Alexander-Haasen Model Research Articles

Study of effects of varying parameters on the dislocation density in 200 mm SiC bulk growth

Physical vapor transport (PVT) is the dominant method for growing 200 mm SiC crystals, and the crystals produced by this method still have dislocations, which affect the performance of the device. In this study, the finite element analysis of 200 mm SiC crystal growth has been conducted to investigate the influencing parameters on the dislocation density. The calculations are based on the model of multiple resistance heating. The transient heat transfer for the crystal growth is first calculated. A dynamic mesh technique is then employed to consider the shape evolution of the crystal during the growth. Finally, the distributions of the internal stress and dislocation density have been calculated based on the Alexander-Haasen model. The comparison among different parameters provides guidance for reducing the thermal stress and dislocation density in the 200 mm SiC crystal growth.

Parametric numerical study of dislocation density distribution in Czochralski-grown germanium crystals

Dislocations strongly affect the quality of Czochralski-grown germanium crystals used for various applications, however, the influence of different growth parameters on the dislocation distribution has not been extensively studied in the literature. In the present work, a simplified parametric crystal growth model is introduced which is suitable for large-scale numerical studies. The time-dependent temperature and dislocation density distributions in the crystal are calculated and analyzed. The Alexander–Haasen model is applied for dislocation modeling. Model parameters are taken from previously published discrete dislocation dynamics simulations, establishing a novel coupling between mesoscopic and macroscopic simulations. A W-shaped radial distribution of the dislocation density is obtained in the simulations. It is shown that the effective ambient temperature and crystallization front deflection have the strongest influence on the dislocation distribution. The results are compared to the experiment, showing a qualitative agreement, but also several deviations. Limitations of the considered dislocation modeling approach and the simplified thermal model are discussed.

Effect of crucible thermal conductivity on dislocation distribution in crystals in a silicon carbide physical vapor transport furnace

The effect of the thermal conductivity of a crucible in a silicon carbide physical vapor transport furnace on the dislocation distribution in crystal was investigated using numerical analysis. The numerical analysis includes stress and dislocation propagation calculations based on the Alexander–Haasen model. Modifying the thermal conductivity of the carbon crucible can reduce the dislocation density. It can also homogenize the temperature distribution in the radial distribution in the crystal. The density of dislocations could be further by reduced using a carbon crucible with low thermal conductivity.

Numerical Analysis of Dislocation Density of c ‐, a ‐, m ‐axes Grown Al 2 O 3 Crystals during Annealing Process

A numerical study of dislocation density is carried out by 3D and time-dependent analysis using the Alexander–Haasen model to estimate growth direction dependence of plastic deformation of Al2O3 single crystals during the annealing process. The direction of crystal growth is set to c-, a-, and m-axes. The calculated results show distribution of dislocation density in the crystal grown in the c-axis has a sixfold symmetry while those in the crystals grown in the a- and m-axes show asymmetric distribution. Dislocation densities in the a- and m-axes Al2O3 crystals are larger than that in the c-axis crystal.

Application of the Alexander–Haasen Model for Thermally Stimulated Dislocation Generation in FZ Silicon Crystals

Numerical simulations of the transient temperature field and dislocation density distribution for a recently published silicon crystal heating experiment were carried out. Low- and high-frequency modelling approaches for heat induction were introduced and shown to yield similar results. The calculated temperature field was in very good agreement with the experiment. To better explain the experimentally observed dislocation distribution, the Alexander–Haasen model was extended with a critical stress threshold below which no dislocation multiplication occurs. The results are compared with the experiment, and some remaining shortcomings in the model are discussed.

Numerical Analysis of Dislocation Density of SiC Crystals Tilted from [0001] Toward [12¯10]$[ {1\bar 210} ]$ and [11¯00]$[ {1\bar 100} ]$ Grown by Physical Vapor Transport

AbstractA 3D and time‐dependent analysis of dislocation density using the Alexander–Haasen model to estimate the plastic deformation of tilted angle growth of 4H–SiC single crystals during crystal growth and cooling processes grown using the physical vapor transport method is performed. The growth direction is set to tilt from [0001] toward and to fabricate wafers to stabilize polytype during crystal growth of the epitaxial layer. The maximum dislocation density in SiC crystals increases as the tilted angle increases, whereas von Mises stress, which is one of the measures of dislocation density, decreases. The calculated results show dislocation density distribution in the crystal grown in [0001] exhibits six‐folded symmetry, whereas that in the crystal grown in the tilt direction from [0001] toward and exhibits asymmetric distribution.

Numerical Simulation of the Process of Preparation of Multisilicon by the Directional Solidification Method

A technique for numerical simulation of the process of directional solidification of multisilicon in a square crucible is considered. The application of 2D axisymmetric geometry constructed for a vertical furnace cross section in the calculations is justified. The mathematical model describes the hydrodynamics of a melt, gas flow, global heat exchange, thermal stress, and evolution of the dislocation density in a growing crystal. The sensitivity of the stress and dislocation density to the Alexander–Haasen model parameters is determined.

