Dielectric Breakdown Model Research Articles

Self-affine nature of dielectric-breakdown model clusters in a cylinder.

Clusters grown with the dielectric-breakdown model (DBM) in cylinder geometry show two growth phases: a scaling regime for cluster heights smaller than the cylinder circumference and a subsequent steady state, which is translationally invariant in the main growth direction. By studying a very simple, exactly solvable growth model involving arrays of sticks, we arrive at an affine framework, suitably describing the scale invariance of its clusters in the scaling regime. Within this framework we derive new relations between various exponents, which are expected to also hold for DBM clusters. Numerical work on DBM in the scaling regime is presented, indicating that the clusters are indeed self-affine in this phase.

Role of fluctuations in fluid mechanics and dendritic solidification

Many recent advances have occurred in our understanding of the role of fluctuations in fluid mechanics and dendritic solidification. Our purpose here is to present a review of some of these. We shall argue that if one understands completely the simple diffusion-limited aggregation (DLA) model or the closely related dielectric breakdown model (DBM), one understands the role of fluctuations in a range of fluid mechanical systems, as well as in dendritic solidification. The detailed descriptions of some such systems requires suitably chosen variants, such as DBM with anisotropy and noise reduction.

Theory of fractal growth

The problem we consider is to understand why nature gives rise to fractal structures. The first step is to formulate models of fractal growth based on physical mechanisms like the Diffusion Limited Aggregation and the more general Dielectric Breakdown Model. They are based on a simple iterative process governed by the Laplace equation and a stochastic field and they give rise to patterns that spontaneously evolve into random fractal structures of great complexity. In addition one would like to achieve a theoretical understanding of these models similar to that provided by the Renormalization Group for Ising-type models. Recently we have introduced a new theoretical framework for intrinsically critical growth models. This method is based on a Fixed Scale Transformation (with respect to the dynamical evolution) that defines a functional iteration for the distribution of elementary configurations that appear in a coarse graining process. This allows to include screening effects in terms of convergent series and to describe the intrinsic fluctuations of the boundary conditions. This approach clarifies the origin of fractal structures in these models and provides a systematic method for the calculation of the fractal dimension and the multifractal properties. It also makes clear why the usual renormalization schemes are not suitable for these problems. Here we describe the basic ideas of this new approach and report about recent developments, including the application to the percolation cluster interpreted as a problem of fractal growth.

Fractal dimension of intersection sets in the dielectric breakdown model

Clusters grown with the dielectric breakdown model (DBM) in the cylinder geometry show two growth phases: a scaling regime for cluster heights smaller than the cylinder circumference and a subsequent steady state, which is translational invariant in the main growth direction. The box-counting dimension of one-dimensional intersection sets of the clusters is studied for six different values of the growth parameter eta and four cylinder circumferences. The author finds that in the scaling regime this dimension depends on the height at which the intersection is made and the clusters are thus inhomogeneous. The results also show that clusters in the steady state are translational invariants in the main growth direction and self-similar in the direction perpendicular to it. A comparison between his findings and the theoretical results obtained from the fixed scale transformation approach by Pietronero et al. (1988), shows good agreement.

Theoretical concepts fpr fractal growth

After the introduction of fractal geometry by Benoit Mandelbrot the key problem is to understand why nature gives rise to fractal structures. This implies the formulation of models of fractal growth based on physical phenomena and the subsequent understanding of their mathematical structure in the same sense as the renormalization group has allowed to understand sing-type models. The models of diffusion-limited aggregation and the more general dielectric breakdown model, based on iterative processes governed by the Laplace equation and a stochastic field, have a clear physical meaning and they spontaneously evolve into random fractal structures of great complexity. From a theoretical point of view however it is not possible to describe them within usual concepts. Recently we have introduced a new theoretical framework for this class of problems. This clarifies the origin of fractal structures in these models and provides a systematic method for the calculation of the fractal dimension and the multifractal properties. Here we summarize the basic ideas of this new approach and report about recent developments.

