PERCOLATION THEORY AND ITS APPLICATION IN MATERIALS SCIENCE AND MICROELECTRONICS (Part II - Experiments and numerical simulations) Andrzej DZIEDZIC Institute of Microsystem Technology, Wroclaw University of Technology Poland Keywords: percolation theory, percolation threstnold, applications, materials science, microelectronics, critical exponents, electric conductivity thermal conductivity, 1/f noise, composite ceramics, electrical thick film resistors, electrical conductive adhesives, VLSI interconnect breal Vc and phase 2 (here ideal insulator) creates random spherical precipitation in continuous phase 1 built from conductive material. The opposite case, i.e. conductive spheres embedded in continuous insulative matrix is called IRV (inverted random void) or ISC /24/. Calculation of critical indices in such models is shown for example in /24/. The distance between spheres in RV medium is equal to 5 (5 « ao). Therefore using the Weak Link Model /3/ for RV or IRV media we have the bridge consisted of resistors with resistances dependent on 5. The critical conductivity index t = tun for IRV and f = ?,„, + 0.5 for RV model (for 3D medium) has been found when /7(5) distribution is uniform if 5/ao 0 /24/. But when we have more general assumption, that /z((^)oc (J -CÜ (3) where w is less then 1, then conductivity is described by new, nonuniversal critical index t given by Eq. (4) t = + (m + čfJ - l)/(l -(ß)-lUl + (ü >1 and f -if«+(0< 1. (4) with u^d-ZIl for RV model and u = dll-\ for IRV model (d - system dimensionality) /25/. The same percolation theory permits to accept wide range values of t - not only near 2 (characteristic for 3D IRV model) or 2.5 (obligatory for 3D RV model). The concept of excluded volume, i.e. volume near the object where it is forbidden to put the centre of the other object with similar shape/26/, permits to explain very small values of percolation threshold Vc in continuum percolation model. Percolation threshold is related to total exclud- ed volume (ü. v^ = 1 - exp , according to formula (5) where Nc - critical volume density of objects in the medium, ß-volume of single object, averaged excluded volume connected with single object and its spatial direction. Value of is dependent on precipitation geometry. The analysis of excluded volume for cylindrical element with length / and diameter 0 terminated by two identical semispheres (Fig. 1) is given below. It was proved in /26/ that 0/1 approximately and this explains very small values of percolation threshold. Fig. 1. Cylindrical element composed of carbon black grains in 3D space Very long chains of individual carbon black grains, kept together by van der Waals forces, are characteristic for high structure carbon black. Replacing such a chain by cylindrical element with I = n0 length, where n - number of grains in the chain and 0-mean diameter of single HSCB grain we get the following equations for Q and = /6 + n0\n-l)/4 3 ' ' 2 and the same Vc is given by formula (8) 4,2^ (6) (7) Vp = 1 - exp 6n (8) A few values of Vc are placed in Table 4. It is worth to note that Vc « 0.075 was found for MSCB-based system (Table 1) and this conform to chain consisting of 10 grains {n = 10). Table 4. Values of critical volume fraction Vc for medium structure carbon black or high structure carbon black used as active phase (calculated on the basis of Eq. (8)) Type of carbon black 0 [nm] n Vc MSCB 40 3 0,2081 MSCB 40 10 0,0748 MSCB 40 30 0,0223 HSCB 6 100 6,9x10-^ HSCB 6 300 2,3x10"^ HSCB 6 1000 7,0x10''^ HSCB 6 3000 2,3x10-'* It is worth to add that Eq. (1), i.e. one of the basic equations for percolation theory, has been applied successfully in semiquantitative analysis of resistivity changes of polymer thick-film resistors during high hydrostatic pressure compression /27/. The increase of pressure causes decrease of resistor volume. But because of significant compressibility differences between carbon black and polymer the effective volume fraction of conductive phase increases with pressure. This fact leads to such resistance decrease that they are in agreement with Eq. (1). The noise intensity C versus carbon black volume fraction vcB and curing temperature Tc is shown in Fig. 2. In general it is visible that increase of active phase amount or increase of curing temperature leads to noise intensity decrease. The below power law C = C,{v-vJ (9) 10 10 D 250'C o 300 "C A 350 10 21 □ O A fj, MSCB + G £ O 10 10 □ ° o A o A IVßCß HSCB o A O □ A O ir 10 o A 0,01 0,1 Fig. 2. Noise intensity C versus volume fraction vcb of carbon blacl< (+ graphite for (MSCB+G)-based system) in carbon black/poiyesterimide