Etched Glass Surfaces, Atomic Force Microscopy and Stochastic Analysis
Etched Glass Surfaces, Atomic Force Microscopy and Stochastic Analysis
G. R. Jafari a,b, M. Reza Rahimi Tabar c,d, A. Iraji zad c, G. Kavei f 1
1a Department of Physics, Shahid Beheshti University, Evin, Tehran 19839, Iran
b Department of Nano-Science, IPM, P. O. Box 19395-5531, Tehran, Iran
c Department of Physics, Sharif University of Technology, P. O. Box 11365-9161, Tehran, Iran
d CNRS UMR 6529, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France
e Material and Energy, Research Center, P.O. Box 14155-4777, Tehran, Iran
The effect of etching time scale of glass surface on its statistical properties has been studied using atomic force
microscopy technique. We have characterized the complexity of the height fluctuation of a etched surface by
the stochastic parameters such as intermittency exponents, roughness, roughness exponents, drift and diffusion
coefficients and find their variations in terms of the etching time.
PACS numbers:
I. INTRODUCTION
The complexity of rough surfaces is subject of a large va-
riety of investigations in different fields of science1,2. Sur-
face roughness has an enormous influence on many important
physical phenomena such as contact mechanics, sealing, ad-
hesion, friction and self-cleaning paints and glass windows,3,4.
A surface roughness of just a few nanometers is enough to re-
move the adhesion between clean and (elastically) hard solid
surfaces3. The physical and chemical properties of surfaces
and interfaces are to a significant degree determined by their
topographic structure. The technology of micro fabrication of
glass is getting more and more important because glass sub-
strates are currently being used to fabricate micro electro me-
chanical system (MEMS) devices5. Glass has many advan-
tages as a material for MEMS applications, such as good me-
chanical and optical properties. It is a high electrical insulator,
and it can be easily bonded to silicon substrates at tempera-
tures lower than the temperature needed for fusion bonding6.
Also micro and nano-structuring of glass surfaces is impor-
tant for the production of many components and systems such
as gratings, diffractive optical elements, planar wave guide de-
vices, micro-fluidic channels and substrates for (bio) chemical
applications7. Wet etching is also well developed for some of
these applications8,9,10,11,12,13,14.
One of the main problems in the rough surface is the scal-
ing behavior of the moments of height h and evolution of the
probability density function (PDF) of h, i.e. P (h, x) in terms
of the length scale x. Recently some authors have been able
to obtain a Fokker-Planck equation describing the evolution
of the probability distribution function in terms of the length
scale, by analyzing some stochastic phenomena, such as rough
surfaces15,16,17, turbulent system18, financial data19, cosmic
background radiation20 and heart interbeats21 etc. They no-
ticed that the conditional probability density of field incre-
ment satisfies the Chapman-Kolmogorov equation. Mathe-
matically, this is a necessary condition for the fluctuating data
to be a Markovian process in the length (time) scales22.
In this work, we investigate the etching process as a
stochastic process. We measure the intermittency exponents
of height structure function, roughness, roughness exponents
and Kramers-Moyal‘s (KM) coefficients. Indeed we consider
the etching time t, as an external parameter, to control the sta-
tistical properties of a rough surface and find their variations
with t. It is shown that the first and second KM‘s coefficients
have well-defined values, while the third and fourth order co-
efficients tend to zero. The first and second KM‘s coefficients
for the fluctuations of h(x), enables us to explain the height
fluctuation of the etched glass surface.
II. EXPERIMENTAL
We started with glass microscope slides as a sample. Only
one side of samples was etched by HF solution for different
etching time (less than 20 minutes). HF concentration was
%40 for all the experiments. The surface topography of the
etched glass samples in the scale (< 5µm) was obtained us-
ing an AFM (Park Scientific Instruments). The images in
this scale were collected in a constant force mode and digi-
tized into 256×256 pixels. A commercial standard pyramidal
Si3N4 tip was used. A variety of scans, each with size L,
were recorded at random locations on the surface. Figure 1
shows typical AFM image with resolutions of about 20nm.
III. STATISTICAL QUANTITIES
A. Multifractal Analysis and the Intermittency Exponent
Assuming statistical translational invariance, the structure
functions Sq(l) =< |h(x + l) − h(x)|q >, (moments of the
increment of the rough surface height fluctuation h(x)) will
depend only on the space deference of heights l, and has a
power law behavior if the process has the scaling property:
Sq(l) =< |h(x + l)− h(x)|q >∝ Sq(L0)(
)ξ(q) (1)
where L0 is the fixed largest length scale of the system,
< ··· > denotes statistical average (for non-overlapping incre-
ments of length l), q is the order of the moment (we take here
q > 0), and ξ(q) is the exponents of structure function. The
http://arxiv.org/abs/0704.1030v1
FIG. 1: AFM surface image of etched glass film with size 5× 5µm2
after 12 minutes.
second moment is linked to the slope β of the Fourier power
spectrum: β = 1 + ξ2. The main property of a multifractal
processes is that it is characterized by a non-linear ξq function
verses q. Monofractals are the generic result of this linear be-
havior. For instance, for Brownian motion (Bm) ξq = q/2,
and for fractional Brownian motion (fBm) ξq ∝ q.
