Controlling surface statistical properties using bias voltage: Atomic
  force microscopy and stochastic analysis
Controlling surface statistical properties using bias voltage: Atomic force microscopy and stochastic
analysis
P. Sangpour,1 G. R. Jafari,1 O. Akhavan,1 A.Z. Moshfegh,1 and M. Reza Rahimi Tabar1, 2
1Department of Physics, Sharif University of Technology, P.O. Box 11365-9161, Tehran, Iran
2CNRS UMR 6529, Observatoire de la Côte d’Azur, BP 4229, 06304 Nice Cedex 4, France
The effect of bias voltages on the statistical properties of rough surfaces has been studied using atomic force
microscopy technique and its stochastic analysis. We have characterized the complexity of the height fluc-
tuation of a rough surface by the stochastic parameters such as roughness exponent, level crossing, and drift
and diffusion coefficients as a function of the applied bias voltage. It is shown that these statistical as well
as microstructural parameters can also explain the macroscopic property of a surface. Furthermore, the tip
convolution effect on the stochastic parameters has been examined.
PACS numbers: 05.10.Gg, 02.50.Fz, 68.37.-d
I. INTRODUCTION
As device dimensions continue to shrink into the deep sub-
micron size regime, there will be increasing attention for un-
derstanding the thin-film growth mechanism and the kinet-
ics of growing rough surfaces in various deposition methods.
To perform a quantitative study on surfaces roughness, an-
alytical and numerical treatments of simple growth models
propose, quite generally, the height fluctuations have a self-
similar character and their average correlations exhibit a dy-
namic scaling form1,2,3,4,5,6. In these models, roughness of
a surface is a smooth function of the sample size and growth
time (or thickness) of films. In addition, other statistical quan-
tities such as the average frequency of positive slope level
crossing, the probability density function (PDF), as well as
drift and diffusion coefficients provide further complete analy-
sis on roughness of a surface. Very recently, it has been shown
that, by using these statistical variables in the Langevin equa-
tion, regeneration of rough surfaces with the same statistical
properties of a nanoscopic imaging is possible7,8.
In practice, one of the effective ways to modify roughness
of surfaces is applying a negative bias voltage during deposi-
tion of thin films10, while their sample size and thickness are
constant. In bias sputtering, electric fields near the substrate
are modified to vary the flux and energy of incident charged
species. This is achieved by applying either a negative DC or
RF bias to the substrate. Due to charge exchange processes
in the anode dark space, very few discharge ions strike the
substrate with full bias voltage. Rather a broad low energy
distribution of ions bombard the growing films.
Generally, bias sputtering modifies film properties such as
surface morphology, resistivity, stress, density, adhesion, and
so on through roughness improvement of the surface, elimi-
nation of interfacial voids and subsurface porosity, creation of
a finer and more isotropic grain morphology, and the elimina-
tion of columnar grains10.
In this work, the effect of bias voltage on the statistical
properties of a surface, i.e., the roughness exponent, the level
crossing, the probability density function, as well as the drift
and diffusion coefficients has been studied. In this regard, we
have analyzed the surface of Co(3 nm)/NiO(30 nm)/Si(100)
structure (as a base structure in the magnetic multilayers, e.g.,
spin valves operated using giant magnetoresistance (GMR)
effect11,12) fabricated by bias sputtering method at different
bias voltages. The behavior of statistical characterizations ob-
tained by nanostructural analysis have been also compared
with behavior of sheet resistance measurement of the films
deposited at the different bias voltages, as a macroscopic anal-
ysis.
II. EXPERIMENTAL
The substrates used for this experiment were n-type Si(100)
wafers with resistivity of about 5-8 Ω-cm and the dimension
of 5×11 mm2 . After a standard RCA cleaning procedure
and a short time dip in a diluted HF solution, the wafers were
loaded into a vacuum chamber. The chamber was evacuated
to a base pressure of about 4×10−7 Torr. To deposit nickel
oxide thin film, first high purity NiO powder was pressed and
baked over night at 1400 ◦C in an atmospheric oven yielded a
green solid disk suitable for thermal evaporation. Before each
NiO deposition, a pre-evaporation was done for about 5 min-
utes. Then a 30 nm thick NiO layer was deposited on the Si
substrate with applied power of about 350 watts resulted in
a deposition rate of 0.03 nm/s at a pressure of 2×10−6 Torr.