thici^-fiim resistors cured at various temperatures where C^ - constant and k - noise critical index is used very often in description of C{v) experimental data. Actually Eq. (9) conform to the first constituent of formula (10) (10) well known for noise intensity of percolation structures above percolation threshold. Values of i/o and K-for HSCB/PEI and MSCB/PEI systems are given in Table 5. The C^ constant is weakly dependent on curing temperature. Kind of active material affects this parameter much stronger - it is about 3 orders smaller for HSCB-based resistors than for MSGS ones. The index K^is decreased when curing temperature is increased. Much larger changes of this parameter are observed for MSCB-based composites. The dependencies of Ra as well as C versus volume fraction of active phase are described by power laws (1) and (9), respectively. This is why 1 /f noise could be presented in the form of C versus Ru plot (Fig. 3). This has the advantage that only clearly electrically measurable quantities appear on both axes and the C versus Ro can be presented as C-(11) and values of index ?] for particular investigated carbon/ polyesterimide systems and various curing temperatures are shown in Table 6. As has been proven in /29/ contin- uum percolation theory gives an explanation of Coc {Ra)'^ dependence and observed values of index r]. The reader especially interested in index 77 values both for other polymer as well as high temperature cermet thick-film resistors should find useful information in /29/, too. 10'^ 10"' 10" ^ 10- 10"^ 10" /0 / / MSCB+G / A / // A / □ p * 7 ° n/^ HSCB /Ž A 250 '0 □ 300 '=0 □ / 0 350"0 10' 10' 10' 10' 10' 10' Ro [Q/0] Fig. 3. Noise intensity C versus sheet resistance Rn for carbon blacl 1 and 11 + Cü< 1 K = K^i,^+{z + l-Cü)/{l-Cü)for u + ü)>l, (12) where d, u, co as in Eq. (3), z ^ z = d/2 for IRV model. c/ - 1/2 for RV model and Theoretical shapes of t(co), k[ü)) and ;](&;) for 3D random void model are shown in Fig. 4. The values of t, k and rj calculated for HSCB/PEI as well as MSCB/PEI systems (Tables 1, 5 and 6) are also placed in Fig. 4. As one could notice the ranges of co values responding to them are different for particular critical indices. Therefore the problem of nonuniversality of critical indices is still open and application of Balberg conception for interpretation of conductivity and 1/f noise mechanisms needs further verification. 6 - -( □ HSC O HSC A HSC ------K MSCB, 7,„„ = 250°C ■CT 2 - -6 -4 -2 Fig. 4. Dependence of critical exponents t, k and i] in 3D random void model as a function of index co characterizing the distribution of distances between hard insulating spheres in continuum conductive phase 2. Dimensional effects in percolative systems Classical model of percolation theory is applied under definite conditions: the film should be considered as infinite (with respect to the size of the individual elements), the particles haveto be spherical, monodispersiveand have an isotropic conductivity. Theoretical, numerical and experimental works have been conducted for cases, when the above assumptions have not been preserved. So far Shklovskii /30/, who considered the critical conductivity behaviour near the percolation threshold in an anisotropic two-component system, and Neimark /31 /, who calculated the electrophysical properties of percolation film with a finite thickness, have presented the most formal analyses. But none of papers took into account both finite film thickness and anisotropic shape of fillers, whereas sometimes (for example in conductive adhesive joints) it is necessary to Include both matters into theoretical analysis. Lets consider 3D medium with L, V\J, H » where L, W, H - length, width and height of structure and - correlation length for 3D system. One should remember that correlation length is the average distance between adjacent nodes and in 3D system (13) According to standard, two-component percolation model when the concentration of "good" conductors, p (with resistivity pi) exceeds the percolation threshold pc (p > pcs) the effective resistivity Pe = Pi - Pc3 )" (14) where ts - universal conductivity index for 3D-system (fs = 2). Theoretical description of film from Fig. 5 (H<^3and L = V\I> ^3) demands replacing of initial LxLxH cuboid by set of proper HxH xH cubes. Fig. 5. L X L X H cuboid with percoiative structure and equivalent 2D system received as a resuit of real-space renormalisation procedure In vertical direction (it corresponds with situation in adhesive joint) all H X /-/ X W cubes are connected in parallel. Knowing the properties of such unit cubes it is possible