Viscous fingers on fractals

We investigate the problem of flow in porous media near the percolation threshold by studying the generelized model of Viscous Fingering (VF) on fractal structures. We obtain analytic expressions for the fractal dimensions of the resulting structures, which are in excellent agreement with existing experimental results, and exact relations for the exponent D t, which describes the scaling of the time it takes the fluid to cross the sample, with the sample size, in terms of geometrical exponents for various experimental situations. Lastly, we discuss the relation between the continuous viscous fingers model and stochastic processes such as dielectric breakdown model (DBM) and diffusion limited aggregation (DLA).

Diffusion-limited aggregation near the percolation threshold

Diffusion-limited aggregation (DLA) and dielectric breakdown (DB) models have been used to simulate growth controlled by a Laplacian field on a square lattice network. A fraction ƒ (near the percolation threshold ƒ c ) of the bonds had a high conductivity (equal to 1), while the others had a low conductivity, equal to R . We used 10 -5 < R < 1 for DLA and R = 10 -8 for DB. We find crossover from growth on an incipient percolation cluster, with fractal dimensionality D ≅ 1.3, for small length scales, to that on a uniform substrate ( D ≅ 1.7), for a large length scales. The crossover length behaves as L R ≈ - a , with the crossover exponent a ≅ 0.25. The results were using a scaling theory.

Theory of Fractal Growth

We introduce a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on Laplace equation and provides a systematic method for the calculation of the fractal dimension. A specific application to the Dielectric Breakdown Model (including therefore DLA) in two dimension is presented. The analogies and differences with respect to the Renormalization Group are discussed.

Linear dielectric-breakdown electrostatics.

The dielectric breakdown of solids is a problem of great practical and theoretical interest. It is the electrical analog of the fracture of solids under applied loads. In the case of fracture, the reigning theory for linear elastic materials is linear elastic fracture mechanics. This paper develops the analogous theory, linear dielectric-breakdown electrostatics, based on a Griffith-like energy-balance calculation applied to a single conducting crack in an isotropic dielectric medium. Results include the development of the critical field-intensity factor, ${K}_{c}^{e}$, and the introduction of a contour-independent line integral, ${J}^{e}$, which is analogous to the J integral of linear and nonlinear elastic fracture mechanics. Some discussion of the relation between these results and recent lattice models of dielectric breakdown is given.

Theory of fractal growth.

In the past few years a great deal of activity has been devoted to the study of fractal structures [3] in relation to physical phenomena [4,5]. The prototype fractal growth model is based on a combination of the Laplace equation and a stochastic field. The first model of this class to be formulated was Diffusion Limited Aggregation (DLA) [6]. A few years later the more general Dielectric Breakdown Model (DBM) [7] was introduced. This model used the relation between the random walk and potential theory and made clear that growth could also occur “from inside”. In addition to their intrinsic theoretical interest, these models are now believed to capture the essential features necessary to describe pattern formation in seemingly different phenomena like electrochemical deposition, deudritic growth, dielectric breakdown, viscous fingering in fluids, fracture propagation and others [4,5].

Theory of Laplacian fractals: Diffusion limited aggregation and dielectric breakdown model

We describe a new theoretical approach that clarifies the origin of fractal structures in irreversible growth models based on the Laplace equation and a stochastic field. This new theory provides a systematic method for the calculation of the fractal dimension D and of the multifractal spectrum of the growth probability (ƒ(α)). A detailed application to the dielectric breakdown model and diffusion limited aggregation in two dimensions is presented. Our approach exploits the scale invariance of the Laplace equation that implies that the structure is self-similar both under growth and scale transformation. This allows one to introduce a Fixed Scale Transformation (instead of coarse graining as in the renormalization group theory) that defines a functional equation for the fixed point of the distribution of basic diagrams used in the coarse graining process. For the calculation of the matrix elements of this transformation one has to consider an infinite, but rapidly convergent, number of processes that occurs outside a considered diagram.

