Dielectric Breakdown Model Research Articles

Simple model for time-dependent dielectric breakdown in inter- and intralevel low-k dielectrics

A simple physical model is applied to time-dependent dielectric breakdown failure in ultralow-k(k=2.3) interlevel dielectrics. The model assumes that failure depends on the probability that an electron will have enough energy to damage the dielectric as it is accelerated in an electric field. It is seen that the characteristic form of the dependence of failure time on voltage or electric field is primarily dependent on the probability of having sufficient energy and not on the precise physical mechanism causing damage. An argument for a log-normal-like failure distribution is also presented.

The strip dielectric breakdown model

This paper reports on the analysis of the strip dielectric breakdown (DB) model for an electrically impermeable crack in a piezoelectric medium based on the general linear constitutive equations. The DB model assumes that the electric field in a strip ahead of the crack tip is equal to the dielectric breakdown strength, which is in analogy with the classical Dugdale model for plastic yielding. Using the Stroh formalism and the dislocation modeling of a crack, we derived the relationship between the DB strip size and applied mechanical and electrical loads, the intensity factors of stresses and electric displacement, and the local energy release rate. Based on the results, we discussed the effect of electric fields on fracture of a transversely isotropic piezoelectric ceramic by applying the local energy release rate as a failure criterion. It is shown that for an impermeable crack perpendicular to the poling direction, a positive electric field will assist an applied mechanical stress to propagate the crack, while a negative electric field will retard crack propagation. However, for an impermeable crack parallel to the poling direction, it is found that the applied electric field does not change the mode I stress intensity factor and the local energy release rate, i.e., the applied electric field has no effect on the crack growth.

Growth by random walker sampling and scaling of the dielectric breakdown model

Random walkers absorbing on a boundary sample the harmonic measure linearly and independently: we discuss how the recurrence times between impacts enable nonlinear moments of the measure to be estimated. From this we derive a technique to simulate dielectric breakdown model growth, which is governed nonlinearly by the harmonic measure. For diffusion-limited aggregation, recurrence times are shown to be accurate and effective in probing the multifractal growth measure in its active region. For the dielectric breakdown model our technique grows large clusters efficiently and we are led to significantly revise earlier exponent estimates. Previous results by two conformal mapping techniques were less converged than expected, and in particular a recent theoretical suggestion of superuniversality is firmly refuted.

Modeling of Tunneling Current and Gate Dielectric Reliability for Nonvolatile Memory Devices

We present a hierarchy of tunneling models suitable for the two- and three-dimensional simulation of logic and nonvolatile semiconductor memory devices. The crucial modeling topics are comprehensively discussed, namely, the modeling of the energy distribution function in the channel to account for hot-carrier tunneling, the calculation of the transmission coefficient of single and layered dielectrics, the influence of quasi-bound states in the inversion layer, the modeling of static and transient defect-assisted tunneling, and the modeling of dielectric degradation and breakdown. We propose a set of models to link the gate leakage to the creation of traps in the dielectric layer, the threshold voltage shift, and eventual dielectric breakdown. The simulation results are compared to commonly used compact models and measurements of logic and nonvolatile memory devices.

Quasicontinuum Monte Carlo: A method for surface growth simulations

We introduce an algorithm for treating growth on surfaces which combines important features of continuum methods (such as the level-set method) and Kinetic Monte Carlo (KMC) simulations. We treat the motion of adatoms in continuum theory, but attach them to islands one atom at a time. The technique is borrowed from the Dielectric Breakdown Model. Our method allows us to give a realistic account of fluctuations in island shape, which is lacking in deterministic continuum treatments and which is an important physical effect. Our method should be most important for problems close to equilibrium where KMC becomes impractically slow.

Dielectric breakdown model for conductor-loaded and insulator-loaded composite materials.

In the present work we generalize the dielectric breakdown model to describe dielectric breakdown patterns in both conductor-loaded and insulator-loaded composites. The present model is an extension of a previous one [F. Peruani et al., Phys. Rev. E 67, 066121 (2003)] presented by the authors to describe dielectric breakdown patterns in conductor-loaded composites. Particles are distributed at random in a matrix with a variable concentration p. The generalized model assigns different probabilities P(i,k-->i('),k(')) to breakdown channel formation according to particle characteristics. Dielectric breakdown patterns are characterized by their fractal dimension D and the parameters of the Weibull distribution. Studies are carried out as a function of the fraction of inhomogeneities, p.

Two breakdown mechanisms in ultrathin alumina barrier magnetic tunnel junctions

Two breakdown mechanisms are observed in magnetic tunnel junctions having an ultrathin alumina barrier. The two breakdown mechanisms manifest themselves differently when considering large ensembles of nominally identical devices under different stress conditions. The results suggest that one type of breakdown occurs because of the intrinsic breakdown of a well-formed oxide barrier that can be described by the E model of dielectric breakdown. The other is an extrinsic breakdown related to defects in the barrier rather than the failure of the oxide integrity. The characteristic of extrinsic breakdown suggests that a pre-existing pinhole in the barriers grows in area by means of dissipative (Joule) heating and/or an electric field across the pinhole circumference.

