1. DEPARTMENT OF ADVANCED MATERIALS ENGINEERING
MICROSCOPIC STUDY OF CONDUCTIVE-INSULATOR
COMPOSITES
By
Iliya Tsvibel
and
Denis Teitelbaum
February 2013

2. DEPARTMENT OF ADVANCED MATERIALS ENGINEERING
MICROSCOPIC STUDY OF CONDUCTIVE-INSULATOR
COMPOSITES
By
Iliya Tsvibel
and
Denis Teitelbaum
Approved by the Advisors: Dr. Doron Azulay Date:___________
Approved by the Chairman,
Department of Advanced Materials Engineering Date:___________
February 2013

3. 3
DECLARATION (‫)הצהרה‬
This work was done under the supervision of Dr. Doron Azulay
in the Hebrew University, Givat Ram.
This exposition presents our personal work,
and it is the part of the requirements for owing bachelor’s degree in engineering.
All text and/or results are based on other researches which fully referenced.
‫העבודה‬‫נעשתה‬‫בהנחיית‬‫ד‬"‫ר‬‫דורון‬‫אזולאי‬,
‫ב‬‫אוניברסיטה‬‫העברית‬,‫גבעת‬‫רם‬.
‫החיבור‬‫מציג‬‫את‬‫עבודתנו‬‫האישית‬
‫ומהווה‬‫חלק‬‫מהדרישות‬‫לקבלת‬‫תואר‬‫ראשון‬‫בהנדסה‬.
‫כל‬‫טקסט‬‫ו‬/‫או‬‫תוצאה‬‫המבוססים‬‫על‬‫עבו‬‫דות‬‫מחקר‬‫אחרות‬,
‫מתועדים‬‫בציון‬‫המקור‬‫המדעי‬.(Fully Referenced)

4. 4
Table of Contents
1. Abstract ..................................................................................................................................... 9
2. Symbols ................................................................................................................................... 11
3. Theoretical Background .......................................................................................................... 12
3.1 Fundamentals of percolation theory................................................................................. 12
3.2 Fractals .............................................................................................................................. 14
3.21 Fractals and its examples in nature............................................................................. 14
3.22 Fractal dimension........................................................................................................ 15
3.3 Fractal structure and fractal analysis of percolation clusters........................................... 16
3.31 Fractal structure of percolation clusters near pc........................................................ 16
3.32 Fractal analysis of percolation clusters. ...................................................................... 17
3.33 Fractal structure of the percolation cluster. ............................................................... 17
3.34 Bond percolation......................................................................................................... 18
3.35 Conductivity properties of the percolation cluster..................................................... 18
3.4 Atomic-Force Microscopy (AFM) ...................................................................................... 19
3.41 General information.................................................................................................... 19
3.42 Principle of operation.................................................................................................. 19
3.43 Operating Modes of the Atomic Force Microscope.................................................... 20
3.44 Image processing......................................................................................................... 22
3.5 Kelvin Probe microscopy Method. .................................................................................... 24
3.51 General information.................................................................................................... 24
3.52 Principle of operation.................................................................................................. 24
3.6 Carbon Black Polymer composite ..................................................................................... 25
3.61 General information about the composite and the percolation mechanism in it...... 25
3.62 Breakdown phenomena of electrical current in CBP.................................................. 25
3.63 Thermal fuse................................................................................................................ 27

5. 5
3.7 Charging effects in Si/SiO2 system.................................................................................... 29
3.71 Nanocrystalline silicon................................................................................................. 29
3.72 Charge storage and cluster statistics in ensembles of Si quantum dots..................... 29
4. Objectives. .............................................................................................................................. 30
5. Experimental Procedures........................................................................................................ 30
5.1 General.............................................................................................................................. 30
5.11 Introduction................................................................................................................. 30
5.12 Evaluation of the mass fractal dimension of the percolation network....................... 30
5.13 Electro-thermal switching effect................................................................................. 31
5.14 Charging effect in Si/SiO2............................................................................................ 31
5.2 Equipment and Analytical Methods.................................................................................. 31
5.21 Atomic Force Microscope............................................................................................ 31
5.22 CBP composites........................................................................................................... 32
5.23 Charging effect in Si/SiO2 system. .............................................................................. 33
5.3 Data processing................................................................................................................. 34
5.31 MATLAB (software). .................................................................................................... 34
6. Results ..................................................................................................................................... 34
6.1 CBP composite................................................................................................................... 34
6.11 Fractal dimension of the percolation cluster. ............................................................. 34
6.12 Microscopic study of current-voltage dependence in CPB composite. ...................... 36
6.2 Charging effect in Si/SiO2 composite................................................................................ 37
6.21 Kelvin probe spectroscopy of the Si/SiO2 samples..................................................... 37
7. Discussion................................................................................................................................ 39
7.1 CBP composites................................................................................................................. 39
7.2 Charging effect in Si/SiO2 composites .............................................................................. 40
8. Conclusion............................................................................................................................... 41

6. 6
8.1 CBP composite................................................................................................................... 41
8.2 Charging effect in the Si/SiO2 composite ......................................................................... 42
9. Acknowledgments................................................................................................................... 42
10. References............................................................................................................................. 43
11. Appendixes............................................................................................................................ 46
11.1 box count dimension...................................................................................................... 46
11.2 Charging effect of Si/SiO2 ............................................................................................... 47
List of Figures
Figure 3.1-1: Site percolation on the square lattice. The small circles represent
the occupied sites for three different concentrations: p=0.2, 0.59, and 0.80. Nearest-
neighbor cluster sites are connected by lines representing the bonds. Filled circles are
used for finite clusters, while open circles mark the large infinite cluster [1].…..……….13
Figure 3.1-2. Schematic diagram of the probability (eg. (1), bold line) and the
correlation length (eg. (2), thin line) versus the concentration p of occupied sites. [1]
…………………………………………………………………………………………………………………………………..14
Figure 3.31-1. Self-similarity of a large percolation cluster on a square lattice at
the critical concentration. Figures 3b, 3c and 3d are magnifications of the center parts
marked by white sguares of their previous figure [1]………………………………………..……….16
Figure 3.35-1. Schematic diagram of the (usual) DC conductivity dc (eq. (3.55-2),
bold line) [1], [3] …………………………………………………………………..…………………………………18
Figure 3.42-1: The principle of operation of the AFM [16] ……………………………….19
Figure 3.42-2: Scheme of relevant spatial distances in AFM. The instantaneous
tip-surface separation d, the instantaneous position of the tip z and the average tip-
surface separation zc are plotted. The position of the tip, z, is defined with respect to
the rest position of the tip, positive values and negative values above and below the
rest position of the cantilever-tip system, respectively [13] ………………………………….…..20

7. 7
Figure 3.43-1: Interatomic force vs. distance curve [17] …………..………………………20
Figure 3.44-1: AFM scanning artifact arising from a tip with a high radius of
curvature with respect to the feature which is to be visualized [13] ……………..…………..23
Figure 3.44-2: Schematic representation of the components of an atomic force
microscope (AFM) [15]…………………………………………………….…………………………………………24
Figure 3.62-1: Temperature dependence of the resistance [9] …………....………….26
Figure 3.62-2: The dependence of the thickness of the sample on the
temperature [9] ……………………………………………………………………………………………………..…26
Figure 3.62-3: The dependence of the resistance on the thickness of the
sample[9] …………………………..……………………………………………………………………………………..27
Figure3.63-1: The Joule heat generated by the current causes the polymer to
locally expand, thus increasing the tunneling distance between the particles. This leads
to a sharp current decrease through junctions that are located at critical positions
along the network ……………………………………………………………………………….……………………28
Figure 3.63-2: Current voltage characteristic of the sample of CBP [10] …………28
Figure 5.21-1: The AFM setup ……………………………………………………………………..…..32
Figure 6.11-1: Topography (a) and friction (b) images that were measured
simultaneously. Some of the CB particles are marked by circles ………………………..…….35
Figure 6.12-1: AFM maps that show the dependence of the conductivity of the
simple as a function of the applied voltage. First, the quantity of conducting regions
increases with the voltage. For bias voltage higher than 5V, the number of conducting
regions drops sharply ………………………………………………………………………………………………..36
List of Graphs
Graph 6.11-1: The fractal dimension of the percolation infinite cluster in two-
dimensional slice. (For better visibility derivatives offset one above the other) ..………35

8. 8
Graph 6.12-1: The change of the conductive areas, depending on the applied
voltage ….…………………………………………………………………………………………………………………..37
Graph 6.21-1: An example of the magnitude of the tip oscillations as a function of
the DC Bias Voltage applied to the AFM tip .………………………………………………………………38
Graph 6.21-2: The charging voltage as a function of the acquisition time ..38
Graph 6.21-3: The charging voltage and the conductivity as a function of the
Si vol. fraction …..………………………………………………………………………………..…………………….39

9. 9
1. Abstract
English
The electrical properties of systems that consist of conducting (or semi-
conducting) particles enclosed in a dielectric matrix are widely studied nowadays
because of their applicable potential. The conductivity of these systems is usually
described by the percolation theory that deals with the geometrical and physical
properties above and below a certain threshold, at which a global connectivity is
formed in the system. In addition, when nano-particles are involved, quantum effects
such as quantum confinement and charging also influence the conductivity. Moreover,
the properties of the interface between particles (or clusters of particles) are also
important.
In our work we studied geometric properties of the percolation network, which
permits electrical conductivity through the sample, by using an atomic force
microscope. Especially, we measured the mass fractal dimension of the percolation
cluster, near the percolation threshold, in system that consists of carbon particles
enclosed in a dielectric polymer (carbon black / polymer). Additional effect that was
investigated in these samples is the “Thermal Switching effect”. Current - voltage
characteristics of the sample show that increasing the voltage above a certain value
leads to a reduction of the current in the system. This property has a lot of applications
such as switches in electric windows in cars etc. This effect is caused by the local Joule
heating generated by the current in the system and the difference between the
thermal expansion coefficient of the polymer and the particles.
In addition, we studied the charging effect in system consisting of silicon nano-
particles that are implanted (with different concentration) in a dielectric matrix
(Si/SiO2), as a function of the silicon concentration and the scanning time of. The
measurements were made by using Kelvin probe spectroscopy. Strong charging effect
can be achieved only near the percolation threshold, when the conductivity of the
sample is very low. The effect decreases as the conductivity of the sample increases
(increasing the silicon concentration) since discharging becomes easier.