B. Roughness and Roughness Exponents
It is also known that to derive the quantitative information
of the surface morphology one may consider a sample of size
L and define the mean height of growing film h and its vari-
ance, σ by:
σ(L, t) = (〈(h − h)2〉)1/2 (2)
where t is etching time and 〈· · · 〉 denotes an averaging over
different samples, respectively. Moreover, etching time is a
factor which can apply to control the surface roughness of thin
films.
Let us now calculate also the roughness exponent of the
etched glass. Starting from a flat interface (one of the possible
initial conditions), it is conjectured that a scaling of space by
factor b and of time by factor bz (z is the dynamical scaling
exponent), rescales the variance, σ by factor bχ as follows1:
σ(bL, bzt) = bασ(L, t) (3)
which implies that
σ(L, t) = Lαf(t/Lz). (4)
If for large t and fixed L (x = t/Lz → ∞) σ saturate. How-
ever, for fixed large L and t ≪ Lz , one expects that corre-
lations of the height fluctuations are set up only within a dis-
tance t1/z and thus must be independent of L. This implies
that for x ≪ 1, f(x) ∼ xβ with β = α/z. Thus dynamic
scaling postulates that
σ(L, t) ∝
tβ , t≪ Lz;
Lα, t≫ Lz.
The roughness exponent α and the dynamic exponent β char-
acterize the self-affine geometry of the surface and its dynam-
ics, respectively.
The common procedure to measure the roughness exponent
of a rough surface is use of the surface structure function de-
pending on the length scale l which is defined as:
S2(l) = 〈|h(x + l)− h(x)|2〉. (6)
It is equivalent to the statistics of height-height correlation
function C(l) for stationary surfaces, i.e. S2(l) = 2σ2(1 −
C(l)). The second order structure function S(l), scales with l
as l2α1.
C. The Markov Nature of Height Fluctuations: Drift and
Diffusion Coefficients
We check whether the data of height fluctuations follow a
Markov chain and, if so, measure the Markov length scale
lM . As is well-known, a given process with a degree of
randomness or stochasticity may have a finite or an infinite
Markov length scale23. The Markov length scale is the min-
imum length interval over which the data can be considered
as a Markov process. To determine the Markov length scale
lM , we note that a complete characterization of the statisti-
cal properties of random fluctuations of a quantity h in terms
of a parameter x requires evaluation of the joint PDF, i.e.
PN (h1, x1; ....;hN , xN ), for any arbitrary N . If the process
is a Markov process (a process without memory), an im-
portant simplification arises. For this type of process, PN
can be generated by a product of the conditional probabili-
ties P (hi+1, xi+1|hi, xi), for i = 1, ..., N − 1. As a nec-
essary condition for being a Markov process, the Chapman-
Kolmogorov equation,
P (h2, x2|h1, x1) =
d(hi)P (h2, x2|hi, xi)P (hi, xi|h1, x1) (7)
should hold for any value of xi, in the interval x2 < xi <
The simplest way to determine lM for homogeneous sur-
face is the numerical calculation of the quantity, S =
|P (h2, x2|h1, x1)−
dh3P (h2, x2|h3, x3)P (h3, x3|h1, x1)|,
for given h1 and h2, in terms of, for example, x3 − x1
and considering the possible errors in estimating S. Then,
lM = x3 − x1 for that value of x3 − x1 such that, S = 0
It is well-known, the Chapman-Kolmogorov equation
yields an evolution equation for the change of the distribu-
tion function P (h, x) across the scales x. The Chapman-
Kolmogorov equation formulated in differential form yields a
l (nm)
50 100 150 200
FIG. 2: Scaling of the structure functions in log-log plot for mo-
ments less than 8. (from bottom to top).
master equation, which can take the form of a Fokker-Planck
equation22,23:
P (h, x) = [−
D(1)(h, x) +
D(2)(h, x)]P (h, x).(8)
The drift and diffusion coefficients D(1)(h, r), D(2)(h, r) can
be estimated directly from the data and the moments M (k) of
the conditional probability distributions:
D(k)(h, x) =
limr→0M
M (k) =
dh′(h′ − h)kP (h′, x+ r|h, x). (9)
The coefficients D(k)(h, x)‘s are known as Kramers-Moyal
coefficients. According to Pawula‘s theorem22, the Kramers-
Moyal expansion stops after the second term, provided that
the fourth order coefficient D(4)(h, x) vanishes22. The forth
order coefficients D(4) in our analysis was found to be about
D(4) ≃ 10−4D(2). In this approximation, we can ignore the
coefficients D(n) for n ≥ 3. We note that this Fokker-Planck
equation is equivalent to the following Langevin equation (us-
ing the Ito interpretation)22:
h(x) = D(1)(h, x) +
D(2)(h, x)f(x) (10)
where f(x) is a random force, zero mean with gaussian statis-
tics, δ-correlated in x, i.e. 〈f(x)f(x′)〉 = 2δ(x−x′). Further-
more, with this last expression, it becomes clear that we are
able to separate the deterministic and the noisy components
of the surface height fluctuations in terms of the coefficients
D(1) and D(2).