Consistent model for the voltage and temperature dependence of the soft breakdown conduction mechanism in ultrathin gate oxides

The voltage and temperature dependence of the soft breakdown conduction mechanism in ultrathin gate oxides in MOS structures was investigated. Measurements performed on p- and n-type substrate samples with different oxide thicknesses were compared, and the results analyzed within the framework of the quantum point contact model for dielectric breakdown. The model has been extended so as to include the thermal vibrations of the atoms that form the constriction’s bottleneck. A new interpretation of the model parameters, in connection with the potential drops distribution, is discussed.

Fracture behaviors of piezoelectric materials

Theoretical analyses and experimental observations of the failure and fracture behaviors of piezoelectric materials are presented. The theoretical analyses are based on the Stroh formalism. A strip dielectric breakdown model is proposed to estimate the effect of electrical non-linearity on the piezoelectric fracture of electrically insulated cracks. The reviewed experiments include the indentation fracture test, the bending test on smooth samples, the fracture test on pre-notched or pre-cracked samples, the environment-assisted fracture test, etc. For electrically insulated cracks, the experimental results show a complicated fracture behavior under combined electrical and mechanical loading. Fracture data are greatly scattered when a static electric field is applied. For electrically conducting cracks, the experimental results demonstrate that static electric fields can fracture poled and depoled lead zirconate titanate (PZT) ceramics. A charge-free zone model is introduced to understand the failure behavior of conducting cracks in the depoled lead zirconate titanate ceramics under electrical and/or mechanical loading. These theoretical and experimental results indicate that fracture mechanics concepts are useful in the study of the failure behaviors of piezoelectric materials.

Dielectric breakdown mechanism of Hf -silicate high-k gate dielectrics.

Dielectric breakdown mechanism of Hafnium (Hf) silicate film has been discussed, on the basis of the temperature dependence of the charge-to-breakdown (Qbd). It has been found that the activation energy of Qbd as a function of reciprocal temperature was around 0.1 eV, irrespective of the injection direction of electrons. The activation energy of Qbd for Hf-slicate gate dielectrics is almost same as that for silicon-dioxide (SiO2). Furthermore, it was found that Qbd for Hf-silicate was remarkably small, compared with that for SiO2. On the basis of the experimental results, the dielectric breakdown model for Hf-slicate dielectrics is proposed in this paper. It is inferred that, in high temperatures, the dominant mechanism for the dielectric breakdown is the hydrogen-related process, irrespective of the material of the gate dielectrics. We consider that a large amount of the defects in Hf-silicate films brings about the relative small life-time, compared with that with SiO2.

Dielectric breakdown model for composite materials

This paper addresses the problem of dielectric breakdown in composite materials. The dielectric breakdown model was generalized to describe dielectric breakdown patterns in conductor-loaded composites. Conducting particles are distributed at random in the insulating matrix, and the dielectric breakdown propagates according to new rules to take into account electrical properties and particle size. Dielectric breakdown patterns are characterized by their fractal dimension D and the parameters of the Weibull distribution. Studies are carried out as a function of the fraction of conducting inhomogeneities, p. The fractal dimension D of electrical trees approaches the fractal dimension of a percolation cluster when the fraction of conducting particles approximates the percolation limit.

Time-reversed dielectric-breakdown model for erosion phenomena.

A time-reversed dielectric-breakdown model in which the annihilating probability of a particle on the surface site (x,h) depends on Laplacian field phi(x,h,t) as P(x,h,t)=| inverted Delta phi(x,h,t)|(kappa)/ Sigma(x,h)| inverted Delta phi(x,h,t)|(kappa) is suggested. This model is shown to be a theoretical model that covers a variety of eroding surfaces from the linear phenomena with dynamic exponent z=1 to those showing nonlinear behavior. phi(x,t) is defined to satisfy the Laplace equation inverted Delta (2)phi=0 with the boundary condition phi=0 on the material and phi=1 far from the material. The model with 0.5 < or = kappa < or = 2 is found to follow the linear growth equation with z=1 as the diffusion-limited erosion, which is also a time-reversed version of diffusion-limited deposition. For small kappa, the dynamical scaling property of the eroding surface belongs to the Kardar-Parisi-Zhang universal class as the time-reversed Eden model. The model with kappa >2.5 does not show any surface roughening behavior.