10. 10
Hebrew
‫תכונות‬‫ההולכה‬‫של‬‫מערכות‬‫המורכבות‬‫מחלקיקים‬‫מוליכים‬(‫או‬‫מוליכים‬‫למחצה‬)
‫המשובצים‬‫במטריצה‬‫מבודדת‬‫נחקרים‬‫באופן‬‫אינטנסיבי‬‫בעשור‬‫האחרון‬‫כתוצאה‬‫מהפוטנציאל‬
‫הגלום‬‫בהם‬‫ליצירת‬‫התקנים‬‫ש‬‫ונים‬.‫המוליכות‬‫במערכות‬‫אלה‬‫מתוארת‬‫בעזרת‬‫תורת‬‫הפרקולציה‬
‫העוסקת‬‫בתיאור‬‫ההתנהגות‬‫של‬‫תכונות‬‫גיאומטריות‬‫ופיזיקליות‬‫מעל‬‫ומתחת‬‫לסף‬‫בו‬‫נוצרת‬
‫קשירות‬(connectivity)‫גלובלית‬‫במערכת‬.‫כאשר‬‫מדובר‬‫בננו‬-‫חלקיקים‬‫באות‬‫לידי‬‫ביטוי‬‫גם‬
‫תופעות‬‫קוונטיות‬‫כגון‬‫כליאה‬‫קוונטית‬‫ואפקטים‬‫של‬‫טעינה‬‫המשפיעות‬‫על‬‫ההולכה‬‫בין‬‫חלקיקים‬.
‫בנוסף‬,‫כאשר‬‫מדובר‬‫בצבירים‬‫של‬‫ננו‬-‫חלקיקים‬‫יש‬‫חשיבות‬‫גדולה‬‫גם‬‫לממשק‬‫בין‬‫שני‬‫חלקיקים‬.
‫בעבודה‬‫זו‬‫נחקרו‬‫באמצעות‬‫מיקרוסקופ‬‫כ‬‫ו‬‫ח‬‫אטומי‬‫תכונות‬‫גיאומטריות‬‫של‬‫רשת‬
‫הפרקולציה‬‫המאפשרת‬‫הולכה‬‫חשמלית‬‫לאורך‬‫הדגם‬.‫בייחוד‬,‫נמדד‬‫המימ‬‫ד‬‫הפרקטלי‬‫של‬
‫המסה‬‫של‬‫צביר‬‫הפרקולציה‬‫ליד‬‫סף‬‫הפרקולציה‬‫במערכת‬‫המורכבת‬‫מחלקיקי‬‫פחמן‬‫המשובצים‬
‫בפולימר‬‫מבודד‬((carbon black/polymer.‫אפקט‬‫נוסף‬‫שנחקר‬‫בדגמים‬‫אלה‬‫הוא‬‫המיתוג‬
‫התרמי‬.‫מדידות‬‫זרם‬‫כתלות‬‫במתח‬(I-V)‫בדגמים‬‫אלה‬‫מראים‬‫שהגדלת‬‫המתח‬‫מעבר‬‫לערך‬
‫מסויים‬‫גורמת‬‫להק‬‫טנה‬‫של‬‫הזרם‬‫במערכת‬.‫לתכונה‬‫זו‬‫שימושים‬‫רבים‬‫בתעשיה‬‫כגון‬‫מתגים‬
‫לחלונות‬‫חשמליים‬‫וכדומה‬.‫תופעה‬‫זו‬‫נגרמת‬‫כתוצאה‬‫מחימום‬‫ג‬'‫אול‬‫מקומי‬‫כתוצאה‬‫מהזרם‬
‫במערכת‬‫וההבדלים‬‫במקדמי‬‫ההתפשטות‬‫התרמיים‬‫של‬‫החלקיקים‬‫והפולימר‬.
‫בנוסף‬,‫נחקר‬‫אפקט‬‫הטעינה‬‫במערכות‬‫המורכבות‬‫מננו‬-‫חלקיקים‬‫ש‬‫ל‬‫סיליקון‬‫המשובצים‬
(‫בריכוזים‬‫שונים‬)‫בתוך‬‫מטריצה‬‫מבודדת‬(Si/SiO2)‫כתלות‬‫בריכוז‬‫שונים‬‫ובזמני‬‫הסריקה‬
‫השונים‬.‫מדידות‬‫אלה‬‫בוצעו‬‫בעזרת‬‫ספקטרוסקופיית‬‫קלוין‬(Kelvin Probe spectroscopy).
‫אפקט‬‫זה‬‫מתקבל‬‫בצורה‬‫חזקה‬‫רק‬‫סמוך‬‫מאוד‬‫לסף‬‫פרקולציה‬‫כאשר‬‫המוליכות‬‫של‬‫הדגם‬‫מאוד‬
‫נמוכה‬‫ודועך‬‫ככל‬‫שמוליכות‬‫הדגם‬‫גדלה‬(‫הגדלת‬‫ריכוז‬‫הסיליקון‬)‫כתוצאה‬‫מפריקת‬‫המטען‬.

11. 11
2. Symbols
AFM Atomic Force Microscope
pc Percolation Threshold.
p Concentration
df Fractal Dimension
z Coordination Number
M Mass of the Percolation Cluster
 Conductivity
CBP Carbone Black Polymer
QD Quantum Dot
AC Alternating Current
DC Direct Current
CPD Contact Potential Difference

12. 12
3. Theoretical Background
3.1 Fundamentals of percolation theory
Percolation theory deals with the onset of global connectivity in systems that are
randomly occupied by conductive sites. Due to its universality, the results do not
depend on concrete model, but general laws of scaling can be deduced. In this section
we give a short introduction to percolation theory and also describe its application to
composites [1]. We start with the structural properties of the percolation clusters and
their substructures.
As an example, we consider a square lattice in which each site is randomly
occupied with probability p, or empty with probability 1-p. Occupied and empty sites
have very different physical properties. For example, let us suppose that the occupied
sites are electrically-conductive zones, the empty sites are insulators, and that the
current might flow only between adjacent conductive sites.
We denote the concentration of the occupied sites as p. When p is low, the
conductive sites are isolated from each other, or form small clusters. Two conductive
sites belong to the same cluster if they are connected by means of conductive path i.e.
they are electrically connected [1].
When p is low, the system is insulating, because there is no conductive pathway
that connects the opposite edges of the lattice. When p is high, multiple pathways are
formed between the opposite edges of the lattice. In these conditions current flows
from one edge to another and the system is conducting. At some threshold
concentration, the first conducting path is formed from one edge of the lattice to the
opposite one, creating at the same time a condition for a current flow. (Figure 3.1-1).

13. 13
Figure 3.1-1: Site percolation on the square lattice. The small circles represent the occupied
sites for three different concentrations: p=0.2, 0.59, and 0.80. Nearest-neighbor cluster sites
are connected by lines representing the bonds. Filled circles are used for finite clusters, while
open circles mark the large infinite cluster [1].
The concentration, in which the first path is formed, is called "percolation
threshold”, [2]. For < only finite small clusters are formed. When the
concentration is increases, the average size of clusters also increases. At the critical
concentration , the size of the larger cluster connects the opposite edges of the
lattice. This cluster is called the infinite cluster, as its size diverges in the
thermodynamic limit. With further increasing the density of the infinite cluster is
growing because more and more sites become part of it, and the average size of the
finite clusters, which do not belong to the infinite cluster, is reduced. At p = 1 all sites
belong to the infinite cluster [3]. The critical concentration depends on the details of
the lattice. We denote the coordination number; the number of nearest neighbours in
the lattice, as z. At a fixed dimension d, the value of increases as the coordination
number z of the lattice decreases. For example, for a triangular lattice z = 6 and pc =
1/2, for a square lattice, z = 4 and pc = 0,592746, and for hexagonal lattice, z = 3 and
= 0,6962. For constant coordination number z, decreases, as the dimension d
increases. For example, the value of of a triangular lattice )z=6( is ½ and of a simple
cubic lattice is 0,3116. The percolation transition is characterized by the geometric
properties of the clusters around pc [1, 2]. The probability that an occupied site
belongs to the infinite cluster is giving by:
(3.1-1) [1], [3]

