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

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

`deal.II`[19] and are freely available to all interested researchers as a new open-source solver package

`MACPLAS`[20].

## 2. Description of Experiment

## 3. Numerical Model

#### 3.1. Heat Transfer

#### 3.1.1. Temperature Field

#### 3.1.2. Heat Induction

#### 3.1.3. Equation Coupling

`deal.II`[19], which has been applied in the past for the modelling of transient temperature, stress, and point-defect fields [24]. The HF EM field was calculated beforehand using inductor current $I=1\mathrm{A}$. During the simulations (each time step), the induced heat sources were scaled by ${I}^{2}$ and updated according to the temperature-dependent skin depth. Since both the temperature Equation (2) and the boundary condition (3) contain non-linearities, Newton’s method was applied. A direct solver was used for the system of linear equations. The developed temperature solver is available in the open-source library

`MACPLAS`[20].

#### 3.2. Dislocation Density Dynamics

#### 3.2.1. Constitutive Equations

#### 3.2.2. Plastic Stress

#### 3.2.3. Equation Coupling

`MACPLAS`[20].

## 4. Results and Discussion

#### 4.1. Heat Transfer and Elastic Stress

- $\sigma \left(T\right)=100\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{10}^{4.247-2924/T}$ (in $\mathrm{S}/\mathrm{m}$) [23];
- $\lambda \left(T\right)=22\left(\right)open="("\; close=")">4.495-7.222T/{T}_{0}+3.728{(T/{T}_{0})}^{2}$ (in $\mathrm{W}/\mathrm{m}/\mathrm{K}$) [21];
- $\epsilon \left(T\right)=\left(\right)open="\{"\; close>\begin{array}{cc}0.46\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}1.39,\hfill & T0.593\phantom{\rule{0.166667em}{0ex}}{T}_{0}\hfill \\ 0.46\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}(1.96-0.96\phantom{\rule{0.166667em}{0ex}}T/{T}_{0}),\hfill & T\ge 0.593\phantom{\rule{0.166667em}{0ex}}{T}_{0}\hfill \end{array}$ (dimensionless) [21].

`Gmsh`[26]. The element size in the vertical direction was $0.5$ $\mathrm{m}$$\mathrm{m}$. A non-uniform size distribution was applied for 15 cells in the radial direction with the minimum element size at the crystal surface of $0.3$ $\mathrm{m}$$\mathrm{m}$. Second-order finite elements (nine-node quadrilaterals) were used, resulting in the total number of degrees of freedom for temperature of 61,000. The same discretization was used also for stress and dislocation density simulations (Section 4.2). A maximum time step of $1\mathrm{s}$ was imposed. The time step cannot be much larger due to highly localized EM power sources (profile full width at half maximum of about 4 $\mathrm{m}$$\mathrm{m}$) and a relatively fast crystal movement velocity ( 5 $\mathrm{m}$$\mathrm{m}$/$\mathrm{min}$). The influence of the mesh size, finite element order, and the time step on the temperature field was checked by comparing the time-dependent maximum crystal temperature and the temperature at four probe points between simulations with different numerical settings and was found out to be very weak (below $0.5$ $\mathrm{K}$).

#### 4.2. Dislocation Density Dynamics

^{−2}and $2\times {10}^{5}$ $\mathrm{c}$$\mathrm{m}$

^{−2}were experimentally measured [18], the value of ${N}_{0}=100\mathrm{c}{\mathrm{m}}^{-2}$ was selected. Although it is known that the influence of ${N}_{0}$ on the results is small for sufficiently low ${N}_{0}$ [12], an additional simulation with ${N}_{0}=1\mathrm{c}{\mathrm{m}}^{-2}$ was carried out.

^{−2}agrees with the experimentally measured values. Finally, high dislocation densities are created after the inductor power has been turned off due to a rapid temperature drop, which causes high stresses.

## 5. Summary and Conclusions

`deal.II`was developed for the simulation of the temperature field and the dislocation density dynamics. It is freely available as a new open-source solver package

`MACPLAS`[20].

