Haasen Model Research Articles

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.

Dislocation analysis of germanium wafers under 1080 nm laser ablation.

COMSOL Multiphysics was employed to establish a dislocation model based on the Alexander and Haasen (AH) model, the heat conduction equation, and Hooke's law for calculating the dislocation distribution of germanium (Ge) under laser irradiation. The numerical simulation results were obtained. A continuous 1080 nm laser was utilized to ablate the monocrystalline Ge wafers to validate the model. The experimental results show that no surface damage appears until the irradiances go up to 234W/cm2 for 100 ms laser ablation. This is consistent with the numerical findings. The initiation times of surface damage by the experiments at 234W/cm2 and above agree well with the numerical results, which means that the model can efficiently predict the dislocation field.

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.

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.

Study on the usage of a commercial software (Comsol-Multiphysics®) for dislocation multiplication model

Elaboration of silicon ingots for photovoltaic application in Directional Solidification furnace leads to formation of dislocations mainly due to thermoelastic stresses, which impact photovoltaic conversion rate. Several research teams have created numerical simulation models using home-made software in order to study dislocation multiplication and predict the dislocation density and residual stresses inside ingots after elaboration.In this study, the commercial software Comsol-Multiphysics® is used to calculate the evolution of dislocation density during the ingot solidification and cooling. Thermo-elastic stress, due to temperature field inside the ingot during elaboration, is linked to the evolution of the dislocation density by the Alexander and Haasen model (A&H model). The purpose of this study is to show relevance of commercial software to predict dislocation density in ingots.In a first approach, A&H physical model is introduced for a 2D axisymmetric geometry. After a short introduction, modification of Comsol® software is presented in order to include A&H equations. This numerical model calculates dislocation density and plastic stress continuously during ingot solidification and cooling. Results of this model are then compared to home-made simulation created by the teams at Kyushu university and NTNU. Results are also compared to characterization of a silicon ingot elaborated in a gradient freeze furnace. Both of these comparisons shows the relevance of using a commercial code, as Comsol®, to predict dislocations multiplication in a silicon ingot during elaboration.

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.

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.

Dislocation Analysis of a New Method for Growing Large-Size Crystals of Monocrystalline Silicon Using a Seed Casting Technique

A new method is proposed for growing large-size crystals of monocrystalline silicon using a seed casting technique. The stress and dislocation distributions for this type of growth during crystallization were simulated. The results indicate that the order of dislocation density is not very large. If the cooling flux of the crystal growth is controlled to be small at the initial stage, and the annihilation and direction effects of dislocation in the Alexander and Haasen model are included, the real dislocation density could be considerably smaller than the calculated value. Therefore, the new growth method for monocrystalline silicon is promising for complete single crystal growth, effectively avoids polycrystalline nucleation, and has a moderate dislocation density.

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.

A visco-plastic model of the deformation of InP during LEC growth taking into account dislocation annihilation

In order to determine which region is the major source of the dislocations which will be present in an LEC grown InP single crystal, a model of dislocation generation in InP has been designed and numerically solved in the actual growth parameters and geometry. The Haasen model of dislocation generation has been improved by projection of the mechanical stresses on the glide planes. This permits to take independently into account the interaction of dislocations situated in the same plane and in different planes (following works by Sumino). An important mechanism, acting principally at high temperatures, is the annihilation of dislocations by pairs. In the proposed model, the associated decrease of dislocation density is proportional to the square of the density of dislocations and of their velocity. The model has been solved numerically with the help of the finite element software MARC ®. It has been validated by comparison with deformation tests at high temperature and the annihilation mechanism permits to explain the fully plastic behaviour of InP above 1250 K. In a second step, this numerical model is interfaced with a global fully time-dependent modelling of heat transfer in the furnace. The model is adjusted by fitting the coefficient of dislocation annihilation in order to get the experimental number of dislocations in the first wafer of the reference crystal. The evolution of dislocation density with time, for a given location in the crystal, can therefore be predicted and is in good agreement, axially and radially, with experimental results. It is shown that dislocations situated in the centre of the crystal are generated close to the solid–liquid interface and that peripheral dislocations, which are the most detrimental, are generated at the surface of the encapsulant.

Experimental and numerical results on the plastic deformation in uniaxial compression experiments at high temperature on Er-doped GaSb samples

High temperature plastic deformation of Er-doped Bridgman-grown GaSb single crystals has been studied. Mechanical testing were carried out at different loads and temperatures around 0.7 T melt. The fits between experimental results and simulation, using the Haasen model, have permitted to obtain the parameters of this model. The comparison between these parameters and the activation energy for pure and Er-doped GaSb indicates a hardening effect on the GaSb host induced by the Er impurities.

Plastic deformation at high temperatures of pure and Mn-doped GaSb

In this work the plastic behavior of GaSb and Mn-doped GaSb at high temperature has been analyzed. Several experiments at different constant load and temperatures around 500 °C were carried out. The parameters used in the Haasen model have been obtained experimentally and compared with the ones obtained from simulations.

New Analysis of the Yield Point of Germanium

The interpretation of the yield point of germanium is still an unsolved question and, for more than a decade, no new information or suggestion on this problem has become known to the authors. In the present paper the available data are again analyzed, together with measurements of the strain rate at the inflection point of the creep curve, and taking into account investigations of the dislocation velocity. A consistent picture can be obtained by assuming that at least two (perhaps three) different regimes are involved. At temperatures above about 600 °C, the lower yield stress obeys Haasen's model with a strain-rate exponent of 1/2.9 and an activation energy of 1.74 eV. At lower temperatures experimental evidence indicates different strain-rate and temperature dependences. A new phenomenological decription is discussed, but a conclusive model is difficult to establish. The underlying activation energy, however, can be derived as approximately 1.85 eV. Arguments are presented which may elucidate the origin of the problems arising for the low-temperature measurements.

Relaxation of thermal stresses by dislocation flow and multiplication in the continuous casting of silicon

A semi-analytical model is proposed for simulating thermal stress generation during experimental continuous casting of polycrystalline silicon billets for photovoltaic applications. The geometry is axi-symmetric and the temperature field is formulated using three parameters. The thermoelastic stress source is given by the plane strain approximation. The Alexander and Haasen model, based on the classical Orowan equation, expresses the plastic relaxation and the associated multiplication of dislocations. This set of approximations allows us to discuss separately the influence of the withdrawal rate and of the thermal parameters on the variation along the billet of both the stress and the dislocation density.