# Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal

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## Abstract

**:**

## 1. Introduction

`CZ-Trans`is used for dynamic simulations of the temperature field, phase boundaries [27] and PD distribution in the crystal [28]. The novelty is that we (i) apply an improved and numerically more precise

`CZ-Trans`model; (ii) simulate 12 different PD parameter sets using the same thermal conditions; (iii) compare simulation results with the experimental PD distribution in the whole crystal, including start and end cones; (iv) adjust PD parameters for 1 position in the crystal and find the parameter sets with the best agreement to experiment.

## 2. Growth Experiment

## 3. Numerical Model

#### 3.1. Heat Transfer and Phase Boundaries

`CZ-Trans`[27] was used to model the transient heat transfer in the Cz system and the changing phase boundaries. The model was axisymmetric, and the temperature field $T(r,z)$ was calculated in the crystal, melt, crucible and heat shield domains while a simple integral model was used for the heater to calculate its time-dependent temperature as a function of the heater power change. Radiative heat transfer between surfaces was considered. Since the crystal was pulled upwards, both the convection and diffusion terms were included in the temperature equation

`CZ-Trans`. The developed temperature solver uses the open-source finite element library

`deal.II`[31]. While the solver supports arbitrary element orders, from a practical point of view a good balance between the accuracy and performance is achieved for second-order elements (9-node quadrilaterals).

`deal.II`, it was introduced for the crystal, melt and crucible domains. Where possible, a structured grid was generated by

`CZ-Trans`, otherwise an unstructured mesh was created using

`Gmsh`[32].

#### 3.2. Point Defects

#### 3.2.1. Governing Equations

**∇**T, from a colder to a hotter crystal part since the drift velocity is given by ${\mathbf{v}}_{\mathrm{drift}}=-D\mathbf{\nabla}({Q}^{\ast}/\left(kT\right))=D{Q}^{\ast}/\left(k{T}^{2}\right)\mathbf{\nabla}T$ [33]. Note that ${Q}^{\ast}<0$ is also a valid value. The definition of the thermal diffusion term (its sign) is not consistent in the literature, therefore some PD parameter sets [18,34,35,36] were converted to match the formulation used in this study.

#### 3.2.2. Numerical Aspects

`CGSim`[40] has been used: a static (unchanged) grid is created in the low-temperature part of the crystal, which is automatically extended during the simulation as the crystal grows.

#### 3.3. Considered Parameter Sets

#### 3.4. Thermal Stresses

## 4. Results and Discussion

#### 4.1. Heat Transfer and Phase Boundaries

`CZ-Trans`simulations were started from the beginning of the cone phase. PID control of the crystal pull rate was applied according to the difference between the actual crystal radius R and its setpoint, which was maintained within a few millimetres. Figure 1 demonstrates a good agreement between the modeled pull rate curve and experimental data. One reason for discrepancies can be due to long-scale temperature fluctuations caused by turbulent melt flow. While the pull rate oscillations are much stronger in the experiment during the cone phase, it practically does not influence the heat transfer, as verified by performing a simulation with non-smoothed experimental data as the pull rate setpoint; this also holds for the point defect distribution.

#### 4.2. Thermal Stresses

#### 4.3. Point Defects

#### 4.3.1. Overview

#### 4.3.2. Original PD Parameters

#### 4.3.3. Adjusted PD Parameters

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

BC | Boundary condition |

FEM | Finite element method |

I | Self-interstitials |

IVB | I-V boundary |

LPS | Lateral photovoltage scanning |

MP | Melting point |

µ-PCD | Microwave photoconductivity decay |

PD | Point defects |

TP | Triple point |

V | Vacancies |

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**Figure 1.**(

**a**) Experimental µ-PCD measurements showing the IVB (part of data is mirrored to show the full crystal diameter) and (

**b**) normalized crystal pull rate curve in the experiment and numerical simulations. The crystal length L is matched between (

**a**,

**b**); zero length corresponds to the beginning of the body phase, $L<0$ is the cone (start cone).

**Figure 2.**Calculated $v/G$ ratio on the axis and at the triple point (TP, left axis) and calculated and experimentally measured crystallization interface deflection ${H}_{C}$ (right axis), defined as the vertical distance between the interface at TP and on the axis.