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AH | Alexander–Haasen |

CRSS | Critical resolved shear stress |

CZ | Czochralski |

EM | Electromagnetic |

FZ | Floating-zone |

HF | High-frequency |

LF | Low-frequency |

## Appendix A. Derivation of 1D Temperature Distribution for HF and LF Heat Induction Models

## References

- Dash, W.C. Growth of silicon crystals free from dislocations. J. Appl. Phys.
**1959**, 30, 459–474. [Google Scholar] [CrossRef] - Rost, H.J.; Menzel, R.; Siche, D.; Juda, U.; Kayser, S.; Kießling, F.M.; Sylla, L.; Richter, T. Defect formation in Si-crystals grown on large diameter bulk seeds by a modified FZ-method. J. Cryst. Growth
**2018**, 500, 5–10. [Google Scholar] [CrossRef] - Menzel, R.; Rost, H.J.; Kießling, F.M.; Sylla, L. Float-zone growth of silicon crystals using large-area seeding. J. Cryst. Growth
**2019**, 515, 32–36. [Google Scholar] [CrossRef] - Stoddard, N.; Russell, J.; Hixson, E.C.; She, H.; Krause, A.; Wolny, F.; Bertoni, M.; Naerland, T.U.; Sylla, L.; von Ammon, W. NeoGrowth silicon: A new high purity, low-oxygen crystal growth technique for photovoltaic substrates. Prog. Photovoltaics Res. Appl.
**2018**, 26, 324–331. [Google Scholar] [CrossRef] - Jordan, A.S.; Caruso, R.; VonNeida, A.R.; Nielsen, J.W. A comparative study of thermal stress induced dislocation generation in pulled GaAs, InP, and Si crystals. J. Appl. Phys.
**1981**, 52, 3331–3336. [Google Scholar] [CrossRef] - Miyazaki, N.; Uchida, H.; Munakata, T.; Fujioka, K.; Sugino, Y. Thermal stress analysis of silicon bulk single crystal during Czochralski growth. J. Cryst. Growth
**1992**, 125, 102–111. [Google Scholar] [CrossRef] - Muižnieks, A.; Raming, G.; Mühlbauer, A.; Virbulis, J.; Hanna, B.; Ammon, W. Stress-induced dislocation generation in large FZ- and CZ-silicon single crystals—Numerical model and qualitative considerations. J. Cryst. Growth
**2001**, 230, 305–313. [Google Scholar] [CrossRef] - Alexander, H.; Haasen, P. Dislocations and plastic flow in the diamond structure. In Solid State Physics; Seitz, F., Turnbull, D., Ehrenreich, H., Eds.; Elsevier: Amsterdam, The Netherlands, 1969; Volume 22, pp. 27–158. [Google Scholar] [CrossRef]
- Dillon, O.W.; Tsai, C.T.; de Angelis, R.J. Dislocation dynamics during the growth of silicon ribbon. J. Appl. Phys.
**1986**, 60, 1784–1792. [Google Scholar] [CrossRef] - Völkl, J.; Müller, G. A new model for the calculation of dislocation formation in semiconductor melt growth by taking into account the dynamics of plastic deformation. J. Cryst. Growth
**1989**, 97, 136–145. [Google Scholar] [CrossRef] - Tsai, C.T.; Dillon, O.W.; de Angelis, R.J. The constitutive equation for silicon and its use in crystal growth modeling. J. Eng. Mater. Technol.
**1990**, 112, 183–187. [Google Scholar] [CrossRef] - Miyazaki, N.; Okuyama, S. Development of finite element computer program for dislocation density analysis of bulk semiconductor single crystals during Czochralski growth. J. Cryst. Growth
**1998**, 183, 81–88. [Google Scholar] [CrossRef] - Dadzis, K.; Behnken, H.; Bähr, T.; Oriwol, D.; Sylla, L.; Richter, T. Numerical simulation of stresses and dislocations in quasi-mono silicon. J. Cryst. Growth
**2016**, 450, 14–21. [Google Scholar] [CrossRef] - Gallien, B.; Albaric, M.; Duffar, T.; Kakimoto, K.; M’Hamdi, M. Study on the usage of a