After that, without breaking the vacuum, a thin Co layer of 3
nm was deposited on the NiO surface by using DC sputtering
technique. During the deposition, a dynamic flow of ultra-
high purity Ar gas with pressure of 70 mTorr was used for
sputtering discharge. The discharge power to grow Co layers
was considered around 40 watts that resulted in a deposition
rate of about 0.01 nm/s. The thickness of the deposited films
was measured by styles technique, and controlled in-situ by
a quartz crystal oscillator located near the substrate. The dis-
tance between the target (50 mm in diameter) and substrate
was 70 mm. Before each deposition, a pre-sputtering was also
performed for about 10 minutes. The deposition of Co layers
was done at various negative bias voltages ranging from zero
to -80 V at the same sputtering conditions. The schematic
http://arxiv.org/abs/0704.1029v1
details about the way of exerting the bias voltage to the Si
substrate can be found in13.
In order to analyze the deposited samples, we have used
atomic force microscopy (AFM) on contact mode to study
the surface topography of the Co layer. The surface topogra-
phy of the films was investigated using Park Scientific Instru-
ments (model Autoprobe CP). The images were collected in a
constant force mode and digitized into 256× 256 pixels with
scanning frequency of 0.6 Hz. The cantilever of 0.05 N m−1
spring constant with a commercial standard pyramidal Si3N4
tip with an aspect ratio of about 0.9 was used. A variety of
scans, each with size L were recorded at random locations on
the Co film surface. The electrical property of the deposited
films was examined by four-point probe sheet resistance (Rs)
measurement at room temperature.
III. STATISTICAL QUANTITIES
A. roughness exponents
It is known that to derive a quantitative information of a
surface morphology one may consider a sample of size L and
define the mean height of growing film h and its roughness w
by the following expressions14:
h(L, t, λ) =
∫ L/2
h(x, t, λ)dx (1)
w(L, t, λ) = (〈(h− h)2〉)1/2 (2)
where t is proportional to deposition time and 〈· · · 〉 denotes
an averaging over different samples, respectively. Moreover,
we have introduced λ as an external factor which can apply
to control the surface roughness of thin films. In this work,
λ ≡ V/Vopt is defined where V and Vopt are the applied and
the optimum bias voltages, so that at λ = 1 the surface shows
its optimal properties. For simplicity, we assume that h = 0,
without losing the generality of the subject. Starting from a
flat interface (one of the possible initial conditions), we con-
jecture that a scaling of space by factor b and of time by factor
bz (z is the dynamical scaling exponent), rescales the rough-
ness w by factor bχ as follows:
w(bL, bzt, λ) = bχ(λ)w(L, t, λ) (3)
which implies that
w(L, t, λ) = Lχ(λ)f(t/Lz, λ). (4)
If for large t and fixed L (t/Lz → ∞) w saturate, then
f(x, λ) −→ g(λ), as x −→ ∞. However, for fixed large
L and t << Lz , one expects that correlations of the height
fluctuations are set up only within a distance t1/z and thus
must be independent of L. This implies that for x << 1,
f(x) ∼ xβg′(λ) with β = χ/z. Thus dynamic scaling postu-
lates that
w(L, t, λ) =
tβ(λ)g(λ) ∼ tβ(λ), t≪ Lz;
Lχ(λ)g′(λ) ∼ Lχ(λ), t≫ Lz.
The roughness exponent χ and the dynamic exponent z char-
acterize the self-affine geometry of the surface and its dynam-
ics, respectively. The dependence of the roughness w to the h
or t shows that w has a fixed value for a given time.
The common procedure to measure the roughness exponent
of a rough surface is use of a surface structure function de-
pending on the length scale △x = r which is defined as:
S(r) = 〈|h(x+ r) − h(x)|2〉. (6)
It is equivalent to the statistics of height-height correlation
function C(r) for stationary surfaces, i.e. S(r) = 2w2(1 −
C(r)). The second order structure function S(r), scales with
r as rξ2 where χ = ξ2/2 [1].
B. The Markov nature of height fluctuations
We have examined whether the data of height fluctuations
follow a Markov chain and, if so, determine 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 infi-
nite Markov length scale9. The Markov length scale is the
minimum length interval over which the data can be consid-
ered as a Markov process. To determine the Markov length
scale lM , we note that a complete characterization of the sta-
tistical 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 pro-
cess 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 stationary or homo-
geneous data 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 that 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
master equation, which can take the form of a Fokker-Planck
equation15:
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 theorem15, the Kramers-
Moyal expansion stops after the second term, provided that
the fourth order coefficient D(4)(h, x) vanishes15. 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.