Theory of dielectric breakdown in metal-loaded dielectrics.

We have examined a very simple model of dielectric breakdown in random mixtures of metal and dielectric.1,2,3 We expect this analysis to be relevant for an class of materials which are composed of a random mixture of metallic particles embedded in a dielectric matrix. An example is solid fuel rocket propellant4 which is a mixture of microscopic aluminum particles (the fuel) in a dielectric matrix composed of oxidizer and rubber binder. The model we analyze is a percolation model in which the bonds of a d-dimensional lattice with lattice spacing a are occupied by conductors with probability p and by capacitors with probability 1-p. The probability p is chosen to be less than the percolation threshold pc so that no conducting path traverses the entire system. The breakdown process is modeled by assuming that the capacitors can withstand a maximum voltage drop of 1 volt. The entire lattice has a size of L lattice spacings. A macroscopic voltage is applied across the lattice. This voltage is raised until the voltage drop across one of the capacitors exceeds 1 volt. This macroscopic voltage is called V1 the initial breakdown voltage. The capacitor which fails is replaced by a conducting element. The process of failing one of the capacitors is repeated until a conducting path is formed across the sample. The maximum value of the applied voltage during this procedure is called the complete breakdown voltage and is denoted Vb.

A stochastic model for dielectric breakdown in thin capacitors

A nontrivial two-dimensional stochastic model for dielectric breakdown within a parallel plate capacitor is presented for the first time. The model has been used to determine geometric properties of parallel plate discharges. Comparisons are made between these properties and known fractal properties of electrostatic discharges within cylindrical geometries. As the spacing between the plates of a capacitor increases, the value of the fractal dimension of the associated discharge structure increases from the minimum value of unity and approaches the limiting value corresponding to the case of infinite spacing. For any given spacing, this fractal exponent is equal to the exponent of first passage time for the discharge pattern to reach a given height. A study of various power law relationships governing the breakdown may provide insight into the breakdown mechanism and electrical insulating quality of various materials. The model is applicable to the breakdown of thin insulating layers of metal-oxide-semiconductor devices.

A model of dielectric breakdown in disordered media

Abstract We present an experimental realization of dielectric breakdown in disordered media by means of a random network made of resistors (concentration p) and diodes (concentration 1–p). The diodes are considered as elements which become conductors for a voltage greater than a specific value. Near the percolation threshold we find that the critical exponent of the breakdown voltage is greater than that of the correlation length. Using light-emitting diodes, we can display the breakdown paths and determine their properties.

Renormalization-group approach to multifractal structure of growth probability distribution in diffusion-limited aggregation.

The scaling structure of the growth probability distribution in the surface layer of the generalized diffusion-limited aggregation model (\ensuremath{\eta} model) is derived by making use of the real-space renormalization-group method. A conductance of the surface layer is defined and renormalized as the growth-bond conductance. The renormalization-group transformation equation is derived for the growth-bond conductance. The equation has a nontrivial solution which is a stable fixed point. The growth probability assigned to each growth bond is represented by a random multiplicative process of the cell's growth probabilities evaluated at the fixed point. A hierarchy of generalized dimensions D(q) is calculated and the \ensuremath{\alpha}-f spectrum is found for diffusion-limited aggregation. The dependence of the \ensuremath{\alpha}-f spectra is found on the parameter \ensuremath{\eta} describing the different dielectric breakdown models.

Role of fluctuations in fluid mechanics and dendritic solidification

Our purpose is to review certain recent advances in understanding the role of fluctuations in fluid mechanics and dendritic solidification; many of these represent joint work of the author and J. Nittmann. If one understands completely the simple Ising model, then one understands virtually all systems near their critical points—although the detailed descriptions of many such systems requires a suitably-chosen variant of the Ising model (such as the XY or Heisenberg model). By analogy, we shall argue here that if one understands completely the simple diffusion-limited aggregation (DLA) model or the closely-related dielectric breakdown model (DBM), then one understands the role of fluctuations in a range of fluid mechanical sytems, as well as in dendritic solidification. The detailed descriptions of some such systems requires suitably-chosen variants, such as DBM with anisotropy and noise reduction.