to apply standard 2D percolation model, where above the percolation threshold (p > pc2) the resistance of equivalent 2D system can be expressed as R = r ({P-Pc2)/ -h Pc2 (15) and /-[ = p^aQ /a^ , pi - resistivity of "good" conductor phase (pi«p2), ao - the minimal scale of the system under consideration (e.g. the bond length in the case of lattice models). However for structures with thickness H less than or of the order of correlation length we have so-called fractal (or 2.5D) regime. Modelling of such 2.5D system as 2D system consists in calculation the dependence of rj on H. Therefore i?2.5=r,(//)( Ph - Pc2yh Pel (16) where R2.5 - resistance of the film in fractal regime, ri(H) -resistance oi IHxHxH cube, Ph - concentration of cubes with r-iiH) resistance, Pc2 - percolation threshold in 2D system, t2 - critical conductivity index for 2D system. This means that for calculation of effective resistance (conductance) in fractal regime it is necessary to know the dependence of ri(H) and probability of proper conductive realisation. Such a study, showing the critical behaviour of effective conductivity {ae} and effective resistivity [pe], averaged over the large numbers of realisation in percolation systems, on the length scale L, has been presented in /32/. Next, the analytical analysis has been widened for systems with weak nonlinearity /33/. To include the shape of metallic fillers into the model it is assumed that all metallic particles are replaced by ellipsoids. The different shape of real particles (e.g. needles, fibres or flakes) can be projected by different ratios of ellipsoid semiaxes a, b, c (Fig. 6). Fig. 6. Shape of ellipsoid representing conductive filler and structure of termination made of isotropically conductive adhesives Moreover, for model simplicity, it is taken that particles inside termination are monodispersive and their semiaxes are parallel to proper coordinate axes (but their centres are distributed randomly inside the termination volume). According to/32/ the quantity rAH) from Eq, (3) is calculated for system within the fractal regime as for percolation system inside smearing region, v^/here this region is described by Tf^ =(H / c)"""'. Inside th one can meet structures associated with percolation above and below percolation threshold, i.e. bridge with resistance R-i (with probability Ph) and interlayer with resistance f?2 (with probability 1 - PH). Omitting the subsequent stages of calculation and based on /30-33/ we can write the final formula for unit conductance as [H Icy"' +r (JoC [{HI -T (17) where x = {p- p ^-i)! Pcs^rid parameter of ellipsoid deformation i.e. geometrical anisotropy of conductive grains a -c/b«1 » When we are in deeplyfractal regime i.e. (HIc) (such situation is characteristic for conductive adhesive joints /34/) we can present the above equation as G where = Oj / o, « 1 and (pa = ts + qs- Considering the week nonlinearity of both components the unit vertical conductance can be written as L^ c H V y H c \ / + H n-H'3 C \ / H V ^ / . r, +r (20) where wj = (ts +V3)/V3, W2 = (3t3 -vsj/v, ws = W4 = (V3 -Qsj/va, Ä(I)H - the voltage drop across the film thickness H and parameter of ellipsoid deformation a = c/b. Based on analogy between voltage susceptibility and noise intensity we have the following formula for effective noise intensity of structure within the fractal region. C! =C,(///c) (4V3-f3+l)/V3 Hicr^-T,) (21) This formula is valid independently on the ellipsoid shape i.e. ratio between ellipsoid semiaxes. The above equations permit to take into consideration the shape of conductive fillers, ratio between the metal particle sizes and termination geometry, ratio between the conductivity of "good" and "bad" conductors and volume concentration of active phase related to percolation threshold. In order to facilitate numerical simulations it is necessary to know values of three basic universal exponents n (characteristic for system geometry), t and q (characteristic for conductivity above and below the percolation threshold) both for 2D and 3D systems. Neimark's analysis and results have been also used by Liang and Li /35/to study the thermal conductance. They shown, that there exist a thickness effect on thermal conductivity of thin layers of disordered composites (similar to electrical conductivity). For limited layers (m - number of layers with unit thickness in vertical (normal) direction) the thermal conductivity in the normal direction is (22) where /<1 is the thermal conductivity of good thermal conducting phase. The thermal conductivity for the in-plane (horizontal) direction is ■Pc)"- (23) This means that the thermal conductivity increases in vertical direction with decreasing thickness while the in-plane conductivity declines. 