Talid Sinno

Point Defect Dynamics and the Oxidation‐Induced Stacking‐Fault Ring in Czochralski‐Grown Silicon Crystals

A model is presented and analyzed for the dynamics of intrinsic point defects, vacancies, and self-interstitials, in single-crystal silicon. Computations and asymptotic analysis are used to describe the appearance of the oxidation-induced stacking-fault ring (OSF ring) created during the cooling of silicon crystals in the Czochralski growth process. The model predicts that the OSF ring separates an inner region supersaturated with vacancies from a self-interstitial rich outer region. The OSF ring corresponds to a region of no net excess of either point defect. Simulations of the dynamics of the OSF ring with changes in the crystal growth rate (V) and the axial temperature gradient at the melt/crystal interface (G) accurately predict experimental data for a wide range of growth conditions when point defect thermophysical properties (equilibrium concentrations and diffusivities) are fit to a single set of experimental data. The point defect properties determined this way are within the range of values reported in the literature. Asymptotic analysis of the point defect dynamics model gives a simple mechanistic picture for the development of the point defect supersaturations and yields a closed-form expression for the critical value of (V/G) for the location of the OSF ring. This expression is in excellent agreement with the predictions of simulation and with the empirical correlation determined from experiments.

Defect engineering of Czochralski single-crystal silicon

Modern microelectronic device manufacture requires single-crystal silicon substrates of unprecedented uniformity and purity, As the device feature lengths shrink into the realm of the nanoscale, it is becoming unlikely that the traditional technique of empirical process design and optimization in both crystal growth and wafer processing will suffice for meeting the dynamically evolving specifications. These circumstances are creating more demand for a derailed understanding of the physical mechanisms that dictate the evolution of crystalline silicon microstructure and associated electronic properties. This article describes modeling efforts based on the dynamics of native point defects in silicon during crystal growth, which are aimed at developing comprehensive and robust tools for predicting microdefect distribution as a function of operating conditions. These tools are not developed independently of experimental characterization but rather are designed to take advantage of the very detailed information database available for silicon generated by decades of industrial attention. The bulk of the article is focused on two specific microdefect structures observed in Czochralski crystalline silicon, the oxidation-induced stacking fault ring (OSF-ring) and octahedral voids; the latter is a current limitation on the quality of commercial CZ silicon crystals and the subject of intense research

Modelling point defect dynamics in the crystal growth of silicon

Abstract The quality of silicon single crystals grown by the Czochralski (CZ) and floating zone (FZ) methods depends on the distribution of microdefects formed by silicon vacancies and interstitials and by impurities, such as oxygen and carbon. This paper describes the first steps of an attempt to model the formation of these defects by combining atomistic-level simulation of the equilibrium, transport, and kinetics of point defects and impurities in silicon, with continuum modelling of defect transport and reaction. The continuum models are written in terms of classical equilibrium, transport and kinetic coefficients, which are estimated using atomistic simulations based on the Stillinger-Weber interatomic potential for describing the interactions of silicon atoms. Atomistic simulations are reported for the equilibrium and transport properties of interstitials and vacancies in pure silicon. Calculations predict that interstitials prefer to form 〈110〉 dumbbells in the diamond lattice and that these point defects become delocalized at elevated temperatures. A model is proposed for the recombination of vacancies and interstitials that leads to a high entropic energy barrier at high temperatures due to this delocalization. Calculations of continuum point defect distributions for a prototype, steady-state crystal growth system predict the transition between vacancy (D-defects) and interstitial (A-defects) dominated precipitation of microdefects as a function of temperature gradient, crystal pull rate and crystal radius. These predictions arein qualitative agreement with experiments for FZ-grown crystals.

Modeling Microdefect Formation in Czochralski Silicon

An internally consistent model is presented for the dynamic formation of microdefects in single‐crystal silicon. The model is built on the dynamics of point defects, vacancies and self‐interstitials, and is extended to include the growth of clusters of these point defects into microdefects. A hybrid finite‐element/finite‐difference numerical method is used to solve the coupled system of partial differential equations, which includes sets of discrete rate equations for small clusters and Fokker‐Planck equations for larger ones. As described previously by a point defect dynamics model [J. Electrochem. Soc., 145, 303 (1998)], the oxidation‐induced stacking fault (OSF)‐ring position delineates the vacancy‐rich region inside from the external interstitial‐rich crystal. In Czochralski silicon, the radial position of the OSF‐ring correlates well with the expression . Simulations are used to explore the formation of voids in the vacancy‐rich region inside the OSF‐ring. Predictions of the total concentration of observable voids and the dependence of this concentration on the cooling rate agree with experiments and point to the importance of the axial temperature profile in the crystal from the melting point (1685 K) down to about 1150 K in setting the number and size of voids. The total number of voids correlates with V 〈G〉 where 〈G〉 is a measure of the temperature gradient in the temperature range 1173 K ≤ T ≤ 1685 K. The appearance of the OSF‐ring is explained qualitatively in terms of the residual vacancy concentration remaining in the crystal after aggregation has ceased.