On mechanical stresses at cracks in dielectrics with application to dielectric breakdown

The problem of an electric field applied to an isotropic dielectric containing a crack is considered as far as induced distortions and mechanical stresses are concerned. Near-tip distortions occur when the crack contains a conducting fluid or the surfaces are lined by a conducting layer. Previous treatments of the problem failed to distinguish between material and Maxwell stresses and an unphysical singularity in stress, which was divergent in elastic strain energy, was permitted. In this work, the material stress is identified, and a model is developed in which only physically realistic stress singularities are present. These results are obtained by treating the crack as a narrow but open ellipse rather than a slit crack as in previous models. A conventional crack-tip stress intensity factor KI characterizes the near-tip mechanical fields in the results obtained in this paper. The magnitude of KI due to an electrostatic field is deduced for a conducting central through crack in plane strain. The results suggest a model for dielectric breakdown caused by mechanical propagation of flaws. Data for dielectric breakdown and fracture of MgO and Al2O3 correlate in a way that supports this concept.

A hierarchical model for scaling structure in generalised diffusion-limited aggregations

Deterministic fractal models are presented to be exactly solvable for the growth probability distribution on the surface of the cluster in a hierarchical lattice. The recursion relations of the electric field on the growth bond are obtained for diffusion-limited aggregation and the dielectric breakdown models. A hierarchy of generalised dimensions D(q) is calculated to describe the growth probability, by using the recursion relations. The partition of (q-1)D(q) into a density of singularities f(q) with singularity strength alpha (q) is made and the alpha -f spectra are studied for different dielectric breakdown models. The scaling of the highest growth probability pmax on the growth bond is analytically derived and the dependence of the fractal dimensions is found on the parameter eta describing the different dielectric breakdown models.

Resistor-network approach to growth probability for dielectric-breakdown models at a surface.

For the dielectric-breakdown model a general method is presented for calculation of the growth probability on the growing-perimeter sites of the aggregate at a surface. The growth probability is given by solving the electrostatic problem for a superconducting aggregate in the semi-infinite normal-resistor network. The electric field at the perimeter site is obtained by solving the resistor-network problem with the lattice bond-bond Green's function. The fractal dimension D is found from the scaling assumption for the cluster-top occupancy probability.

Dc electroluminescence in copper-free ZnS:Mn thin films. II. A dielectric breakdown theory of instability

The occurrence of localized destructive breakdown (LDB) is a feature of direct coupled thin-film electroluminescence (DCTFEL) which has been attributed in the past to variations in applied field or threshold voltage caused by geometrical artifacts introduced during film deposition. However, in a companion paper it has been reported that this view of LDB is inconsistent with observations; instead there are strong indications that both LDB and DCTFEL are natural consequences of dielectric breakdown in ZnS:Mn. In this paper, the O’Dwyer theory of dielectric breakdown is applied to DCTFEL in ZnS:Mn with the aid of a simple computational equilibrium model. It is found that with reasonable assumed values for basic electronic parameters, current-controlled negative resistance (CCNDR) is expected in the normal DCTFEL current density regime. Localized (filamentary) conduction is, of course, a well-known consequence of CCNDR and the profound implications of this for EL are explored in some detail. It becomes clear that a number of poorly understood features of practical DCEL devices arise quite naturally from this dielectric breakdown theory; examples are premature brightness saturation at near maximum efficiency, the coexistence of DCTFEL and LDB, and the factors affecting the balance of that coexistence. It is concluded that while the dielectric breakdown model of DCTFEL presently rests precariously on assumed parameter values, the qualitative predictive power of the theory gives it significant weight. In view of this, some of the implications for DCTFEL device design are explored.