**Figure 3.**Compilation of the calculated crystallization interface shapes plotted each 20 min. Also shown are the interfaces manually extracted from the experimental measurements (red dots). Dimensions are given in millimeters.

**Figure 4.**Mean thermal stress ${\sigma}_{\mathrm{ave}}$ (negative–compressive stresses shown with dashed isolines) for parameter sets I (

**left**side), K and L (

**right**side). In reading order: cone ($L=-50\phantom{\rule{0.166667em}{0ex}}$mm); shouldering (0 mm); body phase (200 mm); body phase (1000 mm); endcone (1250 mm).

**Figure 5.**Comparison of PD distributions with experimental IVB (solid magenta curves—measurements, dashed lines—interpolation) for original PD parameter sets (

**A**–

**L**). Solid black curve—$\Delta =0$, red curve—$T=1000\text{}{}^{\circ}\mathrm{C}$ below which PD are frozen-in, dotted white lines for length reference are plotted every 200 mm starting from $L=0$.

**Figure 6.**The influence of thermal stress on ${\xi}_{\mathrm{crit}}$ for original and adjusted PD parameter sets.

Parameter Set | Ref. | ${\mathit{D}}^{0}\phantom{\rule{0.277778em}{0ex}}({\mathbf{cm}}^{2}/\mathit{s})$ | ${\mathit{H}}^{\mathbf{m}}\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{eV}\right)$ | ${\mathit{C}}^{0}\phantom{\rule{0.277778em}{0ex}}\left({\mathbf{cm}}^{-3}\right)$ | ${\mathit{H}}^{\mathbf{f}}\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{eV}\right)$ | ${\mathit{Q}}^{\ast}\phantom{\rule{0.277778em}{0ex}}\left(\mathbf{eV}\right)$ | ||
---|---|---|---|---|---|---|---|---|

A | 2000-Nakamura | [41] | I | 1.040 × 10^{4} | 2.4 | 1.060 × 10^{22} | 2.4 | |

V | 2.140 | 1.4 | 5.290 × 10^{22} | 2.6 | ||||

B | 2001-Wang | [34,35] | I | 2.101 × 10^{−1} | 0.907 | 3.945 × 10^{26} | 3.943 | −3.66 |

V | 1.000 × 10^{−4} | 0.489 | 2.675 × 10^{23} | 2.848 | −2.66 | |||

C | 2002-Nakamura | [36] | I | 2.459 × 10^{−1} | 0.9 | 6.284 × 10^{26} | 4.05 | −1.01 |

V | 3.513 × 10^{−4} | 0.3 | 3.951 × 10^{26} | 3.94 | 0.0 | |||

D | 2007-Kulkarni-I | [25,42] | I | 1.950 × 10^{−1} | 0.9 | 6.176 × 10^{26} | 4 | |

V | 6.262 × 10^{−4} | 0.4 | 7.520 × 10^{26} | 4 | ||||

E | 2007-Kulkarni-II | [25,42] | I | 4.000 × 10^{−3} | 0.3 | 4.725 × 10^{27} | 4.3492 | |

V | 2.000 × 10^{−3} | 0.38 | 1.200 × 10^{27} | 4.12 | ||||

F | 2007-Sinno | [43] | I | 2.370 × 10^{−1} | 0.937 | 6.365 × 10^{26} | 4.0 | |

V | 7.870 × 10^{−4} | 0.457 | 9.931 × 10^{25} | 3.7 | ||||

G | 2009-Voronkov | [33] | I | 3.667 × 10^{−3} | 0.2 | 1.884 × 10^{9} | 4.95 | 4.5 |