14. 14
Where is a universal exponent. For < the probability is zero. This behavior
is illustrated in Figure 3.1-2.
Figure 3.1-2. Schematic diagram of the probability (eg. (1), bold line) and the correlation
length (eg. (2), thin line) versus the concentration p of occupied sites. [1]
The correlation length characterizes the linear dimension of the finite clusters.
It is defined as the average distance between two points on a finite cluster, and
represents the characteristic length scale in the percolation. When the value of is
close to , increases as:
(3.1-2) | |
with the same exponent above and below the percolation threshold (Figure
3.1-2). While obviously depends on the type of lattice, the critical exponents β and
are universal and they depend only on the dimension d of the lattice and do not
depend on the type of lattice [1],[2].
3.2 Fractals
3.21 Fractals and its examples in nature.
Fractal is a geometrical figure with the property of self-affinity (self-similarity). In
mathematics, fractals are defined as a set of points in Euclidean space that has a
fractional metric dimension (Minkowski dimension or Hausdorff dimension) that in its
turn is different from the topological dimension. [5]. Fractal is a shape which has the
following properties:

15. 15
• It has a nontrivial structure on all scales. If we observe a small fragment of a
regular figure in a very large scale, it will be similar to the fragment of line. For a
fractal, zoom (scale increasing) does not lead to the simplification of the structure; the
same complex picture is observed at all scales.
• It is self-similar or approximately self-similar.
• It has a fractional metric dimension [6].
3.22 Fractal dimension.
Intuitively, we understand the term dimension as the number of coordinates
which is necessary to set a point location in the figure. Thus, every line (for example,
circle or straight line) is one-dimensional. Only one coordinate is enough to exactly
indicate the position of a point [4]. As for a plane and sphere, they are two-
dimensional surfaces. According to Minkowski, we can define the fractal dimension of
a figure in the following way. Suppose that one wants to find the fractal dimension of a
figure F that is located on a plane. The plane, in its turn, is covered by a lattice of
squares with side δ. The number of squares that are crossed with the figure F
(combination of all these squares that contains parts of F) is denoted by N)δ(. Clearly,
the number of squares depends on the size of these squares: the smaller they are,
more squares are needed to cover the figure. If this dependence is expressed by a
power law, i.e. the number N (δ) is proportional to , the dimension of F is
considered to be D which may not be an integer. For non-fractal figures this procedure
gives the same result as the intuitive notion of dimension. For example, let’s calculate
the dimension of a square with side 1 (placing the square on the plane, so that the
square sides lie every time on the lines of the lattice). In this case, N (1) = 1, N (1/2) = 4,
N (1/3) = 9, N (1/4) = 16, etc. It is evident, that in this case D = 2, which means that the
square is two-dimensional. [4], [5]. Many physical objects do not possess an integer-
valued dimension; they have a fractional dimension. For this reason they are called
fractals. In general, the parameters of length, area and volume are inseparably
connected with the notion of dimension. For a segment of a line d = 1, for a planar
object d = 2, for the three-dimensional object d = 3, etc. [5], [6]. For example, for a
planar figure (e.g. circle) in three-dimensional space, the volume is equal zero, the

16. 16
length is infinite (here we do not mean the length of the circumference, we speak
about the length of the closed segment that has no beginning and end, for example,
such a figure as a circle ) and the area is finite and do not equal zero. For fractals, a
similar situation is observed, but between 1 and 2 there are not intermediate integer
values, that is why the dimension of fractal curves should be fractional [5].
3.3 Fractal structure and fractal analysis of percolation clusters.
3.31 Fractal structure of percolation clusters near pc.
In domain (around the percolation threshold, as discussed in paragraph
number 3.1) and also above and below the percolation threshold on scales that are
smaller than , infinite and finite clusters have a self similar appearance. In other
words, if you select a small part of a cluster, increase it to the size of original cluster
and compare them, they will be very similar. This property is shown in Figure. 3.31-1,
where part at the infinite cluster ( = ) is shown in four different magnifications [1].
a) b)
c) d)
Figure 3.31-1 Self-similarity of a large percolation cluster on a square lattice at the critical
concentration. Figures 3b, 3c and 3d are magnifications of the center parts marked by white
sguares of their previous figure [1]
The clusters are characterized by "fractal" dimension because of their non-trivial
self-similarity. This dimension is numerically smaller than the dimension of the lattice,
d. [7].

17. 17
3.32 Fractal analysis of percolation clusters.
Consider the average mass of a cluster inside a circle. This mass increases with
the radius r as:
(3.32-1) [1],
with "fractal dimension" . Over pc domain (near pc), on scales which are larger
than , the infinite cluster can be considered as a homogeneous system, which consists
of many cells, possessing a size equal to the [1]. Thus in this case we don’t expect
fractal dimension.
Mathematically, it can be represented in the following way:
(3.32-2) {
The fractal dimensions is related to other critical exponents and as follow:
[3]
(3.32-3)
where and are universal exponents, and consequently, is also universal.
[1], [3].
3.33 Fractal structure of the percolation cluster.
The percolation cluster consists of some fractal substructures. These structures
can be described in the following way. Consider a voltage drop between two points on
opposite edges of the metal percolation cluster. The backbone of the cluster consists
of the cells in which the current flows through. The topological distance between two
points (also called chemical distance) is the length of the shortest path on the cluster,
connecting these points. [1]. "Dangling ends" are those parts of the cluster which are
connected to the percolation cluster only through one connection. "Red cells" (or
singly connected sites) are sites that carry out the full current transmission. [1], [7].
The fractal dimension of the backbone is smaller than fractal dimension of
the cluster. It reflects the fact that the largest part of the site’s mass is concentrated in

18. 18
dangling ends. In average, the topological length in the path between two points on
the cluster increases with the Euclidean distance r between them as: . The
average number of red cells depends on p, as: ⁄ ⁄
. And the
fractal dimension of the red cells therefore can be considered as: ⁄ [1],[3].
3.34 Bond percolation.
Up to now, we have considered the percolation effect, where lattice sites were
occupied in random order (site-percolation). One can similarly discuss a situation in
which all sites in the lattice are occupied, and the bonds between the sites are
randomly occupied with probability , i.e. bond percolation. In this case, two occupied
sites belong to the same cluster only if they are connected by a path of occupied
bonds. The critical concentration of bonds is the concentration in which an infinite
cluster is formed. For example, in a square lattice = 1/2 and in a simple cubic lattice
= 0,2488. [1], [7], [8].
3.35 Conductivity properties of the percolation cluster.
Below , the DC (direct current) conductivity in a system of conductor-insulator
is zero. Near (from above) the conductivity increases as a power law:
(3.35-1) [27]
where t is a corresponding critical exponent.
Figure 3.35-1. Schematic diagram of the (usual) DC conductivity dc (eq. (3.55-2), bold line)
[1], [3]

19. 19
3.4 Atomic-Force Microscopy (AFM)
3.41 General information.
Atomic-force microscopy (AFM) is a very high-resolution type of scanning probe
microscopy. It is used to determine the surface topography with a resolution of
nanometers. In contrast to the scanning tunneling microscope, AFM can scan both
conductive and non-conductive surfaces [12] [16].
3.42 Principle of operation.
The principle of operation of the atomic force microscope is based on recording
the force interaction between the surface of the scanned sample, sharp tip that is
located at the end of an elastic cantilever [13] (see figures 3.42-1 and 3.42-2). The
force, acting on the probe due to its interaction with the surface, bends the cantilever.
By monitoring the up/down adjustments the system need to make in order to keep
constant force, the topography map of the sample surface is achieved.
Figure 3.42-1: The principle of operation of the AFM [16]

20. 20
Figure 3.42-2: Scheme of relevant spatial distances in AFM. The instantaneous tip-surface
separation d, the instantaneous position of the tip z and the average tip-surface separation
zc are plotted. The position of the tip, z, is defined with respect to the rest position of the tip,
positive values and negative values above and below the rest position of the cantilever-tip
system, respectively [13]
The forces, acting between the probe and the pattern, are Van der Waals. These
forces first of all are the forces of gravity, and only with a further approach they become
repulsive forces [13].
3.43 Operating Modes of the Atomic Force Microscope.
The AFM has three modes of operation that depend on the nature of the force
between the cantilever and the surface (see figure 3.43-1):
*Contact mode
*Non-contact mode
*Semi contact mode
Figure 3.43-1: Interatomic force vs. distance curve [17]

21. 21
Contact mode operation of the AFM
In this mode the tip is in contact with the surface of the scanned sample. This
means that the tip-sample separation is in the range of the repulsive force (see figure
3.43-1) [13]. Usually scanning is realized in a constant force mode when the feedback
system supports the value of the cantilever’s curve constant. When we explore the
samples with amplitude of height in some angstroms it is possible to scan the sample at
the constant average distance between the probe and sample’s surface. In this case, the
cantilever moves at some average altitude over the pattern. The curve of console ΔZ is
proportional to the force acting on the probe from the surface [13],[14]. In this mode,
other measurements such as spreading resistance and local friction can be performed
along with the topography.
Advantages of the method:
*Resistance to the noise is more high in comparison with other methods;
*It is the only mode that allows to achieve atomic resolution;
*Provides the best quality of scanning surfaces with abrupt relief differences.
Disadvantages of the method:
*The presence of artifacts connected with the presence of the lateral forces
acting on the probe from the surface;
*When scanning in the open air, capillary forces can influence the probe.
*Not adequate for studying objects with low mechanical stiffness (organic
materials, biological objects) [13],[17].
Non-contact mode of AFM operating.
When AFM operates in non-contact mode, a piezoelectric vibrator excites the
probe at a certain frequency (usually, its resonance frequency). The force, which exerts
from the surface, leads to displacement of the amplitude-frequency and phase-
frequency characteristics of the probe. A feedback system generally provides a
constant state of oscillation amplitude of the probe. Monitoring the change of the
piezoelectric crystal in order to keep constant amplitude gives information about the

