
for the cumulative probability distribution of the smallest value of
2.
Of interest here is
the solution in which
k
is positive. In this case the distribution
F(z)
is limited and
possesses all moments. The parameter
z
is a normalised variable and can be transformed.
The distribution function of
z
is given in the normal manner by the differential of
as
g(z)
dz
=
kzk'
exp
[(zk)]
dz.
(2)
For
k
less than unity
decays monoronically and for
k
greater than unity the distri
bution is peaked at the value
z
=
[(k

l)/k]"k.
(3)
@
1983 The Institute of Physics
2145
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2146
R
M
Hill and
LA
Dissado
A
general cumulative distribution function of the Weibull form has been postulated
for the process of dielectric breakdown in the form that the probability of breakdown
occurring under the action of an electric field
E,
at a time tis given by
1

P(E,, t)
=
1

exp
(

La b‘t“l(€a/€o)b
dr)
(4)
where
is the probability of a breakdown not occurring,
L
is the thickness of the
sample and
a,
b
and
Eo
are experimentally determinable parameters. Equation
can
be evaluated for particular experimental conditions:
(i) Constant electric field. If
is constant with respect to time the integration is
trivial and leads to
=
exp[

L~’(EJE~)~]
(5)
which is of the Weibull form in both time to breakdown and in the breakdown field at
constant time.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Antonio
 Normal Distribution, Permittivity, Fundamental physics concepts, Cumulative distribution function, Dielectric

Click to edit the document details