3. Percolation model of metal oxide gas sensors Metal oxide gas sensors seem be the simplest type chemical sensors - the sensitive layer of these devices consist of a microcrystalline (or nanocrystalline) metal oxide film. Contrary to simple construction the gas detection mechanism is complex, representing interactions between various gaseous molecules and defects at or near surfaces or grain boundaries. It is based on variations of the charge-carrier concentration within a depletion layer at the grain boundaries in the presence of reducing or oxidizing gases, which leads to changes in the height of the energy barriers for free charge carriers. Except of many phenomeno-logical explanations also percolation theory has been applied for analysis of response of gas sensitive-resistors very recently/36,37/. For example Ulrich et al /31/ show that there are transitions between conducting and insulating stage for some nanocrystalline grains of gas-sensitive layer Such a grain becomes an insulating (is totally exhausted of free charge carriers) when its diameter is below a critical value, Ocr/f-They calculate the net number of free electrons from every grain, which can contribute to the conduction process as (24) where no - density of electrons in grain, 0- grain diame- ter, Nq^y ' ''^'t'sl surface density of oxygen adsorbed at the grain surface, N- surface density of chemisorbed gas species and - function taking into ac- count that after sintering the surface exposed to the ambient atmosphere is smaller than the surface of sphere with diameter (P (value of function f depends on 0, diameter of neighbouring grains fPnw and local coordination number k). For Nfree < 0 grain is insulating, for A/free > 1 is conducting and for 0 < A/free < 1 the number of free electrons Nuee should be considered as probability that the grain is conducting. Moreover, according to the percolation theory in order to connect sensor electrodes the concentration of conducting grains must exceed the percolation threshold Pc. This percolation concentration leads to a detection limit. Therefore the variation of conductance and the same the gas sensitivity is very high for concentration of detected gas p just slightly above pc. As it is seen from Eq. (24) the model of Ulrich et al /37/ connects appearing percolation effects of nanocrystalline metal oxide gas sensors with morphology of sensitive layer very strongly. 4. Reliability of VLSI circuits and percolation It is well known that VLSI chips reliability is determined by the interconnect failure, which consists in the breakdown of the path connectivity. Moreover, the gate oxide breakdown is important in the case of CMOS VLSI circuits. On the other hand much works have been done to explain dielectric or electrical breakdown phenomena of percolative metal-insulator composites (please see for example /38-40/). Therefore it is nothing strange that percolation theory has been applied very recently for reliability analysis of VLSI circuits. For example a biased percolation model /41-43/ is used for simulation of interconnect failures. Thin-film conductors, which create wire connections between particular transistor structures, can be treated as a large two-dimensional network of identical resistor elements deposited on an insulating substrate with temperature To. Because of two different operation modes (constant current and constant voltage) there are two opposite cases when degradation occurs. Single defect corresponds to a zero resistance value of an element (short circuit model) for constant voltage operation mode and to an infinite value of resistor element (open circuit model) for constant current operation mode. The total film degradation is reached when exist one continuous path of defects between lattice contacts. Therefore the degradation is synonymous with the conduc-tor-insulator transition in open circuit model and conduc-tor-superconductortransition for short circuit model. Moreover the biased percolation model assumes that the degradation starts because of spontaneous creation of some initial defects. When the constant current is applied in such resistor network then the creation of defects causes an increase of the current flowing in the neighbourhood resistors, especially those located in the region perpendicular to the contact direction. Therefore a significant extra Joule heat occurs in this region together with a significant increase of the local temperature. The mathematical notation is the