22. 22
surface morphology it is also possible to establish a feedback by maintaining the
frequency of the oscillations constant.
Advantages of the method:
*There is no influence of probe on the investigated surface;
Disadvantages of the method:
*Sensitive to external noises;
*Low lateral resolution;
*Low scanning rate;
*Better work in a vacuum, where there is no adsorbed layer of water on the
surface;
*Particles that attached to the cantilever can change the properties of the
cantilever.[13], [17].
Semi-contact mode
When AFM operates in semi-contact mode the cantilever is excited. In the low
half-cycle of the oscillations the cantilever touches the surface of the sample. This
method is intermediate between contact and non-contact modes.
Advantages of the method:
*The most universal method of AFM, which allows to obtain resolution of 1-5;
*Lateral forces acting on the probe from the surface are eliminated, it simplifies
the interpretation of received images;
Disadvantages of the method:
*Maximum scanning rate is lower than in the contact mode [13],[17].
3.44 Image processing.
In the above description, only Van der Waals forces were referred. In fact, other
forces such as elastic forces and forces of adhesion might be involved and affect the
measurements. For example, the "sticking" of the cantilever to the surface leads to

23. 23
hysteresis which can significantly complicate the process of image acquisition and
interpretation of results. In addition, magnetic and electrostatic forces may also act
between the tip and the surface. Using certain techniques and special probes we can
monitor their distribution over the surface [13].
When AFM microscope scans, there are some drawbacks. For example, for
objects with lateral size of few nanometers the image quality will be determined by
the radius of curvature of the tip. This could lead to artifacts on the received image, as
illustrated in figure 3.44-1:
Figure 3.44-1: AFM scanning artifact arising from a tip with a high radius of curvature with
respect to the feature which is to be visualized [13]
Another disadvantage of the AFM is the thermal drift of the needle. The
temperature linear expansion coefficient of most materials is about 6-10. When the
size of the manipulator is a few centimeters, a temperature change of 0,01 ° leads to a
movement of the needle of 1 Å because of the thermal drift. Therefore, because of the
low scanning velocity of AFM the images can be distorted. Non-linearity, hysteresis
and creep of the piezoceramic scanner are other disadvantages of AFM. They can also
cause a serious distortions of the AFM images. In addition, some part of distortions
occurs because of the mutual parasitic relations existing between X, Y, Z-manipulators
of the scanner. To correct distortions in real time, a modern AFM uses scanners,
equipped by closed-loop servo system consisting of linear measuring devices [13].
Schematic representation of the components of an AFM shows in figure 3.44-2:

24. 24
Figure 3.44-2: Schematic representation of the components of an atomic force microscope
(AFM) [15]
3.5 Kelvin Probe microscopy Method.
3.51 General information
Kelvin probe microscopy is used to measure the potential difference between
the probe and the surface of the sample. It is measured in semi-contact mode. It is
usually based on the two-pass technique. In the first pass the topography is acquired
using standard semi-contact mode (mechanical excitation of the cantilever). In the
second pass this topography is retraced at a set lift height from the sample surface to
detect the electric surface potential Ф(x) [26]. During this second pass, the cantilever
oscillations are excited not mechanically but electrically by applying AC bias voltage to
the probe.
3.52 Principle of operation.
The total tip voltage contains a static and dynamic components:
(3.52-1) [18].
The capacitive force Fcap between the tip and the surface is:
(3.52-2) ( ) ( ) [18]
where C(z) is the probe-sample capacitance. The force operating at the first
harmonic leads to the corresponding oscillations of the cantilever.

25. 25
(3.52-3) [18].
By changing and monitoring its value when become zero namely
when = , the value of is determined. Therefore, mapping of these values of
as a function of the tip position reflects the distribution of the surface potential along
the sample. [18], [30].
3.6 Carbon Black Polymer composite
3.61 General information about the composite and the percolation mechanism in it.
Carbone Black Polymer (CBP) composite represents carbon particles that are
implanted in polyethylene matrix. Inside the matrix, each particle is enveloped by a
thin layer of polymer. At sufficient carbon black (CB) concentration (percolation
threshold) an electrically conductive network that extends across the sample is
formed. According to theoretical prediction, near the percolation threshold (above the
threshold), the mass of a percolation cluster will behave in the following way:
(3.61-1) [1].
Where is the concentration of the CB particles in the composite, is the
concentration at the percolation threshold, and is the fractal dimension of the
percolation cluster mass. In three dimensions is predicted to be 2.51 [1],[11].
3.62 Breakdown phenomena of electrical current in CBP.
CBP composite has a very useful feature to protect electrical circuits from
overload. Specifically, at a certain current value the electrical conductivity of the CBP
composite decreases. This property is used in electronics, and on this basis current and
voltage limiters are created to prevent the device from combustion and electrical
breakdowns. More information about these properties can be found in ref. [9]. The
dependence of the resistance on the temperature in this system is shown in Figure
3.62-1. When the temperature increases from 25°C to 120°C the resistance increases
by factor 10. Then, at a temperature range of 13°C )between 120°C-133°C(, the
resistance increases rapidly by 5 orders of magnitude. From 133°C to 180°C the
resistance increases in a more moderate way [9].

26. 26
Figure 3.62-1: Temperature dependence of the resistance [9]
This behavior is well correlated with the thermal expansion of the polymer, as
can be seen in figure 3.62-2. Between 25°C to 120°C, the thickness of the sample
increases more fluent than linearly. Then, in a temperature range of 13°C (120°C -
133°C), the thickness is rapidly increasing. This temperature range corresponds to the
melting point of the polymer. From 133°C to 180°C the thickness of the polymer
increases linearly.
Figure 3.62-2: The dependence of the thickness of the sample on the temperature [9].
Figure 3.62-3 shows the dependence of the resistance on the polymer thickness.
The resistance increases more than exponentially with the thickness over all the
temperature range [9]. This dependence is relatively smooth and monotonic, without
abrupt changes in the resistance. This suggests that the conductivity mechanism is
similar both below and above fusion point of the polymer. This experimental data also
shows that a simple quantum tunneling model cannot fully explain the behavior of

27. 27
resistance and temperature. In case of quantum tunneling, if all gaps between the CB
particles increase uniformly with the thickness, then the resistance must also increase
more or less linearly (in semi log presentation) with the thickness. However, according
to the experimental data, it does not occur. It can be assumed that the behavior of the
resistance dependence on the temperature (figure 3.62-3) is provoked by the
reduction of the CB concentration. That is, while the polymer is extended the CB
concentration is falling below the percolation threshold. However, the result show that
the behavior of the resistance dependence on the temperature cannot be explained
only by the trivial laws of percolation [10].
Figure 3.62-3: The dependence of the resistance on the thickness of the sample[9]
3.63 Thermal fuse.
Measurements of electrical current as a function of the voltage show that
increasing the voltage above a certain value, causes an abrupt reduction of the current
in the circuit (Figure 3.63-2). This property is widely used in modern technologies. For
example, limit switches for electric windows in cars and fuses for the circuits in
microelectronics.
The switching effect is a consequence of a local Joule heating between particles
while the current flows in the system (figure 3.63-1) [10]. As a consequence of the
difference between the thermal coefficients expansion of PB particles and the
polymer, the distance between CB particles increases. This causes a sharp decrease of
the tunneling current in the circuit [11].