On the importance of dislocation flow in continuum plasticity models for semiconductor materials

Abstract Crystalline defects, such as dislocations, may have a strong negative influence on the performance and quality of electronic devices. Modelling the defect evolution of such systems helps to understand where defects occur, how they evolve and interact, Modelling also might allow to avoid – or at least to reduce – the number of line defects in such materials. A widely used continuum model for predicting the evolution of dislocation density in semiconductors is the Alexander-Haasen (AH) model (Alexander and Haasen, Solid state physics,1969) which describes the evolution of dislocation density through a local dislocation multiplication law; the motion of dislocations, i.e. dislocation flow, is not considered. Within this work, the underlying assumptions for the validity of the AH model are studied and compared to a more complex continuum model of dislocation dynamics (CDD) which explicitly considers dislocation fluxes. We use a simplified 2D scenario of a 4H-SiC crystal at the end of the growth as a benchmark system. Our comparisons show that considering the motion of dislocations can have a significant influence on the evolution and final distribution of dislocation density, which may strongly limit the applicability of the AH model. Based on the CDD model, we then investigate the cooling process and study the effect of the cooling rate on the resulting dislocation microstructure.

Simulations of dislocation density in silicon carbide crystals grown by the PVT-method

Abstract The Alexander-Haasen (AH) model has been applied to analyze the plastic deformation and dislocation generation during the crystal growth process of 4H-SiC (silicon carbide). Plastic parameters are obtained by fitting the predicted curves to the experimental data on the plastic deformation of α-SiC crystals under uniaxial compression. The relationship between the activity energy (Q) and stress exponent (n) is considered when using the AH model. This relationship explicitly represents two deformation mechanisms around the critical temperature. The ratio of the activity energy and stress exponent, Q/n, equals 0.3 eV when the temperature is below the transition temperature, and 1.3 eV when the temperature is above the transition temperature. Then, the model is used to predict the dislocation density and thermal stresses in the crystals. The largest dislocation density is found to occur near the graphite/SiC interface, and the dislocation density gradually decreases with the thickness of the ingot.

Modeling of dislocation dynamics in germanium Czochralski growth

Obtaining very high-purity germanium crystals with low dislocation density is a practically difficult problem, which requires knowledge and experience in growth processes. Dislocation density is one of the most important parameters defining the quality of germanium crystal.In this paper, we have performed experimental study of dislocation density during 4-in. germanium crystal growth using the Czochralski method and comprehensive unsteady modeling of the same crystal growth processes, taking into account global heat transfer, melt flow and melt/crystal interface shape evolution. Thermal stresses in the crystal and their relaxation with generation of dislocations within the Alexander-Haasen model have been calculated simultaneously with crystallization dynamics. Comparison to experimental data showed reasonable agreement for the temperature, interface shape and dislocation density in the crystal between calculation and experiment.

Numerical simulation of stresses and dislocations in quasi-mono silicon

The Alexander–Haasen model is applied for the analysis of dislocation dynamics in quasi-mono crystalline silicon. Model constants are re-calibrated using stress–strain measurements on small silicon samples under uniaxial compression. It is observed that the activation energy may decrease at low temperatures and the hardening parameter generally increases due to the presence of grown-in dislocation clusters. The calibrated model is applied to an idealized cooling process which allows for a discussion of the basic physical mechanisms leading to residual stresses in quasi-mono ingots. Residual stresses can be reduced by minimizing thermal stresses during the elastic–plastic transition, which was observed approximately between 1100°C and 750°C in the present case.

Modeling grown-in dislocation multiplication on prismatic slip planes for GaN single crystals

To dynamically model the grown-in dislocation multiplication on prismatic slip planes for GaN single crystal growth, the Alexander–Haasen (AH) model, which was originally used to model the plastic deformation of silicon crystals, is extended to GaN single crystals. By fitting the model to the experimental data, we found that it can accurately describe the plastic deformation of GaN caused by prismatic slip. A set of unified parameters for the AH model at different temperatures can be found. This model provides a possible method to minimize grown-in dislocations caused due to prismatic slip by optimizing growing and cooling conditions during GaN single crystal growth.

Applicability of the three-dimensional Alexander-Haasen model for the analysis of dislocation distributions in single-crystal silicon

Applicability of the three-dimensional Alexander-Haasen (AH) model for the analysis of dislocation distributions in single-crystal silicon has been estimated. The numerical results obtained from the AH model agree well with the experimental data for both CZ-Si and FZ-Si crystals with the axis in the [001] direction but do not completely agree with the experimental data for the FZ-Si crystal with the axis in the [111] direction. The inapplicability of the AH model in a crystal with the axis in the [111] direction may arise from the neglect of dislocation propagation in this model, because the dislocation propagation in a crystal with the axis in the [111] direction is more active than that in a crystal with the axis in the [001] direction. Therefore, to increase the applicability of the AH model, it is necessary to include the effect of dislocation propagation.

