Bruce W. Roberts

Hysteresis and hierarchies: Dynamics of disorder-driven first-order phase transformations

We use the zero-temperature random-field Ising model to study hysteretic behavior at first-order phase transitions. Sweeping the external field through zero, the model exhibits hysteresis, the return-point memory effect, and avalanche fluctuations. There is a critical value of disorder at which a jump in the magnetization (corresponding to an infinite avalanche) first occurs. We study the universal behavior at this critical point using mean-field theory, and also present results of numerical simulations in three dimensions.

Tight-binding studies of the tendency for boron to cluster in c-Si. II. Interaction of dopants and defects in boron-doped Si

Clusters containing up to five boron atoms were considered as extended defects within a crystalline Si matrix. Tight-binding calculations suggest that a cluster containing two boron atoms occupying substitutional sites is stable, unlike any other small boron cluster that we studied. The formation energy increases when a third and fourth substitutional boron atom is added to the cluster. Estimates of the equilibrium concentration, using tight-binding-derived formation energies and formation entropies from the Stillinger–Weber model, indicate that B2 clusters become important when the boron doping level is ∼1018 cm−3, well below the solubility limit. In contrast, the formation energy of defect clusters involving an interstitial (BnI clusters, n=1–5, in their preferred charge states) decreases with increasing cluster size, down to 0.6 eV for B5I in a −5 charge state. None had formation energies that would lead to stable bound clusters. Several BnI clusters were found to be considerably more stable than isola...

Tight-binding studies of the tendency for boron to cluster in c-Si. I. Development of an improved boron–boron model

A tight-binding model for B–B interactions has been developed to study the stability of small boron clusters in crystalline silicon. The model was produced by fitting to the band structure determined by local-density approximation calculations on periodic supercells. This model is able to reproduce, relatively accurately, the cohesive energy of free boron clusters as determined by self-consistent field and configuration-interaction calculations.

A study of the failure mechanism of a titanium nitride diffusion barrier

TiN has been popularly used as a diffusion barrier between Al and Si to prevent “spiking.” It has, however, been reported that spiking still occurs through TiN at temperatures higher than 500 °C. In this study, we investigated the mechanism of spiking through TiN using high resolution transmission electron microscopy and electron dispersive spectroscopy (EDS). We found TiN to be saturated with Al upon annealing at 550 °C. Si also diffuses through TiN and dissolves into Al. Spikes form upon 550 °C annealing at the Si substrate. EDS analysis revealed the phase of the spikes to be Al3Ti containing a considerable amount of Si. These results indicate that spiking through TiN is due to the formation and growth of Al3Ti after the Al saturation at the bottom of TiN. We discuss these results based on the Al–Ti–Si and Al–Ti–N ternary phase diagrams.

Mass-Extinction: Evolution and the Effects of External Influences on Unfit Species

We present a new model for extinction in which species evolve in bursts or ‘avalanches’, during which they become on average more susceptible to environmental stresses such as harsh climates and so are more easily rendered extinct. Results of simulations and analytic calculations using our model show a powerlaw distribution of extinction sizes which is in reasonable agreement with fossil data. We also see several features qualitatively similar to those seen in the fossil record. For example, we see frequent smaller extinctions in the wake of a large mass extinction which arise because there is reduced competition for resources in the aftermath of a large extinction event, so species which would not normally be able to compete can get a foothold, but only until the next cold winter or bad attack of influenza comes along to wipe them out.

Hysteresis, avalanches, and noise

In our studies of hysteresis and avalanches in the zero-temperature random-field Ising model, a simple model of magnetism, we often have had to do very large simulations. Previous simulations were usually limited to relatively small systems (up to 900/sup 2/ and 128/sup 3/), although there have been exceptions. In our simulations, we have found that larger systems (up to a billion spins) are crucial to extracting accurate values of the critical exponents and understanding important qualitative features of the physics. We show three algorithms for simulating these large systems. The first uses the brute-force method, which is the standard method for avalanche-propagation problems. This algorithm is simple but inefficient. We have developed two efficient and relatively straightforward algorithms that provide better results. The sorted-list algorithm decreases execution time, but requires considerable storage. The bits algorithm has an execution time that is similar to that of the sorted-list algorithm, but it requires far less storage.