following W^ = exp (25) where M/a is probability of local defect generation, Kb is the Boltzmann constant, and T a is the local temperature at the resistor a given by (26) and A is the key parameter responsible for the coupling between current and device degradation (value of A depends on the heat coupling of each resistor to the substrate), Ta and i a are the resistance of a single network element and the current flowing in it, respectively. Subsequent evolution stages of biased percolation model and its application (for example to electromigration in metallic lines) one could find in /44-49/. The breakdown of thin gate oxide layer, which can be defined experimentally as a large increase in conductance, occurs as soon as a critical density of neural electron traps in the oxide is reached. Degraeve et al /50/ simulated breakdown of thin Si02 layers based on percolation approach and verified such simulations with experimental results. Since the breakdown occurs at a critical electron trap density therefore conduction via generated traps is a plausible breakdown mechanism. This phenomenon has been simulated in the following way: a test sample with fixed dimensions has been defined, electron traps have been generated at random positions inside this volume, a sphere with a fixed radius r has been defined around the generated traps, conduction between two neighbouring traps has been possible when their spheres overlapped, the breakdown has appeared when a conducting path has been created from one interface (which has been an infinite set of traps) to the other. Stathis /51 / has modelled oxide breakdown using percolation formalism for very small samples, comparable to the lattice spacing. He shown, that the critical defect density exhibits a strong decrease with thickness below 5 nm, then becomes constant below 3 nm. For the second value the oxide thickness becomes less than the defect size. Therefore a single defect near the oxide centre is sufficient to create a continuous path across the sample in a 3-nm thickness limit. 5. Application of percolation theory in pharmacy Many pharmaceutical tablets are composed of binary inert matrices where water-soluble, finely dispersed drugs are embedded in an insoluble carrier material. Such drugs are released in a patient system by diffusion. Percolation theory is a relatively novel approach to design and characterisation of solid dosage forms and controlled drug release properties of such matrix system. Some papers related to this topic (as e.g. /52-57/) one can find in International Journal of Pharmaceutics. Tablet components usually have quite various electrical properties. For example the difference in electrical conductivity reaches to several orders of magnitude. Therefore the direct resistance measurement of tablets and then resistivity calculation can indicate the presence of percolation threshold /52/. A sudden resistivity drop indicates the presence of infinite clusters of both phases. Moreover such information may provide a valuable tool for explanation of changes observed in dissolution process of matrix tablets. Below the percolation threshold the drug release is incomplete. It has been also proved that so-called "combined percolation threshold" is characteristic for multicom-ponent tablet systems and therefore they can be reduced to a binary one /54/. Such a simple experiment and percolation attempt makes easier more rational design of pharmaceutical solid dosage forms. This is an interesting problem in which manner the percolation theory can help to control the drug release properties. Diffusion and conductivity are very similar because both describe transport processes. It is well known from percolation theory that the normal diffusion laws are not valid below the percolation threshold but above pc the diffusion coefficient D obeys the following power law D^XDo{p-pJ (27) where xDo represents a scaling factor and t - conductivity exponent 121. The references /53,57/ confirm both analytically as well as experimentally that tablet's conductivity and dissolution rate process can be successfully modelled by the same basic equation of percolation theory and that both processes scale in an identical way. 6. Conclusions This paper shows that percolation theory is still alive. As one can notice new problems, for example the role of finite geometry and dimensional effects on the form or characteristic power laws, are solved using novel models of percolation structures. The application range of percolation theory in microelectronics and materials science is very wide. 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