Explanation of stretched exponential growth behavior
Kwangjoon Kim
a)
Research Department, Electronics and Telecommunications Research Institute,
Yusong P.O. Box 106, Taejon 305-600, Korea
Arthur J. Epstein
Department of Physics, The Ohio State University, Columbus, Ohio 43210-1106
͑Received 22 May 1995; accepted for publication 22 August 1995͒
We show that the photoinduced defect population growth of an optically thick film may have
stretched exponential time dependence if the defect does not decay nor diffuse within the time scale
of the experiment. After the formulation of this simple model, low-temperature photoinduced
absorption of polyaniline is described as an example. © 1995 American Institute of Physics.
The stretched exponential time dependence ͑so-called
Kohlrausch-Williams-Watts function͒ behaviors of many
physical quantities have been reported in various systems.
Among them are luminescence in porous silicon,
1
dielectric
relaxation in glassy and polymeric materials,
2
electric bire-
fringence in polymer solutions,
3
charge injection and trap-
ping in amorphous thin-film transistors,
4
and optically in-
duced charge, spin, and/or defect densities in amorphous
silicon nitride,
5
in hydrogenated amorphous silicon,
6
and in
polyaniline,
7
to name a few.
The physical quantities of interest in the systems may
grow or decay as a function of time. In case of growth, the
time dependence of the observable n͑t͒ fits to the stretched
exponential function
n
͑
t
͒
ϭn
max
͕
1Ϫexp
͓
Ϫ
͑
t/
͒

͔
͖
, ͑1͒
where

has value between 0 and 1,
is a time constant, and
n
max
is the upper limit of n.
There have been many attempts to explain this stretched
exponential behavior and most of the theories are based on
the defect diffusion in a disordered system or distribution of
relaxation times.
2,6,8,9
In this letter, we demonstrate that if the
stimulation comes unidirectionally from outside as in optical
pumping and the induced defect does not diffuse nor decay,
the thickness of the sample may cause such stretched expo-
nential time dependence of the induced defect population.
The formulation is straightforward. Let us imagine a film
of thickness d, which is made of a photosensitive material. If
we assume that the photoinduced defect of the material does
not diffuse nor decay when the material is exposed to an
optical pumping, the photoinduced defect density growth
rate will be proportional to the density of the available site
͑which can accommodate the photoinduced defect͒ and to
the optical pumping intensity. Suppose that a continuous and
constant intensity optical pumping to the film starts at time
tϭ0. The photoinduced defect number density
, the absorp-
tion coefficient
␣
, and the pump beam intensity I are func-
tions of time t and position x with 0рxрd. The pump beam
comes along the x axis from negative to positive direction,
with intensity I(x ϭ0,t)ϭI
0
for all tу0. If the maximum or
saturation number density is
max
, their relations can be ex-
pressed as
␣
͑
x,t
͒
ϭ
␣
0
Ϫ
͑
␣
0
Ϫ
␣
1
͒
͑
x,t
͒
max
, ͑2͒
I
͑
x,t
͒
ϭI
0
exp
ͭ
Ϫ
͵
0
x
␣
͑
x
Ј
,t
͒
dx
Ј
ͮ
, ͑3͒
d
͑
x,t
͒
dt
ϭB
͓
max
Ϫ
͑
x,t
͒
͔
I
͑
x,t
͒
. ͑4͒
In Eq. ͑2͒,
␣
0
and
␣
1
are the pristine and fully saturated
sample absorption coefficients at the optical pump wave-
length, respectively. Usually photoinduced bleaching occurs
at the optical pumping wavelength and
␣
1
is smaller than
␣
0
. Due to the absorption, the local pump intensity I(x,t)
depends on position x, as shown in Eq. ͑3͒, and, as men-
tioned, Eq. ͑4͒ implies that the local photoinduced defect
density change will be proportional to both the pumping in-
tensity and the density of available sites. B is a constant
which reflects the efficiency of the creation of photoinduced
defects.
The above equations can be placed into one integral-
differential equation
d
͑
x,t
͒
dt
ϭB
͓
max
Ϫ
͑
x,t
͒
͔
I
0
ϫexp
ͩ
Ϫ
␣
0
xϩ
␣
0
Ϫ
␣
1
max
͵
0
x
͑
x
Ј
,t
͒
dx
Ј
ͪ
. ͑5͒
The total number per unit area of photoinduced defects n͑t͒,
can easily be obtained by the relation
n
͑
t
͒
ϭ
͵
0
d
͑
x
Ј
,t
͒
dx
Ј
. ͑6͒
From Eq. ͑5͒, the sample can be regarded as a collection
of many thin layers perpendicular to the pump beam direc-
tion and the pump intensity is reduced gradually as the pump
beam is partly absorbed by outer layers. When we ignore the
difference between
␣
0
and
␣
1
,atx, the photoinduced defect
number density
͑x,t͒ will satisfy
a͒
Electronic mail: kkim@utopia.etri.re.kr
2786 Appl. Phys. Lett. 67 (19), 6 November 1995 0003-6951/95/67(19)/2786/3/$6.00 © 1995 American Institute of Physics