A bound on the decay of defect-defect correlation functions in two-dimensional complex order parameter equations

Abstract Motivated by generic scale invariance, we examine the behavior of topological defect-defect correlation functions in two-dimensional systems driven out of equilibrium to regimes where they exhibit “defect chaos”. Using the topological nature of the defects, we show that these defect-defect correlations cannot decay as slowly as predicted by generic scale invariance. We also provide numerical calculations that yield defect-defect correlation functions in the defect turbulence regime of the two-dimensional, anisotropic complex Ginzburg-Landau equation. These numerical results, which test the specific regime of broken square symmetry, do not appear to decay as slowly as predicted by the ideas of generic scale invariance. These results are in agreement with the analytical predictions.

Real-space renormalization group for the random-field Ising model.

We present real-space renormalization-group (RG) calculations of the critical properties of the random-field Ising model on a cubic lattice in three dimensions. We calculate the RG flows in a two-parameter truncation of the Hamiltonian space. As predicted, the transition at finite randomness is controlled by a zero-temperature, disordered critical fixed point, and we exhibit the universal crossover trajectory from the pure Ising critical point. We extract scaling fields and critical exponents, and study the distribution of barrier heights between states as a function of length scale.

An order(N) tight-binding molecular dynamics study of intrinsic defect diffusion in silicon

We present calculations of the intrinsic vacancy and interstitial diffusivities, DV and DI, in silicon using order(N) tight-binding molecular dynamics simulations. Vacancy diffusion was found to occur rapidly, with a diffusivity of around 10 ˇ4 cm 2 /s in the temperature range 900‐ 12008C. Interstitial diffusion was found to be a factor of 10‐100 times slower than vacancy diffusion, being on the order of 10 ˇ5 cm 2 /s for the same temperature range. These diffusivities have the same order of magnitude as previous molecular dynamics calculations performed with both classical and Car-Parrinello models. The interstitial diffusion was found to occur via two different paths, one involving motion of a single interstitial down the open (110) channels in the lattice, and another involving an intermediate h110i split interstitial which facilitates the interstitial crossing from one (110) channel to another. Within the tight-binding model we use, the split interstitial path is more important than drift of a single interstitial at higher temperatures (here, above around 10008C). The reverse is true below this temperature, with few, if any, formations of split interstitials and a diffusion dominated by traversal down the open channels of the lattice. New LDA data shows that the energetic advantage of the split interstitial over the tetrahedral interstitial is smaller than previously calculated, lending credence to the tight-binding results. The split interstitials were found to be relatively long-lived (in some cases, lifetimes in excess of 15 ps), even in a potential that favors tetrahedral interstitial formation. In order to perform these calculations, we developed a constant Nelec version of the forces in the Goedecker and Colombo O(N) tight-binding algorithm originally written for systems with a constant chemical potential. Omission of this correction can lead to errors approaching 40% in the forces. # 1999 Elsevier Science S.A. All rights reserved.

Disorder-driven first-order phase transformations: A model for hysteresis

Hysteresis loops in some magnetic systems are composed of small avalanches (manifesting themselves as Barkhausen pulses). Hysteresis loops in other first‐order phase transitions (including some magnetic systems) often occur via one large avalanche. The transition between these two limiting cases is studied, by varying the disorder in the zero‐temperature random‐field Ising model. Sweeping the external field through zero at weak disorder, we get one large avalanche with small precursors and aftershocks. At strong disorder, we get a distribution of small avalanches (small Barkhausen effect). At the critical value of disorder where a macroscopic jump in the magnetization first occurs, universal power‐law behavior of the magnetization and of the distribution of (Barkhausen) avalanches is found. This transition is studied by mean‐field theory, perturbative expansions, and numerical simulation in three dimensions.