28. 28
Figure3.63-1: The Joule heat generated by the current causes the polymer to locally expand,
thus increasing the tunneling distance between the particles. This leads to a sharp current
decrease through junctions that are located at critical positions along the network.
Figure 3.63-2 shows the results of I(V) measurement that was performed on CBP
composite with CB concentration slightly above the percolation threshold at room
temperature.
The figure shows a characteristic which was measured by slow increase of
the voltage V to ensure that the system obtains a steady state. Below , the
curve is linear. For < < , the characteristic bends over, the tangential
conductivity decreases, signaling a modification of the connectivity of the beads.
Maximum current , is achieved at voltage , at which the tangential conductivity
vanishes. With further voltage increasing, the current decreases [10].
Figure 3.63-2: Current voltage characteristic of the sample of CBP [10]

29. 29
3.7 Charging effects in Si/SiO2 system.
3.71 Nanocrystalline silicon.
The Si/Si02 system consists of Si nanocrystals that are embedded in Si/Si02
matrix [25].
In our samples, that were made using co sputtering process, the nanocrystals
concentration and size decrease with the distance from the Si target. The size of the
crystallites varies between 3nm (in the Si-poor side) and 10nm (in the Si-reach
side). These sizes are close to the exciton Bohr radius of Si and therefore effect of
quantum confinement appears in the smaller crystallites.
3.72 Charge storage and cluster statistics in ensembles of Si quantum dots.
Quantum dot is a nanocrystallite of a conductor or semi-conductor, which is
limited in all three spatial dimensions so that the electron wave function is confined to
its volume. Consequently, such materials have electronic properties intermediate
between those of bulk semiconductors and those of discrete molecules [20], [26]. An
electron inside such quantum dot can be described as an electron in a three-
dimensional potential well, with fixed energy levels with a characteristic distance of
between them (were h – is Plank constant, d - a characteristic size of the dot and
m - the effective mass of the electron in the dot). The exact expression for the energy
levels depends on the specific characteristics of the dot [21].
The charging energy of a quantum dot is given by:
(3.72-1) [22],
where is the dielectric constant of the surrounding insulator and is the dot radius.
Accordingly, the charging energy increases as the radius of the nanocrystal
decreases. Theoretically, the capacity of a spherical body is expressed as: . It
is necessary to consider not only the capacitance of the quantum dot but also the
capacity of the electrodes, connected in series with the dot and involved in the

30. 30
charging. In this case one speaks about the common effective capacity ( ) which can
be expressed as: . Equation 3.72-1 can now be written as:
(3.72-2) [22].
A more detailed calculation of an spherical isolated quantum dot in close
proximity to the conductive substrate gives:
(3.72-3) * + [22],
Where is the distance between the center of the dot and the substrate.
Knowing the input data (value of , and ) and using two previous expressions, we
can calculate the energy required to charge the quantum dot.
4. Objectives.
The objectives of the project:
1.) Studying the geometrical properties of the percolation lattice.
2.) Understand the thermal switching effect in CBP composites.
3.) Investigate the charging effect in Si/SiO2 system.
5. Experimental Procedures.
5.1 General.
5.11 Introduction.
Our research can be divided into two main sections: the study of the percolation
effect and the “thermal switch” mechanism of CPB composites and the investigation of
charging effect in Si/SiO2.
5.12 Evaluation of the mass fractal dimension of the percolation network.
1. Scanning the current maps along with the topography in order to identity the
particles that are connected to the percolation cluster.

31. 31
2. Writing a MATLAB routine that calculate the Box-counting dimension.
3. Evaluating the mass fractal dimension.
5.13 Electro-thermal switching effect.
1. Measuring the current maps as a function of the applied bias.
2. Monitoring the number of the conducting areas as a function of the applied
bias.
3. Analyzing the results.
5.14 Charging effect in Si/SiO2.
1. Measure the charging in this system using Kelvin probe spectroscopy as a
function of the Si concentration.
2. Write the appropriate MATLAB routine to analyze the results.
3. Data analysis.
5.2 Equipment and Analytical Methods
5.21 Atomic Force Microscope.
All measurements were performed at room temperature by using a scanning
atomic force microscope (NT-MDT). This microscope allows us to study both
conductive and non-conductive surfaces at a very high resolution of a few nanometers.
Its properties are very suitable to obtain topographic and material sensitive images
that allow us to determine the location of the CB particles, and to map the conductive
areas on the surface of the sample.
To study the surface topography, contact mode was chosen. To study directly the
CB particles that are connected to the percolation cluster we measured the current
distribution using contact mode with a closed electrical circuit. Throughout the
research we used an electrically conductive cantilever.

32. 32
Figure 5.21-1 shows the AFM setup. The sample is attached to the scan head. A
unit that contain the tip, laser and photodiode is placed on top of the sample. The
whole setup is covered by a Faraday cage and hanged by elastic springs to minimize
electrostatic and mechanical noises. In addition a CCD camera was used in order to
adjust the position of the tip over the sample.
Figure 5.21-1: The AFM setup.
5.22 CBP composites.
The measurements were carried out on sample of carbon-black-polymer
composite, with CB concentration close to . A set of current measurements with low
bias voltage where performed in order to evaluate the fractal dimension of
the mass of the percolation network.
In order to study the electro-thermal switching effect, a set of current
measurements at different voltage levels (0.5V, 1V, 2V, 3V, 4V, 5V, 6V and 7V) was
The digital camera
The lamp
The springs
Faraday cage

33. 33
measured. From the results we identify the dependence of the current on the applied
voltage.
5.23 Charging effect in Si/SiO2 system.
The charging effect was measured using Kelvin probe spectroscopy. In every new
location, we measured the topography of the area before the Kelvin probe and then
we measured 9 spectra on area of 1 to average the results. We performed these
measurements starting from the low Si concentration (and small crystallites size) side
to the higher Si concentration (and larger crystallites) side with steps of 0.8 . The
electrostatic interaction energy between the tip and the sample with a charge trapped
in a Si/SiO2 in between is given by:
5.23-1 ,
where is the tip-substrate capacitance, is the tip-nanocrystal capacitance,
is the nanocrystal-substrate capacitance, and total capacitance seen
by the tip. The electrostatic force can be obtained from Eq. 5.23-1 by differentiation
with respect to the tip-sample separation z:
5.23-2 ( ) .
Using Eq. 5.23-2, and the AC modulated bias, different frequency components
of the electrostatic force can be evaluated. When the electrostatic excitation is
measured as a function of the bias, the presence of charge between the tip and the
sample results in a shift of the curve along the bias coordinate. Using algebraic
manipulation, Eq. 5.23-2 can be used to calculate the hysteresis between the force-
voltage curves:
5.23-3 [ ( )⁄ ] .
The voltage is the charging voltage of the trap, = q/ , divided by a
dimensionless factor that quantifies the shielding effect of the tip. Using Eq. 5.23-3, by
measuring the hysteresis at specific location, information about the local density of
states and charging-discharging dynamics can be obtained.

34. 34
5.3 Data processing.
5.31 MATLAB (software).
Most of the data processing was carried out by using MATLAB. In order to
evaluate the fractal dimension of the percolation network we wrote an algorithm that
uses the box counting method. Box counting is a sampling or data gathering process
uses to find several types of . The basic procedure is to systematically lay a series of
grids of decreasing caliber (the boxes) over an image and record data (the counting)
for each successive caliber. Using the box counting method, the fractal dimension is
given by the slope of the log(N) vs. log(r) plot where N is the number of boxes that
cover the pattern, and r is the magnification, or the inverse of the box size. The same
equation is used to define the fractal dimension, D.
5.31-1
The algorithm can be found in paragraph 11.1.
In order to analyze the results of the Kelvin probe spectroscopy measurements,
another MATLAB algorithm was used with an eye to plot graph of the charging voltage
of the “trap” between shifted minimums and the conductivity as a function of the
vol. fraction along the sample, in the SiO2.
6. Results
6.1 CBP composite
6.11 Fractal dimension of the percolation cluster.
Figure 6.11-1 shows topography (a) and friction (b) AFM maps that were
measured on CBP sample that contains of CB particles. We can identify the CB
particles by correlating the topographic image with the material-sensitive friction
image.

35. 35
(a) (b)
Figure 6.11-1: Topography (a) and friction (b) images that were measured simultaneously.
Some of the CB particles are marked by circles.
Graph 6.11-1: The fractal dimension of the percolation infinite cluster in two-dimensional
slice. (For better visibility derivatives offset one above the other)
In order to measure the fractal dimension of the percolation cluster we
measured (in addition to topography and friction) the current images for several low
voltages . In graph number 6.11-1 we show the result of the box counting
dimension algorithm an different images that were measured at different voltages
(below the thermal switching).

36. 36
6.12 Microscopic study of current-voltage dependence in CPB composite.
We studied the dependence of the measured current on the applied bias
voltage. In these measurement, current was detected only when the conductive tip
touched a CB particle that is connected to the percolation cluster.
Figure 6.12-1 shows the change in the number of conducting regions on the CBP
surface depending on the applied voltage between the cantilever and the sample. The
white dots on the pictures represent these areas. All the maps shown were measured
at the same area. Initially, when we increase the voltage, more areas become
conductive as electrons get enough energy to overcome the tunnel burier between
more distant particles. Thus, with increasing the voltage the conductive network
includes new conductive chains. However, for high enough voltage the increase of the
voltage lead to a decrease of the conducting areas.
Figure 6.12-1: AFM maps that show the dependence of the conductivity of the simple as a
function of the applied voltage. First, the quantity of conducting regions increases with the
voltage. For bias voltage higher than 5V, the number of conducting regions drops sharply.