Now, analogous to equation (8), we can write a Fokker-
Planck equation for the PDF of h which is equivalent to the
following Langevin equation (using the Ito interpretation)15:
h(x, λ) = D(1)(h, x, λ) +
D(2)(h, x, λ)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′)〉 = δ(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).
C. The level crossing analysis
We have utilized the level crossing analysis in the context
of surface growth processes, according to19,20. In the level
crossing analysis, we are interested in determining the aver-
age frequency (in spatial dimension) of observing of the defi-
nite value for height function h = α in the thin films grown at
different bias voltages, ν+α (λ). Then, the average number of
visiting the height h = α with positive slope in a sample with
size L will be N+α (λ) = ν
α (λ)L. It can be shown that the
ν+α can be written in terms of the joint PDF of h and its gra-
dient. Therefore, the quantity ν+α carry the whole information
of surface which lies in P (h, h′), where h′ = dh/dx, from
which we get the following result for the frequency parameter
ν+α in terms of the joint probability density function
ν+α =
p(α, h′)h′dh′. (11)
The quantity N+tot which is defined as N
tot =
ν+α dα will
measure the total number of crossing the surface with positive
slope. So, the N+tot and square area of growing surface are
in the same order. Concerning this, it can be utilized as an-
other quantity to study further the roughness of a surface. It is
expected that in the stationary state the N+tot depends on bias
voltages.
IV. RESULTS AND DISCUSSION
To study the effect of the bias voltage on the surface sta-
tistical characteristics, we have utilized AFM method for ob-
taining microstructural data from the Co layer deposited at the
different bias voltages in the Co/NiO/Si(100) system. Figure 1
shows AFM micrographs of the Co layer deposited at various
negative bias voltages of -20, -40, -60, and -80 V, as compared
with the unbiased samples.
For the unbiased very thin Co layer, Fig. 1a shows a colum-
nar structure of the Co grains grown over the evaporated NiO
underlaying surface. However, Fig. 1b shows that by applying
the negative bias voltage during the Co deposition, the colum-
nar growth is eliminated. Moreover, Figs. 1c and 1d show
that by increasing the bias voltage up to -60 V the grain size
of the Co layer is increased which means a more uniform and
smoother surface is formed. But, for the bias voltage of -80 V,
due to initiation of resputtering of the Co surface by the high
energy ion bombardment, we have observed a non-uniform
surface, even at the macroscale of the samples. Therefore,
based on the AFM micrographs, the optimum surface mor-
phology of the Co/NiO/Si(100) system was achieved at the
bias voltage of -60 V for our experimental conditions21.
Now, by using the introduced statistical parameters in the
last section, it is possible to obtain some quantitative infor-
mation about the effect of bias voltage on surface topography
of the Co/NiO/Si(100) system. Figure 2 presents the struc-
ture function S(r) of the surface grown at the different bias
voltages, using Eq. (6). The slope of each curve at the small
scales yields the roughness exponent (χ) of the correspond-
ing surface. Hence, it is seen that the surface grown at the
optimum bias voltage (-60 V) shows a minimum roughness
with χ = 0.60, as compared with the other biased samples
with χ =0.75, 0.70, and 0.64 for V=-20, -40, and -80 V, re-
spectively. For the unbiased sample, we have obtained two
roughness exponent values of 0.73 and 0.36, because of the
non-isotropic structure of the surface (see Fig. 1a). In any
case, at large scales where the structure function is saturated,
Fig. 2 shows the maximum and the minimum roughness val-
ues for the bias voltages of 0 and -60 V, respectively.
It is also possible to evaluate the grain size dependence to
the applied bias voltage, using the correlation length achieved
by the structure function represented in Fig. 2. For the unbi-
ased sample, we have two correlation lengths of 30 and 120
nm due to the columnar structure of the grains. However, by
applying the bias voltage, we can attribute just one correla-
tion length to each curve showing elimination of the columnar
structure in the biased samples. For the bias voltage of -20 V,
the correlation length of r∗ is found to be 56 nm. By increas-
ing the value of the bias voltage to -40 and -60 V, we have
measured r∗=76 and 95 nm, respectively. However, at -80 V,
due to initiation of the destructive effects of the high energy
ions on the surface, the correlation length is reduced to 76 nm.