Driving diffusionless transformations in colloidal crystals using DNA handshaking

Many crystals, such as those of metals, can transform from one symmetry into another having lower free energy via a diffusionless transformation. Here we create binary colloidal crystals consisting of polymer microspheres, pulled together by DNA bridges, that induce specific, reversible attractions between two species of microspheres. Depending on the relative strength of the different interactions, the suspensions spontaneously form either compositionally ordered crystals with CsCl and CuAu-I symmetries, or disordered, solid solution crystals when slowly cooled. Our observations indicate that the CuAu-I crystals form from CsCl parent crystals by a diffusionless transformation, analogous to the Martensitic transformation of iron. Detailed simulations confirm that CuAu-I is not kinetically accessible by direct nucleation from the fluid, but does have a lower free energy than CsCl. The ease with which such structural transformations occur suggests new ways of creating unique metamaterials having structures that may be otherwise kinetically inaccessible.

A systems approach to hemostasis: 2. Computational analysis of molecular transport in the thrombus microenvironment

Hemostatic thrombi formed after a penetrating injury have a heterogeneous architecture in which a core of highly activated, densely packed platelets is covered by a shell of less-activated, loosely packed platelets. In the first manuscript in this series, we show that regional differences in intrathrombus protein transport rates emerge early in the hemostatic response and are preserved as the thrombus develops. Here, we use a theoretical approach to investigate this process and its impact on agonist distribution. The results suggest that hindered diffusion, rather than convection, is the dominant mechanism responsible for molecular movement within the thrombus. The analysis also suggests that the thrombus core, as compared with the shell, provides an environment for retaining soluble agonists such as thrombin, affecting the extent of platelet activation by establishing agonist-specific concentration gradients radiating from the site of injury. This analysis accounts for the observed weaker activation and relative instability of platelets in the shell and predicts that a failure to form a tightly packed thrombus core will limit thrombin accumulation, a prediction tested by analysis of data from mice with a defect in clot retraction.

On the dynamics of the oxidation-induced stacking-fault ring in as-grown Czochralski silicon crystals

The behavior of the oxidation-induced stacking-fault ring (OSF ring) in Czochralski (CZ)-grown silicon crystals is predicted based on the dynamics of point defects during growth. Preexponential constants for the equilibrium point defect concentrations and diffusivities are determined by fitting the predictions of the model to a single set of experimental data for OSF-ring dynamics. Other experimental data is well fit by this model. Moreover, point defect properties used are consistent with other estimates. Asymptotic analysis of the point defect model leads to a closed-form expression for the dependence of the OSF-ring location on processing conditions and thermophysical properties of point defects at the melting temperature. These results indicate that differentiation between defect types in CZ-grown material can be done entirely on the basis of point defect dynamics.

Modeling of transient point defect dynamics in Czochralski silicon crystals

Intrinsic point defects control the formation of grown-in defects in silicon crystals. Under steady state conditions, the type of the prevailing point defect species is exclusively determined by the ratio of pull rate and temperature gradient in the crystal at the interface. In this study, simulations have been performed for transient growing processes where the pulling rate has been abruptly changed. Large reservoirs of interstitials are formed in fast-grown, vacancy-rich crystals near the interface after abruptly reducing the pulling rate for 30 min. During further growth at high pull rate, these interstitial reservoirs are transformed into large ellipsoidal defect patterns. Experimental results are excellently reproduced if equilibrium concentrations are used as boundary conditions for interstitials and vacancies at all crystal surfaces.

Atomistic simulation of point defects in silicon at high temperature

The Stillinger–Weber interatomic potential is used in molecular dynamics simulations to compute estimates of the equilibrium and transport properties of self‐interstitials and vacancies in crystalline silicon at high temperature. Equilibrium configurations are predicted as a 〈110〉 dumbbell for a self‐interstitial, and as an inwardly relaxed configuration for a vacancy. Both structures show considerable delocalization with increasing temperature, which leads to a strong temperature dependence of the entropy of formation, as suggested by diffusion experiments. Diffusion coefficients and mechanisms are predicted as a function of temperature. The predictions are discussed in the context of experiments and first‐principle calculations.

A mechanistic view of binary colloidal superlattice formation using DNA-directed interactions

The use of grafted, single-stranded DNA oligomer brushes with engineered sequences to effect tunable interactions between colloidal particles has now been demonstrated experimentally in both nano- and microscale systems. The versatility of this technology is highly appealing for realizing self-assembly of complex structures. However, ambiguities remain regarding how operating conditions, such as the rate at which the system temperature is reduced, interact with the other system parameters to produce crystalline assemblies with particular structures and defect densities. In this paper, a computational analysis is presented for the crystallization of binary superlattice crystals comprised of sub-micron colloidal spheres using a realistic model for the DNA-mediated interactions. The binary system consists of two populations of identical spheres that differ only in the sequence of the DNA oligomers grafted onto their surfaces. Metropolis Monte Carlo simulations and perturbation theory for free energy estimation are used to construct a detailed mechanistic picture for binary superlattice formation. The analysis reveals several interesting features of this system, particularly the role of kinetics in dictating not only the quality of the superlattice crystals, but also their crystalline structure. Using the results presented here, we make connections to recent experimental findings in similar binary crystallization systems.