2 4 6 8 10
FIG. 3: The results of scaling exponent ξq which is clearly linear vs.
l (nm)
500 1000
6 min
8 min
10 min
12 min
15 min
FIG. 4: Log-Log plot of selection structure function of the etched
glass surfaces.
IV. RESULTS AND DISCUSSION
Now, using the introduced statistical parameters in the pre-
vious sections, it is possible to obtain some quantitative infor-
mation about the effect of etching time on surface topography
of the glass surface. To study the effect of the etching time on
the surface statistical characteristics, we have utilized AFM
imaging technique in order to obtain microstructural data of
the etched glass surfaces at the different etching time in the
HF. Figure 1 shows the AFM image of etched glass after 12
minuets etched. To investigate the scaling behavior of the mo-
ments of δhl = h(x + l) − h(x), we consider the samples
that they reached to the stationary state. This means that their
statistical properties do not change with time. In our case the
-0.2 0 0.2
-0.08
-0.06
-0.04
-0.02
2 min
6 min
8 min
10 min
12 min
15 min
FIG. 5: Drift coefficients of the surfaces at different etching time
less than 20 minutes.
samples with etching time more than 20 minutes are almost
stationary. Figure 2 shows the log-log plot of the structure
functions verses length scale l for different orders of moments.
The straight lines show that the moments of order q have the
scaling behavior. We have checked the scaling relation up to
moment q = 10. The resulting intermittency exponent ξq is
shown in figure 3. It is evident that ξq has a linear behavior.
This means that the height fluctuations are mono-fractal be-
havior. We also directly estimated the scaling exponent of the
linear term lqH/ < (h(x + l)− h(x))q > and obtain the fol-
lowing values for the samples with 20 minuets etching time,
ξ1 = 0.70± 0.04 and ξ2 = 1.40 ± 0.04. This means etching
memorize fractal feature during etching. Therefore using the
scaling exponent ξ2 we obtain the roughness exponent α as
ξ2/2 = 0.70± 0.04. Figure 4 presents the structure function
S(l) of the surface at the different etching time, using equation
(6). It is also possible to evaluate the grain size dependence
to the etching time, using the correlation length achieved by
the structure function represented in figure 4. The correlation
lengths increase with etching time. Its value has a exponen-
tial behavior 448(1 − exp(−0.15t))nm. Also we find that
the dynamical exponent is given by β = 0.6 ± 0.1. Also
we measured the variation of the Markov length with etching
time t (min), and obtain lM = 40 + 3t (nm) for time scales
t < 20min.
Finally to obtain the stochastic equation of the height fluc-
tuations behavior of the surface, we need to measure the
Keramer- Moyal Coefficients. In our analysis the forth order
coefficients D(4) is less than Second order coefficients, D(2),
about D(4) ≃ 10−4D(2). In this approximation, we ignore the
coefficients D(n) for n ≥ 3. So, to discuss the surfaces it just
needs to measure the drift coefficient D(1)(h
) and diffusion
coefficient D(2)(h
) using Eq. (9). Figures 5 and 6 show the
drift coefficient D(1)(h
) and diffusion coefficients D(2)(h
for the surfaces at the different etching time, respectively. It
-0.2 0 0.2
0.001
0.002
0.003
0.004 2 min
6 min
8 min
10 min
12 min
15 min
FIG. 6: Diffused coefficients of the surface at different etching time
less than 20 minutes.
can be shown that the drift and diffusion coefficients have the
following behavior,
D(1)(
, t) = −f (1)(t)
D(2)(
, t) = f (2)(t)(
)2 (12)
The two coefficients f (1)(t) and f (2)(t) increase with the
then is saturated. Using the data analysis we obtain that
they are linear verses time (min): f (1)(t) = 0.005t and
f (2)(t) = 0.0003t for time scales t < 20 min. To better
comparing the parameter of samples we divided the heights
to their variances. In this case, maximum and minimum of
heights are about plus 1 and mines 1, respectively. Compar-
ing samples with etching times 2 and 6 minutes, shows f (1)
increases 300 percent after 4 minutes (from 2 min to 6 min)
from f (1)(t = 2×60) = 0.6 to f (1)(t = 6×60) = 1.8. Also,
f (2) is 0.006 and 0.018 after 2 and 6 minutes, respectively.
V. CONCLUSIONS
We have investigated the role of etching time, as an exter-
nal parameter, to control the statistical properties of a rough
surface. We have shown that in the saturate state the struc-
ture of topography has fractal feature with fractal dimension
Df = 1.30. In addition, Langevin characterization of the
etched surfaces enable us to regenerate the rough surfaces
grown at the different etching time, with the same statistical
properties in the considered scales15.
VI. ACKNOWLEDGMENT
We would like to thank S. M. Mahdavi for his useful com-
ments and discussions and Also P. Kaghazchi and M. Shirazi
for samples preparation.
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http://arxiv.org/abs/astro-phy/0312227