Now, based on the above analysis, if we assume that Vopt=-60
V, then the roughness exponent and the correlation length can
be expressed in terms of the λ as follows, respectively,
χ(λ) = 0.61 + 0.16 sin2(2πλ/3.31 + 1.07) (12)
FIG. 1: (Color online) AFM surface images (all 1× 1µm2) of Co(3
nm)/NiO(30 nm)/Si(100) thin films deposited at the bias voltages of
a) 0, b) -20, c) -40, and d) -60 V. (from top to bottom corresponding
a to d, respectively)
r (nm)
50 100 150 200
FIG. 2: (Color online) Log-Log plot of structure function of the sur-
face at different bias voltages.
r∗(λ) = 53.50 + 40.23 sin2(2πλ/3.00 + 2.62)(nm) (13)
where for λ = 0 with the columnar structure, we have consid-
ered the average values.
To obtain the stochastic behavior of the surface, we need to
measure the drift coefficient D(1)(h) and diffusion coefficient
D(2)(h) using Eq. (9). Figure 3 shows D(1)(h) for the sur-
faces at the different bias voltages. It can be seen that the drift
coefficient shows a linear behavior for h as:
D(1)(h, λ) = −f (1)(λ)h (14)
where
f (1)(λ) = [0.55 + 1.30 sin2(2πλ/3.50 + 1.40)]× 10−4.(15)
The minimum value of f (1)(λ) for the biased samples at
λ = 1 shows that the deterministic component of the height
fluctuations for these samples is lower than the other biased
and unbiased ones. Figure 4 presents D(2)(h) for the differ-
ent bias voltages. At λ = 0, the maximum value of diffusion
has been obtained for any h, as compared with the other cases.
By increasing the bias voltage, the value of D(2) is decreased,
as can be seen for λ = 1/3 and 2/3. The minimum value
of D(2), which is nearly independent of h, is achieved when
λ = 1. This shows that the noisy component of the surface
height fluctuation at λ = 1 is negligible as compared with
the unbiased and the other biased samples. The behavior of
D(2) at λ = 4/3 becomes similar to its behavior at λ = 2/3.
It is seen that the diffusion coefficient D(2) is approximately
a quadratic function of h. Using the data analysis, we have
found that
D(2)(h, λ) = f (2)(λ)h2 (16)
-30 -20 -10 0 10 20 30
-0.04
-0.02
FIG. 3: (Color online) Drift coefficient of the surface at different bias
voltages.
FIG. 4: (Color online) Diffusion coefficient of the surface at different
bias voltages.
where
f (2)(λ) = [3.20 + 3.53 sin2(2πλ/3.33 + 1.34)]× 10−6
Now, using the Langevin equation (Eq. (10)) and the mea-
sured drift and diffusion coefficients, we can conclude that
the height fluctuation has the minimum value at λ = 1 which
means a smoother surface at the optimum condition. More-
over, the obtained equations for the coefficients (Eqs. (14) and
h (A)
-20 0 20
FIG. 5: (Color online) Level crossing of the surface at different bias
voltages.
(16)) can be used to regenerate the rough surfaces the same as
AFM images shown in Fig. 17,8.
To complete the study, roughness of a surface can be also
evaluated by the level crossing analysis, as another procedure.
Figure 5 shows the observed average frequency ν+α as a func-
tion of h for the different bias voltages. As λ is increased from
0 to 1, the value of ν+α is decreased at any height. Once again,
the optimum situation is observed for the bias voltage of -60 V
showing the surface formed at λ = 1 condition is a smoother
surface with lower height fluctuations than the surface formed
at the other conditions. It is seen that, at λ = 4/3, the height
fluctuation of the surface finds a maximum value, as compared
with the other surfaces. The same as the roughness exponent
and the correlation length behavior in terms of λ, the N+tot can
be also expressed as:
N+tot(λ) = [1.20 + 0.17 sin
2(2πλ/3.52 + 1.40)]. (18)
Since the system under investigation has a thin Co layer
which is the only conductive layer, thus, it is obvious that
lower height fluctuation corresponds to smaller electrical re-
sistivity of the surface. Concerning this, we have measured
sheet resistance of the Co surface grown at the different bias
voltages, as shown in Fig. 6. For the bias voltage ranging from
0 to -60 V, the Rs value is reduced from 432 to 131 Ω/sq.. The
minimum value of Rs is measured at the optimum condition
of -60 V (λ = 1) which can be related to modified and smooth
surface roughness. Elimination of interfacial voids as well as
porosities, and reduction of impurities in the Co layer. A sim-
ilar behavior was also observed at Vopt=-50 V for Ta/Si(111)
system18. By increasing the applied bias voltage to values
greater than its optimum value, surface roughness is increased
because of surface bombardment by high energy ions. This
can be seen by the observed increase in the Rs value at the
0 0.5 1 1.5
FIG. 6: (Color online) Sheet resistance measurement of the Co thin
layer as a function of the applied bias voltage.