Reliability of 100 nm silicon nitride capacitors in an InP HEMT MMIC process

A reliability study has been conducted on capacitors made with 100 nm of silicon nitride, in an InP HEMT MMIC fabrication process. Special wafers were fabricated, containing 1482 200 × 200 μm 2 capacitors each, and these were probed automatically. They were subject to ramped-voltage stress and the breakdown voltages recorded. On a typical wafer the vast majority of the breakdown voltages are between 50 and 90 V. In addition, IV curves were measured on a small number of specimens from 0 V up to breakdown. This was done in two regimes: above 25 V with a conventional setup, and below 25 V with an ultra-low-current measurement system. These were done at 25 and 175 °C above 25 V, and at 25 °C only below 25 V. The data were fitted well with a model for the conductivity, consisting of ohmic conduction at low voltages and Frenkel–Poole conduction at high voltages. Parameters of the fits included thermal activation energies, the voltage acceleration factor in the Frenkel–Poole model, and d eff, the effective thickness of the dielectric at the thinnest point. Analysis invoked the time-dependent dielectric breakdown model, which provides the time to failure as a function of the d eff, while d eff can be found from the ramped-voltage measurements. From the 10 wafers that have been probed so far, the mean of the distribution of failure times (at 1.5 V, 40 °C) is above 5 × 10 7 h, and the distribution becomes insignificant below 2 × 10 6 h. Further, the probability of failure in 10 years at 1.5 V, 40 °C is much less than 1 in 14,600. This indicates that 100 nm silicon nitride capacitors in this technology have good reliability.

A multi-defect initialization-based percolation model: a successful scheme to explain dielectric breakdown in MOS devices

We generalize a two-dimensional dynamic percolation model of dielectric breakdown of the SiO 2 in metal-oxide-semiconductor capacitors to include the effect of device area and oxide defect concentration. The strategy allows for dielectric breakdown to start in several points (seeds) of the Si/SiO 2 interface. The results which arise for the behavior of the time to dielectric breakdown ( t BD) with area, and the effect of the number of seeds (defects) at the interface allow for an improved agreement with experimental results.

Diffusion-controlled growth: theory and closure approximations.

We expand upon a new theoretical framework for diffusion-limited aggregation and associated dielectric breakdown models in two dimensions [R. C. Ball and E. Somfai, Phys. Rev. Lett. 89, 135503 (2002)]. Key steps are understanding how these models interrelate when the ultraviolet cut-off strategy is changed, the analogy with turbulence, and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is discussed. We find angular resonances relatable to the cone angle theory, and we are led to predict a new screening transition in the DBM at large eta.

Ferromagnetism in fractal-based complexes

Ferromagnetism in fractal-based complexes, which are generated using the dielectric-breakdown model with appropriate controls of their fractal dimension, is investigated using the standard Monte Carlo simulations. The difference in the fractal dimensions of a nerve-cell-like complex creates a heterotic phase in which the spin-ordered Gibbs state of a somatic nucleus and the spin-disordered Gibbs state of dendritic portions are orchestrated. On the other hand, a nebulalike complex in which many sites are grown on a dendritic substrate behaves as a single ferromagnetic system and is characterized by a particular Curie temperature.

Theory of diffusion controlled growth.

We present a new theoretical framework for diffusion limited aggregation and associated dielectric breakdown models in two dimensions. Key steps are understanding how these models interrelate when the ultraviolet cutoff strategy is changed, the analogy with turbulence and the use of logarithmic field variables. Within the simplest, Gaussian, truncation of mode-mode coupling, all properties can be calculated. The agreement with prior knowledge from simulations is encouraging, and a new superuniversality of the tip scaling exponent is both predicted and confirmed.

Urban growth simulation from "first principles".

General and mathematically transparent models of urban growth have so far suffered from a lack in microscopic realism. Physical models that have been used for this purpose, i.e., diffusion-limited aggregation, dielectric breakdown models, and correlated percolation all have microscopic dynamics for which analogies with urban growth appear stretched. Based on a Markov random field formulation we have developed a model that is capable of reproducing a variety of important characteristic urban morphologies and that has realistic microscopic dynamics. The results presented in this paper are particularly important in relation to "urban sprawl," an important aspect of which is aggressively spreading low-density land uses. This type of growth is increasingly causing environmental, social, and economical problems around the world. The microdynamics of our model, or its "first principles," can be mapped to human decisions and motivations and thus potentially also to policies and regulations. We measure statistical properties of macrostates generated by the urban growth mechanism that we propose, and we compare these to empirical measurements as well as to results from other models. To showcase the open-endedness of the model and to thereby relate our work to applied urban planning we have also included a simulated city consisting of a large number of land use classes in which also topographical data have been used.

Dielectric breakdown model at small eta: pole dynamics.

We consider the dielectric breakdown model in the limit eta-->0(+). This is shown to lead to Sivashinsky's equation. We show that a particular configuration of poles is linearly stable, in analogy to the stability of the 1/2 finger for diffusion limited aggregation, and compute exactly the eigenvalues of the stability matrix.

Tip splittings and phase transitions in the dielectric breakdown model: mapping to the diffusion-limited aggregation model.

We show that the fractal growth described by the dielectric breakdown model exhibits a phase transition in the multifractal spectrum of the growth measure. The transition takes place because the tip splitting of branches forms a fixed angle. This angle is eta dependent but it can be rescaled onto an "effectively" universal angle of the diffusion-limited aggregation branching process. We derive an analytic rescaling relation which is in agreement with numerical simulations. The dimension of the clusters decreases linearly with the angle and the growth becomes non-ractal at an angle close to 74 degrees (which corresponds to eta = 4.0+/-0.3).