V | 1.876 × 10^{−3} | 0.38 | 2.967 × 10^{26} | 3.95 | 29 | |||

H | 2011-Nishimoto | [26,44] | I | 2.590 | 1.18 | 5.992 × 10^{25} | 3.77 | |

V | 9.918 × 10^{−4} | 0.4 | 1.400 × 10^{26} | 3.84 | ||||

I | 2013-Vanhellemont-s | [17] | I | 3.800 × 10^{−2} | 0.88 | 6.400 × 10^{25} | 3.68 | |

V | 1.200 × 10^{−3} | 0.45 | 2.580 × 10^{26} | 3.88 | ||||

J | 2014-Nakamura-s | [18] | I | 2.459 × 10^{−1} | 0.9 | 6.284 × 10^{26} | 4.05 | −1.01 |

V | 3.513 × 10^{−4} | 0.3 | 3.951 × 10^{26} | 3.94 | 0.0 | |||

K | 2016-Kamiyama-s | [20,26] | I | 2.590 | 1.18 | 5.992 × 10^{25} | 3.77 | |

V | 9.918 × 10^{−4} | 0.4 | 1.400 × 10^{26} | 3.84 | ||||

L | 2019-Mukaiyama-s | [21,22] | I | 2.459 × 10^{−1} | 0.9 | 6.284 × 10^{26} | 4.05 | |

V | 3.442 × 10^{−4} | 0.3 | 4.030 × 10^{26} | 3.94 |

Set | $\mathit{D}\phantom{\rule{0.277778em}{0ex}}({10}^{-4}\phantom{\rule{0.166667em}{0ex}}{\mathbf{cm}}^{2}/\mathit{s})$ | ${\mathit{C}}^{\mathbf{eq}}\phantom{\rule{0.277778em}{0ex}}\left({10}^{14}\phantom{\rule{0.166667em}{0ex}}{\mathbf{cm}}^{-3}\right)$ | ${\mathit{C}}_{\mathit{V}}^{\mathbf{eq}}-{\mathit{C}}_{\mathit{I}}^{\mathbf{eq}}$ | ${\mathbf{DC}}^{\mathbf{eq}}\phantom{\rule{0.277778em}{0ex}}\left({10}^{10}\phantom{\rule{0.166667em}{0ex}}{\mathbf{cm}}^{-1}{\mathit{s}}^{-1}\right)$ | |||
---|---|---|---|---|---|---|---|

I | V | I | V | $\left({10}^{14}\phantom{\rule{0.166667em}{0ex}}{\mathbf{cm}}^{-3}\right)$ | I | V | |

A | 6.898 | 1.390 | 7.030 | 8.850 | 1.819 | 48.495 | 12.303 |

B | 4.070 | 0.034 | 6.348 | 8.111 | 1.762 | 25.839 | 0.280 |

C | 5.000 | 0.445 | 4.840 | 6.490 | 1.650 | 24.200 | 2.888 |

D | 3.964 | 0.398 | 6.712 | 8.172 | 1.461 | 26.603 | 3.256 |

E | 5.067 | 1.460 | 4.635 | 5.707 | 1.071 | 23.489 | 8.334 |

F | 3.734 | 0.338 | 6.917 | 8.520 | 1.602 | 25.831 | 2.881 |

G | 9.250 | 1.370 | 2.950 | 4.550 | 1.600 | 27.288 | 6.234 |

H | 7.656 | 0.631 | 3.174 | 4.580 | 1.406 | 24.300 | 2.890 |

I | 0.887 | 0.541 | 6.301 | 6.407 | 0.106 | 5.587 | 3.467 |

J | 5.000 | 0.445 | 4.840 | 6.490 | 1.650 | 24.200 | 2.888 |

K | 7.656 | 0.631 | 3.174 | 4.580 | 1.406 | 24.300 | 2.890 |

L | 5.000 | 0.436 | 4.840 | 6.620 | 1.780 | 24.200 | 2.886 |

**Table 3.**Thermoelastic parameters (Young’s modulus E, coefficient of thermal expansion $\alpha $, Poisson’s ratio $\nu $) at the MP and parameters for the thermal stress influence on the point defects $\mathrm{d}{H}^{\mathrm{f}}/\mathrm{d}\sigma $.