bias voltage of -80 V (λ = 4/3). It is easy to examine that the
variation of Rs as a function of λ can be expressed as below:
Rs(λ) = [135.48 + 307.74 sin
2(2πλ/3.93 + 1.77)] (19)
It behaves similar to the behavior of roughness characteristics
of the surfaces. Therefore, we have shown that the roughness
behavior explained by the statistical characterizations of the
surface, which have been obtained by using microstructural
analysis of AFM, can be related to the sheet resistance mea-
surement of rough surfaces, as a macrostructural analysis.
V. THE TIP CONVOLUTION EFFECT
It is well-known that images acquired with AFM are a con-
volution of tip and sample interaction. In fact, using scanning
probe techniques for determining scaling parameters of a sur-
face leads to an underestimate of the actual scaling dimension,
due to the dilation of tip and surface. Concerning this, Aue
and Hosson22 showed that the underestimation of the scaling
exponent depends on the shape and aspect ratio of the tip, the
actual fractal dimension of the surface, and its lateralvertical
ratio. In general, they proved that the aspect ratio of the tip is
the limiting factor in the imaging process.
Here, we want to study the aspect ratio effect of the tip
on the investigated stochastic parameters. To do this, using
a computer simulation program, we have generated a rough
surface by using a Brownian motion type algorithm23,24 with
roughness and its exponent of 10.00 nm and 0.67, respectively.
We have assumed these roughness parameters in order to have
some similarity between the generated surface and our ana-
lyzed surface by AFM. In the simulation program, the gener-
ated surface has been scanned using a sharp cone tip with an
250 500 750 1000
-1500
-1000
FIG. 7: (Color online) Height profile of a rough generated surface
before dilation (real) and after dilation (tip) caused by a tip with the
aspect ratio of 0.73.
assumed aspect ratio of 0.73 which is also nearly similar to
the applied tip in our AFM analysis with the aspect ratio of
0.9. Moreover, this assumption does not limit the generality
of our discussion, because it is shown that the fractal behav-
ior of a rough surface presents an independent tip aspect ratio
behavior (saturated behavior) for the aspect ratios greater than
about 0.422.
Figure 7 shows a line profile of a generated surface which
is dilated by a tip with the known aspect ratio. It is clearly
seen that the scanned image (the image affected by the tip
convolution) does not completely show the generated surface
topography (real surface). Now, it is possible to study the
dependance of the examined surface stochastic parameters on
the geometrical characteristic of the tip, i.e. aspect ratio.
In this regard, Fig. 8 shows variation of the one-
dimensional structure function of the generated rough surface
due to the tip convolution effect. It can be seen that by increas-
ing the aspect ratio the tip convolution results in obtaining a
surface image whit a decreased roughness. Since the aspect
ratio of the applied AFM tip was around 0.9, so the measured
roughness exponents at the different bias voltages might be
corrected by a 1.07 factor. In other words, the relative change
(the difference between the real and measured values com-
paring the real one) of the roughness exponent is about 7.2%.
Moreover, Fig. 8 shows that the correlation length is increased
by the tip convolution effect. It should be noted that, in our
simulation, we have assumed that the apex of the tip is com-
pletely sharp (the tip radius is assumed zero). However, it is
well-known that the radius of the pyramidal tips is ∼ 20 nm.
Therefore, the real correlation lengths are even roughly 20 nm
larger than the measured ones by the sharp tip.
The same tip convolution effect can be also presented for
the drift and diffusion coefficients. Figure 9 presents the
100 101 102
tip 40
FIG. 8: (Color online) The one-dimensional structure function anal-
ysis, plotting log[S(r)] vs. log(r) in which r is pixel position along
the x-axis. This results in the roughness values of 10 and 7.36 nm
for the generated surface before dilation and after dilation using a tip
with the aspect ratios of 0.73, respectively.
