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- 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.
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- 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
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