Alexander–Haasen Model of Basal Plane Dislocations in Single-Crystal Sapphire

The Alexander–Haasen model was originally used to model the multiplication of the mobile dislocations in crystalline silicon. Here, we extend this model for studying multiplication of basal plane dislocations (BPDs) in single-crystal sapphire. By fitting the Alexander–Haasen model to experimental data, we find that the model accurately describes the plastic deformation of sapphire caused by BPDs. The application of the Alexander–Haasen model to sapphire growth made it possible to minimize the dislocation density and residual stress in growing crystals by optimizing the furnace structure and operation conditions. We apply the Alexander–Haasen model to investigate the dynamical deformation of single-crystal sapphire during the cooling process and examine the effect of the cooling rate on the generation of BPDs and residual stress. Finally, we present the BPD distribution and discuss the main factor that influences the generation of BPDs.

Dislocation-density-based modeling of the plastic behavior of 4H–SiC single crystals using the Alexander–Haasen model

To dynamically model the plastic deformation of 4H–SiC single crystals during physical vapor transport (PVT) growth, the Alexander–Haasen model, originally proposed for the elemental semiconductor, is extended into IV–IV compound semiconductors. By fitting the model parameters to the experimental data, we show that the Alexander–Haasen model can describe the plastic deformation of 4H–SiC single crystals if the activation of the carbon-core partial dislocation is modeled in the high-temperature region (above 1000°C) and the silicon-core partial dislocation is modeled in the low-temperature region (below 1000°C). We then apply the same model to the dynamical deformation process of a 4H–SiC single crystal during PVT growth. The time evolution of the dislocation density is shown, and the effects of the cooling time on the final dislocation density, residual stress and stacking faults are also examined.

Highly efficient and stable implementation of the Alexander–Haasen model for numerical analysis of dislocation in crystal growth

To effectively simulate the time evolution of the dislocation density during the crystal growth process under high stress by the Alexander–Haasen model, a fully implicit and fully coupled implementation of the Alexander–Haasen model has been proposed. Numerical tests on low-stress multicrystalline silicon grown in a small furnace and high-stress seed-cast monocrystalline silicon grown in an industrial-scale furnace have been done. Results indicate that the proposed algorithm is highly efficient, strongly stable and applicable to any stress level. This algorithm provides an effective tool to optimize the crystal growth process and reduce dislocation density in industrial-scale furnaces.

Modelling dislocation generation in high pressure Czochralski growth of InP single crystals: part I. Construction of a visco-plastic deformation model**A major part of the work was performed when SP was at: Department of Mechanical Engineering, Stony Brook University, Stony Brook, NY 11794, USA.

A visco-plastic constitutive model for single crystal InP is proposed and embedded in the ABAQUS finite element software through a user-defined subroutine. The model is a multi-slip generalization of the Alexander–Haasen model, with measures for dislocation densities on each of the known slip systems acting as internal variables. The model is completed and verified through its application to the constant strain rate uni-axial deformation tests and comparison with the experimental data. It is then used to simulate uni-axial deformation tests with different axes of orientation, temperature and strain rates. Multi-slip deformations and high strain rates lead to high dislocation densities, while the temperature has a mixed effect depending on the strain. The model developed here and the insights gained through selected simulations will be used in part II of this paper to study the dislocation development during the growth of single crystal InP in the high pressure Czochralski process.

Deformation of covalent crystals in the vicinity of the yield drop

The theory for description of the stress peak (yield drop) on the stress-strain curves of covalent crystals with low and zero dislocation densities has been developed. The kinetics of the variation of the dislocation density and the shape of the stress peak in the vicinity of the upper yield stress is described analytically within the framework of the generalized Alexander-Haasen model. The character of the elastoplastic transition is analyzed in detail and the model is compared with the experimental data for silicon.

A theory of sharp yield point in low-dislocation crystals

The condition for yield drop existence in stress-strain curves for crystals with an initially low dislocation density is derived. A new theory that radically differs from the well-known Alexander-Haasen model is proposed to describe the stressed state of the crystal near the lower yield point.

Analytical description of the yield drop in the Alexander-Haasen model

The analytical solution of the equations of the Alexander-Haasen model, which describes the shape of the deformation-curve peak (the so-called yield drop), has been obtained for the case of a low initial dislocation density. It is shown that the self-development of the dislocation structure results in the specific kinetic transition with a dramatic decrease of the elastic-deformation rate and an increase of the plastic-flow rate. It is natural to interpret this phenomenon as an elastoplastic transition, and to consider the corresponding stress as the yield stress despite the fact that, being considered in the traditional way, its value does not coincide with either the upper or lower yield stress. The conditions of the existence of the yield drop are studied, and the quantitative criterion of the corresponding change in the deformation-curve shape is proposed.