commercial software (Comsol-Multiphysics®) for dislocation multiplication model. J. Cryst. Growth
**2017**, 457, 60–64. [Google Scholar] [CrossRef] - Yonenaga, I.; Sumino, K. Dislocation dynamics in the plastic deformation of silicon crystals I. Experiments. Phys. Status Solidi A
**1978**, 50, 685–693. [Google Scholar] [CrossRef] - Suezawa, M.; Sumino, K.; Yonenaga, I. Dislocation dynamics in the plastic deformation of silicon crystals. II. Theoretical analysis of experimental results. Phys. Status Solidi A
**1979**, 51, 217–226. [Google Scholar] [CrossRef] - Gao, B.; Jiptner, K.; Nakano, S.; Harada, H.; Miyamura, Y.; Sekiguchi, T.; Kakimoto, K. Applicability of the three-dimensional Alexander-Haasen model for the analysis of dislocation distributions in single-crystal silicon. J. Cryst. Growth
**2015**, 411, 49–55. [Google Scholar] [CrossRef] - Rost, H.J.; Buchovska, I.; Dadzis, K.; Juda, U.; Renner, M.; Menzel, R. Thermally stimulated dislocation generation in silicon crystals grown by the Float-Zone method. J. Cryst. Growth
**2020**, 552, 125842. [Google Scholar] [CrossRef] - Arndt, D.; Bangerth, W.; Davydov, D.; Heister, T.; Heltai, L.; Kronbichler, M.; Maier, M.; Pelteret, J.P.; Turcksin, B.; Wells, D. The deal.II library, version 8.5. J. Numer. Math.
**2017**, 25, 137–145. [Google Scholar] [CrossRef] - MACPLAS: MAcroscopic Crystal PLAsticity Simulator. Available online: https://github.com/aSabanskis/MACPLAS (accessed on 16 November 2021).
- Ratnieks, G. Modelling of the Floating Zone Growth of Silicon Single Crystals with Diameter up to 8 Inch. Ph.D. Thesis, University of Latvia, Riga, Latvia, 2007. [Google Scholar]
- Muhlbauer, A.; Muiznieks, A.; Leßmann, H.J. The calculation of 3D high-frequency electromagnetic fields during induction heating using the BEM. IEEE Trans. Magn.
**1993**, 29, 1566–1569. [Google Scholar] [CrossRef] - Fulkerson, W.; Moore, J.P.; Williams, R.K.; Graves, R.S.; McElroy, D.L. Thermal conductivity, electrical resistivity, and Seebeck coefficient of silicon from 100 to 1300°K. Phys. Rev.
**1968**, 167, 765–782. [Google Scholar] [CrossRef] - Sabanskis, A.; Plāte, M.; Sattler, A.; Miller, A.; Virbulis, J. Evaluation of the performance of published point defect parameter sets in cone and body phase of a 300 mm Czochralski silicon crystal. Crystals
**2021**, 11, 460. [Google Scholar] [CrossRef] - Sabanskis, A.; Virbulis, J. Simulation of the influence of gas flow on melt convection and phase boundaries in FZ silicon single crystal growth. J. Cryst. Growth
**2015**, 417, 51–57. [Google Scholar] [CrossRef] - Geuzaine, C.; Remacle, J.F. Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. Int. J. Numer. Methods Eng.
**2009**, 79, 1309–1331. [Google Scholar] [CrossRef] - Okada, Y.; Tokumaru, Y. Precise determination of lattice parameter and thermal expansion coefficient of silicon between 300 and 1500 K. J. Appl. Phys.
**1984**, 56, 314–320. [Google Scholar] [CrossRef]

**Figure 1.**Infrared image 3 min before the end of heating. The superimposed horizontal white line shows the vertical position of the inductor mid-plane, while the vertical one denotes the location for the vertical temperature distribution $T\left(z\right)$. The main temperature reading of $1180.0$ ${}^{\xb0}\mathrm{C}$ is for a circular region ($5.7$ $\mathrm{m}$$\mathrm{m}$ in diameter) below the inductor.