37. 37
Graph number 6.12-1 shows the change of the number of conductive areas, as a
function the applied voltage. As shown in the figure, the number of the conducting
areas initially increase with the applied bias as expected. As the bias voltage increases,
it helps to overcome more potential barriers in the conducting network and therefore
more conducting routs are revel. However, for V>5V there is a sharp reduction in the
number of the conductive routs due to the electro-thermal switching effect. In this
effect, the current in the system causes a local Joule heating between CB particles (see
paragraph 3.63). As a consequence of the difference between the thermal coefficients
expansion of PB particles and the polymer, the distance between CB particles
increases. This causes a sharp decrease of the tunneling current in the circuit and the
number of the conducting areas declires.
Graph 6.12-1: The change of the conductive areas, depending on the applied voltage
6.2 Charging effect in Si/SiO2 composite.
6.21 Kelvin probe spectroscopy of the Si/SiO2 samples.
Graph 6.21-1 shows an example of the magnitude of the tip oscilations vs. DC BV
(Bias Voltage) during Kelvin Probe measurement. The offser between the two curves
is a result of the charging of the charging of the sample.
80
100
120
140
160
180
200
220
240
260
280
300
0 1 2 3 4 5 6 7 8
Numberofpoints
Voltage (v)

38. 38
Graph 6.21-1: An example of the magnitude of the tip oscillations as a function of the DC Bias
Voltage applied to the AFM tip
The value of depends on the amount of stored charge. Therefore, it will
depends on the electrical conductivity of the system as well as on the time between
measurements. Graph 6.21-2 shows the dependence of on the scan time.
Graph 6.21-2: The charging voltage as a function of the acquisition time
0.0 0.5 1.0 1.5 2.0 2.5 3.0
0
100
200
300
400
500
600
700
800
V(mV)
Time (s)

39. 39
Graph 6.21-3 show the charging voltage (measured with acquisition time of
0.2sec) and the conductivity as a function of the Si concentration.
Graph 6.21-3: The charging voltage and the conductivity as a function of the Si vol.
fraction
7. Discussion.
7.1 CBP composites.
From the results present in graph 6.11-1: we can see that the fractal dimension
of the 2D cut of the infinite percolation cluster varies between 1.44 to 1.56 with an
average value of 1.5. Using the relation [23], [24], we evaluated the fractal
dimension of the 3D percolation cluster to be 2.5 . This value is very close to the
theoretical predicted value, 2.51 [23].
Graph 6.12-1 presents the number of conducting areas as a function of the
applied voltage. Initially, the number of the conducting regions increases with the
applied voltage, as expected, and the magnitude of the current is also increasing.
0.6 0.7 0.8 0.9
0
100
200
300
400
500
600
700
800
Si vol. fraction
V(mV)
1
10
100
1000
Conductivity(cm)
-1

40. 40
However, in the range between 5 to 7 volts, there is a reduction in both the number of
these areas and the magnitude of some of the detected currents.
This effect occurs because of the local Joule heating as a result of electric
current. Due of the difference in the coefficient of thermal expansion of the polymer
and the particles, the distance between adjacent particles increases and therefore the
tunneling probability between them decreases. As a result - the number of conducting
regions is significantly reduced.
7.2 Charging effect in Si/SiO2 composites
Our results regarding the effect of charging of Si nanocrystals embedded in SiO2
matrix presented in graphs 6.21-2 and 6.21-3. These results can be explained by means
of tunneling and percolation. In order to charge an isolated nanocrystal, we need to
apply large enough bias voltage to overcome the tunnel barrier. When the voltage is
reduced, the nanocrystalline will stay charged until the carries tunnel out. Therefore,
the amount of charging of the particle will depend on time as well as on the tunneling
probability of the curriers from the nanocrystal. The latter is closely related to the
particles concentration.
In graph 6.21-2 we see that the value of is increasing while the scan time is
decreasing. It can be explained by the following – for long measurement, there is
enough time for the sample to discharge between successive points. Therefore for
shorter scan time the charging effect is stronger.
The depends of on the crystallites concentration is shown in graph 6.21-3.
The decrease of (and therefore of the charge Q) with increasing the concentration
is due to the fact that the tunneling probability from a charged nanocrystal increases
as the distance between crystallites decreases. Hence, the charged carriers become
more and more delocalized as the concentration increases. This is evident also from
the comparison between the charging and the conductivity as a function of the Si
concentration. As can be seen, the charging is dominant close to , where the
conductivity is small. As the conductivity increases, it enables an easy charg-transfer
and therefore practically eliminate the effect of charging.

41. 41
8. Conclusion.
8.1 CBP composite.
As we noted above, the mass of the percolation cluster is predicted to increase
with the concentration according to Eq. 3.61-1:
.
Where is the concentration of conductive particles, is the threshold
concentration and is the fractal dimension of the mass of percolation cluster. The
theoretical prediction of is 2.51 in three-dimensions [24].
Our results, that were measured with different bias voltage up to 5v showed
indeed a fractal dimension of 2.5 0.055.
For higher voltage )V>5v(, where the thermal switching effect “shut-down” some
of the conducting paths, it is not possible to do this analysis.
Thermal switch effect.
We carried out a microscopic studies of the dependence of the conductivity of
the CBP composite (near ) on the applied voltage. Graph 6.12-1 clearly shows the
thermal switch effect, which appears at a voltage above 5 volts. Our conclusion is that
effect is purely local and not a global one. This is demonstrated in figure 6.12-1, where
the white spots on the images indicate the areas conducting areas. As we described
above, because of the difference between the coefficients of thermal expansion of the
polymer and the CB particles, the Joule heating generated due to the current causes
mostly the expansion of the polymer. Therefore, the distance between adjacent,
current carrying particles becomes larger. This leads to an increasing of the tunnel
barrier between the particles and to the sharp increasing of electrical resistance. On
the other hand, with increasing of the voltage the electrons receive more energy to
overcome the tunneling barrier. However, above 5 volt the effect of the expansion of
the polymer dominates. As a result of the competition between these two mechanisms
we received non-monotonic graph (6.12-1). In addition to this, we can add that the
reduction of the current is not associated with an effective decrease of the
concentration of CB particles as the polymer expands.

42. 42
8.2 Charging effect in the Si/SiO2 composite
Our results, presented in graphs 6.21-2 and 6.21-3, show a decrease of that
is related to the stored charge with increasing the nc-Si concentration (and increasing
the particles size) and scan time. Graph 6.21-2 shows that the amount of stored
charge, depends on the scan duration. The longer the scan, the higher the probability
that the charge will “leak” from the charged particles, and this leads to a decrease of
. In graph 6.21-3 we can see that the sharp increase of the conductivity ( )
corresponds to the sharp decrease of the value of .
In our samples, the particles size also increases with increasing the
concentration of the particles. This also has a role in reducing of the value of . The
capacitance of the particles increases with the particles size and therefore reduces the
value of . However, the main effect in our case is due to the increase in the
conductivity, since in the range of the decrease of , the size of the particles dues
not change much.
9. Acknowledgments
We express our gratitude to Doctor Doron Azulay that guides us through this
project. The performance of this work would not be possible without his scientific
advice and clear and understandable explanations of the quantum physics related
subjects.
We also express our gratitude to Mr. Shoval Silbert for his guidance and
providing us with the MATLAB code.
We express special gratitude to Mrs. Julia Sorokopudova for her participation in
the translation of this work into English.

43. 43
10. References
[1] Armin Bunde and Jan W. Kanterhaldt, Introdaction to Percolation Theory (Part
A), pp. 1-10.
[2] Michael Aizenman and David J Barsky, Sharpness of the Phase Transition in
Percolation Models, Communications in Mathematical Physics, 1987, V 108, pp. 1-3.
[3] S. R. Gallyamov, S. A. Mel’chukov, Percolation Model of Conductivity of Two-
Phase Lattice: Theory and Computer Experiment, Journal Computer Science, 2010, No.
4, pp. 1-10.
[4] Elements of Great Science, [internet]; available from:
http://elementy.ru/posters/fractals/dimensions.
[5] Fractal Analysis, Sect. 9, pp. 1-5. [internet]; available from:
http://www.polybook.ru/comma/1.9.pdf
[6] F.A. Tsicin, Fractal Universe, Journal “Delfis”, 1997, No. 11, pp.
[7] P.V. Moskalev, Analysis of the Structure of the Percolation Cluster, Journal of
Applied Physics, 2009, V. 79, No. 6, pp. 1-7.
[8] Field of Application of the Percolation Theory, Bibliofond, [internet]; available
from: http://bibliofond.ru/view.aspx?id=523870#1
[9] Michael B. Heaney, Electrical Transport Measurements of a Carbon-Black-
Polymer Composite, Physica A, 1997, No. 247, pp. 296-300.
[10] Laurent Lamaignère, Fran¸cois Carmona, and Didier Sornette, Experimental
Realization of Critical Thermal Fuse Rupture, Physical Review Letters, 1996, Vol. 77, No.
13, pp. 2738-2741.
[11] D. Azulay, M. Eylon, O. Eshkenazi, D. Toker, M. Balberg, N. Shimoni, O. Millo,
and I. Balberg, Electrical-Thermal Switching in Carbon-Black–Polymer Composites as a
Local Effect, Physical Review Letters, 2003, Vol. 90, No. 23, pp. 1-4.
[12] Atomic-force microscopy (AFM), Wikipedia, [internet]; available from:
http://en.wikipedia.org/wiki/Atomic_force_microscopy