-1 0 1
-0.05
tip 40
FIG. 9: (Color online) The calculated drift coefficient for the gener-
ated surface before dilation (real) and after dilation with a tip having
aspect ratio of 0.73 (tip).
calculated drift coefficient for the generated surface and the
scanned surface. One can see that the tip convolution results in
decreasing of the drift coefficient, corresponding to decreas-
ing of the surface roughness. This means that after dilation the
correlation length will increase and hence the measured value
for f (1)(λ) would be smaller than its value for the original sur-
-1 0 1
0.0005
0.001
0.0015
0.002
tip 40
FIG. 10: (Color online) The calculated diffusion coefficient for the
generated surface before dilation (real) and after dilation with a tip
having aspect ratio of 0.73 (tip).
-3000 -2000 -1000 0 1000 2000 3000
10000
12000
14000
tip 40
FIG. 11: (Color online) Level crossing analysis of the generated sur-
face before dilation (real) and after dilation (tip).
face. Therefore, the magnitude of slope of the drift coefficient
must decrease after using the tip. For our generated surface,
the measured value of the drift coefficient should be modified
by a factor of around 2.
The variation of the diffusion coefficient of the generated
rough surface due to the tip convolution effect has been also
shown in Fig. 10. The reduction of the diffusion coefficient of
the scanned surface as compared to its values for the generated
surface, due to the tip convolution, can be easily seen. In fact
to compensate the tip effect on the diffusion coefficient, we
should modify its measured values by a factor of about 4, for
the assumed generated surface.
Finally, we remind that the total number of crossing the sur-
face with positive slope (N+tot) has been defined as a parame-
ter describing the rough surfaces. Hence, we have also studied
the effect of the tip convolution on this parameter, as shown
in Fig. 11. It is seen that N+tot decreased due to the tip con-
volution effect. For the assumed generated surface, we have
obtained that the N+tot of the surface before dilation is about
1.7 times larger than its value after the dilation. In this figure,
we have also shown the variation of the average height due to
the tip effect.
One has to note that our generated surface is a pure two-
dimensional one which presents no line-to-line interaction.
So, for this simple model, differently shaped tips with the
same aspect ratio yield the same results. Therefore, for the
three-dimensional case one can expect to obtain a larger dis-
tortion of the surface due to stronger line-to-line interac-
tion leading to an even larger underestimation of the studied
stochastic parameters.
These analysis showed that, although the measured values
of the surface parameters by AFM method are different from
the real ones, the general behavior of these parameters as a
function of the bias voltage are not affected by the tip convo-
lution. Therefore, our general conclusions about the variation
of the studied stochastic parameters by applying the bias volt-
age is intact.
VI. CONCLUSIONS
We have investigated the role of bias voltage, as an exter-
nal parameter, to control the statistical properties of a rough
surface. It is shown that at an optimum bias voltage (λ = 1),
the stochastic parameters describing a rough surface such as
roughness exponent, level crossing, drift and diffusion coeffi-
cient must be found in their minimum values as compared to
an unbiased and the other biased samples. In fact, dependence
of the height fluctuation of a rough surface to different kinds
of the external control parameters, such as bias voltage, tem-
perature, pressure, and so on, can be expressed by AFM data
which are analyzed using the surface stochastic parameters.
In addition, this characterization enable us to regenerate the
rough surfaces grown at the different controlled conditions,
with the same statistical properties in the considered scales,
which can be useful in computer simulation of physical phe-
nomena at surfaces and interfaces of, especially, very thin lay-
ers. It is also shown that these statistical and microstructural
parameters can explain well the macroscopic properties of a
surface, such as sheet resistance. Moreover, we have shown
that the tip-sample interaction does not change the physical
behavior of the stochastic parameters affected by the bias volt-
Acknowledgments
AZM would like to thank Research Council of Sharif Uni-
versity of Technology for financial support of this work. We
also thank F. Ghasemi for useful discussions.
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http://arxiv.org/abs/cond-mat/0411529
	Introduction
	Experimental
	Statistical quantities
	roughness exponents
	 The Markov nature of height fluctuations
	The level crossing analysis
	Results and discussion
	The tip convolution effect
	Conclusions
	Acknowledgments
	References