Parameter Set | E | $\mathit{\alpha}$ | $\mathit{\nu}$ | $\mathbf{d}{\mathit{H}}^{\mathbf{f}}\mathbf{d}\mathit{\sigma}\phantom{\rule{0.277778em}{0ex}}\left(\right)open="("\; close=")">\mathbf{eV}/\mathbf{GPa}$ | ||
---|---|---|---|---|---|---|

$\left(\mathbf{GPa}\right)$ | $\left({10}^{-6}{\mathit{K}}^{-1}\right)$ | $(-)$ | I | V | ||

I | 2013-Vanhellemont-s | 100 | 4.66 | 0.25 | −0.070 | 0.160 |

J | 2014-Nakamura-s | 139.7 | 4.66 | 0.225 | −0.070 | 0.154 |

K | 2016-Kamiyama-s | 166 | 4.66 | 0.217 | −0.0694 | 0.3069 |

L | 2019-Mukaiyama-s | 165 | 4.66 | 0.217 | −0.0694 | 0.3069 |

**Table 4.**Adjustment of ${C}_{I}^{\mathrm{eq}}$ and the corresponding values of ${\xi}_{\mathrm{crit}}$ (in ${10}^{-3}\mathrm{c}{\mathrm{m}}^{2}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{min}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}$) at ${\sigma}_{\mathrm{ave}}=-20\phantom{\rule{0.166667em}{0ex}}$MPa for original and adjusted PD parameters.

Parameter Set | $\Delta {\mathit{S}}_{\mathit{I}}^{\mathbf{f}}/\mathit{k}$ | ${\mathit{C}}_{\mathit{I}}^{\mathbf{eq}}$ Scale | ${\mathit{\xi}}_{\mathbf{crit}}$ orig. | ${\mathit{\xi}}_{\mathbf{crit}}$ adj. | |
---|---|---|---|---|---|

A | 2000-Nakamura | 0.037 | 1.038 | 1.22 | 1.50 |

B | 2001-Wang | −0.1802 | 0.835 | 2.51 | 1.31 |

C | 2002-Nakamura | −0.056 | 0.946 | 1.63 | 1.32 |

D | 2007-Kulkarni-I | −0.0322 | 0.968 | 1.57 | 1.32 |

E | 2007-Kulkarni-II | −0.031 | 0.969 | 1.47 | 1.24 |

F | 2007-Sinno | −0.0041 | 0.996 | 1.35 | 1.32 |

G | 2009-Voronkov | −0.6635 | 0.515 | 2.32 | 1.23 |

H | 2011-Nishimoto | −0.014 | 0.986 | 1.42 | 1.36 |

I | 2013-Vanhellemont-s | 0.0134 | 1.013 | 0.60 | 0.85 |

J | 2014-Nakamura-s | −0.0306 | 0.970 | 1.45 | 1.29 |

K | 2016-Kamiyama-s | 0.0356 | 1.036 | 1.20 | 1.35 |

L | 2019-Mukaiyama-s | 0.0795 | 1.083 | 0.97 | 1.31 |

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**MDPI and ACS Style**

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.
https://doi.org/10.3390/cryst11050460

**AMA Style**

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(5):460.
https://doi.org/10.3390/cryst11050460

**Chicago/Turabian Style**

Sabanskis, Andrejs, Matīss Plāte, Andreas Sattler, Alfred Miller, and Jānis Virbulis.
2021. "Evaluation of the Performance of Published Point Defect Parameter Sets in Cone and Body Phase of a 300 mm Czochralski Silicon Crystal" *Crystals* 11, no. 5: 460.
https://doi.org/10.3390/cryst11050460