44. 44
[13] Ricardo Garcia, Ruben Perez, Dynamic Atomic Force Microscopy Methods ,
Surface Science Reports, 2002, Vol. 47, pp. 197-301.
[14] NT-MDT Company, Guide For Use of the Atomic Force Microscope, Part 3,
pp. 1-129.
[15] Microbial Nanowires, Geobacter project, [internet]; available from:
http://www.geobacter.org/Nanowires
[16] Atomic Force Microscopy, Roilniland’s Blog, [internet]; available from:
http://roilbilad.wordpress.com/2010/11/09/atomic-force-microscopy-afm/
[17] Theory and Simulation of SPM, In-Vsee, [internet]; available from:
http://invsee.asu.edu/srinivas/spmmod/afm.html
[18] Kelvin Probe Force Microscopy, NT-MDT, [internet]; available from:
http://www.ntmdt.com/spm-principles/view/kelvin-probe-microscopy
[19] Wilhelm Melitz, Jian Shena, Andrew C. Kummela, Sangyeob Lee, Kelvin probe
force microscopy and its application, Surface Science Reports, 2011, Vol. 66, pp. 1-27.
[20] I. V. Antonova and M. Gulyaev, Charge storage, photoluminescence, and
cluster statistics in ensembles of Si quantum dots, Physical Review B. 77, (2008), pp. 1-
5.
[21] Reed MA, Randall JN, Aggarwal RJ, Matyi RJ, Observation of discrete
electronic states in a zero-dimensional semiconductor nanostructure. Phys Rev Lett,
Vol. 60 (6), (1988) pp. 535–537.
[22] H. Levi Aroni, Ms.C. Thesis, Hebrew university (2007).
[23] D. Toker, D. Azulay, N. Shimoni, I. Balberg, and O. Millo, Tunneling and
Percolation in Metal-Insulator Composite Materials, Physical Review B 68, 041403 (R)
(2003), pp. 1-4.
[24] Nira Shimoni, Doron Azulai, Isaac Balberg, and Oded Millo, Tomographic-like
reconstruction of the percolation cluster as a phase transition, Physical Review B 66,
020102 (R) (2002), pp. 1-4.

45. 45
[25] H. Levi Aharoni, D. Azulay, O. Millo, and I. Balberg, Anomalous photovoltaic
effect in nanocrystalline Si/SiO2 composites, Applied Physics Letters, Vol. 92, 112109,
(2008), pp. 1-3.
[26] Umut Bostanci, M. Kurtuluş Abak, O. Aktaş and A. Dâna, Nanoscale charging
hysteresis measurement by multifrequency electrostatic force spectroscopy, Applied
Physics Letters, Vol. 92, 093108, (2008), pp. 1-3.

46. 46
11. Appendixes
11.1 box count dimension
Matlab code 1
function [ G, BOXES, SizeBox ] = boxcount( cbinary )
vectorX = [ 2 3 4 5 6 7 8 10 11 13 15 16 17 22 23 30 32 34 39 46 51 64];
sizemax = 512;
for i=1:1:22
sizebox = fix(sizemax/vectorX(i));
a = 1;
p = sizebox;
f = 0;
for first = sizebox:sizebox:sizemax;
b = 1;
q = sizebox;
for second = sizebox:sizebox:sizemax;
D = cbinary( a:p , b:q );
h = (sum(sum(D))>0);
f = f + h;
b = b+sizebox;
q = q+sizebox;
end
a = a+sizebox;
p = p+sizebox;
end
BOXES(i) = f;
if i~=1;
G(i) =(log(BOXES(i)/BOXES(i-1)))/(log(vectorX(i)/vectorX(i-1)));

47. 47
else
G(i) =(log(BOXES(i)))/(log(vectorX(i)));
end
end
BOXES = BOXES(end:-1:1);
SizeBox=fix(512./vectorX(end:-1:1));
11.2 Charging effect of Si/SiO2
Matlab code 2
function [Initial_params,fitted_prms_vec, phi0_avg, phi0_std, RC0_avg, RC0_std,
phi_avg, phi_std, RC_avg, RC_std ]=fit_one_file2Model_half (file_name1, file_name2,
winSz, V_range, Time)
fitted_prms_vec=[Time,N,V0,extrap_phi_2,init_ADC,init_RC,t0,Zamp_offset] ;
%% set initial params
% S Silbert 17/12/09
if nargin<6
% first find minimum and calc V(t)
%------------------------------------
% find min for -3 to +3 Volt
[min_pnt,avg_min1,err_min1,p1,S1,h1,short_fname1]=find_minimum_of_spectroscop
y(file_name1,winSz);
% find min for +2 to -2 Volt
%calculate work function (method 1)
%------------------------------------
max_v=V_range; % default is 2.0 volt at end of range
h11=squeeze(h1);
h22=squeeze(h1);
ratio_vec=h11./h22; %Ratio of the two ends at t=0.

48. 48
phi_val=(ratio_vec-1)*max_v ./( ratio_vec+1); % work function value for all
100 vecs
phi_avg=mean(phi_val); % Work function avg
phi_err=std(phi_val); % work function error
end
% fit nonlinear function for 1st(-2 V to 2 V) and 2nd (2V to -2V) vectors, for all
vectors in file
%-------------------------------------------------------------------------------------------------
[Initial_params,fitted_prms_vec]=Extract_params_for_single_file(short_fname1,
min_pnt,phi_val,Time);
phi=fitted_prms_vec(:,4);
RC=fitted_prms_vec(:,6);
phi_std=std(phi);
phi_avg=mean(phi);
RC_avg=mean(RC);
RC_std=std(RC);
phi0=Initial_params(:,1);
RC0=Initial_params(:,2);
phi0_std=std(phi0);
phi0_avg=mean(phi0);
RC0_avg=mean(RC0);
RC0_std=std(RC0);
%[Initial_params2,fitted_prms_vec2]=Extract_params_for_single_file(short_fna
me2,min_pnt2,phi_val,Time);
end
function [Initial_params_out, fitted_prms_vec_out ] =
Extract_params_for_single_file (short_fname , min_pnt, phi_val, Time)
run(short_fname);
[L,N]=size(Curves);
for ndx=1:L
% set vectors

49. 49
v_vec1=(XScale*[1:N/2]+XShift);
v_vec=[v_vec1 v_vec1]; % half a vec
z_vec=ZScale*[Curves(ndx,1:N)]+ZShift;
% fit linear section
min_pnt(ndx);
phi_init=phi_val(ndx);
Z_min=[ min( z_vec(1:N/2) ), min( z_vec(N/2+1:N) ) ]
Zamp_offset=mean(( Z_min(1)+Z_min(2) )); % find offset of Tip amplitude -
average the two minimas
find(z_vec==Z_min(1))
find(z_vec==Z_min(2))
ndx_lin(1)=min(find(z_vec(1:N/2)==Z_min(1))) % find beginning (minimal
point in 1st curve) of linear section, i.e., get index of min location
ndx_lin(2)=N/2+min(find(z_vec(N/2+1:N)==Z_min(2))) % find beginning
(minimal point in 1st curve) of linear section, i.e., get index of min location
[a,b]=polyfit(v_vec(1:ndx_lin(1)),z_vec(1:ndx_lin(1)),1); % fit linear section
[a,b]=polyfit(v_vec(ndx_lin(1):N/2),z_vec(ndx_lin(1):N/2),1) % fit linear
section
fitted_lin=a(1)*v_vec(ndx_lin(1):N/2)+a(2); % fitted line
[a,b]=polyfit(v_vec(1:ndx_lin),z_vec(1:ndx_lin),1); % fit linear section
fitted_lin=a(1)*v_vec(1:ndx_lin)+a(2); % fitted line
% hold on; plot(v_vec(1:ndx_lin),fitted_lin,'-g');
%debug
extrap_RC_phi=-2*(a(2)/a(1)); % divide the offset by the inclination
(=2*A*DC/DZ*dV/dt)
V0=v_vec(1); % ( the scale is reversed so V(1)<-->V(1000) )
extrap_phi_1=V0-2*(z_vec(N/2)-Zamp_offset)/abs(a(1));
extrap_phi_2=-phi_init
Delta_t=Time/N;
Delta_V=-(4*V0)/N;

50. 50
M=Delta_V/Delta_t;
extrap_RC_1=(extrap_RC_phi+extrap_phi_1)/M % recieve RC
extrap_RC_2=(extrap_RC_phi+extrap_phi_2)/M % recieve RC
% extrapolated from minpoints
phi1=v_vec(ndx_lin(1))+v_vec(ndx_lin(2))
RC1=( v_vec(ndx_lin(1))-v_vec(ndx_lin(2)) )/M
% init params
init_ADC=-a(1)/2
init_RC=RC1
Zamp_offset=min(z_vec(1:N/2))
t0=0;
prm_vec=[Time,N,V0,phi1,init_ADC,init_RC,t0,Zamp_offset] % set initial
params
Initial_params_out(ndx,:)=[phi1,RC1,init_ADC];
prm_vec_const=[true,true,true,false,false,false,true,true];
% define options for fitting and run nl fit
opt=statset;
opt.MaxIter=100;
%v_vec=flipdim(v_vec,2);
% correct vector direction (instead of reversing x vector, reverse z vector
direction)
%z_new(1:N/2)=z_vec( fitted_prms_vec_out(ndx,:) =
nlinfitsome(prm_vec_const,v_vec(1:N/2), z_vec(1:N/2),@amp1stpassfun,prm_vec); %
nonlinearfit
prm_vec
fitted_prms_vec_out
% plot outcome
init_z_vec=amp1stpassfun(prm_vec,v_vec(1:N/2));