**Figure 2.**Time dependence of experimental parameters—measured crystal temperature (average over the circular region in Figure 1), heating (crystal) position, and HF inductor power. Highlighted areas show time intervals at which the crystal is stationary.

**Figure 3.**Comparison of time-dependent temperature with experimental measurements. $T\left(t\right)$ for the HF simulation is not depicted as it practically overlaps with the LF result. Also shown is the maximum crystal temperature ${T}_{max}$ and inductor current I. Highlighted areas show time intervals at which the crystal is stationary.

**Figure 4.**Comparison of vertical temperature distribution at different times with experimental measurements. Black vertical lines show the lower and the upper boundary of the measurement region. $T(z-{z}_{0})$ distribution at $t=125\mathrm{min}$ was symmetric with respect to the inductor, which allowed to obtain more precise measurement point coordinates.

**Figure 5.**Comparison of radial distributions of temperature T and stress $\sqrt{{J}_{2}}$ at the end of heating ($t=215\mathrm{min}$) for LF and HF modes.

**Figure 6.**Simulation results at the end of heating ($t=215\mathrm{min}$); zoom-in at the upper part of the crystal. (

**a**) temperature, (

**b**) maximum temperature during the whole simulation, (

**c**) elastic thermal stress. Left side: HF mode; right side: LF mode. Black horizontal lines in (

**b**) show heating positions at which the crystal is not moving.

**Figure 7.**(

**a**) Shape of dislocated zone in experiment; (

**b**) calculated dislocation density in the whole crystal according to cases A–F in Table 1; (

**c**) zoomed-in simulation results at two positions for cases D and F. Experimentally observed boundary of the dislocated zone is depicted in (

**b**,

**c**) as blue curves. Black horizontal lines show heating positions at which the crystal is not moving.

**Figure 8.**Dislocation density for simulation case F with varied $I\left(t\right)$ and $z\left(t\right)$ curves. (

**a**) Default (stepwise) curves; (

**b**) linear $I\left(t\right)$ curve; (

**c**) linear $z\left(t\right)$ curve; (

**d**) linear $I\left(t\right)$ and $z\left(t\right)$ curves. Experimentally observed boundary of the dislocated zone is depicted as blue curves. Black horizontal lines show heating positions at which the crystal was not moving.

Case | Heat Induction Mode | Dislocation Parameters |
---|---|---|

A | LF | Basic parameter set |

B | HF | Basic parameter set |

C | LF | $Q=2.3eV$ |

D | LF | ${\tau}_{\mathrm{crit}}\left(T\right)$ |

E | LF | ${\tau}_{\mathrm{crit}}\left(T\right)$, ${N}_{0}=1\mathrm{c}{\mathrm{m}}^{-2}$ |

F | LF | ${\tau}_{\mathrm{crit}}\left(T\right)$, $Q=2.3eV$ |

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**MDPI and ACS Style**

Sabanskis, A.; Dadzis, K.; Menzel, R.; Virbulis, J.
Application of the Alexander–Haasen Model for Thermally Stimulated Dislocation Generation in FZ Silicon Crystals. *Crystals* **2022**, *12*, 174.
https://doi.org/10.3390/cryst12020174

**AMA Style**

Sabanskis A, Dadzis K, Menzel R, Virbulis J.
Application of the Alexander–Haasen Model for Thermally Stimulated Dislocation Generation in FZ Silicon Crystals. *Crystals*. 2022; 12(2):174.
https://doi.org/10.3390/cryst12020174

**Chicago/Turabian Style**

Sabanskis, Andrejs, Kaspars Dadzis, Robert Menzel, and Jānis Virbulis.
2022. "Application of the Alexander–Haasen Model for Thermally Stimulated Dislocation Generation in FZ Silicon Crystals" *Crystals* 12, no. 2: 174.
https://doi.org/10.3390/cryst12020174