51. 51
fitted_z_vec=amp1stpassfun(fitted_prms_vec_out,v_vec(1:N/2));
%%%hold on; plot(v_vec(1:N/2) ,init_z_vec(1:N/2) ,'-.m' ,v_vec(1:N/2)
,fitted_z_vec(1:N/2),'-.k'); title(['ADC=' , num2str(fitted_prms_vec_out(5)), '; RC='
,num2str(fitted_prms_vec_out(6)), '; phi fitted= ', num2str(fitted_prms_vec_out(4)),'
vs phi extrapolated=',num2str(extrap_phi_2) ]);
% hold off
% %output params (debug)
% Tau_in=prm_vec(1);
% Tau_out=fitted_prms_vec_out(1);
% N_in=prm_vec(2);
% N_out=fitted_prms_vec_out(2);
% V0_in=prm_vec(3);
% V0_out=fitted_prms_vec_out(3);
% phi_in=prm_vec(4)
% phi_out=fitted_prms_vec_out(4)
% A_DzC_k_in=prm_vec(5)
% A_DzC_k_out=fitted_prms_vec_out(5)
% RC_in=prm_vec(6)
% RC_out=fitted_prms_vec_out(6)
% t0_in=prm_vec(7);
% t0_out=fitted_prms_vec_out(7);
% Zamp_offset_in=prm_vec(8);
% Zamp_offset_out=fitted_prms_vec_out(8);
end
% plot outcome
% figure; plot(v_vec(1:ndx_lin),fitted_lin,'-g'); % plot fit
% init_z_vec=amp1stpassfun(prm_vec,v_vec);
% fitted_z_vec=amp1stpassfun(fitted_prms_vec_out(ndx,:),v_vec);

52. 52
% hold on; plot(v_vec(1:N),z_vec(1:N),'-.r',v_vec(1:N),fitted_z_vec(1:N),'-
.k',v_vec(1:N),init_z_vec(1:N),'-.b');title(['ADC='
,num2str(fitted_prms_vec_out(ndx,5)),'; RC=',num2str(fitted_prms_vec_out(ndx,6)), ';
phi_new= ', num2str(fitted_prms_vec_out(ndx,4)),' vs orig_phi=',num2str(phi_init) ]);
% hold off
end
% Seconday Function - model fit function
function z_vec=amp1stpassfun(prm_vec_in,v_vec)
% Define initial Params
Tau=prm_vec_in(1);
N=prm_vec_in(2);
V0=prm_vec_in(3);
phi=prm_vec_in(4);
A_DzC_k=prm_vec_in(5);
RC=prm_vec_in(6);
t0=prm_vec_in(7);
Zamp_offset=prm_vec_in(8);
Delta_t=Tau/N;
Delta_V=-(4*V0)/N;
M=Delta_V/Delta_t;
t_vec=Delta_t*[0:N-1]+t0;
% Define function for 1st pass
QdivC1=M*t_vec(1:N/2)+(V0-M*RC)*(1-exp(-(t_vec(1:N/2)/RC)));
%%QdivC2=-M*t_vec(1:N/2)+(-V0+M*RC)*(1-exp(-
(t_vec(1:N/2)/RC)))+QdivC1(N/2);
z_vec1=abs(A_DzC_k*(v_vec(1:N/2)-phi+QdivC1))+Zamp_offset;
%%z_vec2=abs(A_DzC_k*(v_vec(N/2+1:N)-phi+QdivC2))+Zamp_offset;
z_vec=[z_vec1];
% % QdivC1=M*t_vec+(V0-M*RC)*(1-exp(-(t_vec/RC)));

53. 53
% % QdivC2 = -M*t_vec(N/2:N-1) + (-V0+M*RC) * (1-exp(-(t_vec(N/2:N-1)/RC)))
+QdivC1(N/2) ;
% % z_vec1=abs(A_DzC_k*(v_vec-phi+QdivC1))+Zamp_offset;
% % z_vec2=abs(A_DzC_k*(v_vec(N/2+1:N)-phi+QdivC))+Zamp_offset;
% % z_vec=[z_vec1 zvec2]
end
Matlab code 3
function
[min_pnt,avg_min,err_min,p,S,h1,short_mFlName]=find_minimum_of_spectroscopy(fi
le_name,winSz)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% This function finds the minumum of a spectroscopy, by fitting the data to
% y(x^2)=x^2.
% INPUT: text file name from MDT software
% OUTPUT: Minimum and error.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Shoval - 30/10/2008
% Shoval - 25/02/2009
% Corrected for MDT software bug- Shoval 3/3/2009
%%% File handling %%%
% Get current directory and file name & copy to new file name (without "_")
curr_dir=cd;
file=dir(file_name)
mFlName=file.name
target_dir=regexprep(file_name,mFlName,'');

54. 54
%% the following has been removed
NewmFlName=remove_beginning_nums(mFlName);
NewmFlName=regexprep(NewmFlName,'_','');
NewmFlName=regexprep(NewmFlName,'M','m');
[flag,err,errID]=copyfile(file_name,[target_dir,NewmFlName]);
%% Open file and remove the word 'end' from the file
cd(target_dir);
fid=fopen(NewmFlName,'r+'); %instead of %fid=fopen(NewmFlName,'r+');
% Change 'end' at eof to '%nd'
fseek(fid,-5,'eof');
fwrite(fid,'%','char');
fseek(fid,-5,'eof');
fread(fid,3,'*char');
fclose(fid);
% Run new file (NewmFlName) (load data to memory)
short_mFlName=NewmFlName(1:end-2);
[target_dir,short_mFlName];
run(short_mFlName);
cd(curr_dir);
% Analyze Curves
[L,W]=size(Curves);
for i=1:L
tmp1=find(min(Curves(i,1:W/2))==Curves(i,1:W/2)); %Find minimas
min_curve(1)=tmp1(1);
tmp2=find(min(Curves(i,W/2+1:W))==Curves(i,W/2+1:W));

55. 55
min_curve(2)=tmp2(1);
curve(1,:)=Curves(i,min_curve(1)-winSz:min_curve(1)+winSz); %Define
window around
curve(2,:)=Curves(i,min_curve(2)+W/2-winSz:min_curve(2)+W/2+winSz);
for k=1:2
poly=curve(k,:).^2 ;
[p,S]=polyfit([-winSz:winSz],poly,2);
min_pnt(i,k)= ( XScale*( -p(2)/(2*p(1))+min_curve(k)-1 )+XShift); %Scale to
real data, and corret for MDT soft. bug (mult. by -1) % Removed - MDT looks OK
end
h1(i)=ZScale*Curves(i,W/2)+ZShift;
h2(i)=ZScale*Curves(i,W)+ZShift;
end
avg_min=mean(min_pnt);
err_min=std(min_pnt);
%% figure;plot(min_pnt,'*r');
%plot sample figure
% % x=-1*(xscale*[1:w/2]+xshift);
% % figure; plot(x,curves(1,1:w/2),'-r',x,curves(1,w/2+1:end),'-b');
% % title(['amp vs bv (v) applied to afm tip']);
% % text(min(x),max(curves(1,:))+0.05*(max(curves(1,:))-
min(curves(1,:))),file_name,'fontsize',8)
% % xlabel('voltage applied to tip (v)'); ylabel('amplitude (na)');
function str_out=remove_beginning_nums(str_in)
str_out=str_in
i=1;
while ~(isempty(str2num(str_in(i))))
str_out(1)=[];

56. 56
i=i+1;
end
Matlab cod 4
function [varargout] = nlinfitsome(fixed,x,y,fun,beta0,varargin)
% "fixed" indicats which values of beta0 should not change
% Get separate arrays of coefficients to fix and to estimate
bfixed = beta0(fixed); beta0 = beta0(~fixed);
% Estimate only the non-fixed ones
[varargout{1:max(1,nargout)}] = nlinfit(x,y,@localfit,beta0,varargin{:});
% Re-create array combining fixed and estimated coefficients
b(~fixed) = varargout{1};
b(fixed) = bfixed;
varargout{1} = b;
% Nested function takes just the parameters to be estimated as inputs
% It inherits the following from the outer function:
% fixed = logical index for fixed elements
% bfixed = fixed values for these elements
% but its input is the
function y=localfit(beta,x)
b(fixed) = bfixed;
b(~fixed) = beta;
y = fun(b,x);
end
end

57. ‫המחלקה‬‫להנדסת‬‫חומרים‬‫מתקדמים‬
‫מ‬ ‫מדידות‬‫מ‬ ‫וליכות‬‫י‬‫קרוסקופיות‬
‫מוליך‬ ‫תרכובות‬ ‫של‬–‫מבודד‬
‫איליה‬ ‫צביבל‬
‫ו‬
‫דניס‬ ‫טייטלבאום‬
‫נבדק‬‫על‬‫ידי‬:‫ד‬"‫אזולאי‬ ‫ר‬‫דורון‬__________________‫תאריך‬:________
‫נבדק‬‫על‬‫ידי‬‫ראש‬‫המחלקה‬‫להנדסת‬‫חומרים‬‫מתקדמים‬,
‫פרופ‬‘‫מנור‬‫איתן‬_____________________________‫תאריך‬:_________
‫פברואר‬1023

58. ‫המחלקה‬‫להנדסת‬‫חומרים‬‫מתקדמים‬
‫מ‬ ‫מדידות‬‫מ‬ ‫וליכות‬‫י‬‫קרוסקופיות‬
‫מוליך‬ ‫תרכובות‬ ‫של‬–‫מבודד‬
‫איליה‬ ‫צביבל‬
‫ו‬
‫דניס‬ ‫טייטלבאום